phd mathematics topics

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

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PhD in Mathematics

Here are the requirements for earning the PhD degree in Mathematics offered by the School of Math. For requirements of other PhD programs housed within the School, please see their specific pages at  Doctoral Programs . The requirements for all these programs consist of three components:  coursework ,  examinations , and  dissertation  in accordance to the guidelines described in the  GT Catalogue .

Completion of required coursework, examinations, and dissertation normally takes about five years. During the first one or two years, students concentrate on coursework to acquire the background necessary for the comprehensive examinations. By the end of their third year in the program, all students are expected to have chosen a thesis topic, and begin work on the research and writing of the dissertation.

The program of study must contain at least 30 hours of graduate-level coursework (6000-level or above) in mathematics and an additional 9 hours of coursework towards a minor. The minor requirement consists of graduate or advanced undergraduate coursework taken entirely outside the School of Mathematics, or in an area of mathematics sufficiently far from the students area of specialization.

Prior to admission to candidacy for the doctoral degree, each student must satisfy the School's comprehensive examinations (comps) requirement. The first phase is a written examination which students must complete by the end of their second year in the graduate program. The second phase is an oral examination in the student's proposed area of specialization, which must be completed by the end of the third year.

Research and the writing of the dissertation represent the final phase of the student's doctoral study, and must be completed within seven years of the passing of comps. A final oral examination on the dissertation (theses defense) must be passed prior to the granting of the degree.

The Coursework

The program of study must satisfy the following  hours ,  minor , and  breadth  requirements. Students who entered before Fall 2015 should see  the old requirements , though they may opt into the current rules described below, and are advised to do so.

Hours requirements.  The students must complete 39 hours of coursework as follows:

• At least 30 hours must be in mathematics courses at the 6000-level or higher.
• At least 9 hours must form the doctoral minor field of study.
• The overall GPA for these courses must be at least 3.0.
• These courses must be taken for a letter grade and passed with a grade of at least C.

Minor requirement.  The minor field of study should consist primarily of 6000-level (or higher) coursework in a specific area outside the School of Math, or in a mathematical subject sufficiently far from the student’s thesis work. A total of 9 credit hours is required and must be passed with a grade of B or better. These courses should not include MATH 8900, and must be chosen in consultation with the PhD advisor and the Director of Graduate Studies to ensure that they form a cohesive group which best complements the students research and career goals. A student wishing to satisfy the minor requirement by mathematics courses must petition the Graduate Committee for approval.  Courses used to fulfill a Basic Understanding breadth requirement in Analysis or Algebra should not be counted towards the doctoral minor. Upon completing the minor requirement, a student should immediately complete the  Doctoral Minor form .

Breadth requirements.  The students must demonstrate:

• Basic understanding of 2 subjects must be demonstrated through passing the subjects' written comprehensive exams.  At least 1 of these 2 exams must be in Algebra or Analysis.
• Basic understanding of the third subject may be demonstrated either by completing two courses in the subject (with a grade of A or B in each course) or by passing the subject's written comprehensive exam.
• A basic understanding of both subjects in Area I (analysis and algebra) must be demonstrated.
• Earning a grade of A or B in a one-semester graduate course in a subject demonstrates exposure to the subject.
• Passing a subject's written comprehensive exam also demonstrates exposure to that subject.

The subjects.  The specific subjects, and associated courses, which can be used to satisfy the breadth requirements are as follows.

• Area I subjects:​
• Area II subjects:​

Special Topics and Reading Courses.

• Special topics courses may always be used to meet hours requirements.
• Special topics courses may be used to meet breadth requirements, subject to the discretion of the Director of Graduate Studies.
• Reading courses may be used to meet hours requirements but not breadth requirements.

Credit Transfers

Graduate courses completed at other universities may be counted towards breadth and hours requirements (courses designated as undergraduate or Bachelors' level courses are not eligible to transfer for graduate credit).  These courses do not need to be officially transferred to Georgia Tech. At a student’s request, the Director of Graduate Studies will determine which breadth and hours requirements have been satisfied by graduate-level coursework at another institution.  

Courses taken at other institutions may also be counted toward the minor requirement, subject to the approval of the Graduate Director; however, these courses must be officially transferred to Georgia Tech.

There is no limit for the transfer of credits applied toward the breadth requirements; however, a maximum of 12 hours of coursework from other institutions may be used to satisfy hours requirements. Thus at least 27 hours of coursework must be completed at Georgia Tech, including at least 18 hours of 6000-level (or higher) mathematics coursework.

Students wishing to petition for transfer of credit from previous graduate level work should send the transcripts and syllabi of these courses, together with a list of the corresponding courses in the School of Math, to the Director of Advising and Assessment for the graduate program.

Comprehensive Examinations

The comprehensive examination is in two phases. The first phase consists of passing two out of seven written examinations. The second phase is an oral specialty examination in the student's planned area of concentration. Generally, a student is expected to have studied the intended area of research but not necessarily begun dissertation research at the time of the oral examination.

Written examinations.  The written examinations will be administered twice each year, shortly after the beginning of the Fall and Spring semesters. The result of the written examination is either pass or fail. For syllabi and sample exams see the  written exams page .

All students must adhere to the following rules and timetables, which may be extended by the Director of Graduate Studies, but only at the time of matriculation and only when certified in writing. Modifications because of leaves from the program will be decided on a case-by-case basis.

After acceptance into the PhD Program in Mathematics, a student must pass the written examinations no later than their fourth administration since the student's doctoral enrollment. The students can pass each of the two written comprehensive exams in separate semesters, and are allowed multiple attempts.

The Director of Graduate Studies (DGS) will be responsible for advising each new student at matriculation of these rules and procedures and the appropriate timetable for the written portion of the examination. The DGS will also be responsible for maintaining a study guide and list of recommended texts, as well as a file of previous examinations, to be used by students preparing for this written examination.

Oral examination.  A student must pass the oral specialty examination within three years since first enrolling in the PhD program, and after having passed the written portion of the comprehensive exams. The examination will be given by a committee consisting of the student's dissertation advisor or probable advisor, two faculty members chosen by the advisor in consultation with the student, and a fourth member appointed by the School's Graduate Director. The scope of the examination will be determined by the advisor and will be approved by the graduate coordinator. The examining committee shall either (1) pass the student or (2) fail the student. Within the time constraints of which above, the oral specialty examination may be attempted multiple times, though not more than twice in any given semester. For more details and specific rules and policies see the  oral exam page .

Dissertation and Defense

A dissertation and a final oral examination are required. For details see our  Dissertation and Graduation  page, which applies to all PhD programs in the School of Math.

PhD Program

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: (i) algebra, (ii) real analysis, (iii) geometry and topology, (iv) applied mathematics.

Current Course Requirements: To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry

• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later: 

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. (The course sequence requirement is discontinued for students starting in Autumn 2023 and later.)

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination, which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

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Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series.  Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

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PhD in Mathematics

The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

Degree Requirements

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin .

• Courses : The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here . Half may be at the MA 500 level or above, but the rest must be at the MA 700 level or above. Students can also request to use courses from other departments to satisfy some of these requirements. Please contact your advisor for more information about which courses can be used in this way. All courses must be passed with a grade of B- or higher.
• Analysis (examples include MA 711, MA 713, and MA 717)
• PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
• Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
• Topology (examples include MA 721, MA 722, and MA 727)
• Geometry (examples include MA 725, MA 731, and MA 745)
• Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
• Applied Mathematics (examples include MA 750, MA 751, and MA 770)
• Comprehensive Examination : This exam has both a written and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. On completion of the paper, the student takes an oral exam given by a three-person committee, one of whom is the student’s advisor. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. A student who does not pass the MA Comprehensive Examination may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year.
• Oral Qualifying Examination: The topics for the PhD oral qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not pass the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year.
• Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree .

Admissions information can be found on the BU Arts and Sciences PhD Admissions website .

Financial Aid

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

More Information

Please reach out to us directly at [email protected] if you have further questions.

Department of Mathematics

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The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and go on to exciting careers in mathematics both inside and outside of academia.

Faculty research interests in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Students making satisfactory progress are guaranteed support for five years. A sixth year is generally possible for students who are on track to complete their Ph.D. and would benefit from the additional year.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the requirements page .

All graduate students are invited to attend weekly research seminars in a variety of topic areas as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

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Ph.D. in Mathematics, Specializing in Applied Math

Table of contents, overview of applied mathematics at the courant institute.

• PhD Study in Applied Mathematics
• Applied math courses

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
• Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
• Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
• Data Science
• Financial Mathematics
• Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
• Materials Science, including micromagnetics, surface growth, variational methods,
• Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

A list of the current research interests of individual faculty is available on the Math research page.

Courses in Applied Mathematics

The following list is for AY 2023/2024:

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(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

-------------------------------------

( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

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Berkeley Berkeley Academic Guide: Academic Guide 2023-24

Mathematics.

About the Program

The Department of Mathematics offers both a PhD program in Mathematics and Applied Mathematics.

Students are admitted for specific degree programs: the PhD in Mathematics or PhD in Applied Mathematics. Requirements for the Mathematics and Applied Mathematics PhDs differ only in minor respects, and no distinction is made between the two in day-to-day matters. Graduate students typically take 5-6 years to complete the doctorate.

Continuing students wishing to transfer from one program to another should consult the graduate advisor in 910 Evans Hall. Transfers between the two PhD programs are fairly routine, but must be done prior to taking the qualifying examination. It is a formal policy of the department that an applicant to the PhD program who has previous graduate work in mathematics must present very strong evidence of capability for mathematical research.

Students seeking to transfer to the department's PhD programs from other campus programs, including the Group in Logic and the Methodology of Science, must formally apply and should consult the Vice Chair for Graduate Studies.

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Admission to the University

Applying for graduate admission.

Thank you for considering UC Berkeley for graduate study! UC Berkeley offers more than 120 graduate programs representing the breadth and depth of interdisciplinary scholarship. A complete list of graduate academic departments, degrees offered, and application deadlines can be found on the Graduate Division website .

Prospective students must submit an online application to be considered for admission, in addition to any supplemental materials specific to the program for which they are applying. The online application can be found on the Graduate Division website .

Admission Requirements

The minimum graduate admission requirements are:

A bachelor’s degree or recognized equivalent from an accredited institution;

A satisfactory scholastic average, usually a minimum grade-point average (GPA) of 3.0 (B) on a 4.0 scale; and

Enough undergraduate training to do graduate work in your chosen field.

For a list of requirements to complete your graduate application, please see the Graduate Division’s Admissions Requirements page . It is also important to check with the program or department of interest, as they may have additional requirements specific to their program of study and degree. Department contact information can be found here .

Where to apply?

Visit the Berkeley Graduate Division application page .

Admission to the Program

Undergraduate students also often take one or more of the following introductory Mathematics graduate courses:

The Math Department admits new graduate students to the fall semester only. The Graduate Division's Online Application will be available in early September at:  http://grad.berkeley.edu/admissions/index.shtml . Please read the information on Graduate Division requirements and information required to complete the application.

Copies of official or unofficial transcripts may be uploaded to your application. Please do  not  mail original transcripts for the review process. 

We require three letters of recommendation, which should be submitted online. Please do  not  mail letters of recommendation for the review process. 

For more information, please review the department's graduate admissions webpage at:  https://math.berkeley.edu/programs/graduate/admissions . We also recommend reviewing our admissions FAQs page at:  https://math.berkeley.edu/programs/graduate/faqs . 

Doctoral Degree Requirements

Prerequisites

The Department of Mathematics offers two PhD degrees, one in Mathematics and one in Applied Mathematics. Applicants for admission to either PhD program are expected to have preparation comparable to the undergraduate major at Berkeley in Mathematics or in Applied Mathematics. These majors consist of two full years of lower division work (covering calculus, linear algebra, differential equations, and multivariable calculus), followed by eight one-semester courses including real analysis, complex analysis, abstract algebra, and linear algebra. These eight courses may include some mathematically based courses in other departments, like physics, engineering, computer science, or economics.

Applicants for admission are considered by the department's Graduate Admissions and M.O.C. Committees. The number of students that can be admitted each year is determined by the Graduate Division and by departmental resources. In making admissions decisions, the committee conducts a comprehensive review of applicants considering broader community impacts, academic performance in mathematics courses, level of mathematical preparation, letters of recommendation, and GRE scores.

Degree Requirements

In outline, to qualify for the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• take at least four courses, two or more of which are graduate courses in mathematics;
• and pass the six-hour written preliminary examination covering primarily undergraduate material. (The exam is given just before the beginning of each semester, and the student must pass it within their first three semesters.)
• Pass a three-hour, oral qualifying examination emphasizing, but not exclusively restricted to, the area of specialization. The qualifying examination must be attempted  within two years  of entering the program.
• Complete a seminar offered by the Math department, giving a talk of at least one hour duration.  Research presentations held at Mathematical Sciences Research Institute (MSRI), or Lawrence Berkeley National Lab (LBNL) are also acceptable. A Math Department faculty member must be present at the talk and sign the seminar form confirming.
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee.
• Meet the University residence requirement of two years or four semesters.

The detailed regulations of the PhD program are as follows:

Course Requirements Students must take and pass at least four 4-unit courses during the first year of the Ph.D. program; at least two courses per semester. At minimum, two of these courses must be graduate courses (200-level) offered by the Department of Mathematics . Two upper division (100-level) undergraduate courses offered by the Department of Mathematics may also be used toward this requirement. Exceptions may also be considered and must be reviewed by the Head Graduate Advisor for approval.

Preliminary Examination The preliminary examination consists of six hours of written work given over a two-day period. Most of the examination covers material, mainly in analysis and algebra, and helps to identify gaps in preparation. The preliminary examination is offered twice a year—during the week before classes start in both the fall and spring semesters. A student may repeat the examination twice. A student who does not pass the preliminary examination within 13 months of the date of entry into the PhD program will not be permitted to remain in the program past the third semester. In exceptional cases, a fourth try may be granted upon appeal to committee omega.

Qualifying Examination To arrange for the qualifying examination, a student must first settle on an area of concentration, and a prospective dissertation supervisor, someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective supervisor, the student forms an examination committee of four members. Committee members must be members of Berkeley's Academic Senate and the Chair must be a faculty member in the Mathematics Department. The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval it is to be circulated to all faculty members of the appropriate sections. The qualifying examination must cover material falling in at least three subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be seen on the Qualifying Examination page on the department website. 

The student must attempt the qualifying examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program.

For a student to pass the qualifying examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student, if asked. The student must obtain an official dissertation supervisor within one semester after passing the qualifying examination or leave the PhD program. For more detailed rules and advice concerning the qualifying examination, consult the graduate advisor in  910 Evans Hall.

Master's Degree Requirements

Eligibility .

At this time, the MA in Mathematics is a simultaneous degree program only offered to students currently enrolled in a doctoral program at UC Berkeley. The doctoral student must be in good standing in their program and have a faculty adviser in the Mathematics Department who is supportive of the addition of the MA in Mathematics and agrees to supervise the MA work. Current doctoral students must apply during the regular admissions cycle for consideration for fall admission. The degree must be completed prior to or in tandem with the PhD degree. Interested students must inquire with the Mathematics Graduate Student Affairs Officer. 

Unit Requirements

Plan I requires at least 20 semester units of upper division and graduate courses and a thesis. At least 8 of these units must be in graduate courses (200 series). These 8 units are normally taken in the Department of Mathematics at Berkeley. In special cases, upon recommendation of the Graduate Adviser and approval of the Dean of the Graduate Division, some of the 8 graduate units may be taken in other departments.

Plan II requires at least 24 semester units of upper division and graduate courses, followed by a comprehensive final examination, the MA examination. At least 12 of these units must be in graduate courses (200 series). These 12 units are normally taken in the Department of Mathematics at Berkeley. In special cases, upon recommendation of the graduate advisor and approval of the dean of the Graduate Division, some of the 12 graduate units may be taken in other departments. All courses fulfilling the above unit requirements must have significant mathematical content. In general, MA students are encouraged to take some courses outside the Department of Mathematics. In many jobs, at least some acquaintance with statistics and computer science is essential; and, for some students, courses in such fields as engineering, biological or physical sciences, or economics are highly desirable.

A breadth requirement consisting of at least one course in each of three fields must be met by all students. Fields include algebra, analysis, geometry, foundations, history of mathematics, numerical analysis, probability and statistics, computer science, and various other fields of applied mathematics. The last category specifically covers courses in a variety of departments, and the graduate adviser may allow more than one such course to count toward the breadth requirement. A depth requirement consisting of a coherent program of three courses all in one of the above fields, at least two of these courses being at the graduate level, must be met. Students interested in a field of applied mathematics are encouraged to take some of these courses outside the department.

• Advancement to Candidacy
• Thesis (Plan I)
• Capstone/Comprehensive Exam (Plan II)
• Capstone/Master's Project (Plan II)

MATH 202A Introduction to Topology and Analysis 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and Rn. Construction of the integral. Dominated convergence theorem. Introduction to Topology and Analysis: Read More [+]

Rules & Requirements

Prerequisites: 104

Hours & Format

Fall and/or spring: 15 weeks - 3 hours of lecture per week

Additional Format: Three hours of Lecture per week for 15 weeks.

Additional Details

Subject/Course Level: Mathematics/Graduate

Grading: Letter grade.

Introduction to Topology and Analysis: Read Less [-]

MATH 202B Introduction to Topology and Analysis 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations. Introduction to Topology and Analysis: Read More [+]

Prerequisites: 202A and 110

MATH 204 Ordinary Differential Equations 4 Units

Terms offered: Fall 2022, Fall 2016, Spring 2016 Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos. Ordinary Differential Equations: Read More [+]

Ordinary Differential Equations: Read Less [-]

MATH 205 Theory of Functions of a Complex Variable 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem. Theory of Functions of a Complex Variable: Read More [+]

Prerequisites: 185

Theory of Functions of a Complex Variable: Read Less [-]

MATH 206 Functional Analysis 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Spectrum of an operator. Analytic functional calculus. Compact operators. Hilbert-Schmidt operators. Spectral theorem for bounded self-adjoint and normal operators. Unbounded self-adjoint operators. Banach algebras. Commutative Gelfand-Naimark theorem. Selected additional topics such as Fredholm operators and Fredholm index, Calkin algebra, Toeplitz operators, semigroups of operators, interpolation spaces, group algebras. Functional Analysis: Read More [+]

Prerequisites: 202A-202B

Additional Format: Three hours of lecture per week.

Functional Analysis: Read Less [-]

MATH 208 C*-algebras 4 Units

Terms offered: Spring 2023, Spring 2022, Spring 2021 Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory. C*-algebras: Read More [+]

Prerequisites: 206

C*-algebras: Read Less [-]

MATH 209 Von Neumann Algebras 4 Units

Terms offered: Spring 2024, Spring 2017, Spring 2014 Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability. Von Neumann Algebras: Read More [+]

Von Neumann Algebras: Read Less [-]

MATH 212 Several Complex Variables 4 Units

Terms offered: Fall 2023, Fall 2021, Fall 2019 Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohomology and Stein manifolds, or the theory of analytic subvarieties and spaces. Several Complex Variables: Read More [+]

Prerequisites: 185 and 202A-202B or their equivalents

Several Complex Variables: Read Less [-]

MATH 214 Differential Topology 4 Units

Terms offered: Fall 2024, Spring 2024, Fall 2022 This is an introduction to abstract differential topology based on rigorous mathematical proofs. The topics include Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor. Differential Topology: Read More [+]

Prerequisites: 202A

Differential Topology: Read Less [-]

MATH 215A Algebraic Topology 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall. Algebraic Topology: Read More [+]

Prerequisites: 113 and point-set topology (e.g. 202A)

Instructors: 113C, 202A, and 214

Algebraic Topology: Read Less [-]

MATH 215B Algebraic Topology 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall. Algebraic Topology: Read More [+]

Prerequisites: 215A, 214 recommended (can be taken concurrently)

MATH C218A Probability Theory 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 The course is designed as a sequence with Statistics C205B/Mathematics C218B with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Probability Theory: Read More [+]

Also listed as: STAT C205A

Probability Theory: Read Less [-]

MATH C218B Probability Theory 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Probability Theory: Read More [+]

Also listed as: STAT C205B

MATH 219 Dynamical Systems 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2022 Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor. Dynamical Systems: Read More [+]

Prerequisites: 214

Dynamical Systems: Read Less [-]

MATH 220 Introduction to Probabilistic Methods in Mathematics and the Sciences 4 Units

Terms offered: Spring 2012, Spring 2011, Spring 2010 Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications. Introduction to Probabilistic Methods in Mathematics and the Sciences: Read More [+]

Prerequisites: Some familiarity with differential equations and their applications

Introduction to Probabilistic Methods in Mathematics and the Sciences: Read Less [-]

MATH 221 Advanced Matrix Computations 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2022 Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions. Advanced Matrix Computations: Read More [+]

Prerequisites: Consent of instructor

Summer: 8 weeks - 6 hours of lecture per week

Additional Format: Three hours of Lecture per week for 15 weeks. Six hours of Lecture per week for 8 weeks.

Advanced Matrix Computations: Read Less [-]

MATH 222A Partial Differential Equations 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces. Partial Differential Equations: Read More [+]

Prerequisites: 105 or 202A

Partial Differential Equations: Read Less [-]

MATH 222B Partial Differential Equations 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor. Partial Differential Equations: Read More [+]

MATH C223A Advanced Topics in Probability and Stochastic Process 3 Units

Terms offered: Fall 2020, Fall 2016, Fall 2015, Fall 2014 The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability. Advanced Topics in Probability and Stochastic Process: Read More [+]

Prerequisites: Statistics C205A-C205B or consent of instructor

Repeat rules: Course may be repeated for credit with instructor consent.

Also listed as: STAT C206A

Advanced Topics in Probability and Stochastic Process: Read Less [-]

MATH C223B Advanced Topics in Probability and Stochastic Processes 3 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability. Advanced Topics in Probability and Stochastic Processes: Read More [+]

Also listed as: STAT C206B

Advanced Topics in Probability and Stochastic Processes: Read Less [-]

MATH 224A Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall. Mathematical Methods for the Physical Sciences: Read More [+]

Prerequisites: Graduate status or consent of instructor

Instructors: 112 or 113C; 104A and 185, or 121A-121B-121C, or 120A-120B-120C.

Mathematical Methods for the Physical Sciences: Read Less [-]

MATH 224B Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Spring 2015, Spring 2014, Spring 2013 Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall. Mathematical Methods for the Physical Sciences: Read More [+]

MATH 225A Metamathematics 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall. Metamathematics: Read More [+]

Prerequisites: 125A and (135 or 136)

Metamathematics: Read Less [-]

MATH 225B Metamathematics 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall. Metamathematics: Read More [+]

MATH 227A Theory of Recursive Functions 4 Units

Terms offered: Spring 2021, Fall 2015, Fall 2013 Recursive and recursively enumerable sets of natural numbers; characterizations, significance, and classification. Relativization, degrees of unsolvability. The recursion theorem. Constructive ordinals, the hyperarithmetical and analytical hierarchies. Recursive objects of higher type. Sequence begins fall. Theory of Recursive Functions: Read More [+]

Prerequisites: Mathematics 225B

Instructor: 225C.

Theory of Recursive Functions: Read Less [-]

MATH 228A Numerical Solution of Differential Equations 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations. Numerical Solution of Differential Equations: Read More [+]

Prerequisites: 128A

Instructor: 128A-128B.

Numerical Solution of Differential Equations: Read Less [-]

MATH 228B Numerical Solution of Differential Equations 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations. Numerical Solution of Differential Equations: Read More [+]

MATH 229 Theory of Models 4 Units

Terms offered: Spring 2019, Spring 2015, Spring 2013 Syntactical characterization of classes closed under algebraic operations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than first-order. Theory of Models: Read More [+]

Prerequisites: 225B

Theory of Models: Read Less [-]

MATH 235A Theory of Sets 4 Units

Terms offered: Fall 2024, Spring 2024, Fall 2018 Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall. Theory of Sets: Read More [+]

Prerequisites: 125A and 135

Instructor: 125A and 135.

Theory of Sets: Read Less [-]

MATH 236 Metamathematics of Set Theory 4 Units

Terms offered: Fall 2021, Fall 2014, Fall 2010 Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity. Metamathematics of Set Theory: Read More [+]

Prerequisites: 225B and 235A

Metamathematics of Set Theory: Read Less [-]

MATH 239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2011, Fall 2008, Spring 2008 Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Discrete Mathematics for the Life Sciences: Read More [+]

Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor

Discrete Mathematics for the Life Sciences: Read Less [-]

MATH C239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2013 Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Discrete Mathematics for the Life Sciences: Read More [+]

Also listed as: MCELLBI C244

MATH 240 Riemannian Geometry 4 Units

Terms offered: Fall 2022, Fall 2021, Fall 2019 Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes. Riemannian Geometry: Read More [+]

Riemannian Geometry: Read Less [-]

MATH 241 Complex Manifolds 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2021 Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem. Complex Manifolds: Read More [+]

Prerequisites: 214 and 215A

Complex Manifolds: Read Less [-]

MATH 242 Symplectic Geometry 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2021 Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds. Symplectic Geometry: Read More [+]

Symplectic Geometry: Read Less [-]

MATH C243 Seq: Methods and Applications 3 Units

Terms offered: Spring 2015, Spring 2014 A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof. Seq: Methods and Applications: Read More [+]

Prerequisites: Graduate standing in Math, MCB, and Computational Biology; or consent of the instructor

Additional Format: <br/>

Instructor: Pachter

Also listed as: MCELLBI C243

Seq: Methods and Applications: Read Less [-]

MATH 245A General Theory of Algebraic Structures 4 Units

Terms offered: Fall 2017, Fall 2015, Spring 2014 Structures defined by operations and/or relations, and their homomorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence- and subalgebra-lattices, or category-theory and adjoint functors, or other aspects. General Theory of Algebraic Structures: Read More [+]

Prerequisites: Math 113

General Theory of Algebraic Structures: Read Less [-]

MATH 249 Algebraic Combinatorics 4 Units

Terms offered: Fall 2024, Spring 2024, Spring 2023 (I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor. Algebraic Combinatorics: Read More [+]

Prerequisites: 250A or consent of instructor

Algebraic Combinatorics: Read Less [-]

MATH 250A Groups, Rings, and Fields 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree. Groups, Rings, and Fields: Read More [+]

Prerequisites: 114 or consent of instructor

Groups, Rings, and Fields: Read Less [-]

MATH 250B Commutative Algebra 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Development of the main tools of commutative and homological algebra applicable to algebraic geometry, number theory and combinatorics. Commutative Algebra: Read More [+]

Prerequisites: 250A

Commutative Algebra: Read Less [-]

MATH 251 Ring Theory 4 Units

Terms offered: Fall 2021, Fall 2016, Spring 2013 Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods. Ring Theory: Read More [+]

Ring Theory: Read Less [-]

MATH 252 Representation Theory 4 Units

Terms offered: Fall 2021, Fall 2020, Fall 2015 Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups. Representation Theory: Read More [+]

Representation Theory: Read Less [-]

MATH 253 Homological Algebra 4 Units

Terms offered: Spring 2023, Fall 2016, Fall 2014 Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules. Homological Algebra: Read More [+]

Homological Algebra: Read Less [-]

MATH 254A Number Theory 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall. Number Theory: Read More [+]

Prerequisites: 250A for 254A; 254A for 254B

Instructor: 250A.

Number Theory: Read Less [-]

MATH 254B Number Theory 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall. Number Theory: Read More [+]

Prerequisites: 254A

MATH 255 Algebraic Curves 4 Units

Terms offered: Fall 2022, Spring 2019, Fall 2014 Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis. Algebraic Curves: Read More [+]

Prerequisites: 250A-250B or consent of instructor

Algebraic Curves: Read Less [-]

MATH 256A Algebraic Geometry 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall. Algebraic Geometry: Read More [+]

Prerequisites: 250A-250B for 256A; 256A for 256B

Algebraic Geometry: Read Less [-]

MATH 256B Algebraic Geometry 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall. Algebraic Geometry: Read More [+]

Prerequisites: 256A

MATH 257 Group Theory 4 Units

Terms offered: Spring 2021, Spring 2018, Spring 2014 Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohomological methods. Group Theory: Read More [+]

Group Theory: Read Less [-]

MATH 258 Harmonic Analysis 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2021 Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor. Harmonic Analysis: Read More [+]

Prerequisites: 206 or a basic knowledge of real, complex, and linear analysis

Harmonic Analysis: Read Less [-]

MATH 261A Lie Groups 4 Units

Terms offered: Fall 2024, Fall 2023, Fall 2022 Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall. Lie Groups: Read More [+]

Instructor: 214.

Lie Groups: Read Less [-]

MATH 261B Lie Groups 4 Units

Terms offered: Spring 2024, Spring 2023, Spring 2022 Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall. Lie Groups: Read More [+]

MATH 270 Advanced Topics Course in Mathematics 2 Units

Terms offered: Spring 2024, Fall 2023, Spring 2023 This course will give introductions to research-related topics in mathematics. The topics will vary from semester to semester. Advanced Topics Course in Mathematics: Read More [+]

Repeat rules: Course may be repeated for credit when topic changes.

Fall and/or spring: 15 weeks - 1.5 hours of lecture per week

Additional Format: One and one-half hours of lecture per week.

Grading: Offered for satisfactory/unsatisfactory grade only.

Advanced Topics Course in Mathematics: Read Less [-]

MATH 272 Interdisciplinary Topics in Mathematics 1 - 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2019 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Interdisciplinary Topics in Mathematics: Read More [+]

Repeat rules: Course may be repeated for credit without restriction.

Fall and/or spring: 15 weeks - 3-3 hours of lecture per week

Interdisciplinary Topics in Mathematics: Read Less [-]

MATH 273 Topics in Numerical Analysis 4 Units

Terms offered: Spring 2022, Spring 2016, Spring 2014 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Numerical Analysis: Read More [+]

Topics in Numerical Analysis: Read Less [-]

MATH 274 Topics in Algebra 4 Units

Terms offered: Fall 2023, Spring 2023, Fall 2022 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Algebra: Read More [+]

Topics in Algebra: Read Less [-]

MATH 275 Topics in Applied Mathematics 4 Units

Terms offered: Spring 2024, Spring 2023, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Applied Mathematics: Read More [+]

Topics in Applied Mathematics: Read Less [-]

MATH 276 Topics in Topology 4 Units

Terms offered: Spring 2021, Fall 2017, Spring 2016 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Topology: Read More [+]

Topics in Topology: Read Less [-]

MATH 277 Topics in Differential Geometry 4 Units

Terms offered: Spring 2023, Fall 2022, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Differential Geometry: Read More [+]

Topics in Differential Geometry: Read Less [-]

MATH 278 Topics in Analysis 4 Units

Terms offered: Fall 2024, Spring 2024, Fall 2021 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Analysis: Read More [+]

Topics in Analysis: Read Less [-]

MATH 279 Topics in Partial Differential Equations 4 Units

Terms offered: Fall 2024, Fall 2023, Spring 2023 Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. Topics in Partial Differential Equations: Read More [+]

Topics in Partial Differential Equations: Read Less [-]

MATH 290 Seminars 1 - 6 Units

Terms offered: Spring 2017, Spring 2015, Fall 2014 Topics in foundations of mathematics, theory of numbers, numerical calculations, analysis, geometry, topology, algebra, and their applications, by means of lectures and informal conferences; work based largely on original memoirs. Seminars: Read More [+]

Fall and/or spring: 15 weeks - 0 hours of seminar per week

Additional Format: Hours to be arranged.

Seminars: Read Less [-]

MATH 295 Individual Research 1 - 12 Units

Terms offered: Summer 2016 10 Week Session, Spring 2016, Fall 2015 Intended for candidates for the Ph.D. degree. Individual Research: Read More [+]

Fall and/or spring: 15 weeks - 1-12 hours of independent study per week

Summer: 3 weeks - 5 hours of independent study per week 6 weeks - 2.5-30 hours of independent study per week 8 weeks - 1.5-60 hours of independent study per week

Grading: The grading option will be decided by the instructor when the class is offered.

Individual Research: Read Less [-]

MATH N295 Individual Research 0.5 - 5 Units

Terms offered: Summer 2022 8 Week Session, Summer 2021 8 Week Session, Summer 2006 10 Week Session Intended for candidates for the Ph.D. degree. Individual Research: Read More [+]

Summer: 8 weeks - 1-5 hours of independent study per week

MATH N297 General Academic Internship 0.5 Units

Terms offered: Prior to 2007 This is an independent study course designed to provide structure for graduate students engaging in summer internship opportunities. Requires a paper exploring how the theoretical constructs learned in academic courses were applied during the internship. General Academic Internship: Read More [+]

Summer: 8 weeks - 2.5 hours of independent study per week

Additional Format: Two and one-half hours of independent study per week for 8 weeks.

General Academic Internship: Read Less [-]

MATH 299 Reading Course for Graduate Students 1 - 6 Units

Terms offered: Fall 2018, Fall 2017, Fall 2016 Investigation of special problems under the direction of members of the department. Reading Course for Graduate Students: Read More [+]

Fall and/or spring: 15 weeks - 0 hours of independent study per week

Summer: 6 weeks - 1-5 hours of independent study per week 8 weeks - 1-4 hours of independent study per week

Reading Course for Graduate Students: Read Less [-]

MATH 301 Undergraduate Mathematics Instruction 1 - 2 Units

Terms offered: Fall 2018, Spring 2018, Fall 2017 May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program. Undergraduate Mathematics Instruction: Read More [+]

Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center

Repeat rules: Course may be repeated for credit up to a total of 4 units.

Fall and/or spring: 15 weeks - 3 hours of seminar and 4 hours of tutorial per week

Additional Format: Three hours of Seminar and Four hours of Tutorial per week for 15 weeks.

Subject/Course Level: Mathematics/Professional course for teachers or prospective teachers

Grading: Offered for pass/not pass grade only.

Undergraduate Mathematics Instruction: Read Less [-]

MATH 302 Teaching Workshop 1 Unit

Terms offered: Summer 2002 10 Week Session, Summer 2001 10 Week Session Mandatory for all graduate student instructors teaching summer course for the first time in the Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis, classroom visitations by senior faculty member. Teaching Workshop: Read More [+]

Summer: 8 weeks - 1 hour of lecture per week

Additional Format: One hour of Lecture per week for 8 weeks.

Teaching Workshop: Read Less [-]

MATH 303 Professional Preparation: Supervised Teaching of Mathematics 2 - 4 Units

Terms offered: Spring 2017, Spring 2016, Fall 2015 Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty. Professional Preparation: Supervised Teaching of Mathematics: Read More [+]

Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor

Fall and/or spring: 15 weeks - 2-4 hours of independent study per week

Additional Format: No formal meetings.

Professional Preparation: Supervised Teaching of Mathematics: Read Less [-]

MATH 600 Individual Study for Master's Students 1 - 6 Units

Terms offered: Summer 2006 10 Week Session, Fall 2005, Spring 2005 Individual study for the comprehensive or language requirements in consultation with the field adviser. Individual Study for Master's Students: Read More [+]

Prerequisites: For candidates for master's degree

Credit Restrictions: Course does not satisfy unit or residence requirements for master's degree.

Fall and/or spring: 15 weeks - 1-6 hours of independent study per week

Summer: 8 weeks - 1.5-10 hours of independent study per week

Subject/Course Level: Mathematics/Graduate examination preparation

Individual Study for Master's Students: Read Less [-]

MATH 602 Individual Study for Doctoral Students 1 - 8 Units

Terms offered: Fall 2019, Fall 2018, Fall 2016 Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree. Individual Study for Doctoral Students: Read More [+]

Prerequisites: For qualified graduate students

Fall and/or spring: 15 weeks - 1-8 hours of independent study per week

Additional Format: One to Eight hour of Independent study per week for 15 weeks.

Individual Study for Doctoral Students: Read Less [-]

Contact Information

Department of mathematics.

970 Evans Hall

Phone: 510-642-6550

Department Chair

Michael Hutchings

949 Evans Hall

Phone: 510-642-4129

[email protected]

Vice-Chair for Graduate Affairs

Sug Woo Shin

901 Evans Hall

[email protected]

Graduate Student Affairs Officer - Academic Advising

Clay Calder

910 Evans Hall

Phone: 510-642-0665

[email protected]

Graduate Student Affairs Officer - Funding & Employment

Christian Natividad

914 Evans Hall

[email protected]

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

Alternatively, use our A–Z index

Attend an open day

Discover more about postgraduate research

PhD Pure Mathematics / Overview

Year of entry: 2024

• View full page

The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).

Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.

Full entry requirements

Apply online

In your application you’ll need to include:

• The name of this programme
• Your research project title (i.e. the advertised project name or proposed project name) or area of research
• Your proposed supervisor’s name
• If you already have funding or you wish to be considered for any of the available funding
• A supporting statement (see 'Advice to Applicants for what to include)
• Details of your previous university level study
• Names and contact details of your two referees.

Programme options

Programme description.

The The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.

Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.

For entry in the academic year beginning September 2024, the tuition fees are as follows:

• PhD (full-time) UK students (per annum): Band A £4,786; Band B £7,000; Band C £10,000; Band D £14,500; Band E £24,500 International, including EU, students (per annum): Band A £28,000; Band B £30,000; Band C £35,500; Band D £43,000; Band E £57,000
• PhD (part-time) UK students (per annum): Band A £2393; Band B £3,500; Band C £5,000; Band D £7,250; Band E 12,250 International, including EU, students (per annum): Band A £14,000; Band B £15,000; Band C £17,750; Band D £21,500; Band E £28,500

Further information for EU students can be found on our dedicated EU page.

The programme fee will vary depending on the cost of running the project. Fees quoted are fully inclusive and, therefore, you will not be required to pay any additional bench fees or administration costs.

All fees for entry will be subject to yearly review and incremental rises per annum are also likely over the duration of the course for Home students (fees are typically fixed for International students, for the course duration at the year of entry). For general fees information please visit the postgraduate fees page .

Always contact the Admissions team if you are unsure which fees apply to your project.

Scholarships/sponsorships

There are a range of scholarships, studentships and awards at university, faculty and department level to support both UK and overseas postgraduate researchers.

To be considered for many of our scholarships, you’ll need to be nominated by your proposed supervisor. Therefore, we’d highly recommend you discuss potential sources of funding with your supervisor first, so they can advise on your suitability and make sure you meet nomination deadlines.

For more information about our scholarships, visit our funding page or use our funding database to search for scholarships, studentships and awards you may be eligible for.

Contact details

Our internationally-renowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality.  Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .

Programmes in related subject areas

Use the links below to view lists of programmes in related subject areas.

• Mathematics

Regulated by the Office for Students

The University of Manchester is regulated by the Office for Students (OfS). The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. More information can be found at the OfS website .

You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .

Guide for Topics for the Qualifying Exams

The following describes the format and scope of Qualifying Exams in each of the six areas of graduate study. It is department policy that qualifiers be based on curriculum from the first year graduate sequences and any undergraduate prerequisites. Students, who have mastered those courses, should be able to pass the exams. Faculty members, who write the exams, are expected to implement this policy, and to adhere conscientiously to the guidelines that follow. Students, in turn, are expected to interpret each exam problem in a reasonable fashion, so as not to trivialize any solution. Copies of past exams and a record of previous passing scores are available from the department by request.

Qualifying Exams (affectionately known as Quals) are given twice a year and typically take place the week or two before classes begin each semester. A precise schedule is posted months in advance. Students are allowed six hours to take the exam. Food can be brought in to help fuel the brain. Faculty, who grade the exams, are expected to release the results before the last date for students to drop or withdraw from courses without receiving a DR or W on their transcripts, and within two weeks in any case.

The books listed for each area below should be more than sufficient to cover topics that will appear on the exam. It should be emphasized, however, that the exams are intended to test general knowledge and competence rather than any particular set of books or courses.

This is an accordion element with a series of buttons that open and close related content panels.

Galois Theory

• Field extensions including: algebraic and transcendental elements, finite/algebraic/Galois/simple/separable/purely inseparable field extensions, separable and inseparable polynomials.
• Splitting fields and algebraic closures.
• The fundamental theorem of Galois theory.
• Examples including: finite fields, polynomials of degree at most 4, composite extensions.
• Primitive elements.

Reference text Dummitt and Foote’s Abstract Algebra book Chapter 13 (exlcuding 13.3) and Chapter 14 (excluding 14.7, 14.8, and 14.9). (110 pages).

General Algebra

You should know the meaning of and be able to give examples and non-examples of:

• Left/right/two-sided ideals, left and right modules, bimodules
• Annihilator of a module
• Matrix ring, quaternion ring, group ring
• Division ring, simple ring, zero-divisor
• Modules: Exact sequences of modules, tensor products, Hom, localization of modules, flat/projective/free modules, support of a module

References text For a commutative ring specifically, see the references below. For a not necessarily commutative ring, see Dummitt and Foote, Chapters 7 and 10-12. (220 pages).

Commutative Algebra

• Rings and ideals: prime/maximal/radical ideals, quotient rings, integral domains, localization of rings, local rings, polynomial rings, zero-divisors, nilpotent elements, nilradical, fraction fields, Nakayama’s Lemma.
• Modules: see the list in “general algebra.”
• Noetherian rings, including chain conditions and the Hilbert Basis Theorem.

Reference text In recent years 742, which concentrates on commutative algebra, has been taught from Altman-Kleiman’s A term of commutative algebra , and the relevant chapters would be 1-5 and 8-13. (55 pages). An alternate source would be Atiyah-MacDonald’s Introduction to Commutative Algebra chapters 1-3 and 6-7. (88 pages).

Group Theory

You should know the meaning of, and be able to give an example and a non-example of the following:

• order (of a group)
• order (of a group element)
• normal subgroup
• quotient group
• abelian group, nilpotent group, lower central series, solvable group, simple group, perfect group
• commutator subgroup, centralizer, normalizer, conjugacy class
• group homomorphism
• group action, orbit, stabilizer, transitive action, faithful action
• free group, finitely presented group
• p-group, symmetric group, permutation group, alternating group, dihedral group, general linear group

You should be able to:

• State and apply the orbit-stabilizer theorem;
• Compute the conjugacy classes of a finite group;
• Work fluently with free groups, matrix groups, and symmetric groups

Reference text Dummit and Foote Chapters 1, 2.1-2.4, 3.1-3.3, 4.1-4.3, 5.1-5.2, 5.4, 6.1, and 6.3.

Linear Algebra

• Eigenvalue, eigenvector, generalized eigenspace
• Jordan normal form
• dual vector space, transpose, bilinear form, Hermitian form
• orthogonal matrix, symplectic matrix
• tensor product of vector spaces

References text Dummit and Foote, chapters 11-12.

The Analysis Qualifying Exam involves the tools from a) advanced calculus, b) Math 721, and c) one of the two courses: Math 722 (Complex Analysis) and Math 725 (Real Analysis). Choose one at the time of exam registration.

The exam usually consists of nine questions and six are to be attempted. There will be at least two from each of a), b) and c), though some problems may involve tools from more than one area. The content of 721, 722, and 725 certainly varies somewhat from instructor to instructor. Questions for 2018-2019 will come from the topics and tools below.

Recommended texts Function theory of one complex variable by Greene and Krantz ; Functions of one complex variable by J.B. Conway . Good sources for additional reading and problems: old qualifying exams , Gamelin’s Complex Analysis,  Rudin’s Real and Complex Analysis, Stein-Shakarchi : Princeton Lectures in Analysis II: Complex Analysis.

• Analytic functions and Cauchy-Riemann equations. Elementary functions, branches and principal branches.
• Line integrals. Cauchy’s theorem and Cauchy’s formula.
• Cauchy’s estimates, Liouville’s theorem, Morera’s theorem, Goursat’s theorem.
• Power series, Laurent series, and isolated singularity. Residue calculus.
• Argument principle, Rouche’s theorem, Hurwitz’s theorem, open mapping theorem.
• Simply connected domains. Normal families and Montel’s theorem.
• Conformal mappings of the unit disc and upper half-plane, fractional linear transformations. Schwarz’s lemma. Elementary conformal mappings. The Riemann Mapping Theorem.
• Harmonic functions, the mean value property and maximum principle, Harnak’s lemma and principle, subharmonic functions.
• Dirichlet problem on the unit disc. Schwarz reflection principle. Perron’s theorem.
• Mittag-Leffler’s theorem, Runge’s theorem.

Recommended texts The principal reference is Folland’s Real Analysis:  Modern Techniques and Their Applications, Chapters 1-5. Good sources for additional problems:  old qualifying exams,  Rudin’s Real and Complex Analysis , Chapters 1-8.  Chapter 2 of Rudin’s Functional Analysis (for problems on the Baire Category Theorem). Stein-Shakarchi: Princeton Lectures in Analysis III: Real Analysis.

Reference Chapters 1 and 2 of Folland.

Reference Chapter 3 of Folland, excluding functions of bounded variation.

• Basic point set topology, commensurate with Chapter 4 of Folland, particularly non-metric topologies, locally compact and locally convex spaces.

Reference Chapter 5 of Folland.

Recommended texts Details are given in the list of topics. Folland, Chapters 6-9. Rudin’s Functional Analysis , Chapters 6-8. (Distribution theory is typically taught at the level of Rudin’s  Functional Analysis , rather than Folland. Stein and Shakarchi’s  Princeton Lectures in Analysis IV: Functional Analysis , Chapter 4. (Good reference and problems for further consequences of the Baire Category Theorem.)

Reference text Chapter 6 of Folland.

Reference text Chapter 5 of Folland.

Chapter 8 of Folland, Chapter 7 in Rudin.

Chapters  1,  6, 7  Rudin.

9.3 of Folland.

Chapter 4 of Stein and Shakarchi (omitted in 2021)

Applied Math

The Applied Mathematics Qualifying Exam consists of six problems, all of which are to be attempted. The exam is based on material usually covered in undergraduate ordinary differential equations, partial differential equations, complex variables, and the first-year graduate sequence in Applied Mathematics (Math 703-704).

References Churchill, Fourier Series and Boundary Value Problems Gelfand and Fomin B, Calculus of Variation Kevorkian, Partial Differential Equations Levinson and Redheffer, Complex Variables Pinsky B, Partial Differential Equations and Boundary Value Problems Stakgold, Green’s Functions and Boundary Value Problems Strang, Introduction to Applied Mathematics Zanderer B, Partial Differential Equations

Computational Math

The Computational Mathematics Qualifying Exam is offered at the beginning of every fall and spring semester. Students have 6 hours to complete the exam of about 5-6 problems. The exam is typically 120 points in total. The material is based on Math/CS 714 and Math/CS 715. The students taking Math 714 / 715 are assumed to have a basic understanding of the undergraduate level of numerical analysis (covered in Math 513 / 514).

Covered Materials for Math Students

All materials that may appear in the exam are listed below. The components that are marked “advanced” are unlikely to appear, but we do not rule out the possibility. (Please contact the most recent instructor of 714/715 for details.)

• Basic ODE Theory: well–posedness
• Explicit and implicit methods, stability Runge-Kutta and multistep methods, stiff problems
• Numerical differentiations, uniform and nonuniform meshes
• Consistency, stability and convergence
• Multidimensional problems: ADI and fractional step methods
• Linear hyperbolic equations and their numerical discretizations
• Basic theory for nonlinear hyperbolic equations: shock formation, weak solution and entropy condition, Riemann problem
• (advanced) Shock capturing methods: Godnov and Roe methods, slope limiters, flux-splitting
• (advanced) Hamilton-Jacobi equations and the level set method for front propagation
• Fast Fourier transform
• Fourier spectral method, pseudospectral methods, Chebyshev method
• Direct and iterative methods for linear systems, eigenvalue problems, sparse matrices
• Conjugate gradient methods, nonlinear algebraic equations
• Variational formulation, Galerkin methods, energy estimate and error analysis, implementation
• (advanced) Discontinuous Galerkin, multigrid methods, boundary element method

References Many textbooks cover similar topics. If textbook (A) is listed under topic (b), that means we believe (A) organizes materials (b) better than other textbooks. However, every student is different. Ultimately please choose textbooks according to your own preferences. We only list recommendations below. Basics: Basic Numerical Analysis Suli and Mayer, An Introduction to Numerical Analysis Finite Difference Methods LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007. Spectral Methods Trefethen, Spectral Methods in MATLAB, SIAM, 2000. Gottlieb and Orzag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977. Finite Element Methods: Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009. Larson and Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013. Advanced: Finite Volume Methods LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. Monte Carlo Methods: Kalos and Whitlock, Monte Carlo Methods, J. Wiley & Sons, New York, 1986.

Geometry/Topology

Logistics of the exam:

When registering for the exam, students must choose either the algebraic topology option or the differential topology option. The algebraic topology option is based on the courses Math 751/752, and the differential topology option is based on Math 751/761.

The exam consists of two parts, Part I and Part II. Each part has three questions. Part I is the same on both exams, and covers material from 751.  Part II of the Algebraic Topology option covers material from 752, and Part II of the Differential Topology option covers material from 761.

Students are asked to answer two questions from Part I and two questions from Part II.

The exam is based on (a) background material usually covered in advanced calculus, undergraduate topology (e.g. 551) and undergraduate algebra courses (e.g. 541), and (b) topics from the first year graduate topology sequence (751, 752, 761), as identified below. Note that familiarity with basic concepts of point set topology (e.g. metric spaces, completeness, connectedness, and compactness) will be assumed, although these may not be treated in 751, 752, 761.

Reference texts: The reference text for 751 and 752 is Allen Hatcher’s Algebraic Topology. The reference text for 761 is John Lee’s Introduction to Smooth Manifolds. Ch 1-6 8 up to Lie brackets 9 up to Lie derivatives 12,14 15 up to Orientations of Manifold 16 up to Stoke’s Theorem 17 up to Homotopy Invariance Additional reference texts for 761 are Frank Warner’s Foundations of Differentiable Manifolds and Lie Groups  and Spivak’s A Comprehensive Introduction to Differential Geometry, Volume I

Description of advanced material covered by the exam.

Part I. The student should be prepared to:

• Work with the standard constructions in algebraic topology, such as homotopies, chain complexes, quotients, products, suspensions, retracts, and deformation retracts.
• Effectively use the fundamental tools of homology, reduced homology, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence for homology.
• Compute the fundamental group of an explicitly given cell complex.
• Compute the fundamental group of a space using the Seifert-Van Kampen Theorem.
• Compute the homology of an explicitly given cell complex using the definition of cellular homology.
• Make use of the standard cell structures of spheres and real and complex projective spaces in all dimensions.
• Know the fundamental and homology groups of spheres and real and complex projective spaces in all dimensions.
• Compute the homology of a space using the Mayer-Vietoris sequence.
• Make use of the long exact sequence in homology to make computations.
• Compute the Euler characteristic of a space.
• Construct finite covering spaces of an explicitly given cell complex.
• Construct covering spaces with prescribed group of deck transformations by constructing a corresponding quotient of the fundamental group.
• Use contractibility of the universal cover to deduce that certain maps are null-homotopic.
• Use local homology to distinguish two spaces.
• Use the Lefschetz fixed point theorem to find a fixed point of a continuous map.
• Combine the above machinery and techniques to solve problems.

Part II.   Algebraic option. The student should be prepared to:

• Compute the cohomology of an explicitly given cell complex using the definition of cellular cohomology.
• Effectively use the fundamental tools of cohomology, reduced cohomology, cup product, cap product, cross product, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence.
• Apply the Universal Coefficient Theorem in computations.
• Compute cup products of cohomology classes.
• Distinguish the homotopy types of two spaces using the Cohomology Ring.
• Make effective use of Poincaré duality.
• Make elementary computations of homotopy groups using the Hurewicz Theorem.
• Know the homotopy groups of the n-sphere through dimension n.
• Know the homotopy groups of the 2-sphere through dimension 3.
• Build continuous maps between cell complexes inductively using high-connectivity of the target: e.g. “Using the fact that Y is k-connected, construct a map from the given X to Y.”
• Make effective use of Whitehead’s Theorem.
• Recognize and construct fiber bundles.
• Use the long exact sequence of homotopy groups of a fibration.
• Know the standard examples of fiber bundles of spheres over spheres arising from the unit spheres in the real division algebras.

Part II. Differential Option. The student should be prepared to:

• Work with the standard concepts in differential topology, including smooth manifolds, local coordinates, transversality, regular values, the Inverse Function Theorem, tubular neighborhoods, vector fields, flows, differential forms, orientation, integration of forms, distributions, basic de Rham cohomology, and Stokes Theorem.
• Perform computations with differential forms, including integration of explicit forms over given submanifolds.
• Perform computations with the Lie derivative.
• Make use of Sard’s theorem.
• Distinguish de Rham cohomology classes given explicit forms on an explicit manifold.
• Show that a given manifold admits a smooth structure.  For example, the student should be able to show that spheres, projective spaces, Grassmannians, the special linear group, and the orthogonal group admit smooth structures.
• Construct trivializations of explicit vector bundles, such as the tangent bundle of the 3-sphere.

The Logic Qualifying Exam will consist of (usually 6) questions based on the content of the two introductory graduate courses: 770 and 773.

Students should be prepared to answer questions on the following topics. Since these topics may be presented in different ways from year to year, the student should read broadly from the references to supplement the course work.

First-order logic syntax and semantics, Completeness and Compactness Theorems, Löwenheim–Skolem Theorem, Incompleteness Theorem, decidable and undecidable theories, basic properties of ordinals and cardinals.

References Ebbinghaus, Flum and Thomas: Mathematical Logic (Chs. 1–6 and 10) Kunen: The Foundations of Mathematics Kunen: Set Theory (1980 Elsevier edition, Chs. 1 and 3)

Computability Theory

Computable sets and (partial) computable functions, Recursion Theorem, computably enumerable sets, halting problem, Turing reducibility, Turing degrees and jump, arithmetical hierarchy, index sets, low and high degrees, Martin’s high domination theorem, Friedberg and Shoenfield jump inversion, minimal degrees, exact pairs, 1-generic, hyperimmune, and hyperimmune-free degrees, diagonally non-computable functions, Π01-classes, PA degrees, low and hyperimmune-free basis theorems, finite injury, Friedberg-Muchnik theorem, Sacks Splitting theorem, priority trees, infinite injury, Sacks jump inversion, computable ordinals, Kleene’s O, hyperarithmetical hierarchy.

References Soare: Recursively Enumerable Sets and Degrees (Chs. 1–8) Ash/Knight: Computable Structures and the Hyperarithmetical Hierarchy (Chs. 4.5-5.3)

Model Theory

Elementary chains and extensions, preservation theorems, ultraproducts, quantifier elimination, model completeness, types, saturated and special models, small theories, countable categoricity, strong minimality, Baldwin-Lachlan characterization of uncountably categorical theories.

References Hodges: A Shorter Model Theory Marker: Model Theory, An Introduction (up to Ch. 6.1) Tent, Ziegler: A Course in Model Theory (Chs. 1-5)

Qualifying Exams

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Doctor of Philosophy (PhD) in Mathematics Education

Graduate Programs

The Ph.D. program emphasizes research and requires a written dissertation for completion. The program is individualized to meet the needs of graduate students. The student must develop, with the guidance from the major professor and committee, a program that is applicable to their background and interest. The average Ph.D. program requires 4-6 years beyond a master’s degree. The program is comprised of coursework in four major areas.

• Mathematics Education
• Mathematics or a related area
• Cognate Area
• Research Core

This residential program has rolling admission Applications must be fully complete and submitted (including all required materials) and all application fees paid prior to the deadline in order for applications to be considered and reviewed. For a list of all required materials for this program application, please see the “Admissions” section below.

• July 1 is the deadline for Fall applications.
• November 15 is the deadline for Spring applications.
• March 15 is the deadline for Summer applications.

This program does not lead to licensure in the state of Indiana or elsewhere. Contact the College of Education Office of Teacher Education and Licensure (OTEL) at [email protected] before continuing with program application if you have questions regarding licensure or contact your state Department of Education about how this program may translate to licensure in your state of residence.

APPLICATION PROCEDURE

Application Instructions for the Mathematics Education PhD program from the Office of Graduate Studies:

In addition to a submitted application (and any applicable application fees paid), all completed materials must be submitted by the application deadline in order for an application to be considered complete and forwarded on to faculty and the Purdue Graduate School for review.

Here are the materials required for this application:

• Transcripts (from all universities attended, including an earned bachelor’s degree from a college or university of recognized standing)
• Minimum undergraduate GPA of 3.0 on a 4.0 scale
• 3 Recommendations
• Academic Statement of Purpose
• Personal History Statement
• Writing Sample
• International Applicants must meet English Proficiency Requirements set by the Purdue Graduate School

We encourage prospective students to submit an application early, even if not all required materials are uploaded. Applications are not forwarded on for faculty review until all required materials are uploaded.

When submitting your application for this program, please select the following options:

• Select a Campus: Purdue West Lafayette (PWL)
• Select your proposed graduate major: Curriculum and Instruction
• Please select an Area of Interest: Mathematics Education
• Please select a Degree Objective: Doctor of Philosophy (PhD)
• Primary Course Delivery: Residential

Program Requirements

I. mathematics education courses (15 – 18 hours).

In mathematics education, students engage in courses that cover topics in the cognitive and cultural theories of learning and teaching mathematics, and the role of curriculum in mathematics education. A three (3) course sequence is required that consists of:

• EDCI 63500 – Goals and Content in Mathematics Education
• EDCI 63600 – The Learning of Mathematics: Insights and Issues
• EDCI 63700 – The Teaching of Mathematics: Insights and Issues

In addition, students are encouraged to take (6 – 9) hours of EDCI 620: Developing as a Mathematics Education Researcher

II. Related Course Work (minimum 6 hours)

All students should have appropriate course work in mathematics, statistics, educational technology, or a related field. Students without a master’s level background in mathematics may be required to take more courses in mathematics. This will be determined by the student’s major professor and advisory committee.

III. Cognate (9 hours)

Students will take three graduate courses in a self-selected cognate area. Cognate area selection should be discussed with the student’s major professor and advisory committee. Possible cognate areas include: mathematics, psychology, philosophy, sociology, technology.

IV. Research Core Courses (15 hours)

All doctoral students in the Department of Curriculum and Instruction must complete five (5) courses from areas in research methodology and analysis before beginning their dissertation:

• EDPS 53300 – Introduction to Research in Education
• EDCI 61500 – Qualitative Research Methods in Education
• MA 51200 – Introductory Statistics
• Advance electives in either quantitative or qualitative methods
• Ackerman Center
• Serious Games
• CnI Online Fac
• Curriculum Studies
• Education for Work and Community
• Elementary Education
• English Education
• English Language Learning
• Learning Design and Technology
• Literacy and Language Education
• Science Education
• Social Studies Education
• Applied Behavior Analysis
• Counseling and Development
• Educational Leadership and Policy Studies
• Educational Psychology and Research Methodology
• Gifted Education
• Special Education

Laura Bofferding

Amber brown, signe kastberg, rachael kenney, jill newton.

Course Registration, payment, drops/withdraws, and removing holds: [email protected] Career accounts: ITaP (765) 494-4000

Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

Request Info about graduate study Visit Apply

The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

STEM-OPT Visa Eligible

Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

Join us for Fall 2024

Many programs accept applications on a rolling, space-available basis.

Learn what you need to apply

The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
• biomedical mathematics
• discrete mathematics
• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
• mathematics of earth and environment systems
• multi-messenger and multi-wavelength astrophysics

Learn more by exploring the school’s mathematics research areas .

Michael Cromer

Basca Jadamba

Moumita Das

Carlos Lousto

Matthew Hoffman

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Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

Current Students: See Curriculum Requirements

Mathematical Modeling, Ph.D. degree, typical course sequence

Concentrations, applied inverse problems, biomedical mathematics, discrete mathematics, dynamical systems and fluid dynamics, geometry, relativity and gravitation, admissions and financial aid.

This program is available on-campus only.

Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
• A recommended minimum cumulative GPA of 3.0 (or equivalent).
• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
• Submit two letters of recommendation .
• Entrance exam requirements: None
• Writing samples are optional.
• Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.

English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

How to Apply   Start or Manage Your Application

Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

Department of Mathematics

Graduate program.

Application deadline is December 15th, 2023.

Test requirements:

GRE Subject Test:         GRE Subject Math Test scores are OPTIONAL.

GRE General Test:      GRE General Test scores are OPTIONAL.  

TOEFL or IELTS:      Scores are REQUIRED (the link below contains answers to common questions on these exams including who has to take them).

Standardized Test Questions:       Yale Graduate School of Arts & Sciences

Fee waiver:  if you wish to apply to waive the application fee (105$) please apply for the waiver here:  Application Fees & Fee Waivers | Yale Graduate School of Arts & Sciences . We recommend to do this as early as possible and, at least, several days before the deadline of January 2, 2023. Please note that the department has no control over the waivers. 

Program in Applied Mathematics . Note that there is a separate program in Applied Mathematics. You cannot apply for both programs. Follow  Welcome | Applied Mathematics Program (yale.edu)  for the general information about that program  and  https://applied.math.yale.edu/graduate-program-0  for the information about admissions, requirements, etc.

Welcome to the Yale graduate program in Mathematics.

The transition from mathematics student to working mathematician depends on ability, hard work and independence, but also on community. Yale’s graduate program provides an excellent environment for this, and we are proud of the talented students who come here and the leading faculty with whom they learn the profession.

In their first two years, students focus on building their general knowledge and passing the qualifying exams , but are also encouraged to use the time to think about their areas of interest, work together to explore them, and begin making connections with faculty advisors. There are few formal requirements and this flexibility allows students to develop independence, formulating and following their own goals.

Mathematics, while requiring intense individual focus, also thrives on collaborative work. Students form study groups and seminars together, and also benefit from our excellent cohort of Gibbs Assistant Professors and other Postdoctoral Fellows, who are a source of fresh mathematical perspectives and camaraderie.

Research, and the contribution of new ideas and results to the body of mathematical knowledge, naturally form the main focus of the next few years, and typically students complete their PhD by the end of the 5th (sometimes 6th) year. During this time they also get to know the faculty better, and continue building intellectual and personal connections, horizontally across the discipline and through time to our shared intellectual history and tradition.

Teaching is an important component of our profession, and the department provides support and training to graduate students. Teaching assignments proceed from individual coaching to classroom teaching, with careful mentoring provided by our dedicated team of lecturers.  The Lang Lunch Seminar, in the second year, provides in-depth training to graduate students before they begin to lecture.

Director of Graduate Studies : Van Vu .

Inquiries concerning the graduate program in mathematics should be sent to Van Vu .

Registrar of Graduate Studies: TBA 

Some useful links:

• The mathematics department page in the Graduate School catalog.
• Graduate school homepage for general information.
• Admissions information from the graduate school.
• Mathematics Graduate Program Advising Guidelines

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+ - Algebraic geometry Click to collapse

Geometric Invariant Theory : Faculty: Santosha Pattanayak

+ - Commutative Algebra Click to collapse

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula. Faculty : A. K. Maloo

+ - Complex Analysis & Operator Theory Click to collapse

I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory. Faculty : P. Muthukumar

+ - Computational Acoustics and Electromagnetics Click to collapse

The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few. Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research: 1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners. 2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools. 3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing. 4. High performance computing. Faculty : Akash Anand , B. V. Rathish Kumar

+ - Computational Fluid Dynamics Click to collapse

Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems. Faculty : B. V. Rathish Kumar , Saktipada Ghorai

+ - Differential Equations Click to collapse

Semigroups of linear operators and their applications, Functional differential equations, Galerkin approximations

Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.

  Faculty : D. Bahuguna  

Homogenization and Variational methods for partial differential equation

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$- convergence.

  Faculty : T. Muthukumar , B.V. Rathish Kumar  

Functional inequalities on Sobolev space

Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated

Asymptotic analysis on changing domains

Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems. Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed. Faculty : Prosenjit Roy , Kaushik Bal + - Functional Analysis & Operator Theory Click to collapse

Banach space theory

Geometric and proximinality aspects in Banach spaces. Faculty: P. Shunmugaraj  

Function-theoretic and graph-theoretic operator theory

The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. Faculty: Sameer Chavan  

Non-commutative geometry

The main emphasis is on the metric aspect of noncommutative geometry. Faculty: Satyajit Guin  

Bounded linear operators

A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations. Faculty: Sasmita Patnaik

+ - Harmonic Analysis Click to collapse

Operator spaces

The main emphasis is on operator space techniques in abstract Harmonic Analysis.

In the Euclidean setting

Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.

  Faculty : Parasar Mohanty  

On Lie groups

Problems related to integral geometry on Lie groups are being studied.

  Faculty : Rama Rawat  

  + - Homological Algebra Click to collapse

Cohomology and Deformation theory of algebraic structures

Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids.

This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.

  Faculty : Ashis Mandal + - Image Processing Click to collapse

TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.

  Faculty : B. V. Rathish Kumar + - Mathematical Biology Click to collapse

There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions.

Mathematical ecology

1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape.

2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants.

Mathematical epidemiology

1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors.

2. Mathematical Modeling of HIV Dynamics in vivo

Bioconvection

Bioconvection is the process of spontaneous pattern formation in a suspension of swimming microorganisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques.

Bio-fluid dynamics

Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.

  Faculty : Malay Banerjee , Saktipada Ghorai , B.V. Rathish Kumar  

Cardiac electrophysiology

Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc

  Faculty : B.V. Rathish Kumar + - Number Theory & Arithmetic Geometry Click to collapse

Algebraic number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.

 Faculty : Sudhanshu Shekhar

Analytic number theory

L-functions, sub-convexity problems, Sieve method

  Faculty : Saurabh Kumar Singh

Number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Selmer groups

  Faculty : Somnath Jha

Number theory, Dynamical systems, Random walks on groups

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme.

  Faculty : Arijit Ganguly + - Numerical Analysis and Scientific Computing Click to collapse

The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high-quality research in the areas that include (but are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, Computational Fluid Dynamics, Computer-Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis, and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Nyström Method, Spline and Wavelet approximations, etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.

  Faculty : B. V. Rathish Kumar , Akash Anand + - Operator Algebra Click to collapse

Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.

  Faculty : Keshab Chandra Bakshi + - Representation Theory Click to collapse

Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)

 Faculty : Santosh Nadimpalli

Representations of finite and arithmetic groups

Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.

  Faculty : Pooja Singla

Representation theory of Lie algebras and algebraic groups

 Faculty : Santosha Pattanayak

Representation theory of infinite dimensional Lie algebras

Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.

 Faculty : Sachin S. Sharma

Representation theory and Invariant theory

Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.

  Faculty : Preena Samuel

Combinatorial representation theory

String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically

  Faculty : Amit Kuber + - Set Theory and Logic Click to collapse

Set theory (MSC Classification 03Exx)

We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/

  Faculty : Ashutosh Kumar

Rough set theory and Modal logic

Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.

  Faculty : Mohua Banerjee + - Several Complex Variables Click to collapse

Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.

  Faculty : Vikramjeet Singh Chandel + - Topology and Geometry Click to collapse

Algebraic topology and Homotopy theory

The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.

  Faculty : Debasis Sen

Algebraic topology, Combinatorial topology

I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.

  Faculty : Nandini Nilakantan

Differential geometry

Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.

  Faculty : G. Santhanam

Low dimensional topology

The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.

 Faculty : Aparna Dar  

Geometric group theory and Hyperbolic geometry

Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored. Faculty: Abhijit Pal

  Faculty : Abhijit Pal

Manifolds and Characteristic classes

We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.

  Faculty : Ajay Singh Thakur

Moduli spaces of hyperbolic surfaces

The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.

  Faculty : Bidyut Sanki

Systolic topology and Geometry

We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths.

Topological graph theory

We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.

  Faculty : Bidyut Sanki + - Tribology Click to collapse

Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.

  Faculty : B. V. Rathish Kumar

Research Areas in Statistics and Probability Theory

Here are the areas of Statistics in which research is being done currently.

+ - Bayesian Nonparametric Methods Click to collapse

Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process.

Faculty member: Minerva Mukhopadhyay

+ - Data Mining in Finance Click to collapse

Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners.

Faculty: Amit Mitra , Sharmishtha Mitra

+ - Econometric Modelling Click to collapse

Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric

Faculty: Shalabh , Sharmishtha Mitra

+ - Entropy Estimation and Applications Click to collapse

Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists.

Faculty: Neeraj Misra

+ - Environmental Statistics Click to collapse

The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission

Faculty: Arnab Hazra

+ - Estimation in Restricted Parameter Space Click to collapse

In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem.

+ - Game Theory Click to collapse

The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems.

Faculty: Soumyarup Sadhukhan

+ - Machine Learning and Statistical Pattern Recognition Click to collapse

Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms.

Faculty: Subhajit Dutta

+ - Markov chain Monte Carlo Click to collapse

Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target.

Faculty: Dootika Vats

+ - Non-Parametric and Robust Statistical methods Click to collapse

Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shaperestricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest.

Faculty: Subhra Sankar Dhar

+ - Optimal Experimental Design Click to collapse

The area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few.

Faculty: Satya Prakash Singh

+ - Ranking and Selection Problems Click to collapse

About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems.

+ - Regression Modelling Click to collapse

The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be auto correlated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area.

Faculty: Shalabh

+ - Robust Estimation in Nonlinear Models Click to collapse

Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of Mestimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models.

Faculty: Debasis Kundu , Amit Mitra

+ - Rough Paths and Regularity structures Click to collapse

The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of `pathwise' solutions of stochastic differential equations and stochastic partial differential equations.

Faculty: Suprio Bhar

+ - Spatial statistics Click to collapse

The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice.

+ - Statistical Signal Processing Click to collapse

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators.

+ - Step-Stress Modelling Click to collapse

Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc.

Faculty: Debasis Kundu , Sharmishtha Mitra

+ - Stochastic Partial Differential Equations Click to collapse

The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few.

+ - Theory of Stochastic Orders and Aging and Applications Click to collapse

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.

DEPARTMENT OF Mathematics & Statistics

INDIAN INSTITUTE OF TECHNOLOGY KANPUR

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MATH 580 A: Current Topics in Mathematics

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Master of Science in Mathematics and Applied Statistics

At California State University Long Beach

The Department of Mathematics & Statistics at CSULB offers four Master of Science programs .

Teaching & Graduate Assistantships provide students with funding and with college teaching experience.

Graduates have found employment in both technical and academic workplaces. Many have obtained tenure-track community college professorships. Others have gone on to PhD programs.

MS in Mathematics, General Option

Study and explore concepts in areas including analysis, algebra, topology, and geometry, as well as the   deep connections between and among these subjects.

MS in Mathematics, Option in Applied Mathematics

Study applied math methods with an emphasis on computational skills.

MS in Mathematics, Option in Mathematics Education for Secondary School Teachers

A flexible program that includes coursework in mathematics and in mathematics education research & theory.

MS in Applied Statistics

Using conceptual foundations and statistical software packages ( SAS , R , and Python ), students are trained to analyze real world data appropriately and communicate their findings effectively. The tools learnt here will open the door for careers in data science and analytics, or prepare you for a PhD in a variety of related fields.

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Dr. John Brevik, Pure Mathematics Graduate Advisor, [email protected]

Dr. Paul Sun, Applied Math Graduate Advisor, [email protected]

Dr. Xuhui Li, Mathematics Education Graduate Advisor, [email protected]

Dr. Kagba Suaray, Statistics Graduate Advisor, [email protected]

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