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Graduate School of Arts and Sciences Programs and Policies 2023–2024

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Mathematics

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10 Hillhouse Avenue, 203.432.7058 http://math.yale.edu M.S., M.Phil., Ph.D.

Chair Wilhelm Schlag

Director of Graduate Studies Ivan Loseu

Professors Richard Beals ( Emeritus ), Jeffrey Brock, Andrew Casson ( Emeritus ), Ronald Coifman, Igor Frenkel, Howard Garland ( Emeritus ), Anna Gilbert, Alexander Goncharov, Roger Howe ( Emeritus ), Peter Jones, Richard Kenyon, Ivan Loseu, Alexander Lubotzky ( Adjunct ), Gregory Margulis ( Emeritus ), Yair Minsky, Vincent Moncrief ( Physics ), Andrew Neitzke, Hee Oh, Nicholas Read ( Physics ; Applied Physics ), Vladimir Rokhlin ( Computer Science ), Wilhelm Schlag, John Schotland, George Seligman ( Emeritus ), Charles Smart, Daniel Spielman ( Computer Science ), Van Vu, Lu Wang, John Wettlaufer ( Earth & Planetary Sciences ; Physics ), Gregg Zuckerman ( Emeritus )

Assistant Professor  Junliang Shen

Fields of Study

Fields include real analysis, complex analysis, functional analysis, classical and modern harmonic analysis; linear and nonlinear partial differential equations; dynamical systems and ergodic theory; probability; Kleinian groups, low dimensional topology and geometry; differential geometry; finite and infinite groups; geometric group theory; finite and infinite dimensional Lie algebras, Lie groups, and discrete subgroups; representation theory; automorphic forms, L-functions; algebraic number theory and algebraic geometry; mathematical physics, relativity; numerical analysis; combinatorics and discrete mathematics.

Special Requirements for the Ph.D. Degree

In order to qualify for the Mathematics Ph.D., all students are required to:

• Complete eight term courses at the graduate level, at least two with Honors grades.
• Pass qualifying examinations on their general mathematical knowledge;
• Submit a dissertation prospectus;
• Participate in the instruction of undergraduates;
• Be in residence for at least three years;
• Complete a dissertation that clearly advances understanding of the subject it considers.

All students must also complete any other Graduate School of Arts and Sciences degree requirements; see Degree Requirements under Policies and Regulations .

The normal time for completion of the Ph.D. program is five years. Requirement (1) normally includes basic courses in algebra, analysis, and topology. A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term. All qualifying examinations must be passed by the end of the second year. There is no limit to the number of times that students can take the exams, and so they are encouraged to take them as soon as possible.

The dissertation prospectus should be submitted during the third year.

The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)–(5) and obtaining an adviser.

In addition to all other requirements, students must successfully complete MATH 991 , Ethical Conduct of Research, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

Honors Requirement

Students must meet the Graduate School’s Honors requirement by the end of the fourth term of full-time study.

Teaching experience is integral to graduate education at Yale. Therefore, teaching is required of all graduate students, typically one term per year. Generally, first-year students work as coaches for calculus classes, meeting with small discussion sections of undergraduates. Second-year students often work as teaching assistants for a linear algebra class ( MATH 222 , MATH 225 , or MATH 226 ), real analysis ( MATH 255 or MATH 256 ), or discrete mathematics ( MATH 244 ); duties usually include holding office hours or leading discussion sections.

In the spring of their second year, graduate students attend the Lang Teaching Seminar ( MATH 827 ). In this lunch seminar, experienced faculty help students understand the challenges of teaching and prepare students to lead their own section of calculus in the following year and beyond.

Students who require additional support from the Graduate School after the fifth year of study must teach additional terms, if needed.

Master’s Degrees

M.Phil. See Degree Requirements under Policies and Regulations .

M.S. Students who withdraw from the Ph.D. program may be eligible to receive the M.S. degree if they have met the requirements and have not already received the M.Phil. degree. For the M.S., students must successfully complete six term courses with at least one Honors grade, perform adequately on the general qualifying examination, and be in residence at least one year.

MATH 500a, Algebra   Ivan Loseu

The course serves as an introduction to commutative algebra and category theory. Topics include commutative rings, their ideals and modules, Noetherian rings and modules, constructions with rings such as localization and integral extension, connections to algebraic geometry, categories, functors and functor morphisms, tensor product and Hom functors, and projective modules. Other topics may be discussed at the instructor’s discretion. Prerequisites: MATH 350 and MATH 370 . TTh 1pm-2:15pm

MATH 515b, Intermediate Complex Analysis   Ebru Toprak

Topics may include argument principle, Rouché’s theorem, Hurwitz theorem, Runge’s theorem, analytic continuation, Schwarz reflection principle, Jensen’s formula, infinite products, Weierstrass theorem; functions of finite order, Hadamard’s theorem, meromorphic functions; Mittag-Leffler’s theorem, subharmonic functions. TTh 2:30pm-3:45pm

MATH 520a, Measure Theory and Integration   Or Landesberg

Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals. MW 11:35am-12:50pm

MATH 525b, Introduction to Functional Analysis   Wilhelm Schlag

Hilbert, normed, and Banach spaces; geometry of Hilbert space, Riesz-Fischer theorem; dual space; Hahn-Banach theorem; Riesz representation theorems; linear operators; Baire category theorem; uniform boundedness, open mapping, and closed graph theorems. After MATH 520 . TTh 11:35am-12:50pm

MATH 526a, Introduction to Differentiable Manifolds   Subhadip Dey

This is an introduction to the general theory of smooth manifolds, developing tools for use elsewhere in mathematics. A rough plan of topics (with the later ones as time permits) includes (1) manifolds, tangent spaces, vector fields and flows; (2) natural examples, submanifolds, quotient manifolds, fibrations, foliations; (3) vector and tensor bundles,  differential forms; (4) Lie derivatives, Lie algebras and groups; (5) embedding, immersions and transversality; (6) Sard’s theorem, degree and intersection. Prerequisites: some multivariable calculus, linear algebra, and topology. TTh 11:35am-12:50pm

MATH 543b / CPSC 543b, Optimal Transport: Theory, Algorithms, and Applications to Data Science   Smita Krishnaswamy

Optimal transport started with Gaspart Monge in the 1700s when he stated the problem of moving a large pile of sand (whose shape is a probability distribution) to a target pile with minimal effort. The optimal transport plan not only gives a coupling between distributions but also a metric between such probability measures, which has found use in everything from modern neural networks to economic resource allocation problems, to shape matching in computer vision. This course covers the theoretical foundations as well as computational aspects of optimal transport starting with the original formulations as maps between discrete measures and extending to general measures as well as the key Kantorovich relaxation as a coupling between measures and its metric properties. We also cover algorithmic foundations of optimal transport using linear programs that have recently been sped-up via entropic regularizations. In addition to the primal form, we cover the dual form and relaxations which lead to integral probability metrics. We vary the ground space of optimal transport from Euclidean, to arbitrary metrics, to graphs. We move from static to dynamic formulations of optimal transport, which can provide paths of flow for dynamics that are energy-constrained. Finally, we cover important extensions such as unbalanced optimal transport which allows for transport between generic measures (without the same volume) and for Gromov-Wasserstein distances between measures on different metric spaces. Prerequisites: MATH 241 , CPSC 202 , CPSC 223 , and CPSC 365 . Knowledge of Python programming is also required. HTBA

MATH 544a, Introduction to Algebraic Topology   Sebastian Hurtado - Salazar

This is a one-term graduate introductory course in algebraic topology. We discuss algebraic and combinatorial tools used by topologists to encode information about topological spaces. Broadly speaking, we study the fundamental group of a space, its homology, and its cohomology. While focusing on the basic properties of these invariants, methods of computation, and many examples, we also see applications toward proving classical results. These include the Brouwer fixed-point theorem, the Jordan curve theorem, Poincaré duality, and others. The main text is Allen Hatcher’s Algebraic Topology, which is available for free on his website. TTh 2:30pm-3:45pm

MATH 619a, Foundations of Algebraic Geometry   Sam Raskin

This course provides an introduction to the language of basic ideas of algebraic geometry. We study affine and projective varieties, and introduce the more general theory of schemes. Our main references are Robin Hartshorne’s book and Ravi Vakil’s lecture notes. Prerequisite: commutative algebra at the level of MATH 500 /501. MW 2:30pm-3:45pm

MATH 640a / AMTH 640a or b / CPSC 640a or b, Topics in Numerical Computation   Vladimir Rokhlin

This course discusses several areas of numerical computing that often cause difficulties to non-numericists, from the ever-present issue of condition numbers and ill-posedness to the algorithms of numerical linear algebra to the reliability of numerical software. The course also provides a brief introduction to “fast” algorithms and their interactions with modern hardware environments. The course is addressed to Computer Science graduate students who do not necessarily specialize in numerical computation; it assumes the understanding of calculus and linear algebra and familiarity with (or willingness to learn) either C or FORTRAN. Its purpose is to prepare students for using elementary numerical techniques when and if the need arises. HTBA

MATH 710a / AMTH 710a, Harmonic Analysis on Graphs and Applications to Empirical Modeling   Ronald Coifman

The goal of this graduate-level class is to introduce analytic tools to enable the systematic organization of geometry and analysis on subsets of RN (data). In particular, extensions of multi-scale Fourier analysis on graphs and optimal graph constructions for efficient computations are studied. Geometrization of various Neural Net architectures and related challenges are discussed. Topics are driven by students goals. TTh 1pm-2:15pm

MATH 713a, Poisson Algebras and Poisson Geometry   Nicholas Ovenhouse

Poisson Geometry is, in some sense, a generalization of symplectic geometry and is the formalism used to describe classical mechanics. We discuss basic definitions, properties, and structural results about Poisson structures and look at many well-known examples. Beyond the basic theory, more advanced topics may include: R-matrix Poisson structures, integrable systems, Poisson structures on character varieties, and connections to cluster algebras. Prerequisites are basic algebra and differential geometry (such as in a first-year graduate course). TTh 11:35am-12:50pm

MATH 727a, Vertex Operator Algebras and Related Structures   Igor Frenkel

Vertex operator algebras (VOA) is an algebraic formulation of two-dimensional conformal field theory. This course is dedicated to general theory of VOAs and fundamental examples related to representation theory of affine Kac-Moody algebras, Virasoro algebra, and the Monster group. Modular forms and functions is an important class of structures that naturally appear in VOA theory. Various realizations of modular forms and functions also suggest the relation of VOA with the universal quantum Teichmuller space and a mathematical construction of three-dimensional quantum gravity. MW 1pm-2:15pm

MATH 728a, Kleinian Groups and Dynamics   Hee Oh

We discuss various topics on dynamics on hyperbolic manifolds. W 1pm-3:15pm

MATH 729a, Topics in Teichmuller Theory and Mapping Class Groups   Yair Minsky

Surfaces and their geometric structures play roles throughout mathematics. Of particular interest in this course are aspects of low-dimensional topology and geometry, as well as geometric group theory, but complex analysis plays a role as well. Depending on participant and lecturer interest, we cover aspects of the “classical” theory (Thurston compactification for example), coarse geometry of mapping class groups (and perhaps generalization to hierarchical hyperbolicity), and perhaps assorted topics like the study of infinite-type surfaces and their mapping class groups. TTh 2:30pm-3:45pm

MATH 827b, Lang Teaching Seminar   Brett Smith and Su Ji Hong

This course prepares graduate students for teaching calculus classes. It is a mix of theory and practice, with topics such as preparing classes, presenting new concepts, choosing examples, encouraging student participation, grading fairly and effectively, implementing active learning strategies, and giving and receiving feedback. Open only to mathematics graduate students in their second year.  MW 11:35am-12:50pm

MATH 991a / CPSC 991a, Ethical Conduct of Research   Inyoung Shin

This course forms a vital part of research ethics training, aiming to instill moral research codes in graduate students of computer science, math, and applied math. By delving into case studies and real-life examples related to research misconduct, students grasp core ethical principles in research and academia. The course also offers an opportunity to explore the societal impacts of research in computer science, math, and applied math. This course is designed specifically for first-year graduate students in computer science, applied math, and math. Successful completion of the course necessitates in-person attendance on eight occasions; virtual participation does not fulfill this requirement. In cases where illness, job interviews, or unforeseen circumstances prevent attendance, makeup sessions are offered.   0 Course cr F 2:30pm-3:20pm

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Applied Mathematics Program

Upcoming Seminars

Department of Statistics and Data Science

Ph.d. program.

Fields of study include the main areas of statistical theory (with emphasis on foundations, Bayes theory, decision theory, nonparametric statistics), probability theory (stochastic processes, asymptotics, weak convergence), information theory, bioinformatics and genetics, classification, data mining and machine learning, neural nets, network science, optimization, statistical computing, and graphical models and methods.

With this background, graduates of the program have found excellent positions in universities, industry, and government. See the list of alumni for examples.

Yale Statistics Ph.D. Program

The Department offers a broad training program comprised of the main areas of statistical theory (with emphasis on foundations, Bayes theory, decision theory, nonparametric statistics), probability theory, stochastic processes, asymptotics, information theory, machine learning, data analysis, statistical computing, and graphical methods.

With this background, graduates of the program have found excellent positions in universities, industry, and government. See the list of Ph.D. alumni for examples.

The Ph.D. program admits only a small number of new students each year. In 2012, we made eight initial offers of admission from a pool of seventy applicants, with an entering class of four.

All applications for this program should be submitted directly to the Yale Graduate School Office of Admissions through the online application page.

• Application requirements and guidelines
• GRE scores for the General Test and for the Subject Test (usually in Mathematics, sometimes in the area of the undergraduate major) should accompany an application.

All applicants should have a strong mathematical background, including advanced calculus, linear algebra, elementary probability theory, and at least one course providing an introduction to mathematical statistics. An undergraduate major may be in statistics, mathematics, computer science, or in a subject in which significant statistical problems may arise.

For those whose native language is not English, the Test of English as a Foreign Language (TOEFL) scores are required. A strong command of English is essential for success at Yale.

• The offer of admission typically includes full tuition ($35,000 in 2012) and a generous stipend ($27,300 for students entering the program in 2012) for five years. Students usually apply for a dissertation fellowship , which replaces all teaching responsibilities, in the fifth year. Consult the Graduate School's financial assistance page for details.

For incoming students whose TOEFL scores fall below the level required to satisfy the Graduate School's Oral English Proficiency Standard, the Graduate School provides a stipend for support during a three week Immersive English Program , held late in summer on the Yale campus.

• Tuition and Living Costs

Course of Study

Twelve courses, chosen in consultation with the DGS, are required before students can be admitted to candidacy after the second year.

• Stat 551 (Stochastic Processes)
• Stat 600 (Advanced Probability)
• Stat 610 (Statistical Inference)
• Stat 612 (Linear Models)
• Stat 625 (Case Studies)
• Stat 661 (Data Analysis)
• We require that students take Stat 626 (Practical work).

Students usually choose the remaining courses from a list of special topics that are offered on an irregular schedule, determined by a combination of student demand and faculty interests. See the course lists for typical offerings. Courses in other departments can also be taken with permission from the DGS; this includes select courses with undergraduate numbers such as MATH 301 and 320, for example.

• There is no foreign language requirement.

Qualifying Examinations

• Practical Exam: a written report on an analysis of a data set. Held during a five day period in December, following the end of classes.
• Theory Exam: a written paper on theoretical statistics. A one-day exam (9:00 am -- 4:00 pm) held in May of the first year. Students who do not pass the exam have the option of a retake in May of the following year.
• Oral Exam: held shortly after completion of the Theory Exam.
• The theory qualifying exam is usually based on a combination of advanced undergraduate material (as covered in Stat 241, 242, and 251/551) and graduate material at the level of Stat 600, 610, and 612.
• For the practical qualifying exam, students are expected to be comfortable with R, and have had experience at working with real data. Most students gain that experience from a combination of Stat 661, 625 and participation in the statistical consulting clinic (Stat 627).
• A typical theory exam . [Look at http://www.stat.yale.edu/dept-private/Exams/ for copies of other old exams.Yale login required].

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The SPEAK test

For further details consult that web site.

Teaching requirements

Students are required to serve as a teaching fellows for four semesters. Usually that translates to appointment as a TF4 in each semester of the third and fourth years. See Programs and Policies [page 537] for a description of the duties of teaching fellows at each level.

In cases where there is mutual agreement between the student and the DGS, one semester of the teaching requirement can be fulfilled during the second year. The 4 TF units can then be split across two courses. The Department encourages students to consider this option, both for the experience of seeing coursework from the teaching side and as a gentle way to revise basic concepts for the theory qualifying exam (if taken at the end of the second year)

The Graduate Teaching Center offers further ways for students to improve their teaching skills. For those seeking an academic career, the ability to teach well is essential.

• See http://www.yale.edu/graduateschool/academics/program.html for more details about the Teaching Fellow Program and the teaching requirement.

Summer support

Conference support, prospectus and dissertation.

Dissertation research in collaboration with one member of the faculty is begun during the third year. A second faculty member is added to the thesis committee during or before the preparation of the dissertation prospectus, a short document that outlines plans for the research and demonstrates familiarity with relevant literature.

A preliminary version of the prospectus should be submitted no later than the first week of March in the third year. The prospectus must be accepted by the department before the end of the third year.

Upon successful completion of the qualifying examination and the prospectus (as well as meeting the Graduate School Honors requirements), the student is admitted to candidacy. Most students complete the dissertation in the fifth year.

See our Alumni page for a sampling of recent Dissertation topics.

Dissertation fellowships

Further information.

Consult the Graduate School's Programs and Policies for general information about Ph.D. study at Yale.

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• Mathematics

Assistant Professor of Therapeutic Radiology; Director of Clinical Informatics, Therapeutic Radiology; Director Medical School Clerkship, Therapeutic Radiology; Medical School Thesis Oversight, Therapeutic Radiology; Radiation Safety, Therapeutic Radiology; Assistant Cancer Center Director, Bioinformatics

• Artificial Intelligence
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Emeritus Faculty

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Associate Professor of Public Health (Health Policy); Affiliated Faculty, Yale Institute for Global Health; Affiliated Faculty, Yale Institute for Foundations of Data Science

• Communicable Disease Control
• Communicable Diseases
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• Costs and Cost Analysis
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• Applied Mathematics

The graduate program in Applied Mathematics comprises the study and application of mathematics to problems motivated by a wide range of application domains. Areas of concentration include the analysis of data in very high-dimensional spaces, the geometry of information, computational biology, mathematical physics (optical and condensed matter physics), and randomized algorithms. Topics covered by the program include classical and modern applied harmonic analysis, linear and nonlinear partial differential equations, inverse problems, quantum optics, imaging, numerical analysis, scientific computing and applications, discrete algorithms, combinatorics and combinatorial optimization, graph algorithms, geometric algorithms, discrete mathematics and applications, cryptography, statistical theory and applications, probability theory and applications, information theory, econometrics, financial mathematics, statistical computing, and applications of mathematical and computational techniques to fluid mechanics, combustion, and other scientific and engineering problems.

• Programs of Study
• PhD - Doctor of Philosophy
• Applied Mathematics Program

Anna Gilbert

Director of Graduate Studies

• [email protected]

Departmental Registrar

• [email protected]
• 203-432-1278

Admission Requirements

Standardized testing requirements.

GRE is optional.

English Language Requirement

TOEFL iBT or IELTS Academic is required of most applicants whose native language is not English.

You may be exempt from this requirement if you have received (or will receive) an undergraduate degree from a college or university where English is the primary language of instruction, and if you have studied in residence at that institution for at least three years.

Academic Information

Program Advising Guidelines

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Academic Resources

Academic calendar.

The Graduate School's academic calendar lists important dates and deadlines related to coursework, registration, financial processes, and milestone events such as graduation.

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Registration Information and Dates

https://registration.yale.edu/

Students must register every term in which they are enrolled in the Graduate School. Registration for a given term takes place the semester prior, and so it's important to stay on top of your academic plan. The University Registrar's Office oversees the systems that students use to register. Instructions about how to use those systems and the dates during which registration occurs can be found on their registration website.

Financial Information

Phd stipend & funding.

PhD students at Yale are normally full-funded for a minimum of five years. During that time, our students receive a twelve-month stipend to cover living expenses and a fellowship that covers the full cost of tuition and student healthcare.

• PhD Student Funding Overview
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Alumni Insights

Below you will find alumni placement data for our departments and programs.

• Field Descriptions

Development has long been one of the strengths of the Yale Economics Department. The field benefits from its rich history of leadership in the profession, and from the institutional support of the  Economic Growth Center . The focus of the group is on empirical microeconomic analysis of development using a wide variety of methods.

Overview of Courses

We offer a two semester sequence. Econ 730 and Econ 731 together cover a wide variety of topics, including household modeling; the organization and analysis of production in firms and farms; informal institutions, including families and networks; aspects of risk; analysis of credit, land and labor markets; savings; education; health; aggregate growth; project evaluation; and technology adoption.

There is a  development economics seminar  that meets weekly at which faculty from other universities and advanced graduate students make presentations. The Economic Growth Center also sponsors a  weekly lunch  at which more informal presentations of work in early stages are made by faculty and graduate students.

Yale has one of the finest research groups in econometrics in the academic world. The department has consistently led international rankings in econometrics over the last three decades. Our faculty have research interests in all the major fields of econometrics, and the department provides a rich training ground and finishing school for aspiring econometricians. Over the past thirty years, the department has nurtured the development of more than 70 econometrics Ph.Ds, many of whom are now prominent econometricians working in universities, government agencies, or the financial industry.  

Yale faculty play leading editorial roles in the major econometrics journals. The journal  Econometric Theory  has been hosted at Yale since its establishment in 1985. Following its longstanding tradition of supporting research in quantitative economics, the Cowles Foundation provides a uniquely supportive environment for econometric work in all its modern manifestations from theory to practice and amidst its growing number of sub-disciplines from time series econometrics and financial econometrics through to microeconometrics and spatial econometrics.

The Cowles Foundation funds a regular influx of short term and long term academic visitors, post-docs, and doctoral students from other institutes, who contribute to the research atmosphere in econometrics and provide an additional intellectual resource for our own graduate students. Many prominent econometricians from around the world visit the department and spend sabbatical terms at Yale.

The Yale econometrics group has close interactions with applied fields, particularly industrial organization, labor, macroeconomics, development, structural microeconomics, and finance. These interactions assist our graduate students in developing applied interests to accompany their research in econometric theory.

The department offers an intensive six-semester sequence of courses in econometric theory and its applications. These courses enable incoming students to cover foundational material in probability theory and econometric methods. Students with strong backgrounds are encouraged to enter the second-year sequence which covers modern asymptotic theory, parametric and nonparametric modeling, time series, panel data methods, and microeconometrics. Further advanced topics courses are available in the following year as well as courses taught by faculty who specialize in empirical work.

Requirements

The primary departmental requirement in econometrics is an Applied Econometrics Paper. This requirement helps students acquire experience in applied econometric work, including the use of econometric software and programming techniques, which are valuable skills for all practicing economists, irrespective of specialization.  Click for further information on this requirement .

The Department runs three weekly workshop meetings in econometrics. A formal  Econometrics Seminar  hosts speakers invited from other universities to report on their latest research and to provide overviews of developing research areas. A less formal  Econometrics Research Workshop  enables students and faculty to discuss their own ongoing work and go over the details of technical proofs in their papers. The Workshop also provides a venue for short-term visitors to discuss extensions and applications of the work presented in the Econometrics Seminar. An informal  Econometrics Prospectus Lunch , funded by the Cowles Foundation, is intended primarily for our graduate students to assist them in moving forward with their own research agendas, to prepare them for writing a dissertation prospectus, and to report on ideas and early findings. The Lunch enables faculty to discuss recent ideas and wider issues of econometrics, including the history of econometric thought. The Lunch is a convenient venue also for our former students who are working in government or industry to report on their work in these sectors.

Other Recommended Courses

Students are encouraged to take advanced courses in Statistics, Mathematics, and Computer Science Departments, where many complementary courses are offered.

Economic history has long been an important part of the core curriculum in the graduate program at Yale. In addition to faculty in the economics department, there are faculty in the history department and other parts of the university with interests in economic history. Find addtional information on the  Program in Economic History page .

    Timothy W. Guinnane     Professor of Economics, Yale University     Contact information & office hours:  https://sites.google.com/site/timothywguinnanec/(link is external)

We offer to graduate-level courses intended for economics PhD students every year. Econ 580 focuses on Europe. Econ 581 deals with the United States. In addition, we typically have one or more long-term visitors in residence, and these people offer courses about other times and places. Economics Ph.D. students doing a field in economic history may also be interested in taking courses offered by the History Department.  These courses do not satisfy the PhD requirement, and they will not be part of the basic course offerings for a field in the Economic Department, but they can be useful additions for students in some areas.  Interested students should consult the economic history faculty in the Economics Department for further information.

There one is the formal weekly  workshop . Most presentations at this workshop are by speakers invited from other universities.

Ph.D. Requirement

Students in the economics Ph.D. program must have completed either Econ 580 or Econ 581 before they can be admitted to candidacy. Note that Econ 585 does not satisfy this requirement.

Economic History in the Oral Exam

Economic history can be either the major or the minor field in a student’s oral examinations. In each case, the student will have taken both Econ 580 and Econ 581. An oral exam with economic history as a minor field will cover the material from these two courses as well as material from a single topic agreed by the students and instructors in economic history. An oral exam with economic history as a major field will cover Econ 580 and Econ 581. In addition, the student will be responsible for a presentation that could lead to a dissertation topic, and for additional readings relevant to that project.

Students intending to write a dissertation in economic history will probably find it useful to take courses offered in the history department or elsewhere in the university. There are no formal requirements of this sort, however, and such additional coursework is agreed after discussion with the relevant faculty.

Microeconomic Theory at Yale is characterized by a large faculty and comprehensive course offerings across all specializations, foremost among them decision theory, game theory, and general equilibrium theory.

The  Cowles Foundation  is supporting a  Research Prorgram in Economic Theory  which hosts every year a number of long-term visitors.

Microeconomic Theory I (Econ 500a) and II (Econ 501b) is a two-course core sequence for all students in the Ph.D. program. Material covered includes consumer and producer theory, choice under uncertainty, general equilibrium theory, game theory, information economics, and mechanism design. The sequence is designed to provide a thorough overview of microeconomic tools that will be used by Ph.D. students in all fields; it also prepares students for the comprehensive exams taken at the end of the first year in the program.

Advanced Microeconomics I (Econ 520a) and II (Econ 521b) is a two-course sequence examining in more depth foundational issues in game theory, information economics, mechanism design, and social choice.

Mathematical Economics I (Econ 530a) and II (Econ 531b) is a two-course sequence focused on issues in general equilibrium theory. Typically, these sequences are taken by Ph.D. students in the second year, including both those who will end up specializing in microeconomic theory and those who will do applied research using advanced tools of microeconomic analysis.

Course Requirements and Descriptions

The first-year sequence (Econ 500a and 501b) is designed for first-year students in the Economics Ph.D. program. Other students should seek the permission of the instructor. The first-year sequence is a prerequisite for the second-year courses (Econ 520a, 521b, 530a, 531b).

     — ECON 500a: Microeconomic Theory I      — ECON 501b: Microeconomic Theory II      — ECON 520a: Advanced Microeconomics I      — ECON 521b: Advanced Microeconomics II      — ECON 530a: Mathematical Economics I      — ECON 531b: Mathematical Economics II

A central element of the research environment in microeconomic theory are the:

     —  Micro Theory Lunch  (Tuesday, 12:00 - 1:00 pm)      —  Microeonomic Theory Workshop  (Wednesday, 2:30 - 4:00 pm)

both of which are an integral and indispensable part of intellectual communication in microeconomic theory.

Recommended Mathematics Courses of Theory Graduate Students

In microeconomic theory, mathematics and statistics represent important languages and techniques to express our ideas. We, therefore, recommend that graduate students in economic theory at Yale take or audit a stream of mathematics and statistics classes. If you take one course a term starting in the second year on, then by the end of the fifth year you will have at least eight important courses to support your ability to develop, analyze, and solve mathematical models. At Yale, there are a number of courses we can recommend and a partial list is the following:

     — AMTH 237a: Optimization and Complexity      — AMTH 462a: Graphs and Networks     — MATH 250a: Vector Analysis      — MATH 260a: Basic Analysis in Function Spaces      — MATH 301a: Introduction to Analysis      — MATH 305b: Real Analysis MATH      — MATH 320a: Measure Theory and Integration MATH      — MATH 325b: Introduction to Functional Analysis

     — STAT 251b: Stochastic Processes      — STAT 330b: Advanced Probability      — STAT 637a: Deterministic and Stochastic Optimization

Yale has played a central role in the development of modern Financial Economics, ever since James Tobin created the Capital Asset Pricing Model and championed the general equilibrium approach to macroeconomics and finance.  John Geanakoplos, Bob Shiller, and Eduardo Davila, together with the faculty in the School of Management, and the faculty in related fields in economics, cover nearly all areas of finance, theoretical and empirical.  Yale is an exciting place to do research in finance.

The core sequences in Microeconomics (Econ 500a and 501b) and Macroeconomics (Econ 510a and 511b) provide our students with the first exposure to core ideas in Financial Economics.  Students interested in Financial Economics as a field can take Econ 530a and Econ 531b, a full second-year sequence that studies frontier topics, with particular emphasis on macro-finance, financial theory, and general equilibium.  Topics covered in Mathematical Economics I and II include general equilibrium with incomplete markets (GEI), collateral equilibrium and the leverage cycle, default and punishment, adverse selection and moral hazard in general equilibrium, monetary equilibrium, asset price bubbles, limits to arbitrage, liquidity and credit crunches, intermediation and banking, financial regulation, dispersed information and learning in financial markets, and financial recessions.

In addition to the sequence offered by the Department, our students have access to the multiple courses offered by the finance group at the School of Management. These courses include: Financial Economics I (MGT 740/Econ 670), Empirical Asset Pricing (MGT 747), Empirical Corporate Finance (MGT 748), Financial Crises (MGT 746/Econ 674), Behavioral Finance (MGT 745/Econ 672), and Household Finance (MGT 744). Our students often work on cutting-edge topics at the intersection of Financial Economics and other fields, including Macroeconomics, Economic Theory, or Industrial Organization, among others.

Visit the  Finance at Yale webpage(link is external)  for updated information regarding course offerings and other finance related information.

There is a weekly Finance Workshop, hosted at the School of Management, in which outside speakers make presentations. There is a weekly Finance Lunch co-sponsored by the department and the School of Management where Yale PhD students and faculty present work in progress. The Macroeconomics and Theory weekly workshops often host Finance related outside speakers. Graduate students interested in Finance often present work in progress in other lunches hosted at the department.

Industrial Organization at Yale is a strong and distinctive group. Over the past 10–20 years, Yale ranks as one of the largest producers of top IO graduate students, and Yale currently places at least one IO student in a faculty position in the “top 20” virtually every year.

Faculty in IO have a set of strongly interrelated research interests in empirical IO, microeconomic policy, and related methodological techniques. This yields a highly collaborative and collegial research environment. Regular formal and informal interactions with the IO-based Economics group at the School of Management (SOM) broadens the group, particularly in areas of policy interest, creating a larger, vibrant IO community. In recent years, the field has benefited greatly from the formation and growth of the Structural Microeconomics program at the Cowles Foundation.

We offer a two-semester sequence (Econ 600 and 601) that covers a broad range of topics. The first semester begins by locating the study of industrial organization within the broader research traditions of economics and related social sciences. Alternative theories of decision making, organizational behavior, and of market evolution are sketched and contrasted with standard neoclassical theories. The semester includes a detailed examination of the determinants and consequences of industrial market structure. The second semester moves on to policy issues including public control of utilities and antitrust regulation, as well as modeling of dynamic oligopoly, collusion, and technological change.

The  IO seminar  meets weekly and hosts leading senior and junior faculty from around the world. There is also an  IO Prospectus Workshop  primarily for Yale graduate students presenting thesis work at various stages of development.

The International Trade group at Yale extends across the department and the School of Management and includes faculty who work on theoretical, empirical, and quantitative trade models, trade policy, political economy, and international finance. The field also benefits from the university’s globalization initiative.

To complete the International Trade sequence, students must take at least two of the following three courses:  720, 721, and 794.  On the theoretical side, these courses cover the theory of international trade, policy, and institutions.  Specifically, the sequence covers a discussion of classical, neo-classical, and imperfect-competition-scale-economies-based static models of trade; dynamic extensions of some of the issues, such as gains from trade, tariffs and quotas, customs unions and free trade areas, and the political economy of trade policymaking.  On the empirical side, the sequence covers a variety of topics in international trade with particular emphasis on current research areas.  Topics include tests of international trade theories; studies of the relationship between international trade, labor markets, and income distribution; recent trade liberalization episodes in developing countries; empirical assessment of various trade policies, such as VERs and Anti-Dumping; productivity (and its relation to international trade liberalization); and exchange rates, market integration, and international trade.”

We have a weekly  International Trade Workshop  at which faculty from Yale and other universities and advanced graduate students make presentations. The department also sponsors a weekly  International Trade Lunch  where Yale faculty and graduate students present work in its early stages.

With Joe Altonji, Costas Meghir, and several very strong junior faculty, Yale has one of the top labor economics groups in the country. The faculty and graduates are active in research on traditional topics such as determinants of wages and human capital accumulation as well as newer areas such as the impact of health care reform, financial literacy, and behavioral economics.

The department offers a two-semester sequence in Labor Economics (630 and 631). The first semester of the sequence includes topics such as static and dynamic approaches to demand, human capital and wage determination, wage income inequality, unemployment and minimum wages, matching and job turnover, implicit contract theory, and the efficiency wage hypothesis, while the second semester covers static and dynamic models of labor supply, firm-specific training, compensating wage differentials, discrimination, household production, bargaining models of household behavior, intergenerational transfers, and mobility.

There is a joint  Labor/Public Economics Workshop  that meets weekly at which faculty from other universities and advanced graduate students make presentations. The department also sponsors a weekly  Labor/Public Economics Prospectus Workshop  at which more informal presentations of work in its early stages are made by faculty and graduate students in the field.

Yale has a long and storied tradition of excellence in macroeconomics and is currently one of the most vibrant macro research groups in the world. It includes faculty leading the field in both theory and empirical research as well as several faculty influencing current policy debates. It consistently trains and places macroeconomists in leading universities and policy institutions around the world.

The graduate macro sequence consists of two core courses (510 and 511) and two advanced courses (525 and 526). The core courses analyze short-run determination of aggregate employment, income, investment, saving, prices, interest rates, asset prices, as well as growth, fiscal and monetary policy. To this purpose, core courses extensively train students in the methodology of modern dynamic economics: Dynamic Programming, Vector Autoregressions, Equilibrium concepts, and computational methods. The advanced courses are topical and track frontier research in macroeconomics. Prominent examples of recently covered topics are heterogeneous agent economics with adjustment costs to capital and labor, wealth inequality in incomplete market economies with financial market imperfections, optimal taxation, and search theory of unemployment.

We have a weekly  Macroeconomics Workshop  at which faculty from other universities and advanced graduate students make presentations. The department also sponsors a weekly  Macro Lunch  where Yale faculty and graduate students present work in its early stages.

Political economy is a relatively new strength at Yale, benefiting from new faculty and increased collaboration with the Yale Political Science department.

The Political Economy sequence consists of two courses: 790 and 791. The first covers basic theoretical models but focuses on empirical political economy. It covers topics such as voting, campaign finance, the behavior of government representatives, incumbency, and corruption. Econ. 791 begins with lectures on Thomas Piketty’s  Capital in the 20th Century  and then studies the debates in egalitarian political philosophy beginning with John Rawls about the nature of distributive justice.  Philosophical proposals are critiqued with the aid of economic models.

Political Economy Research is presented every week in the  Leitner Political Economy Seminar , as well as in several of the regular economics seminars including  Macro ,  International , and  Labor/Public .

Our faculty in public economics come together from several other sub-fields in the department. These include macro faculty who do theoretical public finance, environmental faculty who work on policies to address climate change, and health economists who work on health care policy. And of course, several of our labor and development faculty also work on public policy issues.

The department offers a two-semester sequence in Public Finance (680 and 681). The sequence covers theories of government provision of public goods, moral hazard, and adverse selection. Empirical methodologies will vary from standard reduced-form techniques to structural estimation. Substantive areas will include health economics, taxation, social security, and non-health components of government spending.

There is a joint  Labor/Public Economics Workshop  that meets weekly at which faculty from other universities and advanced graduate students make presentations. The department also sponsors a weekly  Labor/Public Prospectus Economics Workshop  at which more informal presentations of work in its early stages are made by faculty and graduate students in the field.

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Nsf graduate research fellowship program (grfp) awardees at mcdb.

Congratulations to Frances Moore (Bahmanyar Lab), Eddy Tzintzun-Tapia (Nachtergaele Lab), Karina Ascunce (van Wolfswinkel & Carlson Labs), and Sandra Temgoua (Isaacs Lab) for being awarded NSF GRFP Fellowships!  

The NSF GRFP program recognizes and supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based master’s and doctoral degrees at accredited United States institutions.

Visit  www.nsfgrfp.org/  to learn more.

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Bringing an investigator’s eye to complex social challenges

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Anna Russo likes puzzles. They require patience, organization, and a view of the big picture. She brings an investigator’s eye to big institutional and societal challenges whose solutions can have wide-ranging, long-term impacts.

Russo’s path to MIT began with questions. She didn’t have the whole picture yet. “I had no idea what I wanted to do with my life,” says Russo, who is completing her PhD in economics in 2024. “I was good at math and science and thought I wanted to be a doctor.”

While completing her undergraduate studies at Yale University, where she double majored in economics and applied math, Russo discovered a passion for problem-solving, where she could apply an analytical lens to answering the kinds of thorny questions whose solutions could improve policy. “Empirical research is fun and exciting,” Russo says.

After Yale, Russo considered what to do next. She worked as a full-time research assistant with MIT economist Amy Finkelstein . Russo’s work with Finkelstein led her toward identifying, studying, and developing answers to complex questions. 

“My research combines ideas from two fields of economic inquiry — public finance and industrial organization — and applies them to questions about the design of environmental and health care policy,” Russo says. “I like the way economists think analytically about social problems.”

Narrowing her focus

Studying with and being advised by renowned economists as both an undergraduate and a doctoral student helped Russo narrow her research focus, fitting more pieces into the puzzle. “What drew me to MIT was its investment in its graduate students,” Russo says.

Economic research meant digging into policy questions, identifying market failures, and proposing solutions. Doctoral study allowed Russo to assemble data to rigorously follow each line of inquiry.

“Doctoral study means you get to write about something you’re really interested in,” Russo notes. This led her to study policy responses to climate change adaptation and mitigation. 

“In my first year, I worked on a project exploring the notion that floodplain regulation design doesn’t do a good job of incentivizing the right level of development in flood-prone areas,” she says. “How can economists help governments convince people to act in society’s best interest?”

It’s important to understand institutional details, Russo adds, which can help investigators identify and implement solutions. 

“Feedback, advice, and support from faculty were crucial as I grew as a researcher at MIT,” she says. Beyond her two main MIT advisors, Finkelstein and economist Nikhil Agarwal — educators she describes as “phenomenal, dedicated advisors and mentors” — Russo interacted regularly with faculty across the department. 

Russo later discovered another challenge she hoped to solve: inefficiencies in conservation and carbon offset programs. She set her sights on the United States Department of Agriculture’s Conservation Reserve Program because she believes it and programs like it can be improved. 

The CRP is a land conservation plan administered by USDA’s Farm Service Agency. In exchange for a yearly rental payment, farmers enrolled in the program agree to remove environmentally sensitive land from agricultural production and plant species that will improve environmental health and quality.

“I think we can tweak the program’s design to improve cost-effectiveness,” Russo says. “There’s a trove of data available.” The data include information like auction participants’ bids in response to well-specified auction rules, which Russo links to satellite data measuring land use outcomes. Understanding how landowners bid in CRP auctions can help identify and improve the program’s function. 

“We may be able to improve targeting and achieve more cost-effective conservation by adjusting the CRP’s scoring system,” Russo argues. Opportunities may exist to scale the incremental changes under study for other conservation programs and carbon offset markets more generally.  

Economics, Russo believes, can help us conceptualize problems and recommend effective alternative solutions.

The next puzzle

Russo wants to find her next challenge while continuing her research. She plans to continue her work as a junior fellow at the Harvard Society of Fellows, after which she’ll join the Harvard Department of Economics as an assistant professor. Russo also plans to continue helping other budding economists since she believes in the importance of supporting other students.   

Russo’s advisors are some of her biggest supporters. 

Finklestein emphasizes Russo’s curiosity, enthusiasm, and energy as key drivers in her success. “Her genuine curiosity and interest in getting to the bottom of a problem with the data — with an econometric analysis, with a modeling issue — is the best antidote for [the stress that can be associated with research],” Finklestein says. “It's a key ingredient in her ability to produce important and credible work.”

“She's also incredibly generous with her time and advice,” Finklestein continues, “whether it's helping an undergraduate research assistant with her senior thesis, or helping an advisor such as myself navigate a data access process she's previously been through.”

“Instead of an advisor-advisee relationship, working with her on a thesis felt more like a collaboration between equals,” Agarwal adds. “[She] has the maturity and smarts to produce pathbreaking research.

“Doctoral study is an opportunity for students to find their paths collaboratively,” Russo says. “If I can help someone else solve a small piece of their puzzle, that’s a huge positive. Research is a series of many, many small steps forward.” 

Identifying important causes for further investigation and study will always be important to Russo. “I also want to dig into some other market that’s not working well and figure out how to make it better,” she says. “Right now I’m really excited about understanding California wildfire mitigation.” 

Puzzles are made to be solved, after all.

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Rebecca Kramer-Bottiglio wins NSF Alan T. Waterman Award

Creating robots that adapt and evolve to changing conditions

Her multidisciplinary research approach incorporates materials science, robotics, artificial intelligence, biology and art. Her breakthrough design strategy of adaptive morphogenesis for robotics is inspired by biological adaptations, morphogenesis, and evolution, and extends techniques she developed to create shape-shifting and phase-changing materials. When integrated into robots, these new materials enable transformative new capabilities via unprecedented adaptive behaviors.

“Robots are traditionally mechanical assemblies designed to complete a single task in a specific context. In contrast, biological organisms adapt physiologically, behaviorally, and morphologically in response to changing environmental conditions,” Kramer-Bottiglio said. “My research focuses on imbuing next-generation robots with the ability to adapt and evolve like animals. I study multifunctional materials and their utility in soft-material robots capable of evolving their physical structure and corresponding behavior as they encounter new task demands or environments.”

Kramer-Bottiglio developed an amphibious turtle-inspired robot with limbs made from coupled fluidic actuators and variable-stiffness materials that change between hydrodynamic flippers and load-bearing leg shapes depending on the environment. Adapting both the form and function of the robot enabled efficient movement both in water and on land, yielding energetic savings for multi-environment locomotion. With this morphing robot platform, she proved that morphological adaptation is a favorable strategy to expand the future capabilities of robots while simultaneously reducing the energy requirements to do so.

“Breaking through my own conception of what robots can be helped me realize their potential,” she said. “If we discard the notion that robots must be mechanical assemblies of discrete components and instead consider them as material systems with multifunctionalities, we gain access to a host of new materials, techniques and tools that can be used to create the next generation of robots.”

“The NSF has been incredibly important throughout my career, supporting my research in areas such as materials processing, soft robotics and engineering education,” she said. “For instance, one NSF award supported a collaborative research project to test the hypothesis that the implementation of soft robot design experiences improves students’ engineering self-efficacy and interest in engineering careers as compared to traditional robot design experiences. Through a partnership with experts in engineering education, we found evidence to support this claim. The research resulted in a new engineering design curriculum that has been sustainably integrated into an upper high school course with national dissemination. I am unsure that I would have been able to pursue such an interdisciplinary project without the support of the NSF.”

The Waterman Award will enable Kramer-Bottiglio to pursue interdisciplinary and high-risk research directions that hold the potential to unlock new capabilities for robots, enabling them to evolve on demand in response to changing tasks or environments, she said. “Robots that can evolve on demand provide a platform to study physical evolution-based optimization. How and when should a robot adapt its shape when switching between tasks or environments? What are the optimal shapes for specific tasks and environments? Can we use synthetic evolution to discover adaptation strategies that natural life might use to adapt to our changing environment? Can the use of living materials in robots increase their adaptive capabilities? I hope to find out.”

In addition to a medal, awardees each receive $1 million over five years for research in their chosen field of science. The Waterman Award will be presented to the recipients at a ceremony during the National Science Board meeting, which will be held in Washington, D.C., on May 1.

“This substantial funding is a vital investment in the future of science discovery and innovation, empowering the awardees to deepen their research, expand their projects, and explore new frontiers in their field,” Panchanathan said. “More importantly, it enables these exceptional individuals to apply their groundbreaking work for the betterment of their communities and society at large. This award is a seed for future growth and impact, offering the awardees the resources they need to turn their vision into tangible solutions for some of the most pressing challenges of our time.”

The award, established by Congress in 1975, is named for Alan T. Waterman, NSF's first director.

Former MIT researcher who killed Yale graduate student sentenced to 35 years in prison

A former MIT researcher has been sentenced to 35 years in prison for the killing of a Yale University graduate student in Connecticut

NEW HAVEN, Conn. -- A former researcher at the Massachusetts Institute of Technology was sentenced Tuesday to 35 years in prison for the killing of a Yale University graduate student found shot outside his car on a Connecticut street.

Qinxuan Pan, 33, who pleaded guilty to murder in February, apologized during a hearing in a New Haven courtroom packed with family and friends of the victim, Kevin Jiang.

“I feel sorry for what my actions caused and for everyone affected,” Pan said. “I fully accept my penalties.”

Jiang, 26, a U.S. Army veteran who grew up in Chicago and a graduate student at Yale's School of the Environment, had just left his fiancée's apartment in New Haven on the evening of Feb. 6, 2021, when he was shot multiple times by Pan, according to police and prosecutors. The couple had just gotten engaged days earlier.

Several of Jiang's relatives and friends spoke in court before the judge handed down the sentence, which Pan agreed to as part of his plea bargain.

“My son was a remarkable young man who cherished life and held deep (belief) in God. He had a bright future ahead — one that promised to spread God’s love far and wide,” said Jiang's father, Mingchen Jiang.

A motive for the killing was never made entirely clear. Investigators said they discovered that Pan and Jiang's fiancée were connected on social media and had met while at MIT, where both had graduated from and where Pan was working as a researcher at the time of the shooting.

According to the documents, Jiang’s fiancée told authorities she and Pan “never had a romantic or sexual relationship, they were just friends, but she did get a feeling that he was interested in her during that time.”

After the shooting, Pan fled the scene and eluded police for three months before being apprehended in Alabama, where officials said he was caught living under a fake name with $19,000 in cash, a passport and several cellphones.

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Former MIT researcher who killed Yale graduate student sentenced to 35 years in prison

NEW HAVEN, Conn. — A former researcher at the Massachusetts Institute of Technology was sentenced Tuesday to 35 years in prison for the killing of a Yale University graduate student found shot outside his car on a Connecticut street.

Qinxuan Pan, 33, who pleaded guilty to murder in February, apologized during a hearing in a New Haven courtroom packed with family and friends of the victim, Kevin Jiang.

“I feel sorry for what my actions caused and for everyone affected,” Pan said. “I fully accept my penalties.”

Jiang, 26, a U.S. Army veteran who grew up in Chicago and a graduate student at Yale’s School of the Environment, had just left his fiancée’s apartment in New Haven on the evening of Feb. 6, 2021, when he was shot multiple times by Pan, according to police and prosecutors. The couple had just gotten engaged days earlier.

Several of Jiang’s relatives and friends spoke in court before the judge handed down the sentence, which Pan agreed to as part of his plea bargain.

“My son was a remarkable young man who cherished life and held deep (belief) in God. He had a bright future ahead — one that promised to spread God’s love far and wide,” said Jiang’s father, Mingchen Jiang.

A motive for the killing was never made entirely clear. Investigators said they discovered that Pan and Jiang’s fiancée were connected on social media and had met while at MIT, where both had graduated from and where Pan was working as a researcher at the time of the shooting.

According to the documents, Jiang’s fiancée told authorities she and Pan “never had a romantic or sexual relationship, they were just friends, but she did get a feeling that he was interested in her during that time.”

After the shooting, Pan fled the scene and eluded police for three months before being apprehended in Alabama , where officials said he was caught living under a fake name with $19,000 in cash, a passport and several cellphones.