cmu applied math phd

Mathematical Sciences

Mellon college of science, research areas, algebra and number theory, analysis, calculus of variations, pde & applications, combinatorics, computational mathematics, numerical analysis, and optimization, mathematical finance, probability.

• Algebraic topology
• Arithmetic statistics
• Combinatorial and analytic number theory

Theresa Anderson Florian Frick Prasad Tetali

• Diffusion, particle systems, stochastic partial differential equations
• Energy-driven systems
• Fluids and continuum mechanics
• Free and moving boundary problems
• Harmonic Analysis
• Multiscale problems
• Optimal transport
• Materials science
• Physical sciences
• Data science
• Image processing

Theresa Anderson Irene Fonseca William J. Hrusa Gautam Iyer David Kinderlehrer Giovanni Leoni Robin Neumayer Robert Pego (Emeritus) Matthew Rosenzweig Dejan Slepčev Ian Tice Noel J. Walkington

• Extremal combinatorics
• Probabilistic method
• Random Graphs and Randomized Algorithms
• Ramsey theory
• Topological methods in combinatorics

Tom Bohman Boris Bukh Gerard Cornuejols (Emeritus) Florian Frick Alan Frieze Po-Shen Loh Wesley Pegden Prasad Tetali Konstantin Tikhomirov Michael Young

• Data-driven modeling
• Mathematics and computation in data sciences
• Nonsmooth and nonconvex optimization
• Numerical linear algebra
• Numerical methods for nonlinear partial differential equations and variational problems

Dejan Slepčev Shlomo Ta'asan --> Noel J. Walkington

• Applied algebraic topology
• Convexity and polytopes
• Geometric and topological combinatorics
• Geometric functional analysis and concentration of measure
• The Abelian sandpile
• Tiling, packing, and covering problems

Boris Bukh Florian Frick Robin Neumayer Wesley Pegden Tomasz Tkocz

• Lambda-calculus and combinatory logic
• Model theory
• Semantics of programming languages
• Descriptive set theory
• Large cardinals, forcing and inner models
• Type theory

Clinton Conley James Cummings Rami Grossberg Ernest Schimmerling Richard Statman

• Martingale problems and Markov processes
• Equilibrium theory
• Stochastic control
• Stochastic differential equations
• Viscosity solutions of Hamilton-Jacobi-Bellman equations

William J. Hrusa Dmitry Kramkov Martin Larsson Steven E. Shreve (Emeritus) Johannes Wiesel

• Discrete Random Structures
• Markov Chain Mixing Times
• Random Dynamics
• Stochastic Homogenization and Random Media
• Stochastic Partial Differential Equations

Alan Frieze Gautam Iyer Dmitry Kramkov Martin Larsson Wesley Pegden Agoston Pisztora Dejan Slepčev Prasad Tetali Konstantin Tikhomirov Tomasz Tkocz Johannes Wiesel

Best Graduate Schools for Applied Mathematics Best Graduate Schools for Discrete Mathematics and Combinatorics QS World University Rankings

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Introduction

While Mathematics started out as an algorithmic science, mathematicians through the centuries have emphasized structural results rather than algorithmic or constructive ones. In recent years, however, because of both philosophical reasons and the advent of computers, there has been a renewed interest in the computational areas of mathematics. The branches of mathematics that deal with discrete structures, which we broadly classify as Combinatorics, are deriving particular benefit from this phenomenon. Traditionally, Combinatorics has been concerned with the existence, enumeration, and counting of discrete structures with certain properties; in the case of counting, for example, closed form or recurrence formulas were sought. The recent exploration of the computational aspects of Combinatorics has greatly contributed to the development of such areas as graph theory, matroid theory, and polyhedral theory, to name a few.

In the 40's and 50's, the field of Operations Research was born from the need to model real-world problems, other than those found in the natural sciences, in a systematic way, and put them into a mathematical form solvable by algorithms. Optimization under various constraints became the basic paradigm of quantitative methods for management, economics, and decision making in general. Linear and nonlinear programming emerged as new disciplines with pervasive applications. The simplex method proved to be a practical tool for solving large scale problems. Finally, the need for integer variables to deal with the discrete nature of the actions modeled or the indivisibility of the objects involved has led to the development of integer and combinatorial optimization, currently one of the most active research areas in applied mathematics. Examples of popular models in combinatorial optimization are network flows, including transportation and shortest path problems, matchings in graphs, set covering, and the traveling salesman problem.

The field of Computer Science was established in the 60's. In the 70's, computer scientists developed the striking theory of NP-completeness, which reduces to one another a whole range of computational problems from a myriad of areas, many of them dealing with discrete structures such as scheduling, number theory, logic, and graph theory. New research areas in the computational aspects of combinatorics were introduced, such as parallel algorithms and data structures.

In the last couple of decades, researchers in Mathematics, Operations Research, and Computer Science have made breakthroughs in developing polynomial time bounded algorithms for several important problems, including linear programming. In the case of the so-called NP-complete problems, substantial progress has been made in developing algorithms for approximate as well as exact solutions. Further, the probabilistic analysis of algorithms, where the efficiency of an algorithm is measured for an average instance, has been gaining ground. Interest in geometry has been revived through the study of its computational and combinatorial aspects. The geometry of numbers deals with the interplay between integer lattices and convex bodies; computational geometry addresses problems arising in robotics; and the geometry of integer polyhedra plays a central role in combinatorial optimization.

The mathematics used by computer scientists and operations researchers overlap to a large extent. The boundaries between Operations Research and Computer Science have become blurred. Important new theories and whole fields, like polyhedral combinatorics, have been and are being developed jointly by computer scientists, operations researchers, and applied mathematicians who consider themselves a little bit of both. Presentations of new results on graphs and matroid theory can be heard at Operations Research conferences, while papers on linear programming, network flows, and matchings in graphs are frequently presented at Computer Science conferences. The mathematical content of the papers has become greater and more diverse. Yet, in spite of this, few Ph.D students graduate with an equally solid knowledge of all three areas.

The Ph.D program in Algorithms, Combinatorics, and Optimization at Carnegie Mellon is intended to fill this gap. The program brings together the study of the mathematical structure of discrete objects and the design and analysis of algorithms in areas such as graph theory, combinatorial optimization, integer programming, polyhedral theory, computational algebra, geometry, and number theory.

Course Requirements and Research

Besides the core courses, a host of other courses are available for the students to take. Some examples are combinatorial optimization, graph coloring, matroid theory, convex polytopes, location theory, sequencing and scheduling, large scale OR, parallel algorithms, probabilistic analysis of algorithms, approximation algorithms, machine learning theory, cryptography, nonlinear programming, and optimal control theory. In the event that a student has already mastered a core course at the graduate level when entering the program, another course from the same department may be substituted. Approval must be obtained from the ACO Coordinating Committee and is given on a case-by-case basis. In addition, the students are expected to participate in the weekly research seminar on Algorithms, Combinatorics, and Optimization.

At the end of their third semester in residence, the students take a comprehensive qualifying examination based on the material in the above core courses. At this stage (or earlier), they choose a faculty member to supervise their research and dissertation. Approximately a year before the expected graduation date, the student must make a thesis proposal before a thesis committee, composed of the advisor and two or more faculty members of the student's choosing. The final transition point is the thesis defense, which is presented before the same committee. To graduate, the students will also need some teaching experience and must demonstrate adequate programming skill.

Questions about the ACO program can be addressed to any member of the ACO faculty.

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Carnegie Mellon University PhD in Applied Mathematics

The main focus area for this major is Computational Mathematics . For more details on this concentration, visit its profile page.

Applied Mathematics is a major offered under the mathematics and statistics program of study at Carnegie Mellon University. We’ve pulled together some essential information you should know about the doctor’s degree program in applied math, including how many students graduate each year, the ethnic diversity of these students, and more.

If there’s something special you’re looking for, you can use one of the links below to find it:

• Graduate Cost
• Online Learning
• Student Diversity
• Related Majors
• Focus Areas

Featured Programs

Learn about start dates, transferring credits, availability of financial aid, and more by contacting the universities below.

BA in Mathematics

If you have a knack for mathematics and an interest in learning more, study online to achieve your career goals at Southern New Hampshire University. Our mathematics degree can help you enhance your mathematical abilities, including reasoning and problem-solving in three areas: analysis, algebra and statistics.

BA in Mathematics - Applied Mathematics

Put mathematical concepts to work to solve today's most complex real-world problems by studying applied mathematics with this specialized online bachelor's from Southern New Hampshire University.

How Much Does a Doctorate in Applied Math from Carnegie Mellon Cost?

Carnegie mellon graduate tuition and fees.

In 2019-2020, the average part-time graduate tuition at Carnegie Mellon was $633 per credit hour for both in-state and out-of-state students. The average full-time tuition and fees for graduate students are shown in the table below.

Does Carnegie Mellon Offer an Online PhD in Applied Math?

Online degrees for the Carnegie Mellon applied math doctor’s degree program are not available at this time. To see if the school offers distance learning options in other areas, visit the Carnegie Mellon Online Learning page.

Carnegie Mellon Doctorate Student Diversity for Applied Math

Male-to-female ratio.

None of the students who received their PhD in applied math in 2019-2020 were women.

Racial-Ethnic Diversity

Of those students who received a doctor’s degree at Carnegie Mellon in applied math at 2019-2020, none were racial-ethnic minorities*.

PhD in Applied Math Focus Areas at Carnegie Mellon

Applied Mathematics students may decide to major in one of the following focus areas.

Majors Related to a PhD in Applied Math From Carnegie Mellon

You may also be interested in one of these majors related to applied mathematics.

View All Applied Mathematics Related Majors >

*The racial-ethnic minorities count is calculated by taking the total number of students and subtracting white students, international students, and students whose race/ethnicity was unknown. This number is then divided by the total number of students at the school to obtain the racial-ethnic minorities percentage.

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Best Applied Math Programs

Ranked in 2023, part of Best Science Schools

The applied math discipline is geared toward students

The applied math discipline is geared toward students who hope to use their mathematical prowess in business organizations, government agencies and other job sites. These are the best graduate schools for applied math. Read the methodology »

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Undergraduate Admission

Mathematical sciences, help us push the boundaries of our collective mathematical knowledge even further..

"One of my favorite things about being a math major is all of the amazing faculty I have gotten to know! All of my professors have been so friendly, accommodating and incredibly knowledgeable."

Mellon College of Science

Mathematics provides much of the language and quantitative underpinnings of the natural and social sciences — mathematical scientists built the foundation for modern computational and computer science and developed many of the most-used tools in business management. Whether you are drawn to the abstract beauty of theoretical math or the problem-solving elegance of applied math, our Mathematical Sciences program has a place for you. Here, thinkers and challengers are exploring and defining mathematics. Our faculty are advancing the leading edge of science, conducting research in an array of fields. The collaborative nature of CMU means we work across disciplines to support real-world applications. 

Mathematical Sciences Majors and Minors

Choose the path that fits you best. Browse all current Mathematical Sciences curriculums and courses. (opens in new window)

Bachelor of Science Minor

Mathematical Sciences offers wide latitude to tailor your courses to your interests. Within the major, you can choose a concentration that aligns with your goals:

• The Mathematical Sciences concentration is the least structured of our programs, in recognition of the variety of interests that can be productively coupled with the study of mathematical sciences. It can be an appropriate choice for students planning for graduate studies or those seeking to design their curriculum to take a second major from another department in the University.
• The Operations Research and Statistics concentration prepares students to enter operations research. Mathematicians with a background in operations research are especially valuable in such diverse activities as project planning, production scheduling, market forecasting and finance. 
• The Statistics concentration prepares students for areas ranging from experimental design and data analysis in the sciences, medicine, and engineering, to modeling and forecasting in business and government, to actuarial applications in the financial and insurance industries. 
• The Discrete Mathematics and Logic Concentration provides a background in discrete mathematics, mathematical logic, and theoretical computer science. This concentration prepares the student to do research in these and related fields, or to apply their ideas elsewhere.
• The Computational and Applied Mathematics Concentration provides the background needed to support the computational and mathematical analysis needs of a variety of businesses and industries and is well suited to students with an interest in the physical sciences and engineering.

The minor will allow you to take courses across mathematical disciplines and includes space for electives.

Computational Finance

Bachelor of Science

The Mellon College of Science, the Heinz College of Public Policy and Management and the Tepper School of Business jointly offer this degree uniquely designed to prepare you for the quantitative needs of the finance industry. You will develop a deep knowledge of mathematics, probability, statistics, and the applications of these disciplines to finance. Graduates work in finance or other industries where applied mathematics training is appropriate, or pursue advanced degrees in economics, finance or the mathematical sciences.

Mathematical Sciences and Economics

This flexible program allows you to develop depth in both fields of study, providing courses that complement and develop depth of understanding of economic theory, applied economics, and applied mathematics. Students pursuing this degree will be well prepared to begin their research careers in academia, government, and industry. 

Discrete Mathematics and Logic

This minor develops the fundamentals of discrete mathematics and logic necessary to understand the mathematical foundations of many computer-related disciplines.

Department of Mathematical Sciences

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High School Course Requirements

*Four years of mathematics should include at least algebra, geometry, trigonometry, analytic geometry, elementary functions (pre-calculus) and preferably calculus. Advanced mathematics courses are encouraged, especially a course in calculus.

Submit your application for the Mellon College of Science.

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You’re not just one thing. You’re a scientist. An artist. A technologist. A maker. A writer. Carnegie Mellon has been mixing it up for decades, and whatever you want to pursue, we’ve got the right mix for you.

The CMU Experience

How to Major in Mixing-It-Up

Undergraduate Catalog

Department of mathematical sciences courses, about course numbers:.

Each Carnegie Mellon course number begins with a two-digit prefix that designates the department offering the course (i.e., 76-xxx courses are offered by the Department of English). Although each department maintains its own course numbering practices, typically, the first digit after the prefix indicates the class level: xx-1xx courses are freshmen-level, xx-2xx courses are sophomore level, etc. Depending on the department, xx-6xx courses may be either undergraduate senior-level or graduate-level, and xx-7xx courses and higher are graduate-level. Consult the Schedule of Classes each semester for course offerings and for any necessary pre-requisites or co-requisites.

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Machine Learning - CMU

Master's students.

Wenhao Ding

Ph.D. student in Mechanical Engineering

Tsinghua University, B.S. in Electronic Engineering

Research interests: Deep Generative Models, Adversarial Attack, and Causal Inference

Mohsen Ferdosi

PhD student in Computational Biology

Sharif University of Technology, B.S. in Computer Engineering

Research interests: Machine Learning, Algorithm Design for Large Datasets, Computational Biology

PhD student in Biological Sciences

Fudan University, B.S. in Biological Science

Research interests: Interpretable Machine Learning, Computational Biology

Pablo Guarda

PhD student in Civil and Environmental Engineering

Pontifical Catholic University of Chile, B.S. in Industrial Engineering

Research interests: Continuous Optimization, Deep Learning, Transportation Network Analysis

Samantha Horn

PhD student in Behavioral Decision Research

King's College of London, B.A. in Classics; Yale University, M.A. in International and Development Economics

Research interests: Causal Inference and Healthcare Applications

PhD student in Mechanical Engineering

Ohio State University, B.S. in Mechanical Engineering

Research interests: Sequential data analysis, dynamic system prediction, drug discovery

PhD student in Electrical and Computer Engineering

Tsinghua University, B.E. in Electronic Engineering

Research interests: Machine learning (especially Generative Adversarial Networks, or GANs), and the applications in privacy, security, and networking

Keane Lucas

United States Air Force Academy, B.S. in Computer Science

Research interests: Adversarial Machine Learning, Machine Learning on Malware, Reinforcement Learning, AI Safety

Rohan Saxena

BITS Pilani (Pilani Campus), Bachelor of Engineering (honors) in Computer Science

Research interests: Computer Vision, Robotics

University of Toronto, BSc in Biological Physics

Research interests: Computational Biology

Yanhan (Savannah) Tang

PhD student in Operations Management

Tsinghua University, B.Mgmt in Information Management and Information Systems

Research interests: Multimodal Learning, Time Series Analysis, Online Learning, Meta Learning

Xian Jiaotong University, B.S. in Biomedical Engineering

Research interests: Few-shot learning, self-supervised learning

Shuyan Wang

PhD student in Logic, Computation and Methodology

University of Illinois in Urbana-Champaign, B.S. in Chemistry and Psychology

Research interests: Causal Discovery

Yuyang Wang

Tongji University, B.Eng. in Engineering Mechanics

Research interests: Deep Learning, Computational Biology

PhD student in Mathematical Sciences

University of Science and Technology of China, B.S. in Mathematics

Research interests: Statistical Machine Learning

Tsinghua University, B.Eng. in Vehicle Engineering ; B.Eng. in Management

Research interests: Reinforcement Learning, Robotics

PhD student in Chemistry

Nanjing University, B.S. in Chemistry

Research interests: Deep Learning, NLP

Tianming Zhou

Tsinghua University, B.E. in Computer Science and Technology (Yao Class)

Research interests: Hypergraph Neural Network, Probabilistic Graphical Model, Self-Supervised Learning

Michael R. Agaby

McGill University, BSc in Statistics and Computer Science

Research interests: Computer Vision, NLP

Sudeep Agarwal

Georgia Institute of Technology, B.S. in Computer Science

Research interests: Recommender Systems, Privacy-Aware Machine Learning, Natural Language Processing

Neeraj Aggarwal

University of Illinois at Urbana-Champaign, B.S. in Computer Science

Research interests: Secure Machine Learning, Security & Privacy

Dhananjay Ashok

University of Toronto, Honors Bachelors of Science

Research interests: Robust and trustworthy AI, Verification and Compliance of Deep Learning Systems

University of California, Los Angeles in Mathematics

Research interests: Machine Learning and Systems, Computer Vision

Titas Chakraborty

Indian Institute of Technology Bombay, B.Tech in Electrical Engineering

Research interests: Reinforcement Learning, Speech Processing, Image Processing

Pranit Chawla

Indian Institute of Technology Kharagpur, B.Tech in E&ECE

Research interests: Vision and Language, Multi-modal Representation Learning

Tongji University, B.S. in Applied Mathematics

Research interests: Machine Learning, Optimization

Leihao Chen

University of Illinois at Urbana-Champaign, B.S. in Electrical Engineering

Research interests: Machine Learning Systems

Nicholas Chen

University of Illinois, BS in Computer Science

Research interests: Deep learning, finance, healthcare, secure machine learning, computer vision

Peking University, B.S. in Computer Science

Research interests: Deep Learning, Computer Vision, Medical Image Processing, Natural Language

Vikram Duvvur

Research interests: Adversarial Machine Learning, ML for Healthcare, and Computer Vision

Adiel Felsen

Binghamton University, B.S. in Computer Science, B.A. in Mathematics

Research interests: Computer Vision, Deep Learning, Biomedical Applications

Olivier Filion

McGill University, B.Sc. in Mathematics and Computer Science

Research interests: Machine learning applications to finance, security, sports and social good

Richa Gadgil

California Polytechnic State University San Luis Obispo, B.S. in Computer Science

Research interests: Statistical Machine Learning, Bayesian Deep Learning, and Machine Learning Systems and Deployment

Sankalp Garg

Indian Institute of Technology Delhi, B.Tech in Electrical Engineering

Research interests: Reinforcement Learning, NLP and Representation Learning

Indian Institute of Technology, Bombay; B. Tech. in Computer Science and Engineering (with Honors); Minor in Artificial Intelligence

Research interests: Machine Learning, Natural Language Processing, Information Retrieval, Quantitative Finance Applications

University of Chicago, B.S. in Computer Science

Research interests: Computer Vision, Computer Graphics, Deep Learning

Quincy Hughes

Amherst College, BA in Computer Science, Economics

Research interests: ML, Optimization

Pratik Elias Jacob

IIIT Hyderabad, B. Tech (Honours) in Computer Science and Engineering

Research interests: Computer Vision, NLP, Data Mining

Sedrick Scott Keh

Hong Kong University of Science and Technology, BSc in Data Science and Mathematics

Research interests: Deep Learning, Language Understanding and Synthesis, Social Media

Alexandre Kirchmeyer

Ecole Polytechnique, France - MSc in Computer Science

Research interests: Machine Learning, Computer Vision

Soundarya Krishnan

BITS Pilani Goa Campus, B.E. in Computer Science + M.Sc. in Physics

Research interests: Trustworthy Machine Learning and Healthcare Applications

Harsha Vardhan

Vellore Institute of Technology, B.Tech. in Computer Science and Engineering

Research interests: Knowledge Graphs, Deep Generative Models, Computer Vision

Shanghai Jiao Tong University, University of Michigan, B.S.E in Computer Science

Research interests: Deep learning, Data Mining, Computer Vision, Reinforcement Learning

Columbia University, B. S. in Applied Math

Research interests: Deep Learning, Explainable Machine Learning, ML Applications in Healthcare

Vanderbilt University, B.S. in Math and Computer Science

Research interests: Data Visualization, Computer Vision, Geometric Deep Learning

UC Berkeley, B.A.s in Computer Science, Linguistics, and Data Science

Research interests: Few shot learning, information theory, generation, NLP

Rice University, B.A. in Computer Science and Mathematics

Research interests: Efficient, robust, and interpretable Machine Learning systems

Pranav Mani

National Institute of Technology Trichy, B-Tech in ECE

Research interests: Reinforcement Learning, Computer Vision, and Robotics

Hari Manikandan

Vellore Institute of Technology, B.Tech in Computer Science and Engineering

Research interests: Deep Representation Learning, Healthcare Data Analytics, Image and Language Processing

Aboli Marathe

SCTR's Pune Institute of Computer Technology, B.Eng. in Computer Engineering

Research interests: Computer Vision, Reinforcement Learning, Machine Learning

Blake Martin

University of Michigan, B.S.E. in Data Science

Research interests: Computer Vision, Deep Learning, and Human-Centered Applications

Jared Mejia

Pomona College, B.A. in Computer Science

Research interests: Computer Vision, Reinforcement Learning, and Deep Learning

Mukuntha N S

Indian Institute of Technology Patna, B.Tech in Computer Science and Engineering

Research interests: Machine Learning, Natural Language Processing, Computer Vision

Chenhao Niu

Tsinghua University, B.Eng. in Computer Science and Technology

Research interests: Recommendation Systems, Graph Neural Networks, and Generative Models

UC Berkeley, B.A. in Computer Science, B.A. in Applied Mathematics

Research interests: Reinforcement Learning, CV, Generative Modeling, Self-supervised learning

UCLA, B.S. in Computer Science, B.S. in Applied Mathematics

Research interests: Reinforcement Learning, Self-Supervised Learning, Neuroscience and Learning Theory

Kousik Rajesh

Indian Institute of Technology, Guwahati, B.Tech in Computer Science and Engineering

Research interests: Optimization, Scalable Machine Learning, Big Data

Shanghai Jiao Tong University, B.Eng. in Computer Science

Research interests: Deep Learning Theory, Optimization

Manley Roberts

Research interests: Unsupervised Learning, Reinforcement Learning, Deep Learning, ML Systems

Ekanan Settsuwan

Chulalongkorn University, B.Eng. in Petroleum Engineering

Research interests: Data Mining, Machine Learning, and Deep Learning

Ishaan Shah

Brown University, B.S. in Applied Mathematics-Computer Science

Research interests: Meta Reinforcement Learning, Causal Mechanisms for Deep Learning

Saumya Gaurang Shah

Indian Institute of Technology Kanpur, B.Tech. in Electrical Engineering

Research interests: Quantitative Finance, Applications of ML in Learning Disabilities, Education, and Robotics

Xi'an Jiaotong University, B.S. in Information and Computational Science

Research interests: Adversarial Machine Learning, Natural Language Processing

Andrew Shen

University of Melbourne, B. Sci. in Computing and Software Systems

Research interests: Reinforcement Learning, Computer Vision, Uncertainty Aware Models

Arvind Srinivasan

PES University, B.Tech. in Computer Science

Research interests: Deep Learning and Computer Vision

Mashrin Srivastava

Research interests: Health AI, AI for Good, Optimization, Explainability, ML systems, Secure & Private AI

Chloé, Huangyuan Su

Nanyang Technological University, B.S. in Mathematical Sciences

Research interests: Reinforcement Learning, Physics-Informed Machine Learning, and Financial Applications

University of Michigan, B.S.E. in Computer Science

Research interests: Deep Learning, Reinforcement Learning, Computer Vision, and NLP

Varun Ursekar

Wesleyan University, BA in Physics

Research interests: Few-shot learning, Neuroscience and Biology, Natural Language Processing

Sukriti Verma

Delhi Technological University, B. Tech in Computer Science and Engineering

Research interests: Reinforcement Learning, AI Safety, Understanding AI, Explainable AI

Hongjian Wang

Research interests: Foundations of Machine Learning, Optimization

William Wong

University of California, Berkeley, B.A. in Computer Science

Research interests: Deep Learning, Machine Perception, NLP and Robotics

Purdue University, B.S. in Computer Science, Statistics-Math, and Aviation Management

Research interests: Data Mining, Machine Learning Theory, Time Series, Applications in Education & Transportation

Jiannan Xiang

University of Science and Technology of China, B.Eng. in Electronic Information Engineering

Research interests: Natural Language Processing, Computer Vision

Lucien Lu Xu

The Chinese University of Hong Kong, B.Sc in Computer Science

Research interests: Natural Language Understanding, Generative Modeling

Vanderbilt University, BA in Neuroscience and Computer Science

Shanghai Jiao Tong University, B.Eng. in Computer Science and Technology

Research interests: Deep Learning, Natural Language Processing, and Graph Learning

Zhiying Zhu

Zhejiang University, B.Eng. in Computer Science

Research interests: Speech/Music, Multi-modal Machine Learning

Ketong Chen

Carnegie Mellon University, B.S. in Chemical Engineering

Research interests: Deep Reinforcement Learning, Computer Vision, Music Applications

Albert Liang

Carnegie Mellon University, B.S. in Computer Science, B.S. in Computational Finance

Research interests: Statistical Machine Learning, Deep Learning, Time Series, Machine Learning Systems

Carnegie Mellon University, B.S. in Statistics and Machine Learning

Research interests: Bias and Fairness, Time Series Forecasting

Research interests: Explainable AI, AI Alignment

Poojan Palwai

Carnegie Mellon University, B.S. in Artificial Intelligence

Atul Raghunathan

Carnegie Mellon Unversity, B.S. in Computer Science

Research interests: Statistical Machine Learning, Natural Language Processing

Carnegie Mellon University, B.S in Artificial Intelligence

Research interests: Deep Learning, Statistical Machine Learning, and their Finance Applications

Tianyu Zhang

Research interests: Machine Learning System, Recommendation System

Zhenlin Wang

National University of Singapore, B.S. in Applied Mathematics

Research interests: Bayesian Optimization, Bandit, Reinforcement Learning and Recommender System

Ph.D. Program

The degree of Doctor of Philosophy in Applied Mathematics and Computational Science is conferred in recognition of marked ability and high attainment in advanced applied and computational mathematics, including the successful completion of a significant original research project. The program typically takes four to five years to complete, although this length may vary depending on the student. Below, we describe the requirements and expectations of the program. All graduate students require a 3.0 GPA to graduate (no exceptions).

Written Preliminary Exam

Upon entry into the Ph.D. program, students are required to take the Written Preliminary Exam, typically scheduled the week before classes start in the Fall semester. The coverage of the exam is in Linear Algebra, Advanced Calculus, Complex Variables, and Probability at the undergraduate level. Details of the exam can be found here: Preliminary Exam Details

The student must pass the exam to continue as a Ph.D. student. The Written Exam is offered in April and August. If the student fails on the first attempt, two more attempts are granted (three attempts total).

Course Requirements

The student must take the following six core courses:

• Analysis: AMCS 6081/6091 (MATH 6080/6090)
• Numerical Analysis: AMCS 6025/6035
• Probability and Stochastic Processes: AMCS 6481/6491 (MATH 6480/6490)

These six core courses are to be completed in the first and second years of graduate studies.

Ten elective courses (a total of 14 courses) are required for graduation. These elective courses should be chosen according to the interests and/or research program of the student and must contain significant mathematical content. Whether a given course can be counted toward AMCS elective course credit will be decided in consultation with the Graduate Chair. Recent courses approved for elective credit can be discussed with your advisor and the Graduate Group Chair.

Deviations from the above may be necessary or recommended depending on the individual student; such decisions are made with the approval of the graduate chair.

Choosing an Advisor

In the first two years of graduate studies, students must choose their thesis advisor. Some students already have an advisor to whom they have committed upon entry to the program. Other students will typically start working with their prospective advisors in the latter half of the first year or the summer between the first and second year.

The purpose of the oral exam is to assess a student’s readiness to transition into full-time research and eventually write his or her dissertation. This exam will be taken by the end of the third year of graduate study.

First, an oral exam committee must be formed, consisting of three faculty members, two of whom must belong to the AMCS graduate faculty. The student must then produce a document of up to about 20 pages describing the research proposal and background material, which is then approved by the oral exam committee before the exam. In the exam, the student will give an oral presentation to the committee. A discussion with the committee follows this. In the oral exam, the committee may ask the student about the presentation as well as about necessary background material as seen fit by the committee. If the student fails this exam, the student will have one more attempt.

Dissertation and Defense

The dissertation must be a substantial original investigation in the field of applied mathematics and computational science, done under the supervision of a faculty advisor. A Ph.D. Thesis Committee consists of at least three faculty members, including the thesis advisor. When the dissertation is complete, it must be defended in a Dissertation Exam, at which the student will be expected to give a short public exposition of the results of the thesis and to satisfactorily answer questions about the thesis and related areas.

Teaching Assistant

Full-time students admitted to our Ph.D. program who are offered a financial support package for four years of study are required to be teaching assistants during the second year. Students for whom English is not their native language are required to pass a test the “Speak Test” (IELTS) demonstrating proficiency in English. More information can be found on the English Language Programs  web page.

https://www.elp.upenn.edu/institute-academic-studies/requirements

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Ph.D. Program

Introduction.

These guidelines are intended to help familiarize graduate students with the policies governing the graduate program leading to the degrees of Doctor of Philosophy (Ph.D.) in Applied Mathematics. This material supplements the graduate school requirements found on the  Graduate Student Resources  page and the  Doctoral Degree Policies  of the graduate school. Students are expected to be familiar with these procedures and regulations.

The Doctor of Philosophy program

The Doctor of Philosophy (Ph.D.) Degree in Applied Mathematics is primarily a research degree, and is not conferred as a result of course work. The granting of the degree is based on proficiency in Applied Mathematics, and the ability to carry out an independent investigation as demonstrated by the completion of a doctoral dissertation. This dissertation must exhibit original mathematical contributions that are relevant to a significant area of application.

Course requirements for the Ph.D. program

• AMATH 561, 562, 563
• AMATH 567, 568, 569
• AMATH 584, 585, 586
• AMATH 600: two, 2-credit readings, each with a different faculty member, to be completed prior to the start of the student's second year.
• Students must take a minimum of 15 numerically graded courses. At most two of these can be at the 400 level or be cross listed with courses at the 400 level. Graduate level courses previously taken at UW (e.g., during a Master's program) count toward this requirement. Graduate level courses taken outside of UW may count toward the requirement for 15 numerically graded courses with the approval of the Graduate Program Coordinator. The entire course of study of a student and all exceptions to this list must be approved by the Graduate Program Coordinator and the student’s advisor or faculty mentors.

For students who entered the doctoral program autumn 2017 or autumn 2018, please see these degree requirements. For students who entered the doctoral program prior to autumn 2017, please see these degree requirements.  

Faculty mentoring

Upon arrival, incoming students will be assigned two faculty mentors. Until a student settles on an advisor, the faculty mentors aid the student in selecting courses, and they each guide the student through a 2-credit independent reading course on material related to the student’s research interest. The faculty mentors are not necessarily faculty in the Department of Applied Mathematics.

Faculty advisor

By the end of a student’s first summer quarter, an advisor must be determined.  T he advisor provides guidance in designing a course of study appropriate for the student’s research interests, and in formulating a dissertation topic.

A full Supervisory Committee should be formed four months prior to the student’s General Exam. The full Supervisory Committee should have a minimum of three regular members plus the Graduate School Representative , and will consist of at least two faculty members from Applied Mathematics, one of whom is to be the Chair of the Committee. If the proposed dissertation advisor is a member of the Applied Mathematics faculty, then the advisor will be the Chair. The dissertation advisor may be from another department,  or may have an  affiliate  (assistant, associate, full) professor appointment with the Applied Mathematics department  and is then also a member of the Supervisory Committee.

The Dissertation Reading Committee , formed after the General Exam,  is a subset of  at least   three members from the Supervisory Committee   who are appointed to read and approve the dissertation.  Two members of the Dissertation Reading Committee must be from the Applied Mathematics faculty. At least one of the committee members must be a member of the core  Applied Mathematics faculty. It is required that this member is present for both the general and final examination, and is included on the reading committee.

While the principal source of guidance during the process of choosing specialization areas and a research topic is the thesis advisor, it is strongly advised that the student maintain contact with all members of the Supervisory Committee. It is suggested that the student meet with the Supervisory Committee at least once a year to discuss their progress until the doctoral thesis is completed.

Examination requirements for the Ph.D. program

Students in the Ph.D. program must pass the following exams:

• The  qualifying exam
• The  general exam
• The  final exam  (defense)

Satisfactory performance and progress

At all times, students need to make satisfactory progress towards finishing their degree. Satisfactory progress in course work is based on grades. Students are expected to maintain a grade point average of 3.4/4.0 or better. Satisfactory progress on the examination requirements consists of passing the different exams in a timely manner. Departmental funding is contingent on satisfactory progress.   The Graduate School rules regarding satisfactory progress are detailed in Policy 3.7: Academic Performance and Progress .   The Department of Applied Mathematics follows these recommended guidelines of the Graduate School including an initial warning, followed by a maximum of three quarters of probation and one quarter of final probation, then ultimately being dropped from the program.    We encourage all students to explore and utilize the many available  resources  across campus.

Expected academic workload

A first-year, full-time student is expected to register for a full course load, at least three numerically graded courses, typically totaling 12-18 credits. All other students are expected to consult with their advisor and register for at least 10-18 credits per quarter.  Students who do not intend to register for a quarter must seek approved  academic leave  in order to maintain a student status.   Students who do not maintain active student status through course registration or an approved leave request need to request reinstatement to rejoin the program. Reinstatement is at the discretion of the department. Students approved for reinstatement are required to follow degree requirements active at time of reinstatement. 

Annual Progress Report

Students are required to submit an Annual Progress Report to the Graduate Program Coordinator by the second week of Spring Quarter each year. The annual progress report should contain the professional information related to the student’s progress since the previous annual report. It should contain information on courses taken, presentations given, publications, thesis progress, etc., and should be discussed with the student's advisor prior to submission. Students should regard the Annual Progress Report as an opportunity to self-evaluate their progress towards completing the PhD. The content of the Annual Progress Report is used to ensure the student is making satisfactory progress towards the PhD degree.

Financial assistance

Financial support for Doctoral studies is limited to five years after admission to the Ph.D. program in the Department of Applied Mathematics. Support for an additional period may be granted upon approval of a petition, endorsed by the student’s thesis supervisor, to the Graduate Program Coordinator.

Master of Science program

Students in the Ph.D. program obtain an M.Sc. Degree while working towards their Ph.D. degree by satisfying the  requirements for the M.Sc. degree.  

Additional Ph.D. Degree Options and Certificates

Students in the Applied Mathematics Ph.D. program are eligible to pursue additional degree options or certificates, such as the  Advanced Data Science Option  or the  Computational Molecular Biology Certificate .  Students must be admitted and matriculated to the PhD program prior to applying for these options. Option or certificate requirements are in addition to the Applied Mathematics degree requirements. Successful completion of the requirements for the option or the certificate leads to official recognition of this fact on the UW transcript.

Career resources, as well as a look at student pathways after graduation, may be found   here.

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