Applied and Computational Mathematics

Leet Oliver Memorial Hall
http://applied.math.yale.edu
M.S., M.Phil., Ph.D.

Director of Graduate Studies
Ronald Coifman

Professors Ronald Coifman (Mathematics; Computer Science),* Yuval Kluger (Pathology),* Rajit Manohar (Electrical and Computer Engineering),* Vidvuds Ozolins (Applied Physics; Material Science),* Vladimir Rokhlin (Computer Science; Mathematics),* Charles Smart (Mathematics),* Steven Zucker (Computer Science; Biomedical Engineering)*

Fields of Study

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.

Integrated Graduate Program in Physical and Engineering Biology (PEB)

Students applying to the Ph.D. program in Applied Mathematics may also apply to be part of the PEB program. See the description under Non-Degree-Granting Programs, Councils, and Research Institutes for course requirements, and http://peb.yale.edu for more information about the benefits of this program and application instructions.

Special Requirements for the Ph.D. Degree

All students are required to complete at least eight graduate-level courses, at least two with Honors grades. Four courses must meet core requirements. Courses counted toward the eight-course minimum must be full-credit graduate courses with clear focus that are related to applied and computational mathematics in the judgment of the director of graduate studies (DGS) and/or research adviser. Courses such as Special Investigation, Dissertation Research, Master’s Thesis, or Seminar do not count towards the eight-course requirement. Core classes will be selected from four broad areas and students are required to take a minimum of one class from each area: analysis (real, complex, and functional), probability theory and statistics, computing (either numerical analysis or algorithms and complexity theory), and application area (discrete mathematics, science/engineering, data science, ODEs/PDE).  

Additionally, first year students will be required to complete one course on the responsible conduct of research (i.e. MATH 9910) and the Seminar in Applied Mathematics (AMTH 5250).  These courses do not count toward the eight-course minimum requirement.   

Students typically complete most of their core course requirements in the first year, and sufficient progress toward meeting the course requirements is necessary to remain in good standing in the program. By the end of the first year, students must find a research adviser. 

Students must pass a qualifying exam by the end of their second year. The qualifying exam will consist of either an oral presentation to the qualifying committee followed by questions from the committee or a series of written examinations on topics chosen by the adviser. The content of the exam will be tailored to the research interests of the student, emphasizing broad foundational knowledge in mathematics, applied and computational mathematics, and their area of interest, as well as specific topics and current developments, in the chosen area. 

In their third year, students must submit a written research prospectus and pass an oral area examination. The thesis prospectus should summarize the first significant research project undertaken with the adviser and demonstrate familiarity with relevant literature. Further, it should outline plans for the thesis research, contextualize the work, and provide a timeline for completion. 

Upon completing all coursework, finding an adviser, and passing both the qualifying and area examinations, the student is admitted to candidacy. At the latest, the student must be admitted to candidacy by the end of the third year.  

A final oral presentation of the dissertation research is required. 

Teaching is considered an integral part of training at Yale University, so all students are expected to complete two terms of teaching. Students who require additional support from the graduate school will be required to teach additional terms, if needed, after they have fulfilled the academic teaching requirement.

Students are expected to be in residence for at least three years.

Honors Requirement

Students must meet the graduate school’s honors requirement by the end of the fourth term of full-time study.

M.D.-PH.D. STUDENTS

With permission of the DGS, M.D.-Ph.D. students may request a reduction in the program’s academic teaching requirement to one term of teaching. Only students who teach are eligible to receive a university stipend contingent on teaching.

Master’s Degrees

M.Phil. The minimum requirements for this degree are that a student shall have completed all requirements for the Applied Mathematics Ph.D. program as described above except the required teaching, the prospectus, and the dissertation. Students will not generally have satisfied the requirements for the M.Phil. until after two years of study, except where graduate work done before admission to Yale has reduced the student’s graduate course work at Yale. In no case will the degree be awarded after less than one year of residence in the Yale Graduate School of Arts and Sciences. See also Degree Requirements under Policies and Regulations.

M.S. Only 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 seven graduate-level term courses, maintain a High Pass average, and meet the graduate school’s honors requirement.

More information is available on the program’s website, http://applied.math.yale.edu.

Courses

* AMTH 1600b / MATH 1600b / S&DS 1600b, The Structure of Networks  Gilles Mordant

Network structures and network dynamics described through examples and applications ranging from marketing to epidemics and the world climate. Study of social and biological networks as well as networks in the humanities. Mathematical graphs provide a simple common language to describe the variety of networks and their properties.  QR
MW 9am-10:15am

AMTH 2220a or b / MATH 2220a or b, Linear Algebra with Applications  Staff

Matrix representation of linear equations. Gauss elimination. Vector spaces. Linear independence, basis, and dimension. Orthogonality, projection, least squares approximation; orthogonalization and orthogonal bases. Extension to function spaces. Determinants. Eigenvalues and eigenvectors. Diagonalization. Difference equations and matrix differential equations. Symmetric and Hermitian matrices. Orthogonal and unitary transformations; similarity transformations. Students who plan to continue with upper level math courses should instead consider MATH 2250 or 2260. After MATH 1150 or equivalent. May not be taken after MATH 2250 or 2260. May not be counted toward the Math, CPSC + Math, or Econ + Math major.   QR
HTBA

AMTH 2320b / MATH 2320b, Advanced Linear Algebra with Applications  Ian Adelstein

This course is a natural continuation of MATH 2220. The core content includes eigenvectors and the Spectral Theorem for real symmetric matrices; singular value decomposition (SVD) and principle component analysis (PCA); quadratic forms, Rayleigh quotients and generalized eigenvalues. We also consider a number of applications: optimization and stochastic gradient descent (SGD); eigen-decomposition and dimensionality reduction; graph Laplacians and data diffusion; neural networks and machine learning. A main theme of the course is using linear algebra to learn from data. Students complete (computational) projects on topics of their choosing. Prerequisites: MATH 1200 and MATH 2220, 2250, or 2260. This is not a proof-based course. May not be taken after MATH 3400.  QR
MW 11:35am-12:50pm

AMTH 2440a or b / MATH 2440a or b, Discrete Mathematics  Staff

Basic concepts and results in discrete mathematics: graphs, trees, connectivity, Ramsey theorem, enumeration, binomial coefficients, Stirling numbers. Properties of finite set systems. Prerequisite: MATH 1150 or equivalent. Some prior exposure to proofs is recommended (ex. MATH 2250).  QR
HTBA

AMTH 2470b / MATH 2470b, Intro to Partial Differential Equations  Ruoyu Wang

Introduction to partial differential equations, wave equation, Laplace's equation, heat equation, method of characteristics, calculus of variations, series and transform methods, and numerical methods. Prerequisites: MATH 2220 or 2250 or 2260, MATH 2460 or ENAS 1940.  QR
MWF 10:30am-11:20am

AMTH 3220a / CPSC 4844a, Geometric and Topological Methods in Machine Learning  Smita Krishnaswamy

This course provides an introduction to geometric and topological methods in data science. Our starting point is the manifold hypothesis: that high dimensional data live on or near a much lower dimensional smooth manifold. We introduce tools to study the geometric and topological properties of this manifold in order to reveal relevant features and organization of the data. Topics include: metric space structures, curvature, geodesics, diffusion maps, eigenmaps, geometric model spaces, gradient descent, data embeddings and projections, and topological data analysis (TDA) in the form of persistence homology and their associated “barcodes.” We see applications of these methods in a variety of data types.  Prerequisites: MATH 2250 or 2260; MATH 2550 or 2560; MATH 3020; and CPSC 1001 or equivalent programming experience.   QR, SC
TTh 11:35am-12:50pm

AMTH 3610b / S&DS 3610b, Data Analysis  Brian Macdonald

Selected topics in statistics explored through analysis of data sets using the R statistical computing language. Topics include linear and nonlinear models, maximum likelihood, resampling methods, curve estimation, model selection, classification, and clustering. Extensive use of the R programming language.  Experience with R programming (from e.g. S&DS 1060, S&DS 2200, S&DS 230, S&DS 2420), probability and statistics (e.g. S&DS 1060, S&DS 2200, S&DS 2380, S&DS 2410, or concurrently with S&DS 2420), linear algebra (e.g. MATH 2220, MATH 2250, MATH 1180), and calculus is required. This course is a prerequisite for S&DS 4250 and may not be taken after S&DS 4250.  QR
TTh 2:35pm-3:50pm

* AMTH 3620b / CPSC 3620b / ECE 4351b, Decisions and Computations across Networks  A Stephen Morse

For a long time there has been interest in distributed computation and decision making problems of all types. Among these are consensus and flocking problems, the multi-agent rendezvous problem, distributed averaging, gossiping, localization of sensors in a multi-sensor network, distributed algorithms for solving linear equations, distributed management of multi-agent formations, opinion dynamics, and distributed state estimation. The aim of this course is to explain what these problems are and to discuss their solutions. Related concepts from spectral graph theory, rigid graph theory, non-homogeneous Markov chain theory, stability theory, and linear system theory are covered. Although most of the mathematics need is covered in the lectures, students taking this course should have a working understanding of basic linear algebra. The course is open to all students.  Prerequisite: Linear algebra or instructor permission.  SC
MW 2:35pm-3:50pm

AMTH 3640b / ECE 4541b / S&DS 3640b, Information Theory  Yihong Wu

Foundations of information theory in communications, statistical inference, statistical mechanics, probability, and algorithmic complexity. Quantities of information and their properties: entropy, conditional entropy, divergence, redundancy, mutual information, channel capacity. Basic theorems of data compression, data summarization, and channel coding. Applications in statistics and finance. After STAT 241.  QR
TTh 11:35am-12:50pm

AMTH 4200a / MATH 4210a, The Mathematics of Data Science  Gilles Mordant

This course aims to be an introduction to the mathematical background that underlies modern data science. The emphasis is on the mathematics but occasional applications are discussed (in particular, no programming skills are required). Covered material may include (but is not limited to) a rigorous treatment of tail bounds in probability, concentration inequalities, the Johnson-Lindenstrauss Lemma as well as fundamentals of random matrices, and spectral graph theory.  Prerequisite: MATH 3050.   QR, SC
MW 11:35am-12:50pm

AMTH 4441a / APHY 4410a / MENG 4441a / PHYS 4441a, Nonlinear Dynamics  Bauyrzhan Primkulov

This course introduces nonlinear dynamics and chaos in dissipative systems, tailored broadly for undergraduate students in science and engineering. It focuses on simple dynamical models, the mathematical principles underlying their behaviors, their connection to natural phenomena, and techniques for data analysis and interpretation. Key topics include forced and parametric oscillators, phase space analysis, periodic, quasiperiodic, and aperiodic flows, sensitivity to initial conditions, and strange attractors such as the Lorenz attractor. The course also explores phenomena like period doubling, intermittency, and quasiperiodicity, emphasizing nonlinear processes describable by a limited number of time-evolving variables. ENAS 1510 (Multivariable Calculus or equivalent), ENAS 1940 (Differential Equations or equivalent)  SC
TTh 11:35am-12:50pm

AMTH 4470a / MATH 4470a, Partial Differential Equations  Wilhelm Schlag

Introduction to partial differential equations, wave equation, Laplace's equation, heat equation, method of characteristics, calculus of variations, series and transform methods, and numerical methods. Prerequisites: MATH 3050
TTh 11:35am-12:50pm

* AMTH 4820a, Research Project  John Wettlaufer

Individual research. Requires a faculty supervisor and the permission of the director of undergraduate studies. The student must submit a written report about the results of the project. May be taken more than once for credit.
HTBA

* AMTH 4910a, Senior Project  John Wettlaufer

Individual research that fulfills the senior requirement. Requires a faculty supervisor and the permission of the director of undergraduate studies. The student must submit a written report about the results of the project.
HTBA

AMTH 6400a / CPSC 6400a / MATH 6400a, Topics in Numerical Computation  Eric Michielssen

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.
MW 9am-10:15am

AMTH 7100a / MATH 7100a, 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 11:35am-12:50pm

AMTH 9999a, Directed Reading  Anna Gilbert

In-depth study of elliptic partial differential equations.
HTBA