# Applied Mathematics

A. K. Watson Hall, 203.432.1278

http://applied.math.yale.edu

M.S., M.Phil., Ph.D.

**Director of Graduate Studies**

Vladimir Rokhlin

**Professors** Andrew Barron (*Statistics & Data Science*), Joseph Chang (*Statistics & Data Science*), Ronald Coifman (*Mathematics; Computer Science*), Stanley Eisenstat (*Computer Science*), John Emerson (*Adjunct; Statistics & Data Science*), Thierry Emonet (*Molecular, Cellular, & Developmental Biology; Physics*), Michael Fischer (*Computer Science*), Jonathon Howard (*Molecular Biophysics & Biochemistry*), Peter Jones (*Mathematics*), Yuval Kluger (*Pathology*), Nicholas Read (*Physics; Applied Physics; Mathematics*), Vladimir Rokhlin (*Computer Science; Mathematics*), Wilhelm Schlag (*Mathematics*), Martin Schultz (*Emeritus, Computer Science*), Mitchell Smooke (*Mechanical Engineering & Materials Science; Applied Physics*), Daniel Spielman (*Computer Science; Mathematics*), Van Vu (*Mathematics*), Günter Wagner (*Ecology & Evolutionary Biology*), John Wettlaufer (*Earth & Planetary Sciences; Mathematics; Physics*), Huibin Zhou (*Statistics & Data Science*), Steven Zucker (*Computer Science; Biomedical Engineering*)

**Associate Professors** Josephine Hoh (*Public Health*), Sekhar Tatikonda (*Statistics & Data Science*)

**Assistant Professors** Smita Krishnaswamy (*Genetics; Computer Science*), Roy Lederman (*Statistics & Data Science*)

## 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, and randomized algorithms. Topics covered by the program include classical and modern applied harmonic analysis, linear and nonlinear partial differential equations, 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.

## Special Requirements for the Ph.D. Degree

All students are required to: (1) complete twelve term courses (including reading courses) at the graduate level, at least two with Honors grades; (2) pass a qualifying examination on their general applied mathematical knowledge (in algebra, analysis, and probability and statistics) by the end of their second year; (3) submit a dissertation prospectus; (4) participate in the instruction of undergraduates; (5) be in residence for at least three years; and (6) complete a dissertation that clearly advances understanding of the subject it considers. Prior to registering for a second year of study, and in addition to all other academic requirements, students must successfully complete MATH 991, Ethical Conduct of Research, or another approved course on responsible conduct in research. Teaching is considered an integral part of training at Yale University, so all students are expected to complete two terms of teaching within their first two years. 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.

Requirement (1) normally includes four core courses in each of the methods of applied analysis, numerical computation, algorithms, and probability; these should be taken during the first year. The qualifying examination is normally taken by the end of the third term and will test knowledge of the core courses as well as more specialized topics. The thesis is expected to be independent work, done under the guidance of an adviser. An adviser is usually contacted not long after the student passes the qualifying examinations; students are encouraged to find an adviser sooner rather than later. A student is admitted to candidacy after completing requirements (1)–(5) and finding an adviser.

In addition to the above, all first-year students must successfully complete one course on the responsible conduct of research (e.g., MATH 991 or CPSC 991) and AMTH 525, Seminar in Applied Mathematics.

## 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 Math 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. (en route to the Ph.D.) **Applications for a terminal master’s degree are not accepted. Students who withdraw from the Ph.D. program may be eligible for the M.S. degree if they have completed ten graduate-level term courses, maintained a High Pass average, and met the Graduate School’s Honors requirement for the Ph.D. program. Students who are eligible for or who have already received the M.Phil. will not be awarded the M.S.

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

## Courses

**AMTH 525a or b, Seminar in Applied Mathematics** Peter Jones

This course consists of weekly seminar talks given by a wide range of speakers. Required of all first-year students.

HTBA

**AMTH 553a / CB&B 555a / CPSC 553a / GENE 555a, Unsupervised Learning for Big Data** Smita Krishnaswamy

This course focuses on machine-learning methods well-suited to tackling problems associated with analyzing high-dimensional, high-throughput noisy data including: manifold learning, graph signal processing, nonlinear dimensionality reduction, clustering, and information theory. Though the class goes over some biomedical applications, such methods can be applied in any field. Prerequisites: knowledge of linear algebra and Python programming.

TTh 11:35am-12:50pm

**AMTH 640b / CPSC 640b, 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.

MW 2:30pm-3:45pm

**AMTH 666a / ASTR 666a / EPS 666a / MATH 666a, Classical Statistical Thermodynamics** John Wettlaufer

Classical thermodynamics is derived from statistical thermodynamics. Using the multi-particle nature of physical systems, we derive ergodicity, the central limit theorem, and the elemental description of the second law of thermodynamics. We then develop kinetics, transport theory, and reciprocity from the linear thermodynamics of irreversible processes. Topics of focus include Onsager reciprocal relations, the Fokker-Planck equation, stability in the sense of Lyapunov, and time invariance symmetry. We explore phenomena that are of direct relevance to astrophysical and geophysical settings. No quantum mechanics is necessary as a prerequisite.

TTh 1pm-2:15pm

**AMTH 667b / CPSC 576b / ENAS 576b, Advanced Computational Vision** Steven Zucker

Advanced view of vision from a mathematical, computational, and neurophysiological perspective. Emphasis on differential geometry, machine learning, visual psychophysics, and advanced neurophysiology. Topics include perceptual organization, shading, color, and texture.

MW 2:30pm-3:45pm

**AMTH 710a / MATH 710a, Harmonic Analysis on Graphs and Applications** Ronald Coifman

This class covers basic methods of classical harmonic analysis that can be carried over to graphs and data analysis. We cover the fundamentals of nonlinear Fourier analysis, including functional approximations in high dimensions. We intend to cover foundational material useful for data organization and geometries.

MW 11:35am-12:50pm

**AMTH 765b / CB&B 562b / ENAS 561b / INP 562b / MB&B 562b / MCDB 562b / PHYS 562b, Modeling Biological Systems II** Joe Howard, Thierry Emonet, and Jing Yan

This course covers advanced topics in computational biology. How do cells compute, how do they count and tell time, how do they oscillate and generate spatial patterns? Topics include time-dependent dynamics in regulatory, signal-transduction, and neuronal networks; fluctuations, growth, and form; mechanics of cell shape and motion; spatially heterogeneous processes; diffusion. This year, the course spends roughly half its time on mechanical systems at the cellular and tissue level, and half on models of neurons and neural systems in computational neuroscience. Prerequisite: a 200-level biology course or permission of the instructor.

TTh 2:30pm-3:45pm

**AMTH 797b / MATH 797b, Geometry of Data/Unsupervised Learning on Graphs** Ariel Jaffe

Technological developments have enabled the acquisition and storage of increasingly large-scale, high-resolution, and high-dimensional data in many fields. These datasets pose a challenge for classical methods relying on parametric modeling, statistical estimation, or linear methods. However, it has been shown that many real-world data represented in high dimension (audio, images) lie on or near low-dimensional manifolds. This course covers manifold learning: a class of nonparametric kernel-based methods that extract low-dimensional structures from high-dimensional data. The course also reviews the role graphs play in modern signal processing and machine learning: signal processing on graphs and learning representations with deep neural networks. Topics include: linear data analysis (PCA, ICA, CCA); kernel-based methods, kernel PCA; affinity and Laplacian matrices; spectral clustering; manifold learning and nonlinear dimensionality reduction; graph signal processing; and representation learning with neural networks.

HTBA

**AMTH 863a / MATH 863a, Topics in Sparse Analysis**

How do we compress, approximate, and estimate data (whether from physical observations, digital sources, or the results of computer simulations)? This course covers the basic mathematical and algorithmic results in the sparse approximation of data and the sparse signal recovery of information present in signals and data. A prototypical problem is how to design both a set of linear measurements or observations of a signal (or vector or function) and an algorithm so as to reconstruct important information about the signal. We then turn to more specialized topics in streaming, sketching, and sublinear algorithms (including Fourier sampling algorithms). As part of the sketching and sublinear algorithms, we discuss some developments in randomized numerical linear algebra. We end with sparse analysis topics in machine learning, including sparse coding and auto encoders. This course is suitable for (applied) mathematics, computer science, and statistics and data science graduate students.

TTh 9am-10:15am

**AMTH 864a and AMTH 865b / MATH 864a and MATH 865b, Topics in Inverse Problems I, II**

The goal of this course is to study inverse problems and their applications in imaging. The prototypical problem we consider is to recover the coefficients of a partial differential equation from boundary measurements of its solutions. The fundamental theoretical questions concern the uniqueness, stability, and reconstruction of the coefficients. This is a vast subject, and we will only be able to discuss a few of its important aspects. These include: the Radon transform and other ray transforms, the Calderón problem and related problems for elliptic equations, inverse transport problems and optical tomography, and the Gelfand problem and related problems for hyperbolic equations. The necessary tools from partial differential equations, differential geometry, and microlocal analysis are developed as needed.

TTh 2:30pm-3:45pm

**AMTH 999a, Directed Reading** Vladimir Rokhlin

HTBA