# 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 and Data Science*), Yang Cai (*Computer Science*), Joseph Chang (*Statistics and Data Science*), Ronald Coifman (*Mathematics*;* Computer Science*), Thierry Emonet (*Molecular, Cellular, and Developmental Biology*;* Physics*), Michael Fischer (*Computer Science*), Anna Gilbert (*Mathematics*;* Statistics and Data Science*), Jonathon Howard (*Molecular Biophysics and Biochemistry*), Peter Jones (*Mathematics*), Yuval Kluger (*Pathology*), Owen Miller (*Applied Physics*), Nicholas Read (*Physics*;* Applied Physics*;* Mathematics*), Vladimir Rokhlin (*Computer Science*;* Mathematics*), John Schotland (*Mathematics*), Mitchell Smooke (*Mechanical Engineering and Materials Science*;* Applied Physics*), Daniel Spielman (*Computer Science*;* Mathematics*), Van Vu (*Mathematics*), John Wettlaufer (*Earth and Planetary Sciences*;* Mathematics*;* Physics*), Huibin Zhou (*Statistics and Data Science*), Steven Zucker (*Computer Science*;* Biomedical Engineering*)

**Associate Professors** Smita Krishnaswamy (*Genetics*;* Computer Science*), Sekhar Tatikonda (*Statistics and Data Science*)

**Assistant Professor** Roy Lederman (*Statistics and 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, 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: (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 four core topics and specialized topics in consultation with the Director of Graduate Studies) by the end of their second year; (3) submit a dissertation prospectus; (4) participate in the instruction of undergraduates for at least two terms; (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 (i) the methods of applied analysis, (ii) numerical computation or algorithms, and (iii) discrete mathematics or probability or statistics; 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 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. **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 552b / CB&B 663b / CPSC 552b, Deep Learning Theory and Applications** Smita Krishnaswamy

Deep neural networks have gained immense popularity within the past decade due to their success in many important machine-learning tasks such as image recognition, speech recognition, and natural language processing. This course provides a principled and hands-on approach to deep learning with neural networks. Students master the principles and practices underlying neural networks, including modern methods of deep learning, and apply deep learning methods to real-world problems including image recognition, natural language processing, and biomedical applications. Course work includes homework, a final exam, and a final project—either group or individual, depending on enrollment—with both a written and oral (i.e., presentation) component. The course assumes basic prior knowledge in linear algebra and probability. Prerequisites: CPSC 202 and knowledge of Python programming.

HTBA

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

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.

HTBA

**AMTH 631a / S&DS 631a, Optimization and Computation** Zhuoran Yang

An introduction to optimization and computation motivated by the needs of computational statistics, data analysis, and machine learning. This course provides foundations essential for research at the intersections of these areas, including the asymptotic analysis of algorithms, an understanding of condition numbers, conditions for optimality, convex optimization, gradient descent, linear and conic programming, and NP hardness. Model problems come from numerical linear algebra and constrained least squares problems. Other useful topics include data structures used to represent graphs and matrices, hashing, automatic differentiation, and randomized algorithms. Prerequisites: multivariate calculus, linear algebra, probability, and permission of the instructor. Enrollment is limited, with preference given to graduate students in Statistics and Data Science.

TTh 1pm-2:15pm

**AMTH 640a or b / CPSC 640a or b / MATH 640a, 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

**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.

HTBA

**AMTH 710a / MATH 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

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

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