# Mathematics

10 Hillhouse Avenue, 203.432.4172

http://math.yale.edu

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

**Chair**

Igor Frenkel

**Director of Graduate Studies**

Zhiwei Yun

**Professors** Andrew Casson, Ronald Coifman, Igor Frenkel, Alexander Goncharov, Peter Jones, Gil Kalai (*Adjunct*), Alexander Lubotzky (*Adjunct*), Gregory Margulis, Yair Minsky, Vincent Moncrief (*Physics*), Hee Oh, Sam Payne, Nicholas Read (*Physics; Applied Physics*), Vladimir Rokhlin (*Computer Science*), Daniel Spielman (*Computer Science*), Van Vu, John Wettlaufer (*Geology & Geophysics; Physics*), Zhiwei Yun, Gregg Zuckerman

**Assistant Professors **Matthew Durham, Stefan Steinerberger

## 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; geometric analysis; 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; higher Teichmüller theory and cluster varieties; representation theory; automorphic forms, L-functions; algebraic number theory and algebraic geometry; derived algebraic geometry, and periods and motives; tropical algebraic geometry; tomography and integral geometry; mathematical physics, quantum field theory, relativity, numerical analysis; combinatorics and discrete mathematics.

## Special Requirements for the Ph.D. Degree

All students are required to: (1) complete eight term courses at the graduate level, at least two with Honors grades; (2) pass qualifying examinations on their general mathematical knowledge; (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. The normal time for completion of the Ph.D. program is five years. Requirement (1) should be completed by the end of the second year. A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term, at intervals of about one month. All qualifying examinations must be taken by the end of the third term. 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

Teaching experience is integral to graduate education at Yale. Therefore, most Mathematics students are required to assist in teaching during five terms. Students in years one and two serve as tutors and graders in undergraduate mathematics courses during one term per year. The department also offers a required teaching practicum in year two. In years three through five, students normally teach one section of calculus or its equivalent during one term per year. Students receiving external fellowships may petition for a waiver of teaching while receiving external funding in place of University funding, but they are still required to teach one section of calculus or its equivalent for a minimum of two terms over the course of their program.

## Master’s Degrees

**M.Phil.** In addition to the Graduate School’s Degree Requirements (see under Policies and Regulations), a student must undertake a reading program of at least two terms’ duration in a specific significant area of mathematics under the supervision of a faculty adviser and demonstrate a command of the material studied during the reading period at a level sufficient for teaching and research.

**M.S. (en route to the Ph.D.)** A student must 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. The M.S. degree is conferred only en route to the Ph.D.; there is no separate master’s program in Mathematics.

Program materials are available upon request to the Director of Graduate Studies, Mathematics Department, Yale University, PO Box 208283, New Haven CT 06520-8283.

## Courses

**MATH 500a, Modern Algebra I** Zhiwei Yun

A survey of algebraic constructions and theories at a sophisticated level. Topics include categorical language, free groups and other free objects in categories, general theory of rings and modules, artinian rings, and introduction to homological algebra.

MW 1pm-2:15pm

**MATH 501b, Modern Algebra II** Gregg Zuckerman

Topics in commutative algebra: general extension of fields; Noetherian, local, and Dedekind rings. Introduction to valuation theory. Rudiments of algebraic geometry. After MATH 500.

TTh 1pm-2:15pm

**MATH 515b, Intermediate Complex Analysis** Richard Beals

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** Guy Wolf

Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals.

TTh 2:30pm-3:45pm

**MATH 525b, Introduction to Functional Analysis** Manas Rachh

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.

MW 1pm-2:15pm

**MATH 533b, Introduction to Representation Theory** PhilSang Yoo

An introduction to basic ideas and methods of representation theory of finite groups and Lie groups. Examples include permutation groups and general linear groups. Connections with symmetric functions, geometry, and physics.

TTh 2:30pm-3:45pm

**MATH 544a, Introduction to Algebraic Topology I** Matthew Durham

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.

MW 2:30pm-3:45pm

**MATH 573b, Algebraic Number Theory** Alexander Goncharov

Structure of fields of algebraic numbers (solutions of polynomial equations with integer coefficients) and their rings of integers; prime decomposition of ideals and finiteness of the ideal class group; completions and ramification; adeles and ideles; zeta functions.

MW 2:30pm-3:45pm

**MATH 619a, Foundations of Algebraic Geometry** Yuchen Liu

This course provides an introduction to the language of basic ideas of algebraic geometry. We will study affine and projective varieties, and introduce the more general theory of schemes. Our main references are Kenji Uenoâs book and Ravi Vakilâs lecture notes. Commutative algebra at the level of Math 380/381 or 500/501.

MW 1pm-2:15pm

**MATH 620a, Homogeneous Dynamics, Geometry of Numbers, and Number Theory** Gregory Margulis

This seminar covers basic topics in homogeneous dynamics and geometry of numbers, including classical reduction theory. Various applications to number theory are also discussed.

MW 1pm-2:15pm

**MATH 648a, Tropical Scheme Theory** Kalina Mincheva and Samuel Payne

An introduction to tropical scheme theory, exploring a way to endow tropical varieties, which are combinatorial objects (polyhedral complexes), with additional structure, analogous to the additional structure that schemes carry in algebraic geometry, when compared with classical algebraic varieties. We start with basic foundational material such as commutative algebra over idempotent semirings, matroids, and valuated matroids, and some classical tropical geometry. Further topics include, but are not limited to, the construction of tropical schemes from toric embeddings, universal (scheme theoretic) tropicalization, tropical ideals, and various notions of cohomology in the context of tropical varieties and schemes. Prerequisites: familiarity with toric varieties, scheme theory, and basic tropical geometry is very helpful.

TTh 1pm-2:15pm

**MATH 702a / AMTH 702, Numerical Solution of Ordinary and Partial Differential Equations** Vladimir Rokhlin

This course includes (1) review of the classical qualitative theory of ODEs; (2) Cauchy problem: elementary numerical methods, stiff systems of ODEs, Richardson extrapolation and deferred corrections; (3) boundary value problems: elementary theory; (4) introduction to PDES: counterexamples, Cauchy-Kowalevski theorem, classification of second-order PDEs, separation of variables; (5) numerical methods for elliptic PDEs; (6) numerical methods for parabolic PDEs; and (7) numerical methods for hyperbolic PDEs. Prerequisites: advanced calculus; knowledge of FORTRAN or C.

MW 2:30pm-3:45pm

**MATH 731a, Topics in Real and Harmonic Analysis** Stefan Steinerberger

We discuss topics at the interface of real/harmonic analysis and other fields (mathematical physics, partial differential equations, number theory, combinatorics). The main goal is to gain an overview of how analysis can be fruitfully employed as a tool to attack problems from other fields. The precise choice of topics is at least partly driven by student interests and selected from the following: mathematical physics (minimizers in the calculus of variations; the ground state of certain energy functionals; rearrangement techniques; the Pólya-Szegö inequality and applications/implications; eigenvalues of linear operators); partial differential equations (Fourier analysis techniques for dispersive equations; Strichartz estimates and the restriction problem; elliptic partial differential equations and geometry; the Caffarelli-Kohn-Nirenberg, Nash, Poincaré, Sobolev inequalities; implications of the co-area formula; eigenfunctions of the Laplacian); number theory (exponential sum estimates; irregularities of distribution phenomena; selected topics in analytic number theory; the interaction between certain sum-product phenomena and the Kakeya conjecture; the recent work of Bourgain, Demeter, Guth on the Vinogradov conjecture); and combinatorics (Fourier analysis techniques in additive combinatorics; combinatorial problems in the setting of multilinear harmonic analysis).

TTh 2:30pm-3:45pm

**MATH 810a, Hyperbolic Manifolds: Structure and Deformation** Yair Minsky

Hyperbolic manifolds, especially in three dimensions, are a meeting place for a number of rich areas of geometry, analysis, and dynamics. This course surveys various aspects of this field with an eye to developing its central geometric tools and understanding both their more general applicability and their limitations. Topics covered (some in detail, others in survey mode) include geometric structures in general; topology of three-manifolds; deformation spaces; rigidity; geometric limits; quasiconformal deformation and Teichmüller theory; compactness and compactification theorems; bi-Lipschitz models and quantitative rigidity. Some familiarity with topology and basic hyperbolic geometry is assumed.

TTh 11:35am-12:50pm

**MATH 815a, Geometric Langlands and Derived Algebraic Geometry** PhilSang Yoo

Geometric Langlands theory is a surprisingly rich topic, having to do with various subjects ranging from number theory to quantum field theory in many different guises. In this course, we focus on a global unramified categorical conjecture over the complex number field, following Beilinson and Drinfeld’s ideas in the 1990s. There have been many important developments since then, but in its most recent formulation, the conjecture involves heavy use of derived algebraic geometry. We first review some of the old ideas (which tend to be concrete) in the subject; then develop necessary background for derived algebraic geometry and explain the statement of the main conjecture as formulated by Arinkin and Gaitsgory; and finally discuss some of the important recent ideas toward proving the conjecture. The focus of the lectures is not only on formulating and proving the conjecture; we also discuss some of the central topics in geometric representation theory. Some basic knowledge of representation theory, algebraic geometry, algebraic topology, and homological algebra is assumed.

W 9am-11:30am

**MATH 830a, Introduction to Differential Geometry** Ilia Smilga

This is an introduction to some basic notions of differential geometry; smooth manifolds, differentiable maps, vector fields and the tangent bundle; differential forms and various operations you can do with them; integration and Stokesâ theorem; relationship between exact and closed forms, Poincare lemma and de Rham cohomology. We will also give some basic ideas about Lie groups.

WF 1pm-2:15pm

**MATH 991a / CPSC 991a, Ethical Conduct of Research** Vladimir Rokhlin

0 Course cr

HTBA