MATH 500a, Algebra  Junliang Shen

The course serves as an introduction to commutative algebra and category theory. Topics include commutative rings, their ideals and modules, Noetherian rings and modules, constructions with rings such as localization and integral extension, connections to algebraic geometry, categories, functors and functor morphisms, tensor product and Hom functors, and projective modules. Other topics may be discussed at the instructor’s discretion. Prerequisites: MATH 350 and MATH 370.
MW 2:30pm-3:45pm

MATH 515b, Intermediate Complex Analysis  Richard Kenyon

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  Charles Smart

Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals.
MW 11:35am-12:50pm

MATH 525b, Introduction to Functional Analysis  Hanwen Zhang

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.
TTh 11:35am-12:50pm

MATH 526a, Introduction to Differentiable Manifolds  Tamunonye Cheetham-West

This is an introduction to the general theory of smooth manifolds, developing tools for use elsewhere in mathematics. A rough plan of topics (with the later ones as time permits) includes (1) manifolds, tangent spaces, vector fields and flows; (2) natural examples, submanifolds, quotient manifolds, fibrations, foliations; (3) vector and tensor bundles,  differential forms; (4) Lie derivatives, Lie algebras and groups; (5) embedding, immersions and transversality; (6) Sard’s theorem, degree and intersection. Prerequisites: some multivariable calculus, linear algebra, and topology.
TTh 11:35am-12:50pm

MATH 533b, Introduction to Representation Theory  Igor Frenkel

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 536b, Combinatorics  Staff

Combinatorics is a relatively new and very active area of mathematics, focusing on the study of discrete systems. It has applications in all areas of mathematics, from probability and physics to representation theory and algebraic geometry. It also plays an essential role in computing and data science. The course covers the basic topics of combinatorics, including generating functions, partitions, symmetric polynomials, random matrices, probabilistic methods, additive combinatorics, and graph theory. Prerequisite: Math 345.
MW 1pm-2:15pm

MATH 544a, Introduction to Algebraic Topology  Alexander Goncharov

This is 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.
TTh 2:30pm-3:45pm

MATH 640b / 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.
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MATH 665a, Topics in Quantum Algebra  Minh-Tam Trinh

We discuss the relationship between representations of linear groups over finite and p-adic fields, a part of Lie theory, and isotopy invariants of knot and links, a part of geometric topology. The bridge is the theory of Hecke algebras and their cocenters. We begin with the classical references, potentially including works of Jones, Deligne–Lusztig, and Macdonald and, working our way through the categorification program of Frenkel, Khovanov, and others, aim to arrive at recent works about double affine Hecke algebras and algebraic links.
TTh 2:30pm-3:45pm

MATH 666a / AMTH 666a / ASTR 666a / EPS 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, the origin of diffusion, transport theory, and reciprocity from the linear thermodynamics of irreversible processes. Topics of focus include Onsager reciprocal relations, the Fokker-Planck and Cahn-Hilliard equations, stability in the sense of Lyapunov, time invariance symmetry and maximum principles. We explore phenomena cross a range of problems in science and engineering. Prerequisites for Yale College students: PHYS 301, PHYS 410, MATH 246 or similar and/or permission of instructor.
MW 11:35am-12:50pm

MATH 675a / AMTH 675a, Numerical Methods for Partial Differential Equations  Vladimir Rokhlin

(1) Review of the classical qualitative theory of ODEs; (2) Cauchy problem. Elementary numerical methods: Euler, Runge-Kutta, predictor-corrector. Stiff systems of ODEs: definition and associated difficulties, implicit Euler, Crank-Nicolson, barrier theorems. Richardson extrapolation and deferred corrections; (3) Boundary value problems. Elementary theory: finite differences, finite elements, abstract formulation and related spaces, integral formulations and associated numerical tools, nonlinear problems; (4) Partial differential equations (PDEs). Introduction: counterexamples, Cauchy–Kowalevski theorem, classification of second-order PDEs, separation of variables; (5) Numerical methods for elliptic PDEs. Finite differences, finite elements, Richardson and deferred corrections, Lippmann–Schwinger equation and associated numerical tools, classical potential theory, “fast” algorithms; (6) Numerical methods for parabolic PDEs. Finite differences, finite elements, Richardson and deferred corrections, integral formulations and related numerical tools; (7) Numerical methods for hyperbolic PDEs. Finite differences, finite elements, Richardson and deferred corrections, time-invariant problems and Fourier transform.
TTh 2:30pm-3:45pm

MATH 680a, Fourier Analysis and PDEs  Wilhelm Schlag

This course covers some of the finer techniques in Fourier analysis relevant to nonlinear PDEs. We cover some multilinear estimates of the Coifman-Meyer type and the related para-differential calculus. Classical results in time-frequency analysis including the Beurling-Malliavin theorem and its ramifications might also be included. Students should have taken multivariable calculus, Math 305 and 325. In addition, exposure to complex analysis is recommended as well.
TTh 1pm-2:15pm

MATH 685a, Topics in Representation Theory  Igor Frenkel

The course is dedicated to modern directions in representation theory developed in the last several decades and to related subjects. The program largely depends on the interests of the audience. The participants are encouraged to give presentations on the representation theory topics related to their research. The directions covered may include (but are not limited to): representations of infinite-dimensional Lie algebras including Virasoro algebra and quaternionic analysis, vertex operator algebras, geometric representation theory, categorification and Khovanov homology, cluster algebras, and quantum Teichmuller spaces.
MW 1pm-2:15pm

MATH 690a, Introduction to Quantum Invariants of Knots and Three Manifolds  Ka Ho Wong

This course is an introduction to quantum invariants of knots and three manifolds and their relationships with hyperbolic geometry. Topics include the skein-theoretic constructions of the Jones and colored Jones polynomials, the Witten-Reshetikhin-Turaev invariants, the Turaev-Viro invariants, and their underlying Topological Quantum Field Theory. Interactions between quantum topology and hyperbolic geometry, such as the Kashaev-Murakami-Murakami volume conjecture and its generalizations, are also discussed in this course.
MW 2:30pm-3:45pm

MATH 695a, High-Dimensional Probability  Pei-Chun Su

The course introduces the fundamental concepts and advanced techniques of concentration inequalities and classical results like Hoeffding's and Chernoff's inequalities alongside modern developments such as the matrix Bernstein's inequality. Discover the potency of stochastic processes through Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds rooted in VC dimension.
TTh 9am-10:15am

MATH 827b, Lang Teaching Seminar  Brett Smith

This course prepares graduate students for teaching calculus classes. It is a mix of theory and practice, with topics such as preparing classes, presenting new concepts, choosing examples, encouraging student participation, grading fairly and effectively, implementing active learning strategies, and giving and receiving feedback. Open only to mathematics graduate students in their second year. 
MW 11:35am-12:50pm

MATH 991a / CPSC 991a, Ethical Conduct of Research  Inyoung Shin

This course forms a vital part of research ethics training, aiming to instill moral research codes in graduate students of computer science, math, and applied math. By delving into case studies and real-life examples related to research misconduct, students grasp core ethical principles in research and academia. The course also offers an opportunity to explore the societal impacts of research in computer science, math, and applied math. This course is designed specifically for first-year graduate students in computer science, applied math, and math. Successful completion of the course necessitates in-person attendance on eight occasions; virtual participation does not fulfill this requirement. In cases where illness, job interviews, or unforeseen circumstances prevent attendance, makeup sessions are offered.  0 Course cr
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