MATH 5000a, Algebra  Samuel DeHority

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 9am-10:15am

MATH 5150b, Intermediate Complex Analysis  Charles Smart

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.
MW 2:35pm-3:50pm

MATH 5200a, Measure Theory and Integration  Tamunonye Cheetham-West

Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals. Prerequisite: MATH 3050 or equivalent.
MW 1:05pm-2:20pm

MATH 5250b, Introduction to Functional Analysis  Staff

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 5260a, Introduction to Differentiable Manifolds  Lu Wang

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; and (6) Sard’s theorem, degree and intersection. Prerequisites: Multivariable calculus (MATH 1200 or 3020), linear algebra (MATH 2250 or 2260), and topology (MATH 4300).
TTh 2:35pm-3:50pm

MATH 5330b, Introduction to Representation Theory  Ivan Loseu

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. Prerequisite: MATH 3500.
MW 9am-10:15am

MATH 5440a, 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. Prerequisites: Linear algebra (MATH 2250 or 2260), Analysis 1 (MATH 2550 or 2560), abstract algebra (MATH 3500) required. Topology (MATH 430) is strongly recommended, and much of the material is assumed. Analysis 2 (MATH 3020) is recommended. 
MW 2:35pm-3:50pm

MATH 6400a / AMTH 6400a / CPSC 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

MATH 6660a / AMTH 666 / ASTR 6660a / EPS 6660a, 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 6810a, Working Seminar on Harmonic Analysis and PDEs  Wilhelm Schlag

This seminar complements MATH 680 (Fourier Analysis and PDE). This seminar continues with an exploration of topics in harmonic analysis relevant to PDEs that was started in the fall. Topics include the paradifferential calculus and its applications to nonlinear evolution PDEs. The participants are expected to give presentations on topics that come up in the course. Prerequisite: knowledge of basic harmonic analysis such as the Calderon-Zygmund and Littlewood-Paley theorems. Prerequisites for undergrads: the student should have taken multivariable calculus, MATH 305 and 325. Exposure to complex analysis is recommended as well.
W 5pm-6:30pm

MATH 7030a, Stochastic Partial Differential Equations  Charles Smart

An introduction to stochastic partial differential equations. Topics include rough path theory and the multiplicative stochastic heat equation.
TTh 2:35pm-3:50pm

MATH 7100a / AMTH 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

MATH 7120b / ECE 9100b, Topics in Denoising and Structure Recovery from Data  Boris Landa

Recovering signals and underlying structure from noisy observations is a fundamental problem in many areas of science and engineering. Over the past few decades, a rich body of work has emerged to address this challenge across diverse settings. A common guiding principle is to leverage structural assumptions—such as smoothness, sparsity, or low-rankness in the data—alongside models of the noise to enable effective recovery. This course explores both classical and modern approaches to denoising and structural data recovery, blending theoretical foundations with algorithmic and applied perspectives. Particular emphasis is placed on high-dimensional regimes relevant to modern data analysis, where noise can behave in counterintuitive ways, yet also exhibit predictable patterns that can be exploited for denoising. The course introduces tools and results from high-dimensional probability and random matrix theory that underpin many recent advances in this area.
TTh 2:35pm-3:50pm

MATH 7160a, Geometry and Topology of Spaces with Ricci Curvature Bounded From Below  Xinrui Zhao

This topics course introduces the geometry, analysis, and topology of spaces with Ricci curvature bounded from below. We begin with foundational results for smooth Riemannian manifolds, including volume comparison, splitting theorems, and Gromov compactness, and then study the structure of limit spaces and metric measure spaces with synthetic Ricci curvature lower bounds (such as RCD spaces). Emphasis is placed on the interaction between curvature, topology, and analysis, including rigidity, stability, and fundamental group behavior. The course is intended for graduate students with background in differential geometry and analysis.
TTh 11:35am-12:50pm

MATH 7250a, Group Actions on Manifolds  Sebastian Hurtado - Salazar

We discuss various topics related to groups acting on manifolds, including dynamics in dimension one and two (rotation numbers, rotation sets, hyperbolic and elliptic dynamics), groups acting on the circle, connections to mapping class groups, dynamics on character varieties, actions of Lie groups, stationary measures, rigidity and the Zimmer program.
MW 2:35pm-3:50pm

MATH 7370a, Hyperbolic Algebras and Modular Forms  Igor Frenkel

The course is dedicated to the theory of hyperbolic Kac-Moody algebras and its automorphic extensions introduced by Borcherds. The structure and representation theory of these algebras is based on the theory of Siegel modular forms and their generalizations. The latter generalizes the classical theory of modular forms and plays an important role in the course. Students are encouredged to give talks on various related topics.
TTh 2:35pm-3:50pm

MATH 7440a / CPSC 6440a, 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. 
TTh 1:05pm-2:20pm

MATH 7600a, Condensed Mathematics  Sam Raskin

The goal of this course is to develop foundations for homological algebra of functional analytic objects, following Clausen-Scholze. One of the main goals is to setup the theory of so-called liquid vector spaces.
MW 1:05pm-2:20pm

MATH 7720a, Category O and Soergel Theory  Ivan Loseu

Category O is one of the most important categories of representations of semisimple Lie algebras. Soergel theory provides a powerful tool of studying this category by relating it to a category of more combinatorial/ elementary nature. We cover these subjects and then proceed to a more general version of the Soergel theory that allows to prove various category equivalences between realizations of the “Hecke category,” in particular, those of interest for the geometric Langlands program. Prerequisites: structure and finite dimensional representation theory of semisimple Lie algebras over the complex numbers. For the second part of the course, familiarity with algebraic geometry, algebraic topology, and category theory (e.g. derived categories) is useful.
TTh 1:05pm-2:20pm

MATH 8270b, Lang Teaching Seminar  Brett Smith and Laura Seaberg

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 9910a / CPSC 9910a, Ethical Conduct of Research  Staff

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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