Department of Mathematics

MATH 30200-30300-30400. Computability Theory-1; Computability Theory-2; Computability Theory-3.

The courses in this sequence are offered in alternate years.

MATH 30708. Simple Theories. 100 Units.

Simple theories (so called), introduced almost forty years ago, provide a model theoretic framework for studying certain familes of 'random" objects, such as the theories of random graphs and hypergraphs. Very recent work has shown the class to contain a much greater range of complexity than previously thought. This course will cover the fundamental theorems of simple theories along with some of the new developments.

Instructor(s): Maryanthe Malliaris     Terms Offered: Autumn

MATH 30813. Some Logic and Geometry for Mathematicians. 100 Units.

This quarter course will cover three major applications of logic to other branches of mathematics. The first is Tarski's theorem about quantifier elimination and decidability of the first order theory of the reals. This is a cornerstone of real algebraic geometry with major implications in analysis and geometry. The second is the existence of groups with unsolvable word problem and Higman's characterization of finitely generated subgroups of finitely generated groups. This implies that some geometric problems, such as the homeomorphism problem for manifolds, or deciding whether a simplicial complex is a manifold are undecidable (as we hope to explain). And, finally, we will explain why there is no algorithm to decide whether Diophantine equations with integer coefficients have integral solutions, along with some interesting definability questions in this context.

Instructor(s): Shmuel Weinberger, Maryanthe Malliaris     Terms Offered: Spring

MATH 30900-31000. Model Theory I-II.

MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, we study saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero.

MATH 30905. Decidability. 100 Units.

Decision problems in model theoretic algebra, learning and connections to classification theory, and other topics as time permits.

Instructor(s): Maryanthe Malliaris     Terms Offered: Winter

MATH 31200-31300-31400. Analysis I-II-III.

Analysis I-II-III

MATH 31700-31800-31900. Topology and Geometry I-II-III.

Topology and Geometry I-II-III

MATH 32500-32600-32700. Algebra I-II-III.

Algebra I-II-III

MATH 34200. Geometric Literacy-2. 100 Units.

This ongoing course might be subtitled: "what every good geometer should know". The topics will intersperse more elementary background with topics close to current research, and should be understandable to second year students. The individual modules (2-5 weeks each) might be logically interrelated, but we will try to maintain a "modular structure" so that people who are willing to assume certain results as "black boxes" will be able to follow more advanced modules before formally learning all the prerequisites. This years topics might include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov homeomorphisms and Thurston's compactification of Teichmuller space, algebraic geometry for non-algebraic geometers. Prereq: First year graduate sequence.

MATH 34300. Geometric Literacy - 3. 100 Units.

This ongoing course might be subtitled: "what every good geometer should know". The topics will intersperse more elementary background with topics close to current research, and should be understandable to second year students. The individual modules (2-5 weeks each) might be logically interrelated, but we will try to maintain a "modular structure" so that people who are willing to assume certain results as "black boxes" will be able to follow more advanced modules before formally learning all the prerequisites. This years topics might include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov homeomorphisms and Thurston's compactification of Teichmuller space, algebraic geometry for non-algebraic geometers. Prereq: First year graduate sequence.

Instructor(s): Benson Farb     Terms Offered: Spring
Prerequisite(s): First year graduate sequence.

MATH 34500. Topics in Geometry and Topology. 100 Units.

This course will cover various topics ranging from algebraic and differential geometry to algebraic and geometric topology, often with connections to representation theory and number theory. Recent topics have included Hodge theory, Mostow Rigidity, Topology and geometry of K3 surfaces (joint with Eduard Looijenga), "What all the 3-manifolds are", 4-manifold theory: from Seiberg-Witten to the classification of albebraic surfaces (joint with Danny Calegari), and the cohomology of arithmetic groups (joint with Matt Emerton).

Instructor(s): Benson Farb     Terms Offered: Winter
Prerequisite(s): The first year math graduate courses or permission of instructor.

MATH 34600. Topics in Geometry and Topology-2. 100 Units.

This course will cover various topics ranging from algebraic and differential geometry to algebraic and geometric topology, often with connections to representation theory and number theory. Recent topics have included Hodge theory, Mostow Rigidity, Topology and geometry of K3 surfaces (joint with Eduard Looijenga), "What all the 3-manifolds are", 4-manifold theory: from Seiberg-Witten to the classification of albebraic surfaces (joint with Danny Calegari), and the cohomology of arithmetic groups (joint with Matt Emerton).

Instructor(s): Benson Farb     Terms Offered: Spring
Prerequisite(s): The first year math graduate courses or permission of instructor.

MATH 35600. Topics in Dynamical Systems. 100 Units.

This course covers selected topics in dynamical systems. Topics vary and may include: ergodic theory, smooth dynamical systems, statistical properties of dynamical systems, and geometry and dynamics.

Instructor(s): Anne Wilkinson     Terms Offered: Autumn

MATH 35804. Probabilistic theory of dispersive equations. 100 Units.

In the last 20 years, the probabilistic theory of PDEs has seen a huge growth in interest and significant development in techniques. In the context of dispersive equations, it is intimately linked to fundamental physical questions, such as the Gibbs measures in quantum field theory, and the non-equilibrium dynamics in wave turbulence. In this course, I will start with a brief introduction of dispersive equations, and then move on to the probabilistic context, where I will explain the basic techniques of linear-nonlinear and higher order tree expansions, and the more recent developments including the methods of random averaging operators and random tensors. If time permits, I will also discuss connections with quantum field theory and wave turbulence.

MATH 35904. Persistent Homology and its applications. 100 Units.

Persistent Homology was originally introduced as a geometric tool in data analysis, but over the past decade it's been a useful tool in differential geometry, dynamics, spectral geometry and approximation theory. We hope to explain the formalism, its basic theorems, and some of these applications.

MATH 36000. Proseminar: Topology. 100 Units.

This informal proseminar is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): Staff

MATH 36100. Topology Proseminar. 100 Units.

This informal "proseminar" is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): J. Peter May     Terms Offered: Winter

MATH 36200. Topology Proseminar. 100 Units.

The Spring proseminar is a more formal version of the Fall and Winter topology proseminar. It will be taught primarily or completely by May, on topics of interest to the participants.

Instructor(s): J. Peter May     Terms Offered: Spring

MATH 36206. Algebraic Number Theory. 100 Units.

This is a course in basic algebraic number theory. The only prerequisites will be the material covered in the first year algebra graduate sequence.

Instructor(s): Francesco Calegari     Terms Offered: Spring

MATH 36209. Diophantine geometry and number theory. 100 Units.

Course Information: This course will be a introduction to the methods of trancendental number theory, including the study of heights, applications to approximations of algebraic numbers by rational numbers, the unit equation, and related topics.

MATH 36507. Condensed Mathematics. 100 Units.

This course is an introduction to the new subject of condensed mathematics introduced by Clausen and Scholze. We will discuss some of the foundational results in the theory and study some of the growing applications to analytic geometry.

Instructor(s): Akhil Mathew and Matthew Emerton     Terms Offered: Autumn

MATH 36558. Algebra/Topology. 100 Units.

This will be an advanced course on topics in algebra and topology.

Instructor(s): Akhil Mathew     Terms Offered: Spring

MATH 36603. Expander graphs and metric embeddings. 100 Units.

Expander graphs are fundamental objects in pure math and computer science. This course will study examples, constructions and analytic, geometric, and probabilistic properties of these graphs, and later seque to aspects of the theory of metric embeddings of graphs into Euclidean and perhaps other normed spaces

Instructor(s): Shmuel Weinberger     Terms Offered: Winter
Prerequisite(s): One year of undergraduate analysis.

MATH 36611. Topics in Analytic Geometry. 100 Units.

This course will cover some recent developments in analytic geometry arising from the new theory of condensed mathematics developed by Clausen and Scholze.

Instructor(s): Akhil Mathew     Terms Offered: Autumn
Prerequisite(s): Algebra and Geometry sequences. (1st year), some category theory.

MATH 36614. Topics in Analytic Geometry. 100 Units.

The subject of this course will be "Prismatic F-gauges and Galois representations".

Instructor(s): Vadim Vologodsky     Terms Offered: Winter
Prerequisite(s): Aut 24, Math 36611, Topics in Analytic Geometry by Akhil Mathew

MATH 36704. Dynamics and Applications. 100 Units.

The course will provide an introduction to basic results and techniques in dynamical systems and then discuss selected applications.

Instructor(s): Simion Filip     Terms Offered: Winter

MATH 36888. Pseudodifferential Operators with Applications. 100 Units.

In this course I will introduce classical pseudodifferential operators and their calculus and give some applications, including to variable coefficient Schrodinger equations.

Instructor(s): Carlos Kenig     Terms Offered: Autumn
Prerequisite(s): The first year graduate sequence in analysis.

MATH 36918. Min-max Methods in Minimal Surfaces. 100 Units.

Min-max methods in minimal surfaces have produced a series of spectacular results lately and settle old questions. I will develop the Algren-Pitts min-max theory from the beginning and explain how that can be used to prove existence of minimal surfaces.

Instructor(s): Andre Neves     Terms Offered: Spring

MATH 37001. Bernstein center and cocenter. 100 Units.

We discuss the Bernstein center of a p-adic reductive group, the cocenter of the Hecke algebra as a module over the Bernstein center and its completion. We present Langlands' theory of endoscopy in this language. We discussion the stable Bernstein center, the stable cocenter and their relation to Langlands' functoriality.

Instructor(s): Bao Chau Ngo     Terms Offered: Autumn

MATH 37104. Parabolic Equations with Irregular Data and Related Issues. 100 Units.

We present a general theory of existence and uniqueness of linear parabolic equations with Lebesgue/Sobolev regular coefficients and initial conditions. Applications to the theory of stochastic differential equations are also discussed. The course is based on the recent book C. Le Bris/ P.L. Lions, Parabolic Equations with irregular data and related issues. Application to Stochastic Differential Equations, De Gruyter Series in Applied and Numerical Mathematics, Vol. 4, 2019, 165 pages, ISBN 978-3-11-063313-9. For more information on the course: claude.le-bris@enpc.fr

Instructor(s): Claude Le Bris     Terms Offered: Winter

MATH 37105. Topics in Geometric Measure Theory I. 100 Units.

A measure is a way to assign a size to collections of points. Lebesgue measure is the most important example but, depending upon the application, the 'size' of a set may be measured in many different, very interesting ways. The interplay between measure and geometry can be extremely subtle and has given rise to powerful ideas that are used in energy minimisation problems, the theory of partial differential equations and the study of fractal geometry. This is an advanced course on geometric measure theory and its applications.

Instructor(s): Marianna Csornyei     Terms Offered: Autumn

MATH 37111. Quiver Varieties. 100 Units.

Study of quiver varieties.

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 37114. 00-categories and their applications in homotopy theory. 100 Units.

In the course we shall cover the basic theory of 00-categories using The Joyal-Lurie approach via Quasi-Categories. We shall then see how the 00-categorical approach sheds light on the homotopy theory of CW-complexes. If time permits we shall then see how to use these methods to study stable homotopy theory. In particular the study of stable homotopy groups of spheres.

Instructor(s): Tomer Schlank     Terms Offered: Winter

MATH 37202. Positive Scalar Curvature. 100 Units.

Scalar curvature is the weakest of the curvature notions for Riemannian manifolds. We shall investigate its role, and which manifolds have positive scalar curvature metrics (with complete results in dimensions two and three, conjectures in high dimensions - with some theorems, and great mysteries in dimension four).

Instructor(s): Shmuel Weinberger and Andre Neves     Terms Offered: Winter
Prerequisite(s): The first year sequences in geometry and analysis.

MATH 37214. Hecke algebras, Deligne-Lusztig characters, and character sheaves. 100 Units.

We will discuss Hecke algebras, Deligne-Lusztig characters, and character sheaves.

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 37219. Crystalline Differential Operators. 100 Units.

Introduction to crystalline differential operators.

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 37304. Theory of Elliptic PDES. 100 Units.

We will study the theory for existence and regularity of second order elliptic PDE's. After presenting the basic results (energy estimates, Schauder theory, spectral theory), we will do Di Giorgi-Nash estimates and present several methods to find solutions to quasilinear and fully non-linear second order elliptic PDE's.

Instructor(s): Andre Neves     Terms Offered: Winter
Prerequisite(s): Analysis I and II

MATH 37392. Arithmetic Geometry. 100 Units.

I will explain important aspects in arithmetic geometry.

Instructor(s): Kazuya Kato     Terms Offered: Winter
Prerequisite(s): Algebra 1-Algebra 3 of the first year graduate courses.

MATH 37410. Topics in low-dimensional topology. 100 Units.

We will discuss topics in low-dimensional topology.

Instructor(s): Danny Calegari     Terms Offered: Autumn

MATH 37411. 3-manifolds. 100 Units.

The topic will be foliations, order ability, and the L-space conjecture.

Instructor(s): Danny Calegari     Terms Offered: Autumn

MATH 37502. Symplectic Topology and Hamiltonian Dynamics. 100 Units.

Symplectic geometry and topology is a rapidly developing field of mathematics that originated as a geometric tool for addressing problems in classical mechanics. Powerful methods, such as Gromov's theory of holomorphic curves, have led to the discovery of unexpected phenomena related to the geometry and topology of symplectic manifolds and the dynamics of symplectic maps. In this course, after introducing the basic notions of symplectic geometry and topology, we will discuss some of these developments. We will freely use the language of manifolds and differential forms. 1. Linear symplectic geometry. Affine non-squeezing theorem. 2. Symplectic manifolds and Lagrangian submanifolds. Moser stability theorem and homotopy method. 3. Mathematical model of classical mechanics. Hamiltonian dynamics. 4. Group of Hamiltonian difeomorphisms. Hofer's metric. 5. Displacement energy of Lagrangian submanifolds. 6. Holomorphic curves and Lagrangian submanifolds. 7. Rigidity of the Poisson bracket 8. Diameter of the group of Hamiltonian diffeomorphisms. 9. Symplectic fibrations and Hofer's length spectrum. 10. Symplectic, but non-Hamiltonian, diffeomorphisms. Flux. Books: Arnold, V.I., Mathematical methods of classical mechanics. McDuff, D., Salamon, D., Introduction to symplectic topology. Polterovich, L., The geometry of the group of symplectic diffeomorphisms. Polterovich, L., Rosen, D., Function theory on symplectic manifolds.

Instructor(s): Leonid Polterovich     Terms Offered: Autumn

MATH 37801. Configuration spaces in topology, algebraic geometry and topological field theory. 100 Units.

Configuration spaces seem ubiquitous nowadays. In this course we discuss their role in various settings in which they occur: topology (involving among other things the little disk operad and a certain spectral sequence), algebraic geometry and associated mixed Hodge structures (the case of algebraic curves being the most interesting case) and their role in the theory of conformal blocks.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn

MATH 37902. Topics in unique continuation. 100 Units.

The course will deal with selected topics in unique continuation and boundary unique continuation for elliptic equations, with applications.

Instructor(s): Carlos Kenig     Terms Offered: Autumn
Prerequisite(s): First year analysis sequence, undergraduate pde

MATH 37904. Linear and semilinear Schrodinger evolutions, I. 100 Units.

We will develop harmonic analysis tools for the linear Schrodinger evolution that will then be used for the study of semilinear Schrodinger evolutions. In the first quarter we will treat for the semilinear case small data/short time results, while in the second quarter we will study large data for long times, in critical semilinear problems.

Instructor(s): Carlos Kenig     Terms Offered: Autumn
Prerequisite(s): The first year graduate analysis sequence and familiarity with the Fourier transform and introductory partial differential equations.

MATH 37905. Linear and semilinear Schrodinger evolutions, II. 100 Units.

In the second quarter we will study large data for long times, in critical semilinear problems.

Instructor(s): Carlos Kenig     Terms Offered: Winter
Prerequisite(s): The first year graduate analysis sequence and familiarity with the Fourier transform and introductory partial differential equations.

MATH 37907. Hodge Theory and Moduli. 100 Units.

Perhaps the most important tool for the study of moduli of complex algebraic varieties is the period map, which assigns to a variety its Hodge structure. In this course we shall develop some of the general theory of the notions involved here (including mixed Hodge theory), but examples of interest (some classical and others less so) will be at the center, such as hyperkaehler manifolds---this includes K3 surfaces---and hypersurfaces. This will lead us to see how locally symmetric varieties (such as ball quotients) parametrize Hodge structures. We will also touch on the mixed Hodge theory of the fundamental group and the interesting extensions of Hodge structures that it can give rise to.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn
Prerequisite(s): Prerequisites are basic knowledge of manifolds, (co)homology and De Rham theory (as taught in the courses Algebraic Topology and Topology and Geometry II). We shall treat the classical Hodge theorem as a given (but of course state it), which means that we will not get into its proof.

MATH 37908. Topics in Algebraic Geometry-1. 100 Units.

This is in order to develop some basic algebro-geometric literacy. Topics might include moduli spaces, Deligne-Mostow theory, period maps.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn
Prerequisite(s): Introductory course in algebraic geometry.

MATH 38002. Representation theory of p-adic groups. 100 Units.

Discussing representation theory of p-adic groups

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 38005. Decomposition theorem for perverse sheaves and Hodge theory. 100 Units.

Discussing decomposition theorem for perverse sheaves and Hodge theory

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 38010. The Hitchin Morphism. 100 Units.

The Hitchin morphism will be discussed.

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 38420. Mathematics of Quantum Computing. 100 Units.

This course is a gentle introduction to mathematical foundations of quantum computing taught in completely rigorous format: we will completely disregard physical aspects and specific questions pertaining to particular implementations. An (approximate) list of topics: reversible, probabilistic and quantum computation. Quantum complexity classes and relations to their classical counterparts. Fundamental quantum algorithms, notably Grover's search and Shor's factoring algorithm. Quantum (query) complexity theory and quantum communication complexity. Quantum probability, super-operators and non-unitary quantum computation. Basics of quantum information theory and quantum error-correction.

Equivalent Course(s): CMSC 38420

MATH 38515. Symplectic Topology. 100 Units.

This is an introduction to symplectic topology. The purpose is to provide background and details for some of the material covered in Leonid Polterovich's course.

Instructor(s): Danny Calegari     Terms Offered: Winter

MATH 38595. Topics in Complex Dynamics. 100 Units.

An introduction to the theory of complex dynamics in 1 dimension, especially the theory of rational maps and rational correspondences. Foundations of the theory, quasiconformal analysis, no wandering domains, Mandelbrot set and variations.

Instructor(s): Danny Calegari     Terms Offered: Spring

MATH 38599. Introduction to Floer Theories. 100 Units.

An introduction to the use of gauge theoretic methods in 3-manifold topology, including Seiberg-Witten and Heegaard Floer Homology, connections to taut foliations and sutured manifolds. Thurston norm, contact structures, etc.

Instructor(s): Danny Calegari     Terms Offered: Winter

MATH 38803. Tamagawa number conjectures for zeta values. 100 Units.

I explain the Tamagawa number conjecture on zeta values, a general conjecture about the arithmetic of various zeta functions.

Instructor(s): Kazuya Kato     Terms Offered: Winter

MATH 39013. Crystalline Differential Operators. 100 Units.

Crystalline differential operators will be discussed.

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 39902. Topics in Invariant Theory. 100 Units.

We will discuss Topics in Invariant Theory.

Instructor(s): Bao Chau Ngo     Terms Offered: Autumn

MATH 42002. P-adic Hodge Theory. 100 Units.

Basic things in p-adic Hodge theory are explained.

Instructor(s): Kazuya Kato     Terms Offered: Winter
Prerequisite(s): Algebra 1, 2, 3

MATH 42501. Geometric Representation Theory. 100 Units.

We will discuss Geometric Representation Theory.

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 42503. Geometric Satake and beyond. 100 Units.

Geometric Satake and beyond will be discussed.

MATH 47000. Geometric Langlands Seminar. 100 Units.

This seminar is devoted not only to the Geometric Langlands theory but also to related subjects (including topics in algebraic geometry, algebra and representation theory). We will try to learn some modern homological algebra (Kontsevich's A- infinity categories) and some "forgotten" parts of D- module theory (e.g. the microlocal approach).

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Autumn

MATH 47100. Geometric Langlands Seminar. 100 Units.

The seminar is devoted to the Geometric Langlands theory and related subjects, which covers topics in algebraic geometry, algebra, and representation theory.

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Winter

MATH 47200. Geometric Langlands Seminar. 100 Units.

The seminar is devoted to the Geometric Langlands theory and related subjects, which covers topics in algebraic geometry, algebra, and representation theory.

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Spring

MATH 59900. Reading/Research: Mathematics. 300.00 Units.

Readings and Research for working on their PhD

MATH 70000. Advanced Study: Mathematics. 300.00 Units.

Advanced Study: Mathematics