## Fall 2022

- MATH 501: ALGEBRAIC STRUCTURES

Robert Calderbank

TuTh 1:45 PM-3:00 PM

- MATH 531: REAL ANALYSIS I

Logan Stokols

MW 12:00 PM-01:15 PM

- MATH 541: APPLIED STOCHASTIC PROCESSES

Hongkai Zhao

WF 03:30 PM-04:45 PM

- MATH 551: APPLIED PARTIAL DIFFERENTIAL EQUATIONS & COMPLEX VARIABLES

Tarek Elgindi

MW 03:30 PM-04:45 PM

- MATH 553: ASYMPTOTIC & PERTURBATION METHODS

Thomas Witelski

WF 05:15 PM-06:30 PM

- MATH 555: ORDINARY DIFFERENTIAL EQUATIONS

James Nolen

TuTh 01:45 PM-03:00 PM

- MATH 561: NUMERICAL LINEAR ALGEBRA

Hau-Tieng Wu

MWF 08:30 AM-09:20 AM

- MATH 581: MATHEMATICAL FINANCE

Xavier Mela

581.01 MW 12:00 PM-01:15 PM

581.02 MW 10:15 AM-11:30 AM

581.03 MW 03:30 PM-04:45 PM

- MATH 585: INTRO TO ALGORITHMIC TRADING

David Ye

MF 05:15 PM-06:30 PM

- MATH 590-02: ADVANCED SPECIAL TOPICS IN MATHEMATICS
- MATH 590-02.01 QUANTUM MECHANICS AND STRING THEORY

Paul Aspinwall

MW 01:45 PM-03:00 PM

- MATH 590-02.02 QUANTATIVE METHODS FOR BIO-DATA

Jichun Xie

MW 08:30 AM-09:45 AM

- MATH 590-02.01 QUANTUM MECHANICS AND STRING THEORY
- MATH 601: GROUPS, RINGS, AND FIELDS

Kirsten Wickelgren

WF 01:45 PM-03:00 PM

- MATH 611: ALGEBRAIC TOPOLOGY I

Richard Hain

TuTh 12:00 PM-01:15 PM

- MATH 620: SMOOTH MANIFOLDS

Mark Haskins

TuTh 10:15 AM-11:30 AM

- MATH 627: ALGEBRAIC GEOMETRY

Ezra Miller

TuTh 01:45 PM-03:00 PM

- MATH 631: MEASURE AND INTEGRATION

Jianfeng Lu

WF 10:15 AM-11:30 AM

- MATH 635: FUNCTIONAL ANALYSIS

Mark Stern

TuTh 08:30 AM-09:45 AM - MATH 690-05: TOPICS IN NUMBER THEORY

Heekyoung Hahn

TuTh 10:15 AM-11:30 AM

- MATH 771S: TEACHING COLLEGE MATHEMATICS

Shira Viel

MW 12:00 PM-01:15 PM

- MATH 790-90: MINICOURSE IN ADVANCED TOPICS
- 790-90.01 Similarity solutions for singular dynamics in PDE's

Thomas Witelski

8/30-9/27/2022 TuTh 03:30 PM-04:45 PMThe lectures will describe how to study dynamics leading up to singular limits in partial differential equations (finite-time blow-up, rupture, quenching, etc). For many problems, the intermediate asymptotics before singularity formation are given by self-similar solutions. Topics to be covered will include: determining similarity solutions for nonlinear PDEs, 0th, 1st and 2nd kind similarity solutions, stability analysis of self-similar dynamics, geometric effects and conserved quantities, and numerical methods for problems with finite-time singularities.

The material should be accessible to anyone with a basic background in partial differential equations.

Applications include problems in fluid dynamics, nonlinear diffusion, geometric evolution equations and models from mathematical biology.

- 790-90.02 Mapping Class Groups and Moduli Spaces of Curves

Richard HainThe mapping class group Mod(S) of a compact oriented surface S is the group of homotopy classes of orientation preserving homeomorphisms of the surface to itself. Up to isomorphism, it depends only the genus g of S. The moduli space M_g of curves of genus g (g > 1) is the space whose points correspond to the isomorphism classes of complex structures on S. It is a complex algebraic variety (or orbifold) of complex dimension 3g-3. Mapping class groups and moduli spaces of curves are of central importance in geometric topology, algebraic and arithmetic geometry, topological field theory, and dynamical systems.

In this course I will survey (and prove a selection of) the basic results about mapping class groups, moduli spaces of curves and the relationship between them. Students should know basic algebraic topology (MTH 611) and (preferably) complex analysis or differential geometry at the undergraduate level.

This course should be of interest and useful to those studying studying algebraic geometry, differential geometry, topology and some parts of mathematics physics. It should also be of interest to number theorists (both analytic and algebraic) as moduli space of curves are not Shimura varieties, but are close cousins whose arithmetic properties are not at all well understood, even in low genus.

- 790-90.03 Constructible sheaves on compactified locally symmetric spaces

Leslie Saper

9/29-11/1/2022 TuTh 01:45 PM-03:00 PMLocally symmetric spaces and cohomology theories associated to them play an important role in many areas of mathematics. Often such a theory can be realized as the cohomology of a constructible sheaf on a compactification of

the locally symmetric space, often the reductive Borel-Serre compactification. We will describe a combinatorial model, L-modules, of constructible sheaves. Every L-module has a locally computable invariant, the micro-support, which governs the non-vanishing of its cohomology. We will describe several applications of this theory; there are more that are in progress.I will adjust the course depending on the background of the participants. My aim is to make it as accessible to as many people as possible. Of necessity many proofs will be replaced by examples. Topics that may be touched on include some of the following: linear algebraic groups, arithmetic groups, locally symmetric spaces local systems, differential forms, sheaves, derived categories and derived functors, parabolic subgroups, the reductive Borel-Serre compactification, Satake

compactifications, intersection cohomology, weighted cohomology, L^2-cohomology, finite dimensional representation theory of semi-simple Lie algebras, L-modules and their micro-support, a vanishing theorem for cohomology of L-modules,

Rapoport’s conjecture, the Goresky-Harder-MacPherson theorem, Zucker’s conjecture. - 790-90.04 An introduction to contact geometry

Lenhard Ng

11/3-12/6/2022 MWF 10:15 AM-11:05 AM

This minicourse will cover some of the basic ideas and techniques in contact geometry. Contact geometry, often described as the odd-dimensional analogue of symplectic geometry, is a subject whose origins date back to the 19th century in geometric optics; it's now grown into a large and beautiful subject that is closely tied to three-dimensional topology as well as symplectic and complex geometry. I'll try to lay out some of the motivating questions and foundational results of contact geometry, with special emphasis on ties to 3-manifold topology. The minicourse should be accessible to anyone who is reasonably comfortable with smooth manifolds (along the lines of Math 620) and, not as crucially, algebraic topology (Math 611).

Here are some topics that I hope to cover (however briefly). This list is almost certainly too ambitious, but we can focus on specific topics depending on interest.- contact and symplectic structures; Hamiltonian and Liouville vector fields
- Darboux's Theorem and Moser's method
- tight versus overtwisted contact structures
- Legendrian and transverse knots
- Stein and Weinstein manifolds
- open book decompositions; the Giroux correspondence.

- 790-90.05 Instantons and Nahm transforms

Mark Stern

9/28-10/28/2022 MW 08:30 AM-09:45 AMThe goal of this course is to provide tools to study the moduli space of instanton solutions to the Yang-Mills equations. The Yang-Mills equations are nonlinear analogues of Maxwell's equations, which play an important role in high energy particle physics. Their spaces of solutions play an important role in topology, geometry, and physics. In topology, they function like a nonlinear cohomology theory and have had massive impact in knot theory and 4 manifold topology. These moduli spaces are endowed with canonical metrics, which play an important role in high energy particle physics. Moreover, these metrics tend to have special holonomy.

A fundamental tool for probing the structure of these moduli spaces is the Nahm Transform and its kin.The Nahm transform is a correspondence between solutions to the instanton equations on one manifold and solutions of a (possibly) different nonlinear equation on a different space.

The lectures will begin with a naive introduction to the Yang-Mills equations and their instanton solutions, which I hope to make accessible to those without much familiarity with vector bundles. We will then introduce the Dirac operator coupled to these vector bundles and describe basic properties of these first order elliptic operators. We will introduce their associated index bundles^*, but will probably black box the relevant functional analysis. We will derive the conditions that guarantee that data associated to the index bundles satisfy an interesting nonlinear equation. We will give several examples of this phenomenon, and then prove the correspondence with the solutions of the original equation in some simple cases. A first course in Riemannian geometry would be a useful prerequisite.

- 790-90.06 The universality phenomenon in random matrix theory (and beyond)

Nicholas Cook

10/31-12/2/2022 MWF 10:15 AM-11:05 AM

"Universality" is a concept originating in statistical physics, describing a phenomenon wherein "macroscopic" systems (e.g. a container of gas or a refrigerator magnet) exhibit features that are independent of most of the details of their "microscopic" components (gas molecules or iron atoms). In probability theory, a classic example is the Central Limit Theorem, providing the limiting distribution of the appropriately rescaled sum of independent and identically distributed (iid) random variables; the limiting distribution "forgets" all details of the distribution of the summands except the mean and variance.

Recent years have seen enormous progress in our understanding of the universality phenomenon in the context of random matrices. In this course we'll focus on Wigner matrices, which are large symmetric matrices with iid entries on and above the diagonal. One of our main goals will be to establish the Tao–Vu Four Moment Theorem, stating that local spectral statistics asymptotically depend only on the first four moments of the distribution of the matrix entries (one may analogously call the CLT a "two moment theorem").

The universal distributions encountered in random matrix theory extend (conjecturally) far beyond random matrices to physical systems and to objects in number theory. Perhaps the most famous of these is the Montgomery–Dyson pair correlation conjecture, stating that the statistics of spacings between zeros of the Riemann zeta function (averaged over large ranges of the critical axis) are asymptotically the same as for eigenvalue spacings of Wigner matrices.*Tentative list of topics to be covered:*- Warmup: universality in the context of the Central Limit Theorem (including the Lindeberg exchange argument).
- Universality for global spectral statistics: The resolvent method and Wigner's semicircle law.
- Universality for local spectral statistics: Local semicircle law; the Lindeberg exchange argument and Four Moment Theorem; the dynamical approach to universality (Dyson Brownian motion).
- Connections to analytic number theory: Montgomery–Dyson conjecture, GUE hypothesis, Quantum Unique Ergodicity.
- (Time permitting) Anderson localization transition for random Schrödinger operators.

- 790-90.01 Similarity solutions for singular dynamics in PDE's