**Spring 2020** (click to expand/collapse)

### MATH GR6250 – TOPICS IN REPRESENTATION THEORY

**Instructor**: Andrei Okounkov

**Title**: “Equivariant K-theory and enumerative geometry”

**Abstract**: Certain questions in modern high energy physics may be phrased as computations in equivariant K-theory of various moduli spaces of interest in algebraic geometry, in particular, in enumerative geometry. To address these computations, there are certain tools that generalize classical ideas of geometric representation theory. The course will be an introduction to this circle of topics, starting with a review of equivariant K-theory. It should be accessible to first-year PhD students.

### MATH GR6306 – CATEGORIFICATION

**Instructor**: Mikhail Khovanov

**Title**: “Introduction to categorification”

**Abstract**: This course will deal with the lifting of quantum link invariants to homology theories of links and their extensions to tangle and cobordism invariants. It will cover construction of link homology theories via foams and matrix factorizations, categorification of quantum groups and their representations, and related topics in representation theory and low-dimensional topology.

### MATH GR8209 – TOPICS IN GEOMETRIC ANALYSIS

**Instructor**: Sergiu Klainerman

**Title**: “Stability of black holes in General Relativity”

**Abstract**: Here is a list of topics I hope to be able to cover.

- I will start by giving a general introduction to the problem of nonlinear stability of black holes emphasizing its central role in GR today
- The formalism of null horizontal structures and its role in stability results
- Short description of the proof of the nonlinear stability of Minkowski space
- GCM spheres and their role in stability results. I will describe some recent works with J. Szeftel
- Stability of Schwarzschild under axially symmetric polarized perturbations. Recent work with J. Szeftel
- Perspectives on a general stability results for Kerr black holes

### MATH GR8255 – PDE IN GEOMETRY

**Instructor**: Mu-Tao Wang

**Title**: “Einstein equation and spacetime geometry”

**Abstract**: This course will start with an elementary introduction of the mathematical theory of general relativity and then move on to discuss several related topics such as:

- Spacetime geometry
- The Einstein equation
- Black holes
- Mass and angular momentum in general relativity
- Gravitational radiation

### MATH GR8313 – TOPICS IN COMPLEX MANIFOLDS

**Instructor**: Robert Friedman

**Abstract**: The course is an introduction to various aspects of Hodge theory. Topics include: complex manifolds, Kähler metrics, Hodge and Lefschetz decomposition, variation of Hodge structure, Mumford-Tate group, mixed Hodge structures, rational differentials.

### MATH GR8674 – TOPICS IN NUMBER THEORY

**Instructor**: Dorian Goldfeld

**Abstract**: This course will be focused on trace formulae starting with the Selberg Trace Formula for GL(2) in the classical setting. Then I shall consider the Kuznetsov trace formula followed by higher rank generalizations. The emphasis will be on analytic number theory applications.

### MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

**Instructor**: Ivan Corwin

**Abstract**: This course will focus on topics related to integrable probability. In particular, we will consider the (stochastic) six vertex model and its various generalizations and degenerations through the lens of Bethe ansatz, Markov duality, Schur / Macdonald type measures, and Gibbsian line ensembles. Despite the title “topics in stochastic analysis”, there will be very little stochastic analysis, perhaps save a few discussions about stochastic (partial) differential equation limits of some integrable models. Some basic graduate probability will be assumed in this course. This course is intended for graduate students who are working or plan to work in or adjacent to the field of integrable probability.

### MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

**Instructor**: Julien Randon-Furling

**Title**: “Convex hulls of Random Walks”

**Abstract**: This lecture series will cover a range of results on the convex hull of random walks in the plane and in higher dimensions: expected perimeter length in the planar case, expected number of faces on the boundary, expected d-dimensional volume, and other geometric properties of such random convex polytopes.

In the last part of the course, we will discuss related topics such as the convex hull of Brownian motion (in the plane and in higher dimensions) together with connections to one-dimensional results on the greatest convex minorant of random walks, Brownian motion and more general L evy processes.

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**Fall 2019** (click to expand/collapse)

### MATH GR8210 – Partial Differential Equations

**Instructor**: Richard Hamilton

### MATH GR8255 – PDE In Geometry

**Instructor**: Simon Brendle

**Topics**: “Geometry of submanifolds”

**Abstract**: “In this course will discuss analytical questions that arise in submanifold geometry. In particular, we will focus on the minimal surface equation, and its parabolic counterpart, the mean curvature flow. We will also discuss the main techniques used in the study of these equations: this includes the basic monotonicity formulae, the Michael-Simon Sobolev inequality, and arguments based on the maximum principle.”

### MATH GR8480 – Gromov-Witten Theory

**Instructor**: Chiu-Chu Liu

**Abstract**: Introduction to Gromov-Witten theory: moduli of curves, moduli of stable maps, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Genus zero mirror theorems for complete intersections in projective spaces

Stable maps with fields, N-mixed-Spin-P fields, and higher genus Gromov-Witten invariants of quintic Calabi-Yau threefolds

### MATH GR8674 – Topics in Number Theory

**Instructor**: Eric Urban

**Topic**: Eisenstein congruences, Euler systems, and the p-adic Langlands Correspondance

**Abstract**: The goal of this course is to introduce the strategy and some of the ingredients that are used in a new construction of Euler systems for Galois representations attached to p-ordinary cuspidal representations of symplectic groups or unitary groups. These Euler systems are constructed out of congruences between Eisenstein series and cuspidal forms of all level and weights. The theory of Eigenvarieties allows to see that such non trivial congruences exist, and therefore provides non trivial norm compatible Galois cohomology classes. The integrality of these classes follows from deeper arguments using, among other things, the local-global compatibility in the p-adic Langlands correspondance for GL2(Qp). As usual, the first meeting will be devoted to a more detailed presentation and to explain the motivations, the main ideas and the strategy of this construction. I will also give a schedule for the next lectures and a list of useful references.

### MATS GR8260 – Topics in Stochastic Analysis

**Instructor**: Konstantin Matetski

**Abstract**: The goal of the course is to study several classical and modern topics on stochastic partial differential equations (SPDEs). These equations can be used to describe various processes in Statistical Mechanics, Fluid Mechanics, Quantum Field Theory, Mathematical Biology and so on. Depending on properties of SPDEs, they require different approaches to define solutions and to study their properties. Recent breakthroughs in non-linear SPDEs, which includes the theory of regularity structures by M. Hairer and the theory of paracontrolled distributions by M. Gubinelli, P. Imkeller and N. Perkowski, have opened new prospects in the field.

The course will cover the following topics:

Elements of Gaussian measures, including the Cameron-Martin space, white noise and Wiener integral;

Linear SPDEs: notions of solutions, existence and uniqueness of solutions, and their properties;

Elements of rough paths and their usage for non-linear SPDEs;

The theory of regularity structures with applications to rough SPDEs, including the Kardar-Parisi-Zhang (KPZ) and Stochastic Quantization equations.

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**Spring 2019** (click to expand/collapse)

### MATH GR8210 – Partial Differential Equations

**Instructor:** Richard Hamilton

### MATS GR8260 – Topics in Stochastic Analysis

**Instructor:** Ioannis Karatzas

**Abstract: **Starting with Brownian Motion as the canonical example, we present the theory of integration with respect to continuous semimartingales. We then connect this theory to partial differential equations and functional analysis, and develop some of its applications to optimization, filtering and stochastic PDEs, entropy production, optimal transport, and the theory of portfolios.

### MATH GR8429 – Topics in Partial Differential Equations

**Instructor:** Panagiota Daskalopoulos

**Abstract:** The following is a brief outline of the course which may change depending on the students demands and as the course progresses:

(1) Introduction.

(2) Ancient solutions to the semi-linear heat equation.

(3) Liouville theorems for the Navier-Stokes equations.

(4) The classication of ancient compact solutions to curve shortening flow.

(5) Ancient compact non-collapsed solutions to Mean curvature flow.

(6) The construction of ancient solutions to the Yamabe flow.

(7) The classification of ancient compact solutions to the Ricci flow on s^{2}.

**Background:** A basic courses on: (i) elliptic and parabolic PDE and (ii) Differential geometry.

### MATH GR8675 – Topics in Number Theory

**Instructor:** Raphael Beuzart-Plessis

**Abstract:** The global Gan-Gross-Prasad conjectures relate special values of (automorphic) L-functions to explicit integrals of automorphic forms that are called “periods.” The local versions of these conjectures are concerned with certain branching problems for infinite dimensional representations of real or p-adic Lie groups. The two are intimately related through the representation-theoretic point of view of automorphic forms. The aim of this course will be to introduce the necessary background to state these conjectures and to discuss part of the recent progress made on them, essentially following Waldspurger and W. Zhang. A remarkable common feature of these approaches is the use of so-called relative trace formulae (both globally and locally), and a great part of this course will be devoted to the study of these powerful analytic tools in certain special cases.

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**Fall 2018** (click to expand/collapse)

**MATH GR8255 – PDE in Geometry**

**Section 001**: Partial differential equations in geometry

**Instructor: **Simon Brendle**
Abstract**: This course will focus on geometric flows (such as the Ricci flow and the mean curvature flow) and singularity formation. A central issue is to find quantities which are monotone under the evolution. For the Ricci flow, this includes Perelman’s entropy formula, or Perelman’s monotonicity formula for the reduced volume. We will discuss these monotonicity formulas and their consequences.

**MATH GR8674 – Topics in Number Theory**

**Section 001**: Arithmetic of L-functions

**Instructor**: Chao Li

**Abstract**: We will discuss the conjecture of Birch and Swinnerton-Dyer, which predicts deep connections between the L-function of an elliptic curve and its arithmetic, and the vast conjectural generalizations for motives due to Beilinson, Bloch and Kato. In the first half, we will provide necessary background and explain a proof of the BSD conjecture in the rank 0 or 1 case. We will emphasize new tools which generalize to higher dimensional motives. In the second half, we will study recent results on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.

**Section 002**: Topics in Analytic Number Theory

**Instructor**: Dorian Goldfeld

**Abstract**: This course will focus on recent developments in the spectral theory of automorphic forms on GL(n,R) with n > 2. I will begin by reviewing the basic theory of automorphic forms and L-functions on the upper half plane H^n. A reference is my book: *Automorphic forms and L-functions for the group GL(n,R)*, Cambridge University Press (2015). One of the main goals is to present new methods to derive the Fourier expansion of Langlands Eisenstein series twisted by cusp forms of lower rank and then obtain spectral expansions of L^2 automorphic forms into cusp forms, Langlands Eisenstein series, and residues of Langlands Eisenstein series. We will then give applications that have been developed recently to the trace formula, spectral reciprocity, special values of L-functions, and other topics.

**Section 003**: Arithmetic Statistics

**Instructor**: Wei Ho

**Abstract**: We will discuss a range of topics in the field of “arithmetic statistics”, which focuses on understanding distributions of arithmetic invariants in families. A sample question would be: for all degree d number fields with Galois group G, under some reasonable ordering, what proportion of such fields have trivial class group? We will study both heuristics and theorems related to class groups of number fields and invariants for elliptic curves and Jacobians of higher genus curves over Q. As time permits, we may also consider similar questions over function fields. Students are expected to be comfortable with standard graduate courses in algebraic and analytic number theory and algebraic geometry.

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