### Spring 2020

### 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.

Day & Time: MW 1:10 – 2:25PM

Location: Mathematics Hall, room 507

### 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.

Day & Time: TR 4:10 – 5:25PM

Location: Mathematics Hall, room 507

### 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

Day & Time: TR 4:10 – 5:25PM

Location: Mathematics Hall, room 528

### 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

Day & Time: MW 1:10 – 2:25PM

Location: Mathematics Hall, room 507

### MATH GR8313 – TOPICS IN COMPLEX MANIFOLDS

**Instructor**: Robert Friedman

**Title**:

**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.

Day & Time: MW 2:40 – 3:55PM

Location: Mathematics Hall, room 507

### MATH GR8674 – TOPICS IN NUMBER THEORY

**Instructor**: Dorian Goldfeld

**Title**:

**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.

Day & Time: TR 1:10 – 2:25PM

Location: Mathematics Hall, room 307

### MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

**Instructor**: Ivan Corwin

**Title**:

**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.

Day & Time: TR 1:10 – 2:25PM

Location: Mathematics Hall, room 622

### 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.

*Prerequisites*

Basic knowledge of probability theory, random walks, stochastic processes.

Day & Time: TR 2:40 – 3:55PM

Location: Mathematics Hall, room 622

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