### Fall 2023 Minerva Foundation Lectures

**The main goals of these lectures are:**

1. Provide a comprehensive introduction to the proof of the nonlinear stability of slowly rotating Kerr black holes established recently in the sequence of works [Kl-Sz:Kerr], [GKS-2022], [Kl-Sz:GCM1], [Kl-Sz:GCM2] and [Shen], and briefed in [Kl-Sz:review].

2. Discuss the geometric formalism based on non-integrable null horizontal structures used in these works. Derive the main Teukolsky and generalized Regge- Wheeler equations. These follow the material 1 of Part 1 in [GKS-2022].

3. Discuss the proof of the basic hyperbolic estimates, Morawetz and rp-weighted, following Part 2 of [GKS-2022].

4. Discuss open problems related to these topics.

**First lecture: Wednesday, September 6, 2023**

**Meeting on Wednesdays at 2:45 p.m.**

**Room 507, Mathematics Hall**

**2990 Broadway (117th Street)**

### FALL 2023 SAMUEL EILENBERG LECTURES

**Abstract: **Starting from Harris-Taylor and my PHD work on the geometric realization of the local Langlands correspondences to my recent joint work with Scholze, I will explain the new geometric structures that have emerged in the Langlands program. For the first time, ideas of the geometric Langlands program have been imported into the classical Langlands program in characteristic zero. To give life to those ideas we developed with Scholze some new geometry, following my work with Fontaine where we began to give a meaning to the idea of an holomorphic function of the variable p where p is a prime number. I will explain this new geometry and its applications to the Langlands program.”

**First lecture: Tuesday, September 12, 2023**

**Tuesdays at 4:10pm**

**Room 520, Mathematics Hall**

**2990 Broadway (117th Street)**

### Special Lecture Series

**Speaker**: Professor Nikita Nekrasov (Simons Center for Geometry and Physics and Columbia)

**The Count of Instantons**

**Abstract**: Graduate level introduction to modern mathematical physics with the emphasis on the geometry and physics of quantum gauge theory and its connections to string theory. We shall zoom in on a corner of the theory especially suitable for exploring non-perturbative aspects of gauge and string theory: the instanton contributions. Using a combination of methods from algebraic geometry, topology, representation theory and probability theory we shall derive a series of identities obeyed by generating functions of integrals over instanton moduli spaces, and discuss their symplectic, quantum, isomonodromic, and, more generally, representation-theoretic significance.

Quantum and classical integrable systems, both finite and infinite-dimensional ones, will be a recurring cast of characters, along with the other (qq-) characters.

**First lecture: Friday, September 15, 2023**

**Fridays at 1:30pm until 3pm **(except for the following dates: 10/6 , 10/20/, 11/3, 11/17, 11/24, 12/1-12/8 (TBD), and 12/15)

**Room 520, Mathematics Hall**

**2990 Broadway (117th Street)**

Lecture notes: Not split per lecture will be updated as course continues

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