### SPRING 2024 SAMUEL EILENBERG LECTURES

**Title**: Moduli spaces of high dimensional manifolds

**Abstract**: Following influential work of John Harer in the 1980s, Ulrike Tillmann in the 1990s, and Ib Madsen and Michael Weiss in the 2000s, we learned a new approach to the moduli space of Riemann surfaces, and to the diffeomorphism groups and mapping class groups of oriented 2-manifolds. A lesson learned by their work is that patterns emerge in the large-genus limit, another is that these patterns are well expressed in homotopy theoretic terms.

Inspired by these developments in (real) dimension 2, Oscar Randal-Williams and I set out to study moduli spaces of higher-dimensional manifolds in a similar spirit. The goal of this semester’s Eilenberg Lectures will be to present some of our joint work, as well as some background, context, and some very recent developments in high-dimensional manifold theory building on our work.

**First lecture: January 22, 2024 at 4:10pm**

**Room 520 Mathematics**

### SPECIAL LECTURE SERIES

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

**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, February 2, 2024
**

**Room 520, Mathematics Hall**

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

### Universality and Integrability in KPZ

**March 11-15, 2024**

https://sites.google.com/view/universalityintegrabilityinkpz

This workshop will be focused on topics related to the KPZ universality class. This is a broad class of probabilistic models coming mostly from probability theory and mathematical physics (including random interface growth models, directed random polymers and certain interacting particle systems) which share common universal fluctuation behavior. This universal behavior manifests itself in the form of common scaling exponents and a common scaling limit, which are independent of the microscopic description of model.

Apart from the intrinsic physical interest of this class of models, the subject has attracted intense research interest during the last two decades due to the rich behavior it exhibits and the various, deep connections it has with other areas of mathematics in general and probability in particular, including random matrix theory, SPDEs, integrable systems and combinatorics. This has given rise to a proliferation of methods and approaches being used to study different aspects of the field. These different approaches are deeply connected, and the goal of the workshop is to bring experts in them together with young researchers interested in the area, and to give them a broad view of the many important advances that have taken place in the field in recent years.

This workshop will also provide an occasion to celebrate the work of Jeremy Quastel and his 60th birthday.

Scientific Organizers: Ivan Corwin (Columbia), Daniel Remenik (Universidad de Chile)

Local Organizers: Jeanne Boursier (Columbia), Evan Sorensen (Columbia)

**Room 312 Mathematics
2990 Broadway, New York NY 10027**