Summer school: Recent perspectives on geometry and representation theory

The summer school is part of the programme "Representations, Moduli and Duality" and will be held at the Bernoulli Center at EPFL and consists of the following 4 mini-courses:

Mauro Porta: Derived methods in representation theory
In this mini-course, I will provide an introduction to derived algebraic geometry, with special emphasis on the construction of moduli stacks and spaces relevant for geometric representation theory. I will discuss how to obtain homological and motivic invariants, and introduce cohomological Hall algebras as consequence of the formalism of 2-Segal objects of Kapranov and Dyckerhoff. If time permits, I will outline some recent advances of the theory.

Richard Rimanyi: Characteristic classes of singularities and their 3d mirror symmetry
Characteristic classes of singularities provide a concrete and computable way to detect 3d mirror symmetry. This minicourse will introduce characteristic classes associated with singularities arising in various geometric contexts, including singularities of maps, quivers, and differential forms. We will review computational techniques (from resolution or deformation of singularities to Hall algebra-type recursions, and interpolation) and explore key applications. Our focus will be on stable envelopes, as introduced by Maulik–Okounkov, Okounkov, and Aganagic–Okounkov, and their deep connections to geometric representation theory. We will outline a proof of the 3d mirror symmetry statement for elliptic stable envelopes on bow varieties, based on a joint work with T. Botta.

Peng Shan: Vertex algebras, affine Springer fibres and 4d mirror symmetry
We will explain some results and conjectures about relationships between representation theory of simple affine Vertex algebras and the geometry of Hitchin fibrations, which is related to dualities between Higgs and Coulomb branches for 4d N=2 super conformal field theory.

Eric Chen: S-duality in arithmetic and geometric contexts
The seminal work of Kapustin--Witten posits that various Langlands-type equivalences can be organized under the framework of S-duality for a family of 4d gauge theories. Following this assertion, one would optimistcally expect the existence of a category L of boundary conditions which underlies calculations of interest in this family of theories. Following an introduction to these ideas, we will explain their relation to the recently emergent relative Langlands program of Ben-Zvi—Sakellaridis--Venkatesh, and survey some experiments one can conduct to further our understanding of S-duality on the category L. 

Special Lecture by Yiannis Sakellaridis: L-functions and harmonic analysis
 The goal of this talk will be to explain the arithmetic origins of the "relative" Langlands duality presented in the mini-course by Eric Chen. In particular, I will give an overview of the role played by harmonic analysis on spherical varieties and other interesting spaces in the study of (automorphic) L-functions, and explain the relations between numerical and geometric statements. If time allows, I will also discuss possible extensions of this program that we currently do not understand.