Week1

Program (as of 2019.4.1)

Welcome Banquet on 6/5 (wed) 18:00-20:00 at Camphora (campus cafeteria) (info)

Title and Abstract

Sergey Fomin (Michigan)

Cluster algebras from a combinatorial perspective

These lectures will provide an elementary introduction into cluster algebras from a combinatorial standpoint. We will review historical motivations, introduce basic notions and results, and discuss several examples in which cluster mutations naturally appear, including pseudoline arrangements, triangulations of surfaces, and plabic graphs. No prior familiarity with the subject will be assumed.

Alexander Goncharov (Yale)

Quantum geometry of moduli spaces of local systems and representation theory

Let S be a topological oriented surface with punctures and special points on the boundary, modulo isotopy. Let G be a split semi-simple adjoint group. We define and quantize a moduli space Loc(G,S) G-local systems on S, generalising the character variety.

To achieve this, we introduce a new moduli space P(G, S) closely related to Loc(G,S). We prove that it has a cluster Poisson variety structure, equivariant under the action of a discrete group, containing the mapping class group of S. This generalises results of V. Fock and the author, and I. Le.

For any cluster Poisson variety X we consider its Langlands modular double A(X, h). If the Planck constant h is either real or unitary, we equip it with a structure of a *-algebra, and construct its principal series of representations.

Combining this, we get a principal series of representations of the Langlands modular double of the moduli spaces P(G, S) and Loc(G,S).

When S has no boundary, it provides a local system of infinite dimensional vector spaces over the punctured determinant line bundle L(g,n) on the moduli space M(g,n). Assigning to a complex structure on S the space coinvariants of oscillatory representations of W-algebras at the punctures of S, we get another local system on L(g,n). We conjecture that it maps canonically to the former one, providing conformal blocks for the Liouville / Toda theories.

We discuss other applications to representations theory and algebraic geometry.

The lectures are mostly based on our recent work with Linhui Shen.

Bernhard Keller (Paris 7)

Additive categorification of cluster algebras

In these lectures, we will give an introduction to the additive categorification of cluster algebras based on quiver representations. In the first lecture, we will show how indecomposable representations of Dynkin quivers are linked to non initial cluster variables via the Caldero-Chapoton map. We will introduce the cluster category as a means to include the initial variables and the clusters into the picture. In the second lecture, we will sketch how stable categories of representations of preprojective algebras can be used for an analogous categorification of the cluster algebras arising from varieties like Grassmannians and unipotent cells following Geiss-Leclerc-Schroeer. In the third lecture, we will introduce quivers with potentials and describe how they allow to categorify arbitrary cluster algebras of geometric type following work of Derksen-Weyman-Zelevinsky, Plamondon and Nagao. The final lecture will be devoted to an introduction to stability scattering diagrams following Bridgeland.

Maxim Kontsevich (IHES)

Wall-crossing geometry

I will talk about an approach to cluster algebras and canonical bases via scattering diagrams (a.k.a. wall-crossing structures). It will be illustrated by theory of Donaldson-Thomas invariants and by SYZ picture in mirror symmetry for complex integrable systems.

Bernard Leclerc (Caen)

Monoidal categorification of cluster algebras

In these lectures, we will discuss monoidal categorifications of cluster algebras motivated by examples coming from quantum affine algebras and quiver Hecke algebras. These categorifications, when available, give a transparent explanation of the positivity properties of cluster monomials conjectured by Fomin and Zelevinsky. They have also been used (in a quantum setting) to prove another long-standing conjecture saying that cluster monomials of certain cluster algebras coming from Lie theory are part of the dual of Lusztig's canonical basis. Conversely, if a monoidal category is a categorification of a cluster algebra, one can use cluster theory to obtain factorization properties and character formulas for the simple objects categorifying cluster monomials. The lectures will give an introduction to these questions and survey the results obtained by a number of people, including Hernandez, Nakajima, Qin, and Kang-Kashiwara-Kim-Oh.

(last updated: May 17, 2019)