The following mini courses will be organized. The venue is Room 210, Run Run Shaw Building, Main Campus, The University of Hong Kong.

Date: November 4, 5, 7
Time: 9:30am - 10:30am
Speaker: Lev Borisov
Affiliation: Rutgers

Title: Better-behaved GKZ hypergeometric systems and their duality

Abstract: I will talk about the version of the Gel'fand-Kapranov-Zelevinsky hypergeometric systems that is best suited for the setting of local toric mirror symmetry. Starting from simple combinatorial data of a lattice polytope Δ, one can define two vector bundles with flat connections over the space of non- degenerate Laurent polynomials with support Δ. Flat sections of these vector bundles are related to the K-theory of toric Deligne-Mumford stacks defined by triangulations of Δ. I will, in particular, discuss recent work, joint with Zengrui Han, which gives an explicit formula for the long-conjectured duality of these systems. I will try to keep the discussion as elementary as possible, with no familiarity with mirror symmetry or GKZ hypergeometric systems assumed.

Date: November 4, 5, 7
Time: 10:45am - 11:45am
Speaker: Misha Gekhtman
Affiliation: Notre Dame

Title: Poisson geometric approach to exotic cluster structures on simple Lie groups

Abstract: I will discuss a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group G and arbitrary Poisson- homogeneous bracket on G associated with Belavin-Drinfeld data. Given a pair of representatives r, r' from two arbitrary Belavin{Drinfeld classes, we build a rational map from G with the Poisson structure defined by two appropriately selected representatives from the standard class to G equipped with the Poisson structure defined by the pair r, r'. In the An case, we prove that this map is invertible whenever the pair r, r' is drawn from aperiodic Belavin-Drinfeld data and apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair r, r'. A similar construction exists in the case of the dual Poisson Lie groups. The necessary background on Poisson-Lie groups and cluster structures com- patible with Poisson brackets. An emergence of generalized cluster structures in the context will also be discussed. (Based on joint work with M. Shapiro, A. Vainshtein and D. Voloshyn.)

Date: November 8, 11, 12
Time: 9:30am - 10:30am
Speaker: Matt Pressland
Affiliation: Glasgow

Title: Additive categorification for positroid varieties

Abstract: The totally nonnegative Grassmannian is an important object in several stories, including Lusztig's total positivity, and the calculation of scat- tering amplitudes via the amplituhedron. It has a cell decomposition, described by Postnikov, in which each cell is obtained by intersecting the totally non- negative Grassmannian with a particular subvariety of the full Grassmannian: a so-called open positroid variety. A useful tool in studying totally positive spaces is Fomin{Zelevinsky's theory of cluster algebras, and a recent result of Galashin and Lam is that the coordinate ring of (the cone on) an open positroid variety indeed has a natural cluster algebra structure. In this lecture series, I will describe this cluster structure and explain how to use representation-theoretic techniques to understand it, setting up a dictionary between the combinatorics and the algebra. Some of the results here are joint with anak and King. Galashin and Lam actually produce two (isomorphic, but usually not equal) cluster alge- bra structures on each positroid variety, which correspond in algebraic terms to the choice between left and right modules. An application of the categorification is to prove a precise relationship, called quasi-coincidence, between these two cluster algebra structures, originally conjectured by Muller and Speyer in 2017.

Date: November 8, 11, 12
Time: 10:45am - 11:45am
Speaker: Christof Geiss
Affiliation: UNAM

Title: Quivers with relations for symmetrizable Cartanmatrices

Abstract: This series of talks is mainly based on a series of papers which I published between 2016 and 2020 together with Bernard Leclerc and Jan Schrer. Let C be a (generalized) symmetrizable Cartan matrix with a symmetrizer D and an orientation Ω. If C is symmetric, we may assume that D is trivial and we can associate to this data over any field K a path algebra and the corresponding preprojective algebra. However, if C is not symmetric, similar constructions by Dlab and Ringel were only possible over certain fields which are not algebraically closed. In a series of papers we explore the possibility to replace field extensions by truncated polynomial rings, or by rings of formal power series.

Date: November 12, 14, 15
Time: 2:30pm - 3:30pm
Speaker: Gao honghao
Affiliation: Tsinghua

Title: Cluster algebras and symplectic geometry

Abstract: Cluster algebra is a very rich structure that also appears in many other subjects. This lecture series is devoted to the connection between clus- ter algebras and symplectic geometry. Legendrian knots and their Lagrangian fillings are important geometric objects in 4 dimensional symplectic geometry. The aim of the lecture series is to explain how the cluster algebra emerges when studying invariants for Legendrian knots and how it can be used to classify exact Lagrangian fillings by surveying works in literature.

Date: November 13, 14, 15
Time: 9:30am - 10:30am
Speaker: Bao Huanchen
Affiliation: NUS

Title: Introduction to total positivity

Abstract: An invertible n × n real matrix is called totally positive if all its minors are positive. The study of such matrices dates back to Schoenberg and Grantmacher-Krein in 1930s. The theory has been generalized by Lusztig in 1994 to arbitrary split reductive connected groups (from general linear groups). This generalization depends on deep results from the theory of canonical bases arising from quantum groups. Since then, the theory of total positivity has found numerous applications, including cluster algebras, higher Teichmuller theory, the physics of scattering amplitudes, etc. The goal of this course is give an introduction to the theory of total positivity.

Date: November 13, 14, 15
Time: 10:45am - 11:45am
Speaker: Bernhard Keller
Affiliation: Paris Cite

Title: Introduction to additive categorification of cluster algebras with coecients

Abstract: Recall that a cluster algebra is a commutative algebra endowed with a rich combinatorial structure including a set of distinguished generators, the cluster variables, which are grouped into overlapping subsets of fixed (usually finite) cardinality, the clusters. In additive categorification, one tries to link the combinatorics of the cluster variables in a given cluster algebra to the combi- natorics of the rigid indecomposable objects in a suitable triangulated category endowed with a 2-Calabi-Yau structure. In this lecture series, we will start by explaining this link in the case of cluster algebras associated with finite (sim- ply laced) root systems or, equivalently, with Dynkin quivers. Here, we will only need the notion of a quiver representation and basic facts from Lie the- ory and algebraic geometry. In order to lift more of the combinatorics to the categorical level, we will then introduce the cluster category of more general acyclic quivers. In the final part of the series, we will consider cluster algebras with coecients arising in geometric contexts, for example in Fock-Goncharov's approach to higher Teichmuller theory. We will construct additive categorifica- tions of such cluster algebras using Yilin Wu's Higgs categories, which are no longer triangulated but still extriangulated in the sense of Nakaoka-Palu. Time permitting, we will also present an alternative approach due to Merlin Christ.