Topic 0.  A brief introduction to cluster algebras

Abstract: I will explain basic notions of cluster algebras with two explicit examples.



Topic 1.  Reflections in Coxeter groups and cluster algebras I-IV

In this series of talks, we consider dimension vectors of representations of acyclic quivers, roots of (rank 2) Kac--Moody algebras and c-vectors and denominator vectors of cluster algebras, and show that all these vectors are related to certain curves on a Riemann surface in a uniform way. These curves parametrize a special class of reflections in Coxeter groups. We will present recent results and conjectures.

Talk I:  Su Ji Hong
The dimension vectors of rigid indecomposable modules over an acyclic quiver will be considered, and the geometric model of non-self-crossing curves on a Riemann surface will be introduced to describe the dimension vectors.

Talk II: Jeongwoo Yu

The geometric model brings about a class of reflections in a Coxeter group. For a special family of rank 3 Coxeter groups, it will be shown that these reflections are naturally related to roots of rank 2 Kac-Moody algebras

Talk III: Kyu-Hwan Lee
After connecting the result in Talk I with cluster algebras for acyclic quivers, it will be explained how one can (conjecturally) study the non-acyclic case using a linear ordering on the set of vertices of the quiver.

Talk IV: Kyungyong Lee

The dimension vectors of arbitrary modules over an acyclic quiver will be considered, and it will be shown that the geometric model generalizes naturally in the spirit of Mirror Symmetry.



Topic 2.  Wall-crossing in Lagrangian Floer theory and cluster transformations
Speaker:  Han-sol Hong
Given a symplectic manifold equipped with a Lagrangian torus fibration, Strominger-Yau-Zaslow (SYZ in short) conjecture asserts that its mirror manifold should be obtained as a dual torus fibration. Due to singular fibers, Floer theory of torus fibers experiences discontinuities across certain real codimension=1 loci in the base of the fibration, which is referred to as wall-crossing phenomena. As a result, the SYZ mirror construction admits complicated quantum corrections that forces the mirror manifold to possess a structure of a cluster variety. Namely, it decomposes into several algebraic tori glued by transition maps of a particular type, called cluster transformations.

In the first lecture, I will overview Lagrangian Floer theory of torus fibers in the SYZ setup, and explain how the counts of holomorphic disks bounded by torus fibers can be nicely packaged into cluster data. I will examine some concrete examples of log Calabi-Yau surfaces in the second lecture, whose associated cluster structures can be more explicitly computed. 



Topic 3.  Seeds many Lagrangian fillings for Legendrian links
We will introduce Legendrian links of finite and affine type, and then argue that there are at least as many Lagrangian fillings as seeds in the corresponding cluster structure.

The main ingredients are N-graphs developed by Casals-Zaslow, and cluster structures by Fomin-Zelevinsky.

Talk I:  Youngjin Bae 

We review the definition and basic properties of Legendrians in contact spaces, especially in dimension 3 and 5.
In order to present Legendrian links and surfaces up to Legendrian isotopy, we will introduce N-graphs and their moves.
Flag moduli adapted to Legendrians will be discussed to give Legendrian isotopy invariants.

Talk II:  Eunjeong Lee 

We investigate the flag moduli of Legendrian obtained by rainbow closure of positive braids.
More precisely, we recall from [Shen-Weng] that the flag module becomes a double Bott-Samelson cell which admits a cluster structure.
We consider the Coxeter mutation on the seed pattern of finite/affine Dynkin type which is essential in our construction.

Talk III:  Byung Hee An 

In order to connect N-graphs and cluster structure in a more concrete way, we construct seeds in cluster structure from N-graphs and Flag moduli.
 We also introduce Legendrian realization of Coxeter mutation in N-graphs.
With these tools and terminologies, we finally show that there are at least seeds many Lagrangian fillings.


Topic 4.   Generalized skein algebra and decorated Teichmuller space

Speaker:  Han-Bom Moon

I will explain how the skein theory in geometric topology is related to the quantization of the Teichmuller space of a closed surface and how this picture has been extended to the case of surfaces with punctures. And I will describe how we could naturally meet cluster algebra in this picture. In a sense, this talk is a kind of reminiscence -- how I (a non-expert of cluster algebra) met cluster algebras in my research. This talk is based on ongoing joint work with Helen Wong.



Topic 5.

Title:  Fock-Goncharov duality conjecture and tropical integer points of cluster A-varieties


I will briefly review Fock-Goncharov's moduli spaces of G-local systems on surfaces, where G is an algebraic group.

I will review their cluster structures, and the duality conjectures.

One step toward this conjecture is to understand the tropical integer points of Fock-Goncharov's cluster A-moduli spaces in a geometric manner.

I will review how this is done for the cases when G is SL2 and SL3, using some kinds of laminations on surfaces, based on Kuperberg's webs.


Title: SL3-PGL3 Fock-Goncharov duality conjecture

I will describe a solution to Fock-Goncharov's duality conjecture for their moduli spaces of G-local systems on a punctured surface S, when G is a group of type A2, namely SL3 and PGL3. I will describe the duality map which assigns to each SL3-lamination on S a regular function on the cluster X moduli space X_{PGL3,S}. I will sketch proofs of the properties of this map, and also mention a quantization by the SL3 quantum trace map if time allows.



Topic 6.  Categorification and cluster algebras / Monoidal categorification of quantum cluster algebras and quantum affine algebras

Speaker: 박의용 

Abstract: In this talk, I will talk about the notion of categorification and explain with it with quiver Hecke algebras and quantum (affine) groups.

Then I will talk about monoidal categorification for studying cluster algebras, and explain an advantage of this approach. 


Speaker: 오세진 

Title: Monoidal categorification of quantum cluster algebras and quantum affine algebras.


In this talk, we will briefly review the notion of monoidal categorification of quantum cluster algebras via quiver Hecke algebras.

Then we compare the differences between categories over quiver Hecke algebras and quantum affine algebras in aspect of categorification. 

By introducing new $\Z$-invaraints arising from $R$-matrices of quantum affine algebras, we will see the very recent result of Kashiwara-Kim-O-Park.

Combining results of Hernadez-Leclerc, Fujita-O, Fujita-Hernandez-O-Oya and Kashiwara-Kim-O-Park. I will present the result that various "quantized" categories over quantum affine algebras provide monoidal categorifications of "quantum" cluster algebras of anti-symmetric type, which is not written explicitly in any of those papers. This talk is basically based on the joint work with Kashiwara-Kim-Park.