Cluster Structures in Algebra and Geometry

February 2–6, 2026

경북대학교 인재원, Pohang, Korea

hosted by
Department of Mathematics Education, Kyungpook National University
sponsored by
Kyungpook National University / National Research Foundation of Korea / QSMS Center for Quantum Structures in Modules and Spaces
Cluster

Organizers

  • Byung Hee An (KNU)
  • Youngjin Bae (INU)
  • Yunhyung Cho (SKKU)

Invited Speakers

  • Byung Hee An (KNU)
  • Youngjin Bae (INU)
  • Cheol-Hyun Cho (POSTECH)
  • Yunhyung Cho (SKKU)
  • Wonbo Jeong (Sogang University)
  • Jongmyeong Kim (SNU)
  • Yoosik Kim (PNU)
  • Sukjoo Lee (IBS-CGP)
  • Yoonjae Nho
  • Rak-Kyeong Seong (UNIST)
  • Philsang Yoo (SNU)

Venue

Talk Schedule

February 2 (Mon) February 3 (Tue) February 4 (Wed) February 5 (Thu) February 6 (Fri)
10:00 – 10:50 Arrival
and
Registration		 P. Yoo Y. Bae S. Lee W. Jeong
10:50 – 11:10 Break
11:10 – 12:00 Y. Kim R-K. Seong Y. Cho J. Kim
12:00 – 14:00 Opening Remark
(13:50–14:00)		 Lunch Group Photo (12:00–12:10)
Lunch		 Lunch Closing Remark (12:00–12:10)
Lunch
14:00 – 14:50 C-H. Cho (I) B. An (I) Free Discussion Y. Noh (I) Free Discussion
and
Departure
14:50 – 15:10 Break Break
15:10 – 16:00 C-H. Cho (II) B. An (II) Y. Noh (II)
16:00 – 18:00 Free Discussion Free Discussion
18:00 – Workshop Dinner Workshop Dinner

* The schedule may vary.

Title & Abstract

Abstracts

Speaker
Byung Hee An (KNU)
Title
Cluster structures on cabling (I)
Abstract
It is known that both augmentation varieties and moduli spaces of constructible sheaves admit $\mathcal{A}$ and $\mathcal{X}$ cluster structures, respectively, for a certain class of Legendrian knots and links. The first known such a class is a positive braid closure by Shen-Weng, which can be seen as a cabling (or a satellite) along the boundary of the Lagrangian disk properly embedded in the four-ball. In the first talk, we generalize this viewpoint to the class of cablings along the boundary of compact orientable surfaces $L$ regarded as a Lagrangian in its cotangent bundle and consider algebraic and geometric objects associated with those Legendrians such as augmentation varieties $Aug_L(\beta)$, moduli spaces of sheaves $Sh_L(\beta)$, and so on. These objects are only depending on the topological data of a given surface $L$ and the cabling information (or the satellite pattern) $\beta$. This is a joint work in progress with Youngjin Bae(INU).
Title
Cluster structures on cabling (II)
Abstract
For a compact orientable surface $L$, which is an embedded exact Lagrangian filling of a positive braid closure $\lambda$, called a companion link, one may define augmentation varieties and moduli spaces of sheaves associated to the cabling (or the satellite) $S(\lambda,\beta)$ with a given cabling data (or a satellite pattern) $\beta$. In the second talk, we show that both $Aug_L(\beta)$ and $Sh_L(\beta)$ admit cluster structures under the certain conditions, and discuss how $Aug_L(\beta), Aug(S(\lambda,\beta))$ and $Aug(\lambda)$ (or $Sh_L(\beta), Sh(S(\lambda,\beta))$ and $Sh(\lambda)$), and their cluster structures are related to each other. In particular, we also provide an algorithmic way in terms of $N$-graphs to describe the relationship between Lagrangian fillings of $\beta$ over $L$, and $S(\lambda,\beta), \lambda$ over the disc. This is a joint work in progress with Youngjin Bae(INU).
Speaker
Youngjin Bae (INU)
Title
Monotone Lagrangian tori and exact Lagrangian fillings
Abstract
Monotone Lagrangian submanifolds play a central role in the study of closed symplectic manifolds, particularly in the contexts of mirror symmetry and cluster algebras. On the other hand, exact Lagrangian fillings of Legendrian submanifolds serve as important objects in the study of Weinstein manifolds. In this talk, we explore a correspondence between monotone Lagrangian tori and exact Lagrangian fillings by examining smoothed divisor complements of toric 4-manifolds. We also compare almost toric mutation and Legendrian mutation in concrete examples. This is a joint project with Orsola Capovilla-Searle.
Speaker
Cheol-Hyun Cho (POSTECH)
Title
Exceptional sequences and cluster theory
Abstract
We give introductory lectures on the role of exceptional sequences in cluster theory. Topics will include non-crossing partitions, braid group action and tilting and silting objects.
Speaker
Yunhyung Cho (SKKU)
Title
Mutation of Fano simplices and invariants
Abstract
A combinatorial mutation is a procedure that produces a new Fano polytope from a given one in such a way that the associated toric Fano varieties are deformation equivalent. In this talk, I will introduce this notion through various examples and discuss several algebraic and combinatorial invariants that are preserved under mutations.
Speaker
Wonbo Jeong (Sogang University)
Title
A symplectic geometric model of A type cluster structure
Abstract
In this talk, I will introduce a $(n+1)$-gon model for A type cluster structure. Then, I will explain its symplectic geometric interpretations using cluster category language. This talk is based on several joint works with Hanwool Bae, Cheol-Hyun Cho, Beom-Seok Kim, and Jongmyeong Kim.
Speaker
Jongmyeong Kim (SNU)
Title
Orlov functors for topological Fukaya categories and their applications to the categorical entropy of wrap-once functors
Abstract
This talk explores Orlov functors for topological Fukaya categories and their applications to categorical dynamical systems. Specifically, we will discuss how to compute the categorical entropy of wrap-once functors using Orlov functors (equivalently, the shifted Serre functors of the corresponding gentle algebras). This recovers and generalizes a recent result of Chang and Schroll. Joint work in progress with Kyoungmo Kim.
Speaker
Yoosik Kim (PNU)
Title
Cluster algebras and monotone Lagrangian tori
Abstract
In this talk, we discuss an application of cluster algebras to symplectic geometry and mirror symmetry. We outline an expected framework for realizing cluster duality from a symplectic-geometric perspective. As a first step toward this goal, we explain how to construct and distinguish monotone Lagrangian tori in flag manifolds using the A-cluster algebra structure on the unipotent cell parametrizing the dual canonical basis. The mirror charts associated with these tori are expected to give rise to the corresponding X-cluster variety.
Speaker
Sukjoo Lee (IBS-CGP)
Title
$P=W$ Phenomena
Abstract
We study filtrations on the cohomology of a non-compact space that measure different aspects of its non-compactness. One approach is to compactify the space and record how cohomology behaves with respect to the boundary, leading to the weight filtration. Another approach is to analyze the space via a map to a lower-dimensional base with compact fibers and measure complexity by stratifying the fibers; this gives rise to the perverse filtration. The relationship between these two filtrations turns out to be surprisingly rich. In certain geometric settings, they appear to encode the same information in very different ways, a phenomenon known as the $P=W$ phenomena. In this talk, I will give an informal overview of these ideas and discuss how they may appear in the context of cluster varieties.
Speaker
Yoonjae Nho
Title
Weaves and networks (I)
Abstract
Recently, it became understood that many moduli spaces of local systems over surfaces can be realized as the moduli space of Lagrangian fillings. In this expository talk, while focusing on the $SL_2$ case, I explain how the theory of Legendrian weaves and networks provide a dictionary between symplectic topology and theories of class S.
Title
Weaves and networks (II)
Abstract
In this talk, I will explain how the story extends in the higher rank case. Focusing on the moduli space of $SL_3$ local systems over disks, I will explain how one obtains the quiver, the filling and the network. This is based on the recent joint work with Roger Casals.
Speaker
Rak-Kyeong Seong (UNIST)
Title
From Brane Tilings to Brane Brick Models: An Overview
Abstract
This talk provides an introduction to dimer models, also known as brane tilings. These bipartite graphs on a 2-torus have been extensively studied in connection with a wide range of topics, including enumerative combinatorics, toric Calabi-Yau geometry and mirror symmetry, cluster algebras and quivers, integrable systems, and supersymmetric gauge theories arising in string theory. We place particular emphasis on recent developments in the field, where mutations and deformations of dimer models have led to new insights into associated invariants and dualities. While dimer models are traditionally associated with toric Calabi-Yau 3-folds, they have more recently been generalized to higher-dimensional tessellations of a 3-torus. These structures, known as brane brick models, correspond to toric Calabi-Yau 4-folds. The talk concludes by discussing how phenomena familiar from dimer models extend to brane brick models, including the appearance of an order-3 mutation known as triality.
Speaker
Philsang Yoo (SNU)
Title
A Rosetta stone for cluster X-varieties
Abstract
In this pedagogical talk, we review a dictionary between homological algebra, combinatorics, and geometry underlying cluster X-varieties. Time permitting, we briefly outline expected connections with moduli spaces of Bridgeland stability conditions, following Gaiotto-Moore-Neitzke and Bridgeland.

Accommodation

경북대학교 인재원

Getting to 인재원(Pohang)

You may refer here(Korean).

Financial Support

Financial support for invited speakers and overseas participants is available within a limited budget. Participants who require financial support should contact the organizer in advance.

Contact

  • Byunghee An (anbyhee_at_knu.ac.kr / ban2_at_ncsu.edu)
  • Youngjin Bae (yjbae_at_inu.ac.kr)
  • Yunhyung Cho (yunhyung_at_skku.edu)
KNU
NRF
QSMS