Tokyo-Seoul Conference in Mathematics 2025

Enumerative Geometry and Representation Theory

Invited Speakers (alphabetical order)

Sarunas Kaubrys (Kavli IPMU)

Kyoung-Seog Lee (Postech)

Woonam Lim (Yonsei University)

Kota Murakami (Kyoto University)

Anya Nordskova (Kavli IPMU)

Jeongseok Oh (Seoul National University)

Hyeonjun Park (KIAS)

Shunya Saito (University of Tokyo)

Registration

https://forms.gle/8mWpkFwqU2TDorsD9

Schedule 

December 12 (Fri)

10:00-11:00  Hyeonjun Park

11:30-12:30  Sarunas Kaubrys

14:30-15:30  Kyoung-Seog Lee

16:00-17:00  Shunya Saito

December 13 (Sat)

10:00-11:00   Kota Murakami

11:30-12:30   Woonam Lim

14:30-15:30   Anya Nordskova

16:00-17:00   JeongSeok Oh

Titles and Abstracts

Speaker: Hyeonjun Park

Title: Symplectic pushforwards and Lagrangian classes

Abstract: In the first half of the talk, I will introduce a general operation of producing shifted symplectic stacks from given ones. Basic examples like cotangent bundles, critical loci, and Hamiltonian reduction can be understood as special cases of this operation. Moreover, this unification enables us to provide an etale local structure theorem for shifted symplectic Artin stacks.

In the second half of the talk, I will provide a generalization of virtual classes that can be used to define following enumerative invariants: (1) cohomological field theories for gauged linear sigma models; (2) cohomological Hall algebras for 3-Calabi-Yau categories; (3) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors; (4) refined surface counting invariants for Calabi-Yau 4-folds. This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.

Speaker: Sarunas Kaubrys

Title: Joyce vertex algebras with potential 

Abstract: Joyce has defined a structure of (co)vertex algebra on the (co)homology of the moduli of objects of an abelian or derived category. I will recall this construction and explain how we can extend this construction to define a covertex algebra on the critical cohomology of a quiver with potential. I will then explain how this structure is related to the localised coproduct of Davison and explain how the covertex structure is compatible with the Cohomological Hall Algebra structure. If time permits, I will explain two applications, one to cohomological integrality for symmetric quivers and one to deformed Drinfeld coproducts on Yangians of ADE type. This is joint work in progress with Alexei Latyntsev and Shivang Jindal.

Speaker: Kyoung-Seog Lee

Title: Ulrich bundles and curves on minimal Fano 3-folds

Abstract: Ulrich bundles form an interesting class of vector bundles on projective varieties. Since they were introduced by Eisenbud and Schreyer, there have been intensive works studying these bundles from various aspects. Especially, Ulrich bundles on Fano 3-folds are drawing a lot of attention these days. It turns out that studying Ulrich bundles on a Fano 3-fold is also closely related to the geometry of curves on it. In this talk, I will discuss Ulrich bundles and curves on some minimal Fano 3-folds via representations of quivers. The last part of this talk is based on several joint works in progress with Woohyuck Choi and Kyeong-Dong Park.

Speaker: Shunya Saito

Title: Classifying KE-closed subcategories over a commutative noetherian ring

Abstract: This talk is based on joint work with Toshinori Kobayashi (Meiji University). Classifying subcategories is an active topic in the representation theory of algebras. In particular, several subcategories of the module category of a commutative noetherian ring have been classified so far, and they are described in terms of the prime spectrum.

In this talk, we give a classification of KE-closed subcategories (additive subcategories closed under extensions and kernels) for a commutative noetherian ring. For this, we introduce a class of functions on the spectrum, called n-Bass functions, and establish a bijection between KE-closed subcategories and 2-Bass functions under mild assumptions.

Speaker: Kota Murakami

Title:  On rigid modules and graded preprojective algebras

Abstract: Hernandez-Leclerc studied certain graded analogue of preprojective algebra  from view points of mutation theory. They studied a generating function of Euler characteristics of submodule Grassmannians of certain rigid module over graded preprojective algebra, which recovers a known solution of functional equality called T-system in Lie theory. In this talk, we discuss such generating functions and equalities determined by related rigid modules over the graded preprojective algebra. This talk is based on an ongoing joint work with Bernard Leclerc. 

Speaker: Woonam Lim

Title: Chern filtration of moduli spaces of sheaves

Abstract: The cohomology ring of moduli spaces of sheaves has been a central topic in enumerative geometry due to its rich additional structures, such as tautological generators and relations, Hall algebra, perverse filtration, etc. In this talk, I will introduce a relatively new structure, called the Chern filtration, and explain why it is interesting and how it interacts with existing structures. The main focus will be on moduli spaces of one-dimensional sheaves on surfaces and moduli spaces of bundles on curves. This is based on joint works with Y. Kononov, M. Moreira, W. Pi.

Speaker: Anya Nordskova

Title: Non-commutative crepant resolutions of affine cones over del Pezzo surfaces

Abstract: Non-commutative crepant resolutions (NCCRs), introduced by Van den Bergh, serve as a natural analogue of classical crepant resolutions in algebraic geometry. The main result of the talk is the classification of NCCRs for (the completions of) the anticanonical cones over smooth del Pezzo surfaces. These are standard examples of canonical non-terminal Gorenstein singularities. More precisely, we prove that any NCCR of the completed anticanonical cone over a del Pezzo surface X arises from a full exceptional collection on X satisfying certain conditions. 

We conjecture that in this situation all NCCRs are related by sequences of mutations in the sense of Iyama and Wemyss. In our setting, Iyama–Wemyss mutations can be interpreted both as mutations of graded quivers encoding 3-Calabi–Yau algebras and as sequences of mutations of exceptional collections. I will discuss the progress toward proving this conjecture, as well as some connections to earlier works of Bridgeland and Stern.  

The talk is based on joint work with Michel Van den Bergh (partially in progress).

Speaker:  JeongSeok Oh

Title: TBA

Abstract: TBA

Organizing committee  (alphabetical order)

Osamu Iyama (University of Tokyo)

Young-Hoon Kiem (KIAS)

Yukinobu Toda (Kavli IPMU)

Organized by

Graduate School of Mathematical Sciences, The University of Tokyo 

and KIAS

Supported by

Inamori Research Institute for Science