Past talks

2024 (recent one first)

• Volker Genz (Institute for Basic Science - Center for Geometry and Physics (IBS-CGP), Korea)
  Jun. 27 (Thu),  2024,        10:30-12:00 (Korea time, UTC+9)
  in person, KIAS 1423   (also broadcasted by Zoom)
  title: Crystals and Cluster Algebras
  abstract:
  Crystal operators on canonical bases as introduced by Kashiwara/Lusztig provide in particular a toolbox to compute within the category of finite dimensional representations of finite dimensional simple Lie algebras. Motivated by this we introduce certain operators on the lattice of tropical points of mirror dual A- and X-cluster spaces. In particular, this yields a crystal-like structure on the canonical basis due to Gross-Hacking-Keel-Kontsevich.

• Kyoung-Seog Lee (POSTECH, Korea)
  May 24 (Fri),  2024,        11:30-13:00 (Korea time, UTC+9)
  in person, KIAS 1424
  title: An introduction to moduli spaces of vector bundles on an algebraic curve
  abstract:
  Moduli spaces of vector bundles on an algebraic curve are fundamental objects in modern mathematics. They play important roles in many areas of mathematics, e.g., algebraic geometry, differential geometry, mathematical physics, number theory, topology to name a few. In this introductory talk, I will briefly review the history of the subject and discuss how the moduli spaces of vector bundles (and their variants) appear in several different fields of mathematics. 

• Dmitriy Voloshyn (Institute for Basic Science - Center for Geometry and Physics (IBS-CGP), Korea)
  May 3 (Fri), 2024,        10:30-12:00 (Korea time, UTC+9)
  in person, KIAS 1423 (also broadcasted by Zoom)
  recorded video:  https://youtu.be/Ddvv5glu0qw
  title: Cluster algebras and Poisson geometry
  abstract:
    Cluster algebras are commutative rings with distinguished sets of generators characterized by a remarkable combinatorial structure. Discovered by S. Fomin and A. Zelevinsky in the early 2000s, these algebraic structures have found applications across diverse mathematical fields, including integrable systems, total positivity, Teichmüller theory, Poisson geometry, knot theory and mathematical physics.
    Fomin and Zelevinsky conjectured that numerous varieties in Lie theory are equipped with a cluster structure. Early examples include double Bruhat cells, Grassmannians and simple complex algebraic groups. M. Gekhtman, M. Shapiro and A. Vainshtein observed that cluster algebras in these examples are compatible with certain Poisson brackets. Specifically, for any given cluster 𝑥1,𝑥2,...,𝑥n, there exist constants 𝟂ij such that {𝑥i,𝑥j} = 𝟂ij 𝑥i 𝑥j. This observation led to a program aiming to construct cluster algebras by addressing the inverse problem: given a Poisson bracket in the coordinate ring of an algebraic variety and a collection of regular functions (𝑥1,𝑥2,...,𝑥n) satisfying {𝑥i,𝑥j} = 𝟂ij 𝑥i 𝑥j, does there exist a compatible cluster algebra? The research initiative led to the formulation of the GSV conjecture: for a given simple complex algebraic group and a Poisson bracket from the Belavin-Drinfeld class, there exists a compatible cluster structure.
    The plan for the talk is as follows. First, we will discuss an example of a cluster structure on GL3(ℂ). Then we will explore the connection between cluster algebras and Poisson geometry, as well as discuss how to construct a cluster structure compatible with a Poisson bracket. After that, we will discuss the recent results on the three main families of objects: simple connected simple complex algebraic groups, their Drinfeld doubles and their Poisson duals.

• Sunghyuk Park (Harvard University, US)
  Apr. 30 (Tue), 2024,        10:00-11:30 (Korea time, UTC+9)
  online (Zoom)
  recorded video: https://youtu.be/5Ril0JioNo8
  title: 3d quantum trace map
  abstract:
  I will speak about my recent joint work with Sam Panitch constructing the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.
  
  

• Valentin Buciumas (POSTECH, Korea)
  Apr. 25 (Thu), 2024,        10:30-12:00 (Korea time, UTC+9)
  in person, KIAS 1424 (also broadcasted by Zoom)
  recorded video:  https://youtu.be/JO0ojvpcurc
  title: Hecke algebras, Whittaker functions and quantum groups
  abstract:
  I will give a brief overview of the Satake isomorphism and the Casselman-Shalika formula, which are basic tools in the representation theory of p-adic groups. These two results essentially state that the spherical Hecke algebra and the spherical Whittaker functions on a p-adic group can be understood in terms of the representation theory of the dual group.
  When passing from p-adic groups to their metaplectic covers, it was conjectured by Gaitsgory and Lurie (recently proved in different settings by Campbell-Dhillon-Raskin and Buciumas-Patnaik) that the dual group gets replaced by a certain quantum group at a root of unity. I will try to explain the conjecture of Gaitsgory-Lurie and if time permits some of the ideas of the proof in the algebraic setting, as well as some interactions to combinatorics and number theory.

2023 (recent one first)

• Ben Davison (The Univ. of Edinburgh, UK)
  Dec. 5 (Tue), 2023,        14:00-15:30 (Korea time, UTC+9)
  in person, KIAS 1424  (also broadcasted by Zoom)
  recorded video:  https://youtu.be/RPYAKBrBQgk
  title: Strong positivity for quantum cluster algebras
  abstract:
  Quantum cluster algebras are quantizations of cluster algebras, which are a class of algebras interpolating between integrable systems and combinatorics. These algebras were originally introduced to study positivity phenomena arising in the study of quantum groups, and so one of the key questions regarding them (and their quantum analogues) is whether they admit a basis for which the structure constants are positive. The classical version of this question was settled in the affirmative by Gross, Hacking, Keel and Kontsevich. I will present a proof of the quantum version of this positivity for skew-symmetric quantum cluster algebras, due to joint work with Travis Mandel, based on results in categorified Donaldson-Thomas theory and scattering diagrams.

• Sin-Myung Lee (KIAS, Korea)
  Oct. 23 (Mon), 2023,        11:00AM-12:30PM (Korea time, UTC+9)
  in person, KIAS 1424  (also broadcasted by Zoom)
  recorded video: https://youtu.be/V8r90Xk24n8
  title: Representations of quantum affine (super)algebras from the R-matrix's point of view
  abstract:
  One of the major problems in representation theory of quantum affine algebras is to understand the tensor product structure, for which it has been recognized that (normalized) R-matrices and their poles play a crucial role. In this talk, we first give a brief survey on representations of quantum affine (super)algebras from this perspective. Then we will explain a new approach motivated from the super duality for Lie superalgebras, which is an ongoing project with Jae-Hoon Kwon and Masato Okado.
  
  

• Hyunbin Kim (Yonsei Univ., Korea)
  Oct. 12 (Thu), 2023,        10:30AM-12:00PM (Korea time, UTC+9)
  in person, KIAS 1424  (also broadcasted by Zoom)
  title: Morse Superpotential and Blowups of Surfaces
  abstract:
  Through the framework of tropical geometry, we analyze the critical behavior of the Landau-Ginzburg mirror of toric/non-toric blowups of possibly non-Fano toric surfaces. After a brief review on the mirror construction for log Calabi-Yau surfaces, we introduce a method for identifying the precise location of critical points of the superpotential (or equivalently, non-displaceable fibers). We further show their non-degeneracy for generic parameters, proving closed and open string mirror symmetry.
  
  

• Yoosik Kim (Pusan National University, Korea)
  Jul. 6 (Thu), 2023,        10:30AM-12:00PM (Korea time, UTC+9)
  in person, KIAS 1424
  title: Infinitely many monotone Lagrangian tori in flag manifolds
  abstract:
  In symplectic topology, constructing monotone Lagrangian tori that are not Hamiltonian isotopic to each other is interesting. In this talk, I discuss how to use cluster structures for constructing infinitely many monotone Lagrangian tori and distinguishing them in complete flag manifolds of arbitrary type (except in some low-dimensional cases).
  
  

• Wataru Yuasa (Division of Mathematics and Mathematical Sciences, Kyoto University, Japan)
  Jun. 29 (Thu), 2023,        10:30AM-12:00PM (Korea time, UTC+9)
  online (Zoom)
  recorded video:  https://www.youtube.com/watch?v=FY5WcGcRPRA
  title:  State-clasp correspondence for skein algebras
  abstract:
  We introduce the stated and the clasped sp_4-skein algebras for an oriented surface with marked points on the boundary. Moreover, we show that the reduced version of the stated g-skein algebra is isomorphic to the boundary-localization of the clasped g-skein algebra for g=sl_2, sl_3, or sp_4. This isomorphism is a quantum counterpart of the two descriptions of the cluster algebra of the surface associated with g in terms of the matrix coefficients of Wilson lines and cluster variables, respectively. This talk is based on a joint work with Tsukasa Ishibashi (Tohoku Univ.).
  
  

• Tsukasa Ishibashi (Mathematical Institute, Tohoku University, Japan)
  May. 11 (Thu), 2023,        10:30AM-12:00PM (Korea time, UTC+9)
  online (Zoom)
  recorded video: https://www.youtube.com/watch?v=mwkB0U08_LU
  title:  Moduli space of decorated G-local systems and skein algebras
  abstract:
  The moduli space of decorated (twisted) G-local systems on a marked surface, originally introduced by Fock–Goncharov, is known to have a natural cluster K_2 structure. In particular, it admits a quantization via the framework of quantum cluster algebras, due to Berenstein—Zelevinsky and Goncharov—Shen.
  In this talk, I will explain its (in general conjectural) relation to the skein algebras. This talk is based on joint works with Hironori Oya, Linhui Shen and Wataru Yuasa.
  
  

• Shunsuke Kano (Research Alliance Center for Mathematical Sciences, Tohoku University, Japan)
  Jan. 12 (Thu), 2023,        2PM-3:30PM (Korea time, UTC+9)
  online (Zoom)
  recorded video ->  please contact Hyun Kyu Kim or Shunsuke Kano to request the link.
  title: Unbounded sl(3)-laminations and their shear coordinates
  abstract: Fock--Goncharov pointed out the space of unbounded laminations on a marked surface gives the set of tropical valued points of the moduli space of the framed PGL_2 local systems on the surface. The key point of this identification is that the shear coordinate of the space of unbounded laminations gives the tropicalized cluster structure of the moduli space.
  In this talk, we introduce the space of unbounded sl(3) laminations (with pinnings) and define the "shear coordinate" on it as a generalization of the sl(2) case.
  If time permits, we discuss the graphical basis of the Ishibashi--Yuasa sl(3) skein algebra.
  This talk is based on a joint work with Tsukasa Ishibashi.

2022 (recent one first)

• Hironori Oya (Tokyo Institute of Technology, Japan)
  Dec. 22 (Thu), 2022,        10AM-11:30AM (Korea time, UTC+9)
  online (Zoom)
  recorded video:  https://www.youtube.com/watch?v=w_o97W2TVQM
  title: Wilson lines on the moduli space of $G$-local systems on a marked surface
  abstract:  For a marked surface $\Sigma$, there are two kinds of extensions of moduli spaces of local systems on $\Sigma$, written as $\mathcal{A}_{\widetilde{G}, \Sigma}$ and $\mathcal{P}_{G, \Sigma}$, where $\widetilde{G}$ is a connected simply-connected complex simple algebraic group and $G=\widetilde{G}/Z(\widetilde{G})$ its adjoint group. These are introduced by Fock--Goncharov and Goncharov--Shen respectively, and it is known that the pair $(\mathcal{A}_{\widetilde{G}, \Sigma}, \mathcal{P}_{G, \Sigma})$ forms a cluster ensemble.
    In this talk, we formulate a class of $\widetilde{G}$ or $G$-valued morphisms defined on these moduli spaces, which we call Wilson lines. I explain their basic properties and application. In particular, we give an affirmative answer to the $\mathrm{A}=\mathrm{U}$ problem for the cluster algebras arising from the cluster $K_2$-structures on $\mathcal{A}_{\widetilde{G}, \Sigma}$ under some assumptions on $G$ and $\Sigma$.
    This talk is based on a joint work with Tsukasa Ishibashi and Linhui Shen.
  
  

• Kyoung-Seog Lee (IMSA, Univ. of Miami, US)
  Dec. 8 (Thu), 2022,        9:30AM-11:00AM (Korea time, UTC+9)
  place: KIAS 8309 (offline only)
  title: Parabolic bundles on curves and their applications
  abstract: The notion of parabolic bundle was introduced by Seshadri and further studied by Mehta and Seshadri. After its discovery, parabolic bundle plays important roles in the theory of vector bundles on algebraic varieties. In this talk, I will give a brief introduction to the theory of parabolic bundles on algebraic varieties. Then I will discuss their various applications.
  
  

• Daniel Douglas (Yale University, US)
  Oct. 24 (Mon), 2022,        10AM-11:30AM (Korea time, UTC+9)
  online (Zoom)
  recorded video: https://youtu.be/ZVgJIeu2aF8
  title: Dimers, webs, and local systems
  abstract: For a planar bipartite graph G equipped with a SLn-local system, we show that the determinant of the associated Kasteleyn matrix counts “n-multiwebs” (generalizations of n-webs) in G, weighted by their web-traces. We use this fact to study random n-multiwebs in graphs on some simple surfaces. Time permitting, we will discuss some relations to Fock-Goncharov theory. This is joint work with Rick Kenyon and Haolin Shi.
  
  

• Dylan Allegretti (Yau Mathematical Sciences Center at Tsinghua University, China)
  Oct. 17 (Mon), 2022,        2PM-3:30PM (Korea time, UTC+9)
  online (Zoom)
  recorded video: https://youtu.be/h4iGcoArA0w
  title: Teichmüller spaces, quadratic differentials, and cluster coordinates
  abstract:
  In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe a generalization of their result. I will explain how, by replacing holomorphic differentials by meromorphic differentials, one is naturally led to consider an object called the enhanced Teichmüller space. The latter is an extension of the classical Teichmüller space which is important in mathematical physics and the theory of cluster algebras.