Tohoku Cluster Seminar

2024

• Zachary Greenberg (Heidelberg University)
  Jun. 24 (Mon), 19:00 -- 20:30 (JST), Online
  Title: Finding Polylogarithm Relations in Cluster Algebras
  Abstract: Polylogarithms are a class of functions inductively defined from the standard logarithm by repeated integration. They have found uses across mathematics and physics, including calculating volumes in hyperbolic spaces and expressing scattering amplitudes in N=4 Super Yang Mills theory. Our focus will be on finding the functional relations satisfied by polylogarithms. We will explain some connections between the known functional relations and cluster algebras and even use the structure of cluster algebras to construct new some relations in type D_n.

• Aaron Chan (Nagoya University)
  Feb. 5 (Mon), 17:00 -- 18:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Topological model of torsion classes of gentle algebras
  Abstract: Gentle algebras were introduced by Assem and Skowronski in the 80s as a generalisation of the path algebra of Dynkin and affine type A.  Recent advances in cluster theory and Floer theory unvealed that these algebras, along with their representations and homological algebra, are encoded by geometry of topological surface.  In this talk, we explain how laminations of a surface can be used to classify torsion classes of the gentle algebra corresponding to the surface. 

• Rei Inoue (Chiba University)
  Jan. 15 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Innovation Center for Creation of a Resilient Society, Conference room (307-308)
  Title: Quantum cluster mutations and 3D integrablity
  Abstract: The tetrahedron equation was proposed by Zamolodchikov in 1980, and the 3D reflection equation was proposed by Isaev and Kulish in 1997, to describe the integrability of 3D lattice models. In this talk I introduce new solutions to these equations by using quantum cluster mutations, based on joint works with Atsuo Kuniba and Yuji Terashima. We develop the method introduced by Sun and Yagi in [arXiv:2211.10702], where they attach three types of quivers to wiring diagrams, and for each of them the quantum mutations yield the solution of the tetrahedron equation in terms of quantum dilogarithm functions.
  In this talk we focus on one of the three, which corresponds to Fock-Goncharov quiver. We embed the noncommuting algebra generated by quantum y-variables in the q-Weyl algebra, and construct adjoint operators R and K that realize Yang-Baxter transformation and reflection transformation respectively.

2023

• Tsukasa Ishibashi (Tohoku University)
  Nov. 13 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: The quantum dilogarithm and representations of quantum cluster varieties (after Fock—Goncharov)
  Abstract: I will review the paper by Fock—Goncharov [FG08] of the same title.
  It is a remarkable feature of the quantum cluster theory (compared to the usual deformation quantization) that it provides not only the non-commutative deformation of the function ring of the cluster variety but also its representation on a Hilbert space (which we call the FG representation). When the seed data is associated with the moduli space of G-local systems on a marked surface, this Hilbert space is expected to be identified with the space of conformal blocks of the Liouville-Toda CFT (Modular Functor Conjecture [GS19+]).
  The purpose of this talk is to demonstrate the construction of the FG representation, including the slight modifications/developments provided by the subsequent works. Main ingredients of the construction are the symplectic double of the cluster variety, its quantization and the quantum dilogarithm functions.

References:

• [FG08] V. V. Fock and A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), no. 2, 223-–286.

• [FG16] V. V. Fock and A. B. Goncharov, Symplectic double for moduli spaces of G-local systems on surfaces, Adv. Math. 300 (2016), 505–-543.

• [GS19+] A. B. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory, arXiv:1904.10491v3.

• Toshiya Yurikusa (University of Versailles Saint-Quentin and Tohoku University)
  Nov. 6 (Mon), 14:00 -- 15:30 (JST), online
  Title: Denominator and dimension vectors from intersection numbers
  Abstract: In a categorification of cluster algebras, each cluster variable corresponds with a module over a Jacobian algebra. Buan, Marsh and Reiten (2009) characterized when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this talk, we give an analogue of their result for cluster algebras defined from triangulated surfaces. For that, we mainly use two kinds of intersection numbers of tagged arcs because they induce the desired vectors. 

• Yasuaki Gyoda (The University of Tokyo)
  Oct. 15 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Exchange quivers of cluster algebras and exchange quivers of root systems
  Abstract: Finite type cluster algebras were classified by Fomin-Zelevinsky in the early 2000s based on finite type root systems or their Dynkin diagrams. In their paper, they constructed a bijective map between cluster variables and almost positive roots, associating the denominator vectors of cluster algebras with coefficients of roots. Subsequent research has revealed that this map induces a bijective map preserving orientations of quivers known as the "exchange quivers." In this lecture, I will provide an explanation of this bijective map with specific examples. 

• Hiroaki Karuo (Gakushuin University)
  Sep. 25 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Classification of representations of skein algebras using quantum cluster algebras
  Abstract: Recently, some results on Azumaya loci of quantum cluster algebras are given by Muller--Nguyen--Trampel--Yakimov. This helps us understand representations of quantum cluster algebras. For unpunctured marked surfaces, their quantum cluster algebras are equal to their reduced stated skein algebras under a mild condition. In the talk, I show some results on a classification of finite dimensional representations of reduced stated skein algebras of punctured marked surfaces by attributing to quantum cluster parts and elementary parts. This is a joint work with J. Korinman (Montpelier University).

• Kazuhiro Yana (Tohoku University)
  Jul. 24 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Stability of F-polynomial
  Abstract: A F-polynomial is a polynomial that is generated by a framed quiver.  To research the limit of a mutation loop, it is useful to study the stability of the series of F-polynomial.  In the talk, based on the previous research by R. Eager and S. Franco and a preprint written by G. Zhang, I will talk about how to calculate the limit of transformed F-polynomial in detail. 

• Kazuhiro Hikami (Kyushu University)
  Jun. 26 (Mon), 14:00 -- 15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Skein algebra, cluster algebra, DAHA
  Abstract: I will talk about skein algebra on once-punctured torus and 4-punctured sphere, and show a relationship with cluster algebra and double affine Hecke algebra.

• Keiho Matsumoto (Osaka University)
  Jun. 5 (Mon), 10:30 -- 12:00 (JST), in person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 201
  Title: On L-functions for dg-categories and Coxeter matrices
  Abstract: The question of whether L-function and zeta function can be constructed as categorical invariants have been studied by Kurokawa, Kontsevich and others.In this talk, we construct L-function of dg-categories by using non-commutative p-adic Hodge theory, and explaine its properties. And also, I prove that the Coxeter matrix can describe the L-function of the cluster dg category obtained from a quiver. 

• Shunsuke Kano (MathCCS, Tohoku University)
  Mar. 29 (Mon), 14:00 -- 15:30 (JST), in person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Train track combinatorics and cluster algebras 
  Abstract: The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, I will translate some concepts of train tracks into the language of cluster algebras using the Goncharov--Shen's potential function. If time permits, I will explain the sign stability of the general pseudo-Anosov mapping classes by using the translation.

• Yuma Mizuno (Chiba University)
  Apr. 10 (Mon), 15:00 -- 16:30 (JST), in person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: q-Painlevé systems and cluster algebras
  Abstract: I will review the realization of q-Painlevé systems in cluster theory. I will explain both the combinatorial and the geometric aspects.