Tohoku Cluster Seminar

2025

• Changyu Zhou (Tohoku University)
  Jul. 14 (Mon), 14:00--15:30 (JST), Online
  Title: Quasi-Homogeneous Integrable Systems: Free Parameters, Kovalevskaya Exponents, and the Painlevé Property
  Abstract: This work explores quasi-homogeneous integrable systems, focusing on their resonance condition, Frobenius manifold structure, and Hamiltonian nature. Motivated by Kovalevskaya exponents in four-dimensional Painlevé equations, we derive an arithmetic resonance condition governing the emergence of fractional powers and the breakdown of the Painlevé property. We construct a Frobenius manifold structure on the parameter space, which becomes conformal when the weights coincide. Furthermore, we show that the induced flow preserves a symplectic form and deduce the pairing property of the Kovalevskaya exponents. Additionally, we clarify the interaction between the deformation and original flow exponents via the parameter space, revealing a structural link between their singularity behaviors. These results advance the understanding of deformation theory and singularity structures in integrable systems.

• Masayasu Goto (Tohoku University)
  Jun. 16 (Mon), 15:00--16:00 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: The tropical analogue of cluster transformations and surfaces in 3-manifolds
  Abstract: I will talk about a connection between cluster transformations and 3-manifolds. Their relation was first discovered by Nagao-Terashima-Yamazaki; they showed that the gluing equations for an ideally triangulated pseudo Anosov mapping torus can be written in y-variables. I will show that the tropical analogue of cluster transformations is related to, so called, twisted square surfaces, which were invented by Tomoyoshi Yoshida to construct incompressible surfaces in 3-manifolds. They are related in a similar way to that discussed in the paper of Nagao-Terashima-Yamazaki, but do not exactly correspond. To fill the gap, we need to add a correction term for each transformation.
  
  

• Antoine de Saint Germain (University of Hong Kong)
  May 27 (Tue), 20:00--21:30 (JST), online
  Title: Fixed points of DT transformations and their applications
  Abstract: In the first part of this talk, we will discuss statements and open questions on the existence and uniqueness of « totally positive » fixed points of DT transformations in cluster varieties. In the second part, we will discuss two applications of these statements. From the fixed points we will
  1) introduce cluster exponents in cluster varieties of cluster-finite type, and give a cluster algebra interpretation of degrees of Weyl groups; and
  2) give a criterion for testing finiteness in acyclic cluster algebras.
  This talk is partly based on joint work with Prof. Jiang-Hua Lu, available at arXiv: 2503.11391.

• Patrick Kinnear (Waseda University; University of Hamburg)
  Apl. 28 (Mon), 14:00--15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Skein theory as a gerbe over the character theory
  Abstract: The skein module of a manifold has an action of the coordinate ring of the character variety, so defines a quasicoherent sheaf on this variety. In recent years, results have appeared describing the locus where the skein module of a 3-manifold is a line bundle (0-gerbe) over the character variety, and where the skein algebra of a surface is Azumaya (i.e. a 1-gerbe). In this talk I will describe results of mine which allow us to describe skein-theoretic gerbes categorically, via the powerful technology of factorization homology. This  suggests that there is a local description of the invertibility which is a feature of skein theory.

• Masaki Ogawa (Tohoku University, MathCCS)
  Apl. 14 (Mon), 14:00--15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Legendrian Weaves and N-Graph
  Abstract: This talk is a survey of the paper written by Casals and Zaslow, titled Legendrian Weaves: N-Graph Calculus, Flag Moduli, and Applications. An N-graph is a graph on a surface, and it allows us to describe Legendrians in the 1-jet bundle of the surface. Moreover, Legendrian isotopies can also be described using N-graphs.
  Recently, some authors have discovered exotic Lagrangian surfaces. In this talk, we will introduce the relationship between such phenomena and Legendrians described by N-graphs.

2024

• Dongjian Wu (Tsinghua University)
  Jan. 27 (Mon), 21:00--22:30 (JST), Online
  Title: Introduction to Bridgeland Stability Conditions
  Abstract: In this talk, I will introduce the theory of Bridgeland stability conditions, based on some questions within the field. The main topics to be covered include:
  1): The existence and the construction of stability conditions for projective varieties;
  2): The contractibility of the space of stability conditions;
  3): Relative stability conditions;
  4): Connections between spaces of stability conditions and the moduli of differentials, as well as other related topics.

• Serban Matei Mihalache (Saga University)
  Jan. 20 (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: On the equivalence of two quantum invariants of 3-manifolds
  Abstract: In 1997, G. Kuperberg constructed invariants of framed 3-manifolds (called Kuperberg invariants) for a Hopf algebra. On the other hand, in 2018, F. Costantino, N. Geer, B. Patureau-Mirand, and V. Turaev constructed topological invariants of 3-manifolds (called CGPT invariants) for a non-semisimple category equipped with a modified trace. It is known that the CGPT invariants coincide with the Turaev-Viro invariants and the Kuperberg invariants when using semisimple Hopf algebras. Furthermore, it has been shown that certain classes of Hopf algebras yield categories equipped with modified traces, and it has been conjectured that the CGPT invariants and the Kuperberg invariants coincide in these cases.
  In this talk, I will discuss my results regarding the equivalence of these invariants. 

• Takumi Otani (Yau Mathematical Sciences Center)
  Jan. 6 (Mon), 14:00--15:30 (JST), In person (hybrid) @ Tohoku University, Aobayama Campus, Mathematics building 517
  Title: Twist automorphism for a generalized root system of affine ADE type
  Abstract: A generalized root system (GRS) was introduced by Kyoji Saito in the study of Milnor fibers.
  Mirror symmetry suggests that (some of) invariants of a GRS correspond to those of a certain triangulated category.
  In this talk, I will focus on a GRS of affine ADE type.
  Dubrovin and Zhang introduced an extended affine Weyl group for a GRS of affine ADE type, defining it as the monodromy group of the Frobenius manifold associated with the GRS.
  I will give an intrinsic definition of the extended affine Weyl group by introducing a twist automorphism.
  I will discuss the (conjectural) relationships among various invariants of a GRS of affine ADE type, the invariants of the Frobenius manifold constructed by Dubrovin-Zhang, and the invariants of triangulated categories associated with an affine ADE quiver.

• Dylan G. L. Allegretti (Yau Mathematical Sciences Center)
  Dec. 23 (Mon), 14:00--15:30 (JST), Online
  Title: Skein algebras and quantized Coulomb branches
  Abstract: Character varieties of surfaces are fundamental objects in modern mathematics, appearing in low-dimensional topology, representation theory, and mathematical physics, among other areas. Given a reductive algebraic group G, the G-character variety of a surface is a moduli space parametrizing G-local systems on the surface.
  Character varieties of surfaces are expected to arise in physics as Coulomb branches of certain quantum field theories. A Coulomb branch is a kind of moduli space that was recently given a precise mathematical definition in the work of Braverman, Finkelberg, and Nakajima.
  In this talk, I will focus on the SL(2,C)-character variety of a surface. It has a quantization given by a noncommutative algebra called the Kauffman bracket skein algebra. I will describe a precise relationship between skein algebras and quantized Coulomb branches, confirming the physics prediction in some cases. This is joint work with Peng Shan.

• Sicheng Lu (University of Science and Technology of China)
  Dec. 16 (Mon), 14:00--15:30 (JST), Online
  Title: Shearing coordinates and mapping class group orbit counting
  Abstract: Shearing coordinates, or cataclysm coordinates, was introduced by Thurston to describe a kind of deformation of hyperbolic surfaces. Given an oriented surface with punctures, the shearing coordinates of any ideal triangulation is a global parametrization of Teichmuller space. We study the asymptotic growth of mapping class group orbits with respect to standard Euclidean norm. The result is based on works of Mirzakhani and relations between the shears and lengths of simple closed curves. An important concept is asymptotically piecewise linear functions, which exhibits algebraic properties. This is a joint work with Weixu Su.

• Davide Dal Martello (Rikkyo University)
  Dec. 9 (Mon), 14:00--15:30 (JST), Online
  Title: Crystals, generalized DAHA, and local systems
  Abstract: Generalized double affine Hecke algebras (GDAHA) are quantum with respect to fundamental crystallographic groups in the plane. After introducing this family of algebras through such deformation-driven point of view, with a détour on decorative arts, we will discover a new approach to the GDAHA representation theory based on the higher Teichmüller toolkit. The talk will culminate in the first explicit representation of the GDAHA of type p3.

• Tsukasa Ishibashi (Tohoku University)
  Oct. 21 (Mon), 14:00--15:30 (JST), Online
  Title: Cyclic quantum Teichmüller theory
  Abstract: Incarnating the ideas of Kashaev, we explicitly construct a finite-dimensional projective representation of the dotted Ptolemy groupoid when the quantum parameter is a root of unity, which correctly gives the central charge of the SU(2)-WZW theory. We call it the cyclic quantum Teichmüller theory, as its central ingredient is the cyclic quantum dilogarithm.
  We give a new interpretation of the parameter relation involved in the pentagon identity in terms of the mutation of coefficients, so that we can use the cyclic quantum dilogarithm to describe the quantum cluster transformations at roots of unity. We also clarify its relation to the Kashaev’s invariant of links.

• Dani Kaufman (University of Copenhagen)
  Jul. 29 (Mon), 20:00 -- 21:30 (JST), Online
  Title: Special Folding of Grassmannian Cluster Algebras
  Abstract: In a recent paper I defined a notion of a “special folding” of a cluster algebra. A usual folding of a cluster algebra is one that can be represented with a skew-symmetrizable exchange matrix, like the folding representing a type C_n algebra from a type A_(2n-1) algebra. Special foldings, which cannot be represented by a skew symmetrizable matrix, can have surprising new properties, like producing a finite exchange complex from an infinite one. In this talk I will define this notion, give several examples coming from cyclic symmetries of Grassmannian cluster algebras, and discuss some conjectures about the cluster structure underlying general cyclic symmetry loci of Grassmannians. 

• HyunKyu Kim (Korea Institute for Advanced Study)
  Jul. 22 (Mon), 14:00--15:30 (JST), Online
  Title: Modular functor conjecture from quantum Teichmüller theory
  Abstract: H Verlinde suggested in 1980's to use quantization of the Teichmüller spaces of surfaces to study the spaces of conformal blocks for the Liouville conformal field theory. This suggestion initiated and stimulated the development of quantum Teichmüller theory, and the first major steps were taken by Kashaev and by Chekhov and Fock in 1990's, where the Chekhov-Fock quantization is generalized later by Fock and Goncharov to quantization of cluster varieties. The modular functor conjecture asserts that these quantum theories of Teichmüller spaces indeed yield a 2-dimensional modular functor, which can be viewed as one axiomatization of conformal field theory. The core part of the conjecture says that, for each punctured surface S and an essential simple loop in S, the Hilbert space associated to S by quantum Teichmüller theory should decompose into the direct integral of the Hilbert spaces associated to the surface obtained by cutting S along the loop and shrinking the holes to punctures. I will give an introduction to this story and present some recent developments, including 2001.06802 and 2405.14727.

• 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.