Details of the Researcher

PHOTO

Tsukasa Ishibashi
Section
Graduate School of Science
Job title
Assistant Professor
e-Rad No.
30881718

Research History 2

  • 2022/02 - Present
    Tohoku University　Graduate School of Science Department of Mathematics　Assistant Professor

  • 2020/04 - 2022/01
    Kyoto University　Research Institute for Mathematical Sciences　Researcher

Education 1

  • The University of Tokyo　Graduate School of Mathematical Sciences

    2017/04 - 2020/03

Professional Memberships 1

  • Mathematical Society of Japan

Research Interests 4

  • 数理物理

  • スケイン代数

  • Cluster algebra

  • Teichmüller theory

Research Areas 1

  • Natural sciences / Geometry /

Awards 2

  1. The 2024 MSJ Takebe Katahiro Prize

    2024/09　Mathematical Society of Japan　Teichmüller theory based on cluster algebras

  2. Dean's award (Master course)

    2017/03　Graduate School of Mathematical Sciences, the University of Tokyo

Papers 16

  1. Unbounded 𝔰𝔩3-laminations and their shear coordinates Peer-reviewed

    Tsukasa Ishibashi, Shunsuke Kano

    Algebraic & Geometric Topology　25　(3)　1433-1500　2025/06/20

    Publisher: Mathematical Sciences Publishers

    DOI： 10.2140/agt.2025.25.1433 　

    ISSN： 1472-2747

    eISSN： 1472-2739

  2. Unbounded sl(3)-laminations around punctures Peer-reviewed

    Tsukasa Ishibashi, Shunsuke Kano

    Mathematische Zeitschrift　310　(4)　2025/06/11

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00209-025-03773-z 　

    ISSN： 0025-5874

    eISSN： 1432-1823

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    Abstract We continue to study the unbounded $$\mathfrak {sl}_3$$ -laminations [16], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $$\mathfrak {sl}_3$$ . After giving a classification of signed $$\mathfrak {sl}_3$$ -webs around a puncture, we describe the tropicalization of the Goncharov–Shen’s Weyl group action in detail. We also clarify the relationship with several other approaches by Shen–Sun–Weng [28] and Fraser–Pylyavskyy [10]. Finally, we discuss a formulation of unbounded $$\mathfrak {g}$$ -laminations for a general semisimple Lie algebra $$\mathfrak {g}$$ in brief.

  3. Skein and cluster algebras of unpunctured surfaces for sp(4) Peer-reviewed

    Tsukasa Ishibashi, Wataru Yuasa

    Advances in Mathematics　465　110149-110149　2025/04

    Publisher: Elsevier BV

    DOI： 10.1016/j.aim.2025.110149 　

    ISSN： 0001-8708

  4. Skein and Cluster Algebras with Coefficients for Unpunctured Surfaces Peer-reviewed

    Tsukasa Ishibashi, Shunsuke Kano, Wataru Yuasa

    International Mathematics Research Notices　2025　(4)　2025/02/18

    Publisher: Oxford University Press (OUP)

    DOI： 10.1093/imrn/rnaf024 　

    ISSN： 1073-7928

    eISSN： 1687-0247

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    Abstract We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma ,\mathbb{W } }^{A}$ of a walled surface$(\Sigma ,\mathbb{W})$, and prove that it has a quantum cluster structure. The walled surfaces naturally generalize the marked surfaces with multi-laminations, which have been used to describe the cluster algebras of geometric type for marked surfaces by Fomin–Thurston [ 11]. As a result, this paper gives a simultaneous generalization of Fomin–Thurston and Muller [ 20], incorporating both quantization and (not necessarily normalized) coefficients. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [ 14] between these quantum cluster algebras.

  5. Quantum Duality Maps, Skein Algebras and their Ensemble Compatibility Peer-reviewed

    Tsukasa Ishibashi, Hiroaki Karuo

    Communications in Mathematical Physics　2024/10

    DOI： 10.1007/s00220-024-05119-y 　

  6. Earthquake Theorem for Cluster Algebras of Finite Type Peer-reviewed

    Takeru Asaka, Tsukasa Ishibashi, Shunsuke Kano

    International Mathematics Research Notices　2024　(8)　7129-7159　2024/02/23

    Publisher: Oxford University Press (OUP)

    DOI： 10.1093/imrn/rnae027 　

    ISSN： 1073-7928

    eISSN： 1687-0247

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    Abstract We introduce a cluster algebraic generalization of Thurston’s earthquake map for the cluster algebras of finite type, which we call the cluster earthquake map. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb {R}^{\textrm {trop } }$- and $\mathbb {R}_{>0}$-valued points of the cluster $\mathcal {X}$-variety. For those of type $A_{n}$ and $D_{n}$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to “continuous deformations” of the Fock–Goncharov fan.

  7. A=U for cluster algebras from moduli spaces of G-local systems Peer-reviewed

    Tsukasa Ishibashi, Hironori Oya, Linhui Shen

    Advances in Mathematics　431　109256-109256　2023/10

    Publisher: Elsevier BV

    DOI： 10.1016/j.aim.2023.109256 　

    ISSN： 0001-8708

  8. Wilson lines and their Laurent positivity Peer-reviewed

    Tsukasa Ishibashi, Hironori Oya

    Mathematische Zeitschrift　305　(2)　2023/09/28

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00209-023-03355-x 　

    ISSN： 0025-5874

    eISSN： 1432-1823

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    Abstract For a marked surface $$\Sigma $$ and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism $$g_{[c]}:{\mathcal {P} }_{G,\Sigma } \rightarrow G$$ associated with the homotopy class of an arc c connecting boundary intervals of $$\Sigma $$, which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra $$\mathcal {O}({\mathcal {P} }_{G,\Sigma })$$ when $$\Sigma $$ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [18], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of $$\Sigma $$. We show that the matrix coefficients $$c_{f,v}^V(g_{[c]})$$ give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of $$\Sigma $$, for suitable f and v.

  9. Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$ Peer-reviewed

    Tsukasa Ishibashi, Wataru Yuasa

    Mathematische Zeitschrift　303　(3)　2022/11/05

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00209-023-03208-7 　

    ISSN： 0025-5874

    eISSN： 1432-1823

  10. Algebraic entropy of sign-stable mutation loops Peer-reviewed

    Tsukasa Ishibashi, Shunsuke Kano

    Geometriae Dedicata　214　(1)　79-118　2021/10

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s10711-021-00606-1 　

    ISSN： 0046-5755

    eISSN： 1572-9168

  11. Cluster realizations of Weyl groups and higher Teichmüller theory Peer-reviewed

    Rei Inoue, Tsukasa Ishibashi, Hironori Oya

    Selecta Mathematica　27　(3)　2021/07

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00029-021-00630-9 　

    ISSN： 1022-1824

    eISSN： 1420-9020

  12. Geometry of cluster modular groups Peer-reviewed

    Tsukasa Ishibashi

    Ph.D. thesis in the University of Tokyo　2020/03

  13. Presentations of cluster modular groups and generation by cluster Dehn twists Peer-reviewed

    Tsukasa Ishibashi

    SIGMA　16　(025)　22 pages　2020/03

  14. Geometric description of lattice defects via singular affine manifolds, Peer-reviewed

    Tsukasa Ishibashi

    2019

  15. On a Nielsen-Thurston classification theory for cluster modular groups Peer-reviewed

    Tsukasa Ishibashi

    Annales de l'Institut Fourier　69　(2)　515-560　2019

  16. On a Nielsen-Thurston type classification theory on cluster modular groups Peer-reviewed

    Tsukasa Ishibashi

    Master thesis in the University of Tokyo　2015/03

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Research Projects 3

  1. Geometry of the quantum higher Teichmuller theory

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Early-Career Scientists

    Institution: Tohoku University

    2024/04/01 - 2029/03/31

  2. Combinatorial aspects of the Teichmuller theory

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Research Activity Start-up

    2020/09/11 - 2023/03/31

  3. Geometry of cluster modular groups

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for JSPS Fellows

    Institution: The University of Tokyo

    2018/04/25 - 2020/03/31