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Tilting theory of gentle algebras via surface combinatorics

Research Project

Project/Area Number 19K23401
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionNagoya University

Principal Investigator

チャン アーロンケイヤム  名古屋大学, 高等研究院(多元), 特任助教 (50845039)

Project Period (FY) 2019-08-30 – 2022-03-31
Project Status Granted (Fiscal Year 2020)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsgentle algebra / surface topology / lamination / tilting theory / torsion theory / marked surface / Fukaya category / stability condition / torsion class / Tilting theory / Gentle algebras / Surface combinatorics
Outline of Research at the Start

Modern algebra is about the study of manipulating a given set of rules. Representation theory is about turning such kind of systems into something we can calculate by hand, or with the help of a computer, using the so-called linear algebra. This project aims to establish a connection between representation and certain spaces associated to surfaces; the ingredient used involve classifying the so-called torsion classes of representations over gentle algebras, and the relation between gentle algebras and topological surface combinatorics.

Outline of Annual Research Achievements

Over last year, I have finished the first article on the topic of this research grant, that is, on the study of torsion classes for gentle algebras and their relation to surface combinatorics. It sets out the foundational work for several forthcoming research projects and results, namely, we established the correspondence between torsion classes of gentle algebras with a certain combinatorial tool called maximal noncrossing sets of strings. This combinatorial tool provides a medium that can be translated to the combinatorics surfaces, namely, (a refinement of) the notion of laminations. Beside its topological significance, our work provides a breakthrough in the classification problem of torsion classes of finite-dimensional. Explicit classification were only known to very cases before and showing only limited phenomenon, and we have now extended to a much larger classes where previously unseen phenomenon occur.

I have also completed another project, in collaboration with several other researchers, on the study of periodicity of trivial extension algebras. This connects trivial extension construction with fractional Calabi-Yau property. This contributes to a new advance in attacking the so-called Periocity Conjecture of self-injective algebras.

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

Impeachment of research environment due to ongoing global pandemic, on top of main collaborator leaving academia.

Strategy for Future Research Activity

At least one sequel article is in the making. It demonstrates the power of our result in concrete examples, relates several interesting phenomenon on the once-punctured torus with our study, as well as establishes new reduction techniques. Next, we will concentrate on writing up the surface interpretation of the simple-projective duality phenomenon that appears in torsion classes.

Report

(2 results)
  • 2020 Research-status Report
  • 2019 Research-status Report

Research Products

(9 results)

All 2021 2020 2019 Other

All Int'l Joint Research (2 results) Journal Article (2 results) Presentation (3 results) (of which Invited: 1 results) Remarks (2 results)

  • [Int'l Joint Research] University of Stuttgart(ドイツ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of Stuttgart(ドイツ)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Irreducible representations of the symmetric groups from slash homologies of p-complexes2021

    • Author(s)
      Chan Aaron、Wong William
    • Journal Title

      Algebraic Combinatorics

      Volume: 4 Pages: 125-144

    • DOI

      10.5802/alco.153

    • Related Report
      2020 Research-status Report
  • [Journal Article] On simple-minded systems and τ-periodic modules of self-injective algebras2020

    • Author(s)
      Chan Aaron、Liu Yuming、Zhang Zhen
    • Journal Title

      Journal of Algebra

      Volume: 560 Pages: 416-441

    • DOI

      10.1016/j.jalgebra.2020.05.024

    • Related Report
      2020 Research-status Report
  • [Presentation] Torsion classes of gentle algebras2020

    • Author(s)
      Aaron
    • Organizer
      Oberwolfach meeting on Representation Theory of Quivers and Finite Dimensional Algebras
    • Related Report
      2019 Research-status Report
  • [Presentation] Recollement of comodule categories over coalgebra object2019

    • Author(s)
      Aaron Chan
    • Organizer
      The 8th China-Japan-Korea International Symposium on Ring Theory
    • Related Report
      2019 Research-status Report
  • [Presentation] Torsion classes of gentle algebras2019

    • Author(s)
      Aaron Chan
    • Organizer
      Workshop in memory of Mitsuo Hoshio
    • Related Report
      2019 Research-status Report
    • Invited
  • [Remarks] Personal webpage

    • URL

      http://aaronkychan.github.io/

    • Related Report
      2019 Research-status Report
  • [Remarks] Summer School on DG theory and Derived Categories

    • URL

      https://sites.google.com/site/dgschooljp

    • Related Report
      2019 Research-status Report

URL: 

Published: 2019-09-03   Modified: 2021-12-27  

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