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2019 年度 実施状況報告書

Tilting theory of gentle algebras via surface combinatorics

研究課題

研究課題/領域番号 19K23401
研究機関名古屋大学

研究代表者

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)

研究期間 (年度) 2019-08-30 – 2021-03-31
キーワードtilting theory / gentle algebra / marked surface / Fukaya category / stability condition / torsion class
研究実績の概要

Our proposed project is to study certain homological structure arising from gentle algebras. Our approach is inspired by the connection between gentle algebras and marked surface, where indecomposable modules of these algebras can be represented by curves on surfaces. So far, we have achieved the first goal we set out to do. That is, we can classify the so-called torsion classes of a gentle algebras using certain collections of curves. Such a collection is somewhat a generalisation of the classical notion of lamination of marked surface. Potential application of such a result includes a deeper understanding of other geometric structures arising from marked surface - such as quadratic differentials and stability conditions of the associated derived categories.

Beside research done for the proposed project, I have also engaged in the research community by taking part in conferences, workshops, as well as research visits in Europe. I have also co-organised a rather successful summer school on differential graded theory, which is a fundamental theory in studying homological behaviour of mathematics structures - such as derived categories. In terms of publication, I have submitted two other pieces of works on homological algebra of representations - one on the study of stable module categories of self-injective algebras, and one on the study of a certain p-complexes of permutation modules over the symmetric groups.

現在までの達成度 (区分)
現在までの達成度 (区分)

2: おおむね順調に進展している

理由

The progress of research is roughly as expected. On the other hand, writing up takes more time than expected. This is due to the complexity of producing graphical presentations of examples in a professional manner. Another reason is the difficulty in scheduling discussion with involved international collaborators due to the progression of the COVID-19 pandemic.

今後の研究の推進方策

We will continue the research project as proposed.

次年度使用額が生じた理由

(1) To organise follow-up workshop on DG theory.
(2) To purchase equipment necessary for research purpose, if any.
(3) To purchase equipment needed to tackle any obstacles in international collaboration in the case when research visit is not allowed (say, due to the effect of the pandemic).
(4) To attend domestic/international workshops and conferences, if any.

  • 研究成果

    (6件)

すべて 2020 2019 その他

すべて 国際共同研究 (1件) 学会発表 (3件) (うち招待講演 1件) 備考 (2件)

  • [国際共同研究] University of Stuttgart(ドイツ)

    • 国名
      ドイツ
    • 外国機関名
      University of Stuttgart
  • [学会発表] Torsion classes of gentle algebras2020

    • 著者名/発表者名
      Aaron
    • 学会等名
      Oberwolfach meeting on Representation Theory of Quivers and Finite Dimensional Algebras
  • [学会発表] Recollement of comodule categories over coalgebra object2019

    • 著者名/発表者名
      Aaron Chan
    • 学会等名
      The 8th China-Japan-Korea International Symposium on Ring Theory
  • [学会発表] Torsion classes of gentle algebras2019

    • 著者名/発表者名
      Aaron Chan
    • 学会等名
      Workshop in memory of Mitsuo Hoshio
    • 招待講演
  • [備考] Personal webpage

    • URL

      http://aaronkychan.github.io/

  • [備考] Summer School on DG theory and Derived Categories

    • URL

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

URL: 

公開日: 2021-01-27  

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