2019 Fiscal Year Research-status Report
Tilting theory of gentle algebras via surface combinatorics
| Project/Area Number |
19K23401
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| Research Institution | Nagoya University |
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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)
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| Project Period (FY) |
2019-08-30 – 2021-03-31
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| Keywords | tilting theory / gentle algebra / marked surface / Fukaya category / stability condition / torsion class |
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| Outline of Annual Research Achievements |
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.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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.
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| Strategy for Future Research Activity |
We will continue the research project as proposed.
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| Causes of Carryover |
(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.
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