Tilting theory of gentle algebras via surface combinatorics
| Project/Area Number |
19K23401
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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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 – 2022-03-31
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| Project Status |
Granted (Fiscal Year 2020)
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| 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)
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| Keywords | gentle algebra / surface topology / lamination / tilting theory / torsion theory / marked surface / Fukaya category / stability condition / torsion class / Tilting theory / Gentle algebras / Surface combinatorics |
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| 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.
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| 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.
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| 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.
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| 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.
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Report
(2 results)
Research Products
(9 results)
