| Project/Area Number |
15K04870
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Shinshu University |
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Principal Investigator |
Tamaki Dai 信州大学, 学術研究院理学系, 教授 (10252058)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 小圏 / 分類空間 / Morse理論 / 離散モース理論 / stratified space / 2-category / 2圏 |
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| Outline of Final Research Achievements |
A discrete version of Morse theory, called discrete Morse theory, has been developed for finite regular CW complexes since 1990's. We extend and refine discrete Morse theory by using classifying spaces of small categories. More concretely, we first introduced the notion of flow path for a discrete Morse function f on a finite regular CW complex X, which generalizes that of gradient vector field due to Forman. And then we constructed a category C(f) whose objects are critical cells of f and whose morphisms are flow paths. The set of flow paths can be made into a partially ordered set in such a way that C(f) becomes a category enriched over the category of posets. The main result is that the original space X can be recovered from the discrete Morse function f as the classifying space of this category C(f).
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は, 連続あるいは滑らかな対象を離散的対象で置き換えることにより, 組み合せ論的手法を適用できるようにする, という大きな研究の流れの一部と考えることができる。本研究の主結果は, Morse理論の離散化に関することであるが, 元になったMorse理論として, Cohen-Jones-SegalによるFloer理論への応用を念頭に置いて導入されたものを用いている。その完全な離散化が得られると同時に, 勾配ベクトル場の離散化としてFormanが提案したものよりずっと精密なものが得られた。この手法は, 今後, 様々な離散化に応用できることが期待される。
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