2019 Fiscal Year Final Research Report
Combinatorial homotopy theory of spaces and applications to sensor network using categories
| Project/Area Number |
17K14183
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Shinshu University |
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Principal Investigator |
Tanaka Kohei 信州大学, 学術研究院社会科学系, 助教 (70708362)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | 組合せ的ホモトピー論 / 単体複体 / 圏 / 半順序集合 / LS category / Topological complexity / オイラー標数 / センサーネットワーク |
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| Outline of Final Research Achievements |
This study focused on developement of the combinatorial homotopy theory of simplicial complexes and posets (categories) and its applications. The combinatorial homotopy theory is based on removing points, unlike the classical homotopy theory of spaces based on continuous deformations. Such a descrete operation is compatible with design of algorithms, and we can expect practical applications.
This study developed the combinatorial homotopy theory, and considered applications to computation of topological invariants and sensor network theory with respect to Euler characteristic.We showed that the numerical invariants LS and TC of finite simplicial complexes can be calculated essentially by finite discrete operations and barycentric subdivisions.Moreover, we computed some Euler characteristic of the quotient categories by group actions.
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| Free Research Field |
代数的位相幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は点の消去に基づく組合せ的ホモトピー論の発展を後押しするとともに，それを用いて，ロボットモーション設計やセンサーネットワーク上の数え上げ理論など応用的な分野にアプローチしたものである．空間の複雑さを表す指標としていくつかの位相不変量を組合せ的に計算する方法を導出した．これらは，空間上のロボットの動きを制御するためのアルゴリズムが最低何種類必要かどうかという問題に密接に関わる不変量である．また，周期性や対称性を持つセンサーネットワーク上での効率的なターゲット数え上げ理論についても，いくつかのケースで有効な方法を発見した．
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