Research on homotopy theory and reconstruction of spaces using categories

2016 Fiscal Year Final Research Report

• Project/Area Number: 15K17535
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Shinshu University
• Principal Investigator: TANAKA Kohei 信州大学, 学術研究院社会科学系, 助教 (70708362)
• Research Collaborator: TAMAKI Dai 信州大学, 学術研究院理学系, 教授 (10252058) ; NANDA Vidit The University of Oxford
• Project Period (FY): 2015-04-01 – 2017-03-31
• Keywords: 圏 / ホモトピー論 / センサーネットワーク / オイラー積分 / LS カテゴリー

Outline of Final Research Achievements

In this research, we established the combinatorial homotopy theory and its application based on categories, which consists of points and arrows. Given an acyclic flow on a cell complex, we obtained a category from the critical cells and reconstructed the information of the original space. On the other hand, we introduced a discrete invariant of partially ordered sets as a tool to classify them.
As an application of these theory, we considered the counting problem for targets lying on a sensor network. By using the integration with respect to combinatorial Euler characteristic, we showed that we can compute the number of targets lying on the network graph.

Free Research Field

代数的位相幾何学