Constructing geometric representations of finite groups through equivariant topology

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03304
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyushu University
• Principal Investigator: Kaji Shizuo 九州大学, マス・フォア・インダストリ研究所, 教授 (00509656)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 同変トポロジー / 計算トポロジー / ワイル群 / 旗多様体

Outline of Final Research Achievements

The most interesting result obtained during the research period is the determination of the action of the Weil group on the cohomology of the real toric manifolds associated with root systems. As a byproduct, a topological realisation of a combinatorial object known as the generalised Euler's zigzag numbers were obtained.
Throughout the research period, a number of algorithms for computing concrete examples regarding flag manifolds, real toric manifolds, and Weyl groups were developed.
Also, computer programmes were produced and made publicly available as open-source software, and they have been used in various fields outside of mathematics.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

旗多様体や実トーリック多様体といった空間のトポロジーと組合せ論を協調させて解析する実例を複数提供したことが，同変トポロジー分野における学術的意義といえる．
また，計算アルゴリズムを開発しその実装を公開したことで，今後具体例から新たな知見が得られることが期待される．
応用として，機械学習や画像解析の手法を開発し，こちらもその実装を公開している．実際に医療・材料・データ解析などに利用されており，社会還元がなされている．