Topology, algebraic geometry, and representation theory in GKM theory

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14537
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Naruto University of Education (2020-2023); Osaka City University (2019)
• Principal Investigator: Yamanaka Hitoshi 鳴門教育大学, 大学院学校教育研究科, 准教授 (90725011)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: GKM理論 / トーラス同変コホモロジー / GKMグラフ / トーラスグラフ / 同変Chern類

Outline of Final Research Achievements

This research project belongs to a field called transformation group theory, which studies the properties of spaces by exploiting the symmetries of various spaces. The research field involves various fields, such as topology, algebraic geometry and representation theory, and accordingly, various considerations can be made.
Our study focuses on what is called GKM theory within transformation group theory, and the main object is torus equivariant cohomology, which is an object with a structure of operations such as addition and multiplication. The object c is constracted from a space with a group action and it is known to contain many properties related to the space. In this research project, this has been successfully refined from the point of view of the equivariant rigidity.

Free Research Field

変換群論

Academic Significance and Societal Importance of the Research Achievements

変換群論はそれ自身1つの分野として確立されているが、可換代数、組み合わせ論、トポロジー、代数幾何、表現論といった諸分野とも自然かつ密接に関係している。