New developements in localization principles in differential geometry and noncommutative geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14307
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyoto University
• Principal Investigator: Yamashita Mayuko 京都大学, 理学研究科, 准教授 (30866249)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 数理物理学 / 代数トポロジー / ホモトピー論 / 場の理論

Outline of Final Research Achievements

My initial plan for this project was the purely mathematical study in differential geometry and topology, but my research interest has extended also to include algebraic topology. Accordingly I found the potential application and relation with theoretical physics, and it has led me to collaboration with physicists. Especially we had worked on Topological Modular Forms and Segal-Stolz-Teichner program, which is one of the deepest and most important topic connecting homotopy theory and physics. Throughout the research period, we were able to produce various results both in mathematics and physics.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

代数トポロジーやホモトピー論は抽象化が進んだ結果、純粋数学的には面白いものであってもごく最近までは理論物理学への還元がほとんどなされてこなかったといえる。しかし本研究はホモトピー論の深い結果を理論物理学に実際に応用した先駆的な研究と言える。例えば本研究で構築した量子異常の分類に関する一般論は, 量子異常の解析やトポロジカル物性における相の分類に応用されている。また、量子異常の消滅を示した結果は理論物理学から大きな反響があった. それに続く研究も進んでいる.