New development of topological K-theory pioneered by physics

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04871
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology (2019); Shinshu University (2015-2018)
• Principal Investigator: Gomi Kiyonori 東京工業大学, 理学院, 教授 (00543109)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 位相的K理論 / トポロジカル絶縁体 / ベクトル束 / 特性類

Outline of Final Research Achievements

Since theoretical predictions and experimental confirmations of topological insulators in condensed matter physics, topological K-theory has been recognized as effective of their classifications. In this research, I clarified mathematical nature of twisted equivariant K-theory necessary for such classifications, and applied it to the classifications of topological insulators subject to crystalline symmetry through actual computations. Related to this application, I also investigated and got some results above invariants of vector bundles equipped with certain structures as well as topological T-duality useful for computations of topological K-theory.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

当研究の成果として, トポロジカル絶縁体という現実の物質の分類にために, 位相的K理論という純粋数学で以前から研究されてきた道具が有効に活用できることを示したことが挙げられる. これには, 新奇な性質を持つ物質の存在を具体的に示唆できるという意義に加えて, 純粋数学という基礎研究で培われた概念が, 思いもよらない応用を持つ一例を与えたという意義がある.