Integration of homotopical and analytical methods in the frame work of diffeology

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03279
• 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: Okayama University
• Principal Investigator: Shimakawa Kazuhisa 岡山大学, 自然科学研究科, 特命教授 (70109081)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 微分空間 / モデル圏 / シュワルツ超関数 / ド・ラームカレント

Outline of Final Research Achievements

Based on the notion of smooth homotopy, we introduced on the category of diffeological spaces a model category structure, which turns out to be Quillen equivalent to the standard Quillen model structure on the category of topological spaces. It is proved that there hold analogies to the theorems of Whitney and J. H. C. Whitehead for "smooth cell complexes" associated with the construction of our model category. We then constructed on an arbitrary diffeological space an algebra of generalized functions (called "asymptotic functions") equipped with properties similar to Schwartz distributions, and extended it to a space of morphisms (called "asymptotic maps") between diffeological spaces. The resulting category of diffeological spaces and asymptotic maps is cartesian closed, and is furnished with nice properties enabling us to construct on every diffeological space an exterior algebra containing de Rham currents as its subspace.

Free Research Field

代数的位相幾何学

Academic Significance and Societal Importance of the Research Achievements

多様体の概念の高度な一般化として，幾何学の分野で注目すべき成果を挙げつつある微分空間の概念と，数理科学のみならず，物理科学や工学等の幅広い分野で重要な役割を果たしているシュワルツ超関数や，その一般化であるコロンボー代数の理論を融合発展させた研究対象を創出することによって，それ自体の理論的興味に留まらず，幅広い科学分野で新たな応用研究の発展推進に貢献することが期待できる。