| Project/Area Number |
18K03279
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Okayama University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| 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 / コロンボー代数 / カレント / de Rham複体 / ホモトピー構造 |
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| 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.
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| 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。
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