2022 Fiscal Year Final Research Report
How is singularity theory applied to mathematics such as surface theory
| Project/Area Number |
19K03486
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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 | Saitama University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | 特異点論 |
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| Outline of Final Research Achievements |
One of our goals was to establish a normal form theorem for singular surface called fronts by applying singularity theory, and to use it for differential geometrical studies of singular surfaces. This was established for singular points called cuspisdal edges and swallow tails in three-dimensional Euclidean space, and the results were published as a paper in Osaka Math J.
As for the problem of describing the singularity of polynomial mappings, we have succeeded to obtain a concrete description of the locus of non-properpoints of polynomial mappings using Newton diagrams. This is a joint work with Takeki Tsuchiya, and the results have been published in Arnold Math J as a paper.
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| Free Research Field |
数学
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| Academic Significance and Societal Importance of the Research Achievements |
特異点論が多くの現象を記述することは，特異点論が極値問題の一般化と捉えれば自明の事である．本研究では，カスプ辺やツバメの尾と呼ばれる特異点を持つ曲面の研究，特にその局所有限不変量の決定，並びに多項式写像の非固有点軌跡のニュートン図形を用いた具体的記述などが成果であり，学術的な意義は高い． さらに，カスプ辺の研究は微分幾何学者が興味を持つ時空のカスプ辺の研究へ繋がり，新たな研究の展開を見せている．特異点論と微分幾何学というフィールドをつなぐ社会的意義もあると判断している．
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