| Project/Area Number |
11440017
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Saitama University |
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Principal Investigator |
FUKUI Toshizumi Saitama University, Faculty of Science, Professor, 理学部, 助教授 (90218892)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAI Fumio Saitama University, Faculty of Science, Professor, 理学部, 教授 (40036596)
MIZUTANI Tadayoshi Saitama University, Faculty of Science, Professor, 理学部, 教授 (20080492)
YANO Tamaki Saitama University, Faculty of Science, Professor, 理学部, 教授 (10111410)
IZUMI Shuzo Kinki University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80025410)
KOIKE Satoshi Hyogo University of Teacher Education, Associate Professor, 学校教育学部, 助教授 (60161832)
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| Project Period (FY) |
1999 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥13,500,000 (Direct Cost: ¥13,500,000)
Fiscal Year 2002: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2001: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2000: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1999: ¥3,600,000 (Direct Cost: ¥3,600,000)
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| Keywords | Singularities of maps / Thom-Boardman manifolds / Cohen-Macaaulay property / Representation of general linear group / blow-analytic map / modified analytic trivialization / Newton filtration / 曲面の臍点 / 改変後解析性 / 特異曲面 / 写像度 |
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| Research Abstract |
Our starting point of this research is the following fact : There are two main directions in the research of singularities of maps, which are (1) Investigation of generic perturbation of maps after classification generic singularities, (2) Geometric understanding of desingularization and investigating equisingular problem.
In the first direction, we investigate Thom-Boardmabn strata in the jet space and discuss their Cohen-Macaulay property, which become important in the context of intersection theory. We show a complete result for the problem whether Morin's ideal, which regular locus is the Thom-Boardman strata, determines Cohen-Macaulay spaces, or not. We also discuss Ronga's desingularization and construct complecies supported at Σ^<n-p+1,1> strata. This is related with a generic perturbation of a map germ f : (C^n,0) → (C^2,0). We obtain an explicit description of equations for Σ^<n-p+1,1> locus when n = 2,3,4 and conclude an explicit formula for the number of cusps appeared in a generic deformation in these cases.
We also discuss the use of Newton filtration in the computation of invariants of singularities.
In the second direction, we discuss blow-analytic maps. We showed a weighted version of blow-analytic triviality theorem, clarification of the behavior of blow-analytic invariant which was defined by Fukui, investigation of blow-analytic isomorphisms, and and work about blow-analytic map with biLipschitz property.
Stratification theory is a tool to construct topological triviality. But the usual regularity condition in stratification theory is too strong for topological equivalence. By the above result, we can expect a weaker version of the regularity condition. We define a weighted version of regularity condition and construct a weighted version of stratification theory.
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