| Project/Area Number |
18540084
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Hyogo University of Teacher Education |
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Principal Investigator |
KOIKE Satoshi Hyogo University of Teacher Education, Graduate School of Eduration, Professor (60161832)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOTA Masahiro Nagoya Universiy, Graduate School Mathmatics, Professor (00027385)
FUKUI Toshizumi Saitama Universiy, Faculty of Science, Professor (90218892)
OHMOTO Toru Hokkairo Unicrosity, Graduate Shool of Science, Asscoiate Professor (20264400)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,920,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥420,000)
Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | Rlow-Nash triviality / blow-analitici / tree model / minimal resolution / cascade equivalence / blow-semialeebraic triviality / Fukui invariant / Lipschitz equivalence / モチーフ型不変量 / 実ツリーモデル |
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| Research Abstract |
A singular point is defined in Mathematics as a point at which the space is not smooth or a point at which the map is not regular. Equisingularity Problem is a problem to ask whether the singular points (resp. the families of singular points) of the spaces or maps are the same under some desirable equivalence relation (resp. triviality). It naturally becomes a problem there to introduce some equivalence relation, to ask if the equivalence relation is natural, to analyze the relation between the equivalence and the other equivalences, or to classify the singularities by the equivalence. In order to solve those problems, it is very important to establish the triviality theorem, to introduce some invariants, or to give characterizations for the equivalence. Concerning these things, I have got the following results as equisingularities of real algebraic singularities and real analytic singularities:
(1) Let us consider the finiteness problem on some triviality for a family of zero-sets of Nash mappings defined over a not necessarily compact Nash manifold. The main results on this problem are:
(i) Finiteness theorem holds on Blow-Nash triviality when the zero-sets have isolated singularities.
(ii) Finiteness theorem holds on Blow-semialgebraic triviality in the non-isolated singularity case when the dimension of the zero-sets is 2 or 3.
(iii) Finiteness theorem holds on the existence of Nash trivial simultaneous resolution without the assumptions on the isolated singularity or the dimension of the zero-sets.
(2) I have got some necessary and sufficient conditions with Adam Parusinski for two variable real analytic functions to be blow-analytically equivalent. More precisely, two real analytic function germs of two variables are blow-analytically equivalent if and only if they have weakly isomorphic minimal resolutions, their real tree models are isomorphic, or they are cascade equivalent.
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