2002 Fiscal Year Final Research Report Summary
Differential Equations on Manifolds and Their Singularities
| Project/Area Number |
13640059
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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 | Akita University |
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Principal Investigator |
KAWAKAMI Hajime Faculty of Engineering and Resource Science associate professor, 工学資源学部, 助教授 (20240781)
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| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Mahito Faculty of Engineering and Resource Science associate professor, 工学資源学部, 助教授 (10261645)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Holder continuity / C^∞ smoothing / diffusion equation / inverse problem / Gaussian curvature / stable map / discriminant / computer network |
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| Research Abstract |
Head investigator Kawakami has studied the following. In the first year:
He and Prof. Tsuchiya (Kanazawa Univ.) proved that any Cr,α manifold (manifold pair) has a C∞ smoothing by using a method of J.R. Munkres.
He and Dr. Murayama (Shobi Univ.) and others studied about a means of giving teaching-materials of mathematics through a computer network.
In the second year:
He and Prof. Tsuchiya (Kanazawa Univ.) have studied a generalization of "Kurt Bryant and Lester F. caudill Jr., Inverse Problem 14 1429-1453 (1998)". They proved that the data in a finite time-interval uniquely determine the shape of the back surfice.
He conjectured that the Gauss-Bonnet formula gives a necessary and sufficient condition for the existence of a metric deformation to obtain a positive/negative Gaussian curvature on a disk. He gave a partial answer of the conjecture.
Investigator Kobayashi worked on studying the curious relation of generic maps to their discriminants. The main results in the first year are;
a characterization of the 'folding into four' action in general dimensions by the discriminant of the folding map,
finding of an infinite to one correspondence of maps of a fixed closed 4-manifolds to their discriminants,
providing a family of discrimiants of stable maps of closed manifolds.
Those in the second year are;
a characterization of plane curves which are the critical value set of a generic projection of a closed surface into the plane;
study of planar projections of sphere bundles over spheres.
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