1998 Fiscal Year Final Research Report Summary
On the construction and classification of the finite geometry
| Project/Area Number |
09640306
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Fukuoka University |
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Principal Investigator |
ODA Nobuyuki Fukuoka Univ., Fac.of Science, Prof., 理学部, 教授 (80112283)
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| Co-Investigator(Kenkyū-buntansha) |
AKIYAMA Kenji Fukuoka Univ., Fac.of Science, Prof., 理学部, 教授 (70078575)
AKITA Toshiyuki Fukuoka Univ., Fac.of Science, Assistant, 理学部, 助手 (30279252)
KUROSE Takashi Fukuoka Univ., Fac.of Science, Assoc.Prof., 理学部, 助教授 (30215107)
INOUE Atsushi Fukuoka Univ., Fac.of Science, Prof., 理学部, 教授 (50078557)
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| Project Period (FY) |
1997 – 1998
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| Keywords | geometry / homotopy / algebra / cohomology / plane / Schur ring |
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| Research Abstract |
Geometrical constructions in homotopy sets were studied. We obtained results on the GAMMA-Whitehead product and the GAMMA-Hopf construction. We introduced the transformation between pairings and copairings and showed its applications. We obtained a formula for the smash product. We obtained a generalization of the Hardie-Jansen product and studied its properties. Dual results are also studied.
For geometrical construction in operator algebras, Tomita-Takesaki theory was studied. We obtained results on unbounded C^*seminorms on *-algebra and standard weights which enable us to develop unbounded Tomita-Takesaki theory.
We constructed explicit examples of surfaces in affine spaces of dimension three and four. We gave a necessary and sufficient condition on surfaces in a three-dimensional affine space to be metric when the surfaces have non-zero constant Gauss-Kronecker curvature.
The cohomology of mapping class groups was studied. We obtained a relation among periodic automorphisms of closed surfaces and the eta-invariant of their mapping tori. We also obtained various vanishing theorems of mod 2 Morita-Mumford classes.
The Schur ring of product type was characterized by the existence of a subgroup of a collineation group. The existence of a Schur ring of produt difference set type is characterized by a finite projective plane of order n with a collineation group of order n(n - 1).
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