1993 Fiscal Year Final Research Report Summary
STUDIES OF TOPOLOGY FROM VARIOUS VIEW POINTS
| Project/Area Number |
04302002
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | TOKYO INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
FUKUDA Takuo Tokyo Institute of Tech.Dept.Math.Professor, 理学部, 教授 (00009599)
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| Co-Investigator(Kenkyū-buntansha) |
NOGURA Tsugunori Ehime Univ.Dept.Math.Professor, 理学部, 教授 (00036419)
MATSUMOTO Takao Hiroshima U.Dept.Math.Professor, 理学部, 教授 (50025467)
MATSUMOTO Yukio Tokyo Univ.Inst.Math.Sci.Professor, 大学院・数理科学研究科, 教授 (20011637)
NISHIDA Goro Kyoto Univ.Dept.Math.Professor, 理学部, 教授 (00027377)
KAWAKUBO Katsuo Osaka Univ.Dept.Math.Professor, 理学部, 教授 (50028198)
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| Project Period (FY) |
1992 – 1993
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| Keywords | Manifold / Topology / Dynamical System / Homology / Floer Homology / Gauge Theory / Conformal Field Theory / Knot |
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| Research Abstract |
In this decade various deep relations between topology of manifolds and other mathematical fields and mathematical physics such as the one between Physics and the knot theory as well as the theory of low-dimensional manifolds were discovered. Since then these new researches on topology of manifolds have been quite remarkablely developped. The japanese school of topology has made a big contribution to the development. Among them we have the following distiguished researches done under the support of this Grant-in Aid :
(1)In the theory of complex dynamical systems where one studies dynamics of iterations of complex holomorphic functions, Mitsuhiro SHISHIKURA proved D.Sullivan's conjecture that the Hausdorf dimension of the boundary of the Mandelbrot set is 2. The Mandelblot set itself is 2-dimensional. It may sound bery curious that the boundary of a 2-dimensional set is also of 2-dimension. Shishikura's result shows us complexity of dynamical systems.
(2)Concerning the topology of low-dimensional manifolds and Mathematical Physics, we have remarkable deep researches done by T.Yoshida, K.Fukaya, T.Kohno and S.Morita. On the Floer homology which is an invariant of 3-manifolds based on Gauge theory, Yoshida affirmatively answered to Atyah' conjecture that the Floer homology will be isomorphic to the Lagrangean Floer homology. Among other motivations, inspirerd by Yoshida's work, Fukaya is now constructing a magnificient theory which intends to unify lowdimensional Gaude theory and by which one can regard the Floer homology as the topological field theory. T.Kohno obtained quite interesting results on applications of 2-dimensional conformal field theory to 3-dimensional manifolds. S moritagave conclusive results on mapping class group of surfaces. Using these results, he constructed characteristic classes of surface bundles in a natural way and he succeeded in defining second characteristic classes as well.
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