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1993 Fiscal Year Final Research Report Summary

STUDIES OF TOPOLOGY FROM VARIOUS VIEW POINTS

Research Project

Project/Area Number 04302002
Research Category

Grant-in-Aid for Co-operative Research (A)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionTOKYO INSTITUTE OF TECHNOLOGY

Principal Investigator

FUKUDA Takuo  Tokyo Institute of Tech.Dept.Math.Professor, 理学部, 教授 (00009599)

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)
Project Period (FY) 1992 – 1993
KeywordsManifold / Topology / Dynamical System / Homology / Floer Homology / Gauge Theory / Conformal Field Theory / Knot
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.

  • Research Products

    (13 results)

All Other

All Publications (13 results)

  • [Publications] Mitsuhiro Shishikura: "The Hausdorf dimension of the boundary of the Mandelbrot set and Julia sets" Annals of Mathematics,to appear.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeyuki Morita: "Abelian quotients of subgroups of the mapping class group of surfaces" Duke Mathematical Journal. 70. 699-726 (1993)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kenji Fukaya: "Floer homolgy for connected sum of homology 3-sheres" Topology,to appear.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mikio Furuta: "Morse theory and Thon-Gysin exact sequence" Lecture Notes in Pure and Applied Math.145. 69-77 (1993)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toshitake Kohno: "Monodromy representations of conformal field theory and their applications" Geometry,Topology and Field Theory,Proc.of Taniguchi Workshop and Symp.to appear.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi Matsuoka: "The Brauer representation of the braid group and the Nielsen-Thurston classification" Contemp.Math.157. 229-248 (1993)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mitsuhiro Shishikura: "The Hausdorf dimension of the boundary of the mandelbrot set and Julia sets" Annals of Math.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeyuki Morita: "Abelian quotients of subgroups of the mapping class group of surfaces" Duke Math.J., 70. 699-726 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kenji Fukaya: "Floer homology for connected sum of homology 3-spheres" Topology. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mikio Furuta: "Morse theory and Thom-Gysin exact sequence" Lecture Notes in Pure and Appl.Math.145. 69-77 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshitake Kohno: "Monodromy representation of conforal field theory and their applications" Geometry, Topology and Field Theory, Proc.of Taniguchi Workshop and Symp.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tomoyoshi Yoshida: "A formulation of Symplectic Floer homology for Heegaard Splittings" ibid. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Matsuoka: "The brauer representation of the braid group and the Nielsen-Thurston classification" Contemp.Math.157. 229-248 (1993)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1995-03-27  

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