2003 Fiscal Year Final Research Report Summary
Mapping Class Group of Surfaces and Geometry of Moduli Spaces
| Project/Area Number |
13440017
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | The University of Tokyo |
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Principal Investigator |
MORITA Shigeyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Hiroaki Okayama University, Faculty of Sciences, Professor, 理学部, 教授 (60217883)
KAWAZUMI Nariya The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (30214646)
FURUTA Mikio The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50181459)
MURAKAMI Jun Waseda University, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (90157751)
AKITA Toshiyuki Hokkaido University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30279252)
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| Project Period (FY) |
2001 – 2003
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| Keywords | mapping class group / Riemann surface / Floer homotopy type / Grothendieck-teichmuller group / 3-dimensional manifold / Volume conjecture / non-commutitire geometry / 非可換幾何 |
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| Research Abstract |
In this project, we focussed on the study of the structure of the mapping class group of surfaces (m.c.g. for short) as well as the moduli space of compact Riemann surfaces, together with various problems closely related with this. They include the following thema : cohomology group of m.c.g., the theory of the Floer homotopy types, topological invariants based on gauge theory, construction of the harmonic Magnus expansion of m.c.g., structure of the Grothendieck-Teichm＼"uller group, the volume conjecture, non-commutative geometry in dimensions 3,4, finite subgroups of m.c.g., the Jones representation of m.c.g., relation between m.c.g. with 4-dimensional topology. From the interactions of these thema, we found new directions of research such as the relation between the geometry of m.c.g. and the symplectic topology as well as the comparaison between m.c.g. and the outer automorphism group of free groups.
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