2002 Fiscal Year Final Research Report Summary
Geometry and Topology of 3-Manifolds
| Project/Area Number |
12440015
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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 | Tokyo Institute of Technology |
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Principal Investigator |
KOJIMA Sadayoshi Tokyo Institute of Technology, Mathematical and Computing Sciences, Professor, 大学院・情報理工学研究科, 教授 (90117705)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Shigeyuki University of Tokyo, Department of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
OHTSUKI Tomotada University of Tokyo, Department of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50223871)
YOSHIDA Tomoyoshi Tokyo Institute of Technology, Department of Mathematics, Professor, 大学院・理工学研究科, 教授 (60055324)
SOMA Teruhiko Tokyo Denki University, Department of Mathematical Sciences, Professor, 理工学部, 教授 (50154688)
MATSUMOTO Shigenori Nihon University, Department of Mathematics, Professor, 理工学部, 教授 (80060143)
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| Project Period (FY) |
2000 – 2002
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| Keywords | hyperbolic geometry / cone-manifold / 3-dimensional topology / secondary characteristic class / lamination / foliation / geometric structure / volume conjecture |
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| Research Abstract |
This project was aimed to develop the interdisciplinary study of 3-manifolds which interacts geometry and topology based on the connection between several structures related mainly with hyperbolic geometry. We have in fact promoted our project in conjunction with the activity of the Topology Research Congress.
We made certain significant progresses during these three research years which turned out to be rather surprising than what we had expected. The study of myself together with Mizushima and Tan on circle packings on surfaces with projective structures has clarified an expected global structure of their moduli space in the light of the deformation theory of hyperbolic 3-manifolds. The study of Yoshida on SU(2) conformal field theory has established successfully an explicit description of a basis of conformal blocks, and has approached to the fundamental connection between the geometry and topology of 3-manifolds suggested for instance by the volume con-jecture. Also, the global diagram in the 3-manifold topological invariant world suggested by Ohtsuki was completed quite recently in the most universal way by Habiro and Le. In addition, there have been down-to-earth progresses by other collaborators such as Morita's study on the mapping class group of surfaces, Matsumoto's on foliations, Sakuma's on knots and geometric structures, Soma's on bounded cohomology,
In conclusion, our research has clarified the object on which we should now focus for finding the mathematical principle behind the interaction between many structures observed in the 3-manifold theory. Note in addition, the theme was fortunately funded for further study as the part II.
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