| Project/Area Number |
16204004
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 |
TSUBOI Takashi The University of Tokyo, Mathematical Sciences, Professor (40114566)
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| Co-Investigator(Kenkyū-buntansha) |
OHSHIKA Ken'ichi Osaka University, Sciences, Professor (70183225)
KAWAZUMI Nariya The University of Tokyo, Mathematical Sciences, Associate Professor (30214646)
FUJIWARA Koji Tohoku University, Information Sciences, Professor (60229078)
MATSUMOTO Shigenori Nihon University, Science and Technology, Professor (80060143)
MORITA Shigeyuki The University of Tokyo, Mathematical Sciences, Professor (70011674)
中山 裕道 広島大学, 大学院・理学研究科, 准教授 (30227970)
林 修平 東京大学, 大学院・数理科学研究科, 准教授 (20247208)
皆川 宏之 (皆川 広之) 山形大学, 教育学部, 助教授 (30241300)
足助 太郎 東京大学, 大学院・数理科学研究科, 助教授 (30294515)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥36,140,000 (Direct Cost: ¥27,800,000、Indirect Cost: ¥8,340,000)
Fiscal Year 2007: ¥9,360,000 (Direct Cost: ¥7,200,000、Indirect Cost: ¥2,160,000)
Fiscal Year 2006: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Fiscal Year 2005: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Fiscal Year 2004: ¥8,580,000 (Direct Cost: ¥6,600,000、Indirect Cost: ¥1,980,000)
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| Keywords | infinite groups / transformation grouns / geometric theory of groups / diffeomorphism groups / classifying spaces / characteristic classes / dynamics of group actions / invariant sets / 幾何学 / 多様体 / 無限変換群 / 葉層構造 / 曲面束 / 幾何的群論 / 横断構造 / 接触構造 / クライン群 / 極小集合 / 離散群 / アノソフ流 / 区分線形構造 / 等長変換群 |
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| Research Abstract |
Transformation groups acting on manifolds have been intensively investigated since the establishment of the notion of manifolds. In this research, we look at infinite groups of transformations, that is, infinite groups acting smoothly on manifolds. In this subject, there are still many unsolved problems. The aim of our research was to study the research techniques on infinite groups of transformations, to develop new techniques and to apply it to solve the problems. More specifically, it was stated as follows (1) On the action of finitely presented infinite groups such as the fundamental groups of 2 or 3 dimensional manifolds, we clarify the relationship between the nature of the actions and the quantities coming from geometric group theory. We clarify the relation between the action of groups and the bounded cohomology of groups. (2) We clarify the topology of the classifying spaces of various infinite transformation groups and the invariants for infinite transformation groups. In par
… More
ticular, we look at the topology of the classifying space for the group of symplectomorphisms or contactomorphisms. (3) We study the dynamics of infinite transformation groups, define invariants associated with invariant sets for the transformation groups and study the properties of the invariants. We also study the ergodic properties of transformation groups. (4) We classify several important infinite group actions on manifolds. In particular, we look at the rigidity of the actions on manifolds of discrete subgroups of Lie groups. We also investigate the invariants of complex analytic actions of mapping class groups of surfaces. (5) We look at the relationship among the study on the geometry of infinite groups, on classifying spaces of infinite groups, on dynamics of infinite transformation groups, on rigidity of actions, … We clarify this relation for the bundle transformation groups of fiber bundles, the area preserving diffeomorphism groups of surfaces, the complex analytic transformation groupoid of complex manifolds.
To perform these researches, we maintained the network among the researchers to promote the interaction and the collaborations. We planned timely meetings, sent researchers to other institute, gave presentations in the international meetings, asked being foreign specialists to review our research. With the collaboration of researchers, we performed the research from a global point of view.
These researches were done successfully with their own results in the line listed above. In particular, our high research level was shown by 5 special invited lectures in -the meeting of the Mathematical Society of Japan (including 2 Geometry Prize of the Mathematical Society of Japan awards).
By these research activities, the direction of new development for the next project became clear. Less
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