1999 Fiscal Year Final Research Report Summary
Study of contact structures and foliations on 3-manifolds
| Project/Area Number |
09640130
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Chuo University |
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Principal Investigator |
MITSUHATSU Yoshihiko Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (70190725)
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| Co-Investigator(Kenkyū-buntansha) |
MIZUTANI Tadayoshi Saitama University, Faculty of Science, Professor, 理学部, 教授 (20080492)
TAKAKURA Tatsuru Chuo University, Faculty of Science & Eng., Lecturer, 理工学部, 講師 (30268974)
MATSUYAMA Yoshio Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (70112753)
KANOA Yutaka Hokkaido University, Faculty of Science, Instructor, 理学部, 助手 (30280861)
ONO Kaoru Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (20204232)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Contact Structure / foliation / symplectic structure / bi-contact structure / projectively Anosoo flow / symplectic filling / geometric quantization / tight contact structure |
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| Research Abstract |
Mitsumatsu and Mizutani studied and constructed many examples of bi-contact structures with a research group of foliations. Especially, they constructed a bicontact structure on the 3-sphere which consists both of over-twisted ones. Still the realization problem of homotopy class of plane fields as such structures remains to be studied.
Ono has established in a colaboration with Fukaya foundamental theory in applying the J-curves to symplectic topology, overcoming the notorious problem of negative multiples. Major consequences from this are the definition of Gromov-Witten invariants for general symplectic manifolds and the positive solution for a version of the Arnold conjecture for the same class. He and Kanda also worked on applying Seiberg-witten theory to contact topology, colaborating with Ohta, and got toplogical constraints on the symplectic filling 4-manifolds around simple singularities. This streem of works continues and is expected to make a further progress, especially in relation with the last subject of study below.
Kanda studied contact structures in more toplogical way, and classified tight contact structures on 3-torus and showed nonexactness of Bennequin's inequality.
Takakura and Mitsumatsu have been searching for the formalism to study contact topology by using Lagrangian/Legendrian torus, instead of looking at J-curves in the symplectization. This is based on the theory of geometric quantization, on which Takakura has been working. They found that most of major concepts in the theory of algebraic functions in one variable can be suitably translated and planted in this framework. However, studying contact topology through this remains as the next step of research to go.
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