1988 Fiscal Year Final Research Report Summary
| Project/Area Number |
62460001
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY |
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Principal Investigator |
SUZUKI Haruo DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 教授 (80000735)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Asao DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 講師 (80134807)
GIGA Yoshikazu DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 助教授 (70144110)
KAMISHIMA Yoshinobu DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 講師 (10125304)
IZUMIYA Shyuichi DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 助教授 (80127422)
NISHIMORI Toshiyuki DEPARTMENT OF MATHEMATICS, HOKKAIDO UNIVERSITY, 理学部, 助教授 (50004487)
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| Project Period (FY) |
1987 – 1988
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| Keywords | MANIFOLD / FOLIATION / SINGULARITY / C^*-ALGEBRAS / GROUP ACTION / PERIODIC ORBIT / DIFFERENTIAL EQUATION / 指数 |
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| Research Abstract |
THE PURPOSE OF THE PROJECR WAS TO INVESTIGATE GLOBAL PROPERTIES OF MANIFOLDS FROM VIEWPOINT OF GEOMETRY AND ANALYSIS. CONTENTS OF RESULTS OBTAINED THROUGH 1987-1988 ARE SUMMARIZEF AS FOLLOWS.
1. FOLIATIONS AND SINGULARITIES: (1). THE THEORY OF DEFFERENTIABLE SINGULAR COHOMOLOGY OF FOLIATIONS IS ESTABLISHED. (2). HOLONOMY MAPS AND HOLONOMY GROUPOIDS ARE CONSTRUCTED FOR FOLIATIONS WUTH SINGULARITIES. IT CAN BE APPLICABL TO LOCALLY NON-SIMPLE FOLIATIONS. THIS IS ITS STRONG POINT. ITS CONTRIBUTIONS TO C^*-ALGEBRAS OF FOLIATIONS WITH SINGULARITIES AND INDEX THECRY ARE EXPECTED. (3). A NOTION OF SINGULAR SOLUTIONS OF ORDINARY OR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER IS FORMULATED BY THE LEGENDRIAN SINGULARITY AND THEIR UNFOLDING THEORY IS ESTABLISHED.
2. DYNAMICAL SYSTEMS AND GROUP ACTIONS: (1). BY MAKING USE OF BRAID GROUPS, A CLASSIFICATION OF PERIODIC ORBITS OF A PART OF HENON MAPS:IS MADE. (2). A SUFFICIENT CONDITION FOR A MANIFOLD WITH LOCAL STRUCTURE OF A SIMPLY CONNECTED HOMOGENEOUS SPACE TO BE QUOTIENT MANIFOLD IS OBTAINED.
3. SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS: (1). A RECTIFIABILITY OF SINGULARITY OF A SYMMETRIC MATRIX OF RADON MEASURE WHICH IS A HESSIAN OF A FUNCTION IS INVWSTIGATED AND IS APPLIED TO SINGULARITIES OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS. (2). OPERATORS OF KAHLER-DIRAC TYPE ARE DEFINED IN AN ABSTRACT INFINITE DIMENSIONAL BOSON-FERMION FOCK SPACE AND A PATH INTEGRAL REPRESENTATION OF THEIR INDEX IS ESTABLISHED.
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