Project/Area Number  09640231 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Nagoya Institute of Technology 
Principal Investigator 
NATSUME Toshikzau Nagoya Institute of Technology, Faculty of Engineering, Professor of Mathematics, 工学部, 教授 (00125890)

CoInvestigator(Kenkyūbuntansha) 
YAMADA Osanobu Ritsumeikan University, College of Science and Engineering, Professor of Mathema, 理工学部, 教授 (70066744)
NAKAJIMA Kazufumi Ritsumeikan University, College of Science and Engineering, Professor of Mathema, 理工学部, 教授 (10025489)
NAKAMURA Yoshihiro Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of M, 工学部, 助教授 (50155868)
OHYAMA Yoshiyuki Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of M, 工学部, 助教授 (80223981)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  pseudodifferential operator, / Fredholm index, / AtiyahSingertype index theorem, / Weyl quantization, / C^*algebraic deformation quantization, / KTheory / 擬微分作用素 / フレドホルム指数 / アティヤ・シンガー型指数定理 / ワイル量子化 / C^*環的変型量子化 / K理論 / C^*環的変形量子化 / AtiyahSinger型指数公式 / Weyl量子化 / C^*環 
Research Abstract 
The aim of this project is to show an AtiyahSingertype index formula for pseudodiffer ential operators on open manifolds, particularly on simply connected hyperbolic manifolds. The project is divided into two steps. The first is to isolate a class of pseudodifferential operators that have FredhoIm indices. The second is to actually prove the index theorem. In the first year of this research grant, we aimed at completing the first step. A prelim inary study strongly indicated a difficulty in doing so, due to the lack of general spectral theory on those manifolds. This suggested us to study the geometry of hyperbolic mani folds. Having had discussions with researchers of various fields in order to get information pertinent to our project, we managed to isolate a class of pseudodifferential operators that possibly have Fredholm indices and hoped to complete the first step. However, it was post poned till the second year of the grant to prove that the class we isolated is the right on
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e. This goal turned out too ambitious due to technical obstacles, and unfortunately we could not comp1ete the first step of the project. While working on the objective discussed above, we worked at the same time on devel oping basic tools needed for the proof of the index theorem. One of them is the notion of C^*algebraic deformation quantizations of symplectic manifolds. In this direction, a significant progress has been made. In a joint paper with R.Nest of the University of Copenhagen (Paper 3 in Item 11 of this report) we studied the closed Riemannian sur faces, which are primary examples of symplectic manifolds, and showed the existence of C^*algebraic deformation quantizations. Generalizing this result, in a joint project with R.Nest and I.Peter of Muenster University, under a certain topological condition, we showed the existence of C^*algebraic deformation quantization for any symplectic mani fold. We plan further investigations into this direction. In particular, it is the upcoming project to study C^*algebraic deformation quantization for Poisson manifolds, which are generalization of symplectic manifolds. As for the initial target of the research, that is, an index formula for pseudodifferential operators, we certainly intend to continue working on it. We will hopefully complete the project within a year or so. Less
