The Atiyah-Singer index theorem on hyperbolic spaces and noncommutative geometry
| Project/Area Number |
17540192
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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 |
Global analysis
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| Research Institution | Nagoya Institute of Technology |
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Principal Investigator |
NATSUME Toshikazu Nagoya Institute of Technology, Graduate School of Engineering, Professor of Mathematics, 工学研究科, 教授 (00125890)
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| Co-Investigator(Kenkyū-buntansha) |
ADACHI Toshiaki Nagoya Institute of Technology, Graduate School of Engineering, Professor of Mathematics, 工学研究科, 教授 (60191855)
NAKAMURA Yoshihiro Nagoya Institute of Technology, Graduate School of Engineering, Associate Professor of Mathematics, 工学研究科, 助教授 (50155868)
MORIYOSHI Hitoshi Keio University, Faculty of Science and Engineering, Associate Professor of Mathematics, 理工学部, 助教授 (00239708)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2005: ¥1,600,000 (Direct Cost: ¥1,600,000)
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| Keywords | pseudo-differential operator / C^*-algebra / Atiyah-Singer index theorem / K-theory / cyclic cohomology / quantization / Fredholm index |
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| Research Abstract |
The purpose of the project is to generalize the main result of the joint paper "The Atiyah-Singer index theorem as a passage to classical limit in quantum mechanics" (Com-munications in Mathematical Physics, 182 (1996), 505-533) with G.A. Elliott of the University of Toronto and R. Nest of the University of Copenhagen. In this paper, we studied a certain class of pseudo-differential operators on flat spaces. Employing noncommutative geometric methods we proved an Atiyah-Singer-type index theorem. A crucial property of flat spaces behind the proof is that for given arbitrary two points there exists a unique line segment (geodesic) joining those two points. This property is enjoyed by not only flat spaces but also simply connected hyperbolic spaces, for instance the Poincare disk. Noncommutative geometric methods can be applied to hyperbolic spaces.
The first crucial step to achieve the purpose is to isolate a class of pseudo-differential operators, on hyperbolic spaces, that have Fredholm indices.
In the first year we studied the most important case. That is the Poincare disk. On the flat spaces, the pseudo-differential operators studied are "modelled" on the harmonic oscilators. We constructed the harmonic oscilator on the Poincare disk, what is the Laplacian perturbed by a lower degree term. We showed that the harmonic oscilator, described just above, has compacts resolvents, in particular has a Fredholm index. In the second year, while preparing the paper, a gap in the proof was found, and the most of time was spent on fixing the proof. As a result, unfortunately the goal of the project was not reached in time. The paper on the spectrum of the harmonic oscilator on the Poincare disk will be available some time soon.
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Report
(3 results)
Research Products
(21 results)
