2000 Fiscal Year Final Research Report Summary
Fundamental Research of Discrete Geometric Analysis and Its Applications
| Project/Area Number |
10304011
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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 |
Global analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
SUNADA Toshikazu Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20022741)
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| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Koji Tohoku University, Graduate School of Science, Assistant Prof., 大学院・理学研究科, 講師 (30208483)
SAITO Kazuyuki Tohoku University, Graduate School of Science, Associate Prof., 大学院・理学研究科, 助教授 (60004397)
KOTANI Motoko Tohoku University, Graduate School of Science, Associate Prof., 大学院・理学研究科, 助教授 (50230024)
URAKAWA Hajime Tohoku University, Graduate School of Information Science, Prof., 大学院・情報科学研究科, 教授 (50022679)
KUROKI Gen Tohoku University, Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (10234593)
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| Project Period (FY) |
1998 – 2000
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| Keywords | Discrete Laplacian / Crystal Lattice / Harper Operator / Central Limit Theorem |
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| Research Abstract |
We have handled both geometric and analytic aspects of discrete Laplacians on infinite graphs which are main objects in discrete geometric analysis and show up in various fields of pure and applied mathematics, say the theory of discrete groups communication networks and Markov chains. Especially we obtained interesting results on large time asymptotic behaviors of transition probabilities of random walks on crystal lattices. One is the local central limit theorem, and another is asymptotic expansions. In our study, we made use of the notions of Albanese tori and Albanese maps which have the origin in algebraic geometry. In connection with this, we developed the theory of harmonic maps from graphs into Riemannian manifolds. Albanese maps is defined as a harminic maps from a finite graph into a flat torus. In this project, we have also studied the spectral properties of discrete magnetic Schroedinger. operators (Harper operators) on crystal lattices. The central limit theorem for Harper operators was established. We investigated twisted group C^* algeblas associated with Harper operators which is a generalization of non-commutative tori. As a byproduct of our research, we gave a rigorous treatment of quantized theory of lattice vibrations.
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