Details of the Researcher

PHOTO

Tatsuya Tate
Section
Graduate School of Science
Job title
Professor
Degree

  • 博士（理学）（東北大学）

  • 修士（理学）（東北大学）

Professional Memberships 1

  • Mathematical Society of Japan

Research Interests 3

  • 量子ウォーク

  • Geometric Asymptotics

  • Geometric Analysis

Research Areas 1

  • Natural sciences / Geometry / Geometric Analysis

Papers 17

  1. Eigenvalues of Quantum Walks of Grover and Fourier Types Peer-reviewed

    Takashi Komatsu, Tatsuya Tate

    JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS　25　(4)　1293-1318　2019/08

    DOI： 10.1007/s00041-018-9630-6 　

    ISSN： 1069-5869

    eISSN： 1531-5851

  2. Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for 3-and 4-valent graphs Peer-reviewed

    Toshiaki Omori, Hisashi Naito, Tatsuya Tate

    ELECTRONIC JOURNAL OF COMBINATORICS　26　(3)　2019/07

    ISSN： 1077-8926

  3. Eigenvalues, absolute continuity and localizations for periodic unitary transition operators Peer-reviewed

    Tatsuya Tate

    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS　22　(2)　2019/06

    DOI： 10.1142/S0219025719500115 　

    ISSN： 0219-0257

    eISSN： 1793-6306

  4. Up and down Grover walks on simplicial complexes Peer-reviewed

    Xin Luo, Tatsuya Tate

    LINEAR ALGEBRA AND ITS APPLICATIONS　545　174-206　2018/05

    DOI： 10.1016/j.laa.2018.01.036 　

    ISSN： 0024-3795

    eISSN： 1873-1856

  5. Quantum walks in low dimension

    Tatsuya Tate

    Trends in Mathematics　261-278　2016

    DOI： 10.1007/978-3-319-31756-4_21 　

    ISSN： 2297-0215

    eISSN： 2297-024X

  6. The Hamiltonians Generating One-Dimensional Discrete-Time Quantum Walks Peer-reviewed

    TATE Tatsuya

    Interdisciplinary Information Sciences　19　(2)　149-156　2013/09/19

    Publisher: Graduate School of Information Sciences, Tohoku University

    DOI： 10.4036/iis.2013.149 　

    ISSN： 1347-6157

  7. An algebraic structure for one-dimensional quantum walks and a new proof of the weak limit theorem Peer-reviewed

    Tatsuya Tate

    Infinite Dimensional Analysis, Quantum Probability and Related Topics　16　(2)　2013/06

    DOI： 10.1142/S0219025713500185 　

    ISSN： 0219-0257

  8. Asymptotic behavior of quantum walks on the line Peer-reviewed

    Toshikazu Sunada, Tatsuya Tate

    Journal of Functional Analysis　262　(6)　2608-2645　2012/03/15

    DOI： 10.1016/j.jfa.2011.12.016 　

    ISSN： 0022-1236

    eISSN： 1096-0783

  9. Asymptotic Euler-Maclaurin formula over lattice polytopes Peer-reviewed

    Tatsuya Tate

    JOURNAL OF FUNCTIONAL ANALYSIS　260　(2)　501-540　2011/01

    DOI： 10.1016/j.jfa.2010.08.011 　

    ISSN： 0022-1236

    eISSN： 1096-0783

  10. A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions Peer-reviewed

    Tatsuya Tate

    ASYMPTOTIC ANALYSIS　67　(1-2)　101-123　2010

    DOI： 10.3233/ASY-2009-0973 　

    ISSN： 0921-7134

  11. Bernstein measures on convex polytopes Peer-reviewed

    Tatsuya Tate

    SPECTRAL ANALYSIS IN GEOMETRY AND NUMBER THEORY　484　295-319　2009

    ISSN： 0271-4132

  12. Spectral Analysis in Geometry and Number Theory - Contemporary Mathematics Peer-reviewed

    M. Kotani, H. Naito, T. Tate

    484　1-342　2009

  13. Asymptotics of matrix integrals and tensor invariants of compact Lie groups Peer-reviewed

    Michael Stolz, Tatsuya Tate

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　136　(6)　2235-2244　2008

    DOI： 10.1090/S0002-9939-08-09039-4 　

    ISSN： 0002-9939

  14. Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers Peer-reviewed

    Tatsuya Tate, Steve Zelditch

    Journal of Functional Analysis　217　(2)　402-447　2004/12/15

    DOI： 10.1016/j.jfa.2004.01.004 　

    ISSN： 0022-1236

  15. Weyl pseudo-differential operator and Wigner transform on the Poincaré disk Peer-reviewed

    Tatsuya Tate

    Annals of Global Analysis and Geometry　22　(1)　29-48　2002

    DOI： 10.1023/A:1016253829938 　

    ISSN： 0232-704X

  16. Some remarks on the off-diagonal asymptotics in quantum ergodicity Peer-reviewed

    Tatsuya Tate

    Asymptotic Analysis　19　(3-4)　289-296　1999/04

    ISSN： 0921-7134

  17. Quantum ergodicity at a finite energy level Peer-reviewed

    Tatsuya Tate

    Journal of the Mathematical Society of Japan　51　(4)　867-885　1999

    DOI： 10.2969/jmsj/05140867 　

    ISSN： 0025-5645

    eISSN： 1881-1167

Show all ︎Show first 5

Misc. 1

  1. A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions (vol 67, pg 101, 2010)

    Tatsuya Tate

    ASYMPTOTIC ANALYSIS　68　(3)　187-188　2010

    DOI： 10.3233/ASY-2010-0990 　

    ISSN： 0921-7134

Research Projects 23

  1. Theory for convergence of discrete surfaces with conformal structures

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Tohoku University

    2023/04/01 - 2027/03/31

  2. Geometric analysis for unitary transition operators

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Tohoku University

    2018/04/01 - 2023/03/31

  3. Discrete surface theory for optimal mass transportation in porous materials

    Kotani Motoko

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)

    Category: Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)

    Institution: Tohoku University

    2017/06/30 - 2022/03/31

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    "Discrete geometric analysis" is a mathematical framework, which enables us to describe and analyze the correlation between microscopic structures and macroscopic properties of various materials. In this project, both of the "forward problems" predicting materials properties from their structure and the "inverse problems" suggesting reasonable structures from desired properties are approached by developing such a framework cooperating with materials scientists. It was shown that mathematical findings on "curved surface" in the world of atoms and molecules work also for predicting and understanding the structures and properties of real materials such as nano carbon. A novel approach for design of the next generation materials was successfully proposed.

  4. The development of discrete geometric analysis

    Sunada Toshikazu

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Meiji University

    2015/04/01 - 2019/03/31

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    In this research, I treated several topics in mathematical crystallography. Especially motivated by the recent development in systematic design of crystal structures, I discussed interesting relationships among seemingly irrelevant subjects; say, standard crystal models, tight frames in the Euclidean space, rational points on Grassmannian, and quadratic Diophantine equations. The central object in this study is what I call crystallographic tight frames, which are considered a generalization of root systems. I also made a remark on the connections with tropical geometry, a relatively new area in mathematics, specifically with combinatorial analogues of the Abel-Jacobi map and Abel's theorem. What is more, I explained how the idea of Riemann sum is linked to other branches of mathematics; for instance, some counting problems in elementary number theory and the theory of quasicrystals, the former having a long history and the latter being an active field still in a state of flux.

  5. Quantum walks from a view point of discrete geometric analysis

    Tate Tatsuya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Tohoku University

    2013/04/01 - 2019/03/31

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    The notion of quantum walks is a quantum counterpart of the notion of random walks and it has many applications to computer sciences. For example, the Grover walk, which is one of typical quantum walks, is used to improve Grover's quantum search algorithm as discussed originally by Ambainis et al. In this research program, we focused on rather qualitative aspects of quantum walks because it would be very important to understand its qualitative aspects to find its further applications and develop its theory further in mathematics. One of importances of random walks are its applications to discrete group theory. Also random walks plays important roles in the theory of crystal lattices. Concerning with these facts, we mainly considered discrete geometric analytical problems for quantum walks in this program and we obtained results on such as algebraic structures of 1-dimensional homogeneous quantum walks and setting up of quantum walks on crystal lattices and their spectral behavior.

  6. Discrete Geometric Analysis for Quantum Spin System

    Kotani Motoko, Obata Nobuaki, Tate Tatsuya, Miyaoka Reiko

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Tohoku University

    2012/04/01 - 2017/03/31

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    We study mathematical framework for Quantum spin system. Physics on topological Insulator and topologically protected surface/edge state is formulated in K-theory. By using the non-commutative geometry, we generalized it to disordered systems. We also develop discrete surface theory to study the relation of microscopic structural data and macroscopic properties and apply it to carbon networks.

  7. Development of the index theorem on foliated manifolds

    Moriyoshi Hitoshi, NATSUME TOSHIKAZU, MAEDA YOSHIAKI, MITSUMATSU YOSHIHIKO, ONO KAORU, MIYAZAKI NAOYA, TAKAKURA TATSURU, TATE TETSUYA

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Nagoya University

    2013/04/01 - 2016/03/31

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    First, we extended the index theorem to fractal sets such as the Cantor set and the Sierpinski gasket. Second, by exploiting the framework of Noncommutative Geometry we generalized the Atiyah-Patodi-Singer index theorem to a Galois covering of compact manifold with boundary, which gives a formula for the pairing between K-group and cyclic cohomology. Third, we clarified the relation of the Dixmier-Douady class and the Godbillon-Vey class, which respectively appears as a characteristic class for Gerbe and foliated circle bundles. It turned out that they are connected via the Cheeger-Chern-Simons invariant. As a byproduct we succeeded to describe the universal central extension of circle diffeomorphism group in terms of the Calabi invariant.

  8. Development of Discrete Geometric Analysis

    Sunada Toshikazu, TATE Tatsuya, HIGUCHI Yusuke, AHARA Kazushi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Meiji University

    2012/04/01 - 2016/03/31

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    Motivated by the recent development in systematic design of crystal structures by both mathematicians and crystallographers, we studied interesting relationships among seemingly irrelevant subjects; say, standard crystal models, tight frames in the Euclidean space, rational points on Grassmannian, and quadratic Diophantine equations. Thus our view is quite a bit different from the traditional one in mathematical crystallography. The central object in this study was what we call crystallographic tight frames, which are, in a loose sense, considered a generalization of root systems. We also studied near quasi crystals pf Poisson's type.

  9. Spectral and scattering theory of Schroedinger equations

    NAKAMURA Shu, ADACHI Tadayoshi, NAKANO Fumihiko, FUJIIE Setsuro, TAMURA Hideo, YAJIMA Kenji, ISOZAKI Hiroshi, IWATSUKA Akira, MINAMI Nariyuki, UEKI Naomasa, NOMURA Yuji, DOI Shin-ichi, TATE Tatsuya, MINE Takuya, ITO Kenichi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: The University of Tokyo

    2009/04/01 - 2014/03/31

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    We have investigated the spectral theory, scattering theory and the structure of solutions for Schrodinger equations, using the functional analysis, microlocal analysis, real analysis and probability theory methods. The research results include results on: 1. Microlocal singularities for solutions of Schrodinger equations, 2. Scattering theory for Schrodinger operators on manifolds, 3. Scattering theory for multi-particles in electric/magnetic fields, 4. Semiclassical analysis for Schrodinger equations, 5. Random Schrodinger equations and charge transport. These results are published as 39 publications (including in press, at present), and several projects are still in progress.

  10. Problems on geometric asymptotic analysis over convex polytopes

    TATE Tatsuya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Category: Grant-in-Aid for Young Scientists (B)

    Institution: Nagoya University

    2009 - 2012

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    A convex polytope is said to be a lattice polytope if each vertex is a lattice point. An asymptotic expansion formula for the Riemann sums of general smooth functions over lattice polytopes was obtained. This formula is regarded as a generalization of the so-called local Euler-Maclaurin formula due to Berline-Vergne. Furthermore, an explicit formula for the third term in the expansion was obtained in case where the polytope is Delzant.

  11. Universality of Random Matrices and Semiclassical Quantum Theory

    NAGAO Taro, TATE Tatsuya, SAITO Keiji

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Nagoya University

    2008 - 2012

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    The eigenvalue correlation functions for the matrix ensemble continuously interpolating between the asymmetric real and symmetric real random matrices were evaluated and the asymptotic behavior in the limit of large matrix dimension was derived. The method of semiclassical diagrammatic expansion was applied to classically chaotic quantum systems and the universal properties were reproduced. The random matrix methods were applied to the theory of complex networks and the eigenvalue densities of the matrices describing the networks were shown to be analytically evaluated in the limit of large average degree.

  12. Development and applications of discrete geometric analysis

    SUNADA Toshikazu, TATE Tatsuya, HIGUCHI Yusuke, AHARA Kazushi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Meiji University

    2009 - 2011

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    The primarily purpose of this project was to provide a mathematical insight into the modern crystallography, a typical practical science that originated in the classification of the observed shapes of crystals. The tools we employed are adopted from algebraic topology, a field in pure mathematics cultivated during the first half of the last century. More specifically the elementary theory of covering spaces and homology is effectively used in the study of 3D networks associated with crystals. Further we formulate a minimum principle for crystals in the framework of discrete geometric analysis, which provides us with the concept of standard realizations, a canonical way to place a given crystal structure in space so as to produce the most symmetric microscopic shape. In spite of its pure-mathematical nature, this concept combined with homology theory turns out to fit with a systematic design and enumeration of crystal structures, an area of considerable scientific interest for many years. Meanwhile, standard realizations show up in asymptotic behaviors of random walks on topological crystals, the abstraction of crystal structures, and are closely related to a discrete analogue of Abel-Jacobi maps in algebraic geometry.

  13. Study of Geometry of a discrete space through randomness

    KOTANI Motoko, SHIOYA Takashi, ARAI Hitoshi, KUMAGAI Takashi, IZEKI Hiroyasu, NAYATANI Shin, TATE Tatsuya, ISHIWATA Satoshi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Tohoku University

    2008 - 2011

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    Aim of this research proposal is to develop new methods to study geometric objects with singularities, or discrete spaces, which are not accessible by traditional differential geometrical technics. Our idea is to apply probability theory to those geometric objects. Some results are obtained and published from international journals.

  14. Integrated study of probability and representation theory towards harmonic analysis on huge groups

    HORA Akihito, OKADA Soichi, TATE Tatsuya, HIRAI Takeshi, OBATA Nobuaki, SHIMOMURA Hiroaki, KAWAZOE Takeshi, YAMADA Hirofumi, ARAI Hitoshi, NISHIYAMA Kyo, ISHI Hideyuki, MATSUMOTO Sho, INAHAMA Yuzuru

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Nagoya University

    2007 - 2010

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    Towards developing harmonic analysis on huge groups, we did integrated studies of probability theory and group representations. Harmonic analysis is a discipline which seeks deep mathematical structures by looking at symmetries of the objects and develops analysis relying on them. In this study, we are led to huge groups describing the symmetries because our objects are so big as to have an infinite degree of freedom. Main results among the ones we obtained are (i) classification and explicit formulas of the characters which are building blocks of harmonic analysis, and (ii) a series of theorems which construct a bridge between asymptotic behavior of representations of groups and probabilistic limit theorems.

  15. Development from Poisson Geometry to Noncommutative Differential Geometry via Integrating of Geometry and Physics

    MAEDA Yoshiaki, MORIYOSHI Hitoshi, SHIMOMURA Shun, ISHII Ippei, KAMETANI Yukio, MIYAZAKI Takuya, TAMURA Yozo, IKEDA Kaoru, KURIHARA Masato, OHTA Katsuhiro, ATSUJI Atsushi, TOSE Nobuyuki, OMORI Hideki, FUKAYA Kenji, TATE Tatsuya, WATAMURA Satoshi, KONNO Hiroshi, KOHNO Toshitake, GUEST Martin, MIKAMI Kentaro, ONO Kaoru, YAMAGUCHI Keizo, MITSUMATSU Yoshihiko, MIYAOKA Reiko, FUTAKI Akitp, MATSUO Yuyaka, MIZUTANI Tadayoshi, OHTA Hiroshi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Keio University

    2006 - 2009

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    We obtain many results through the cooperation researches between noncommutative geometry and theoretical physics, namely for proposals on noncommutative gauge theory, p-adic Iwasawa theory, algebraic integrability for quantum Toda lattice, Bernstein measure, string theory and generalized complex structure, quantum cohomology and Frobenius manifolds, symplectic topology and contact topology. The results has been presented by the international conferences and published as monographs in 「Noncommutative Geometry and Physics」, 「Advaned Studies in Pure Mathematics 55」, 「Translations of Mathematical Monographs, 237」 and also published by individual researchers.

  16. Problems in geometric asymptotic analysis from a viewpoint of quantum statistics

    TATE Tatsuya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Category: Grant-in-Aid for Young Scientists (B)

    Institution: Nagoya University

    2006 - 2008

  17. Statistical Laws in Geometry

    KOBAYASHI Ryoichi, TATE Tatsuya, MABUCHI Toshiki, ENOKI Ichiro, FUTAKI Akito, WENG Lin, YAMANOI Katsutoshi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Nagoya University

    2005 - 2008

  18. 非可換解析を基礎とする非可換微分幾何学の構築と超弦理論への展開

    前田 吉昭, 森吉 仁志, 亀谷 幸生, 楯 辰也, 大森 英樹, 綿村 哲, 宮崎 琢也, 鈴木 由紀

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業

    Category: 萌芽研究

    Institution: 慶應義塾大学

    2003 - 2005

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    本研究を主導する研究代表者は、収束する変形量子化の構成から新しい幾何学的概念を展開した。特に、ジャーブ理論との深い関係が解明され、これを詳しく調べた。第二には、Lie環を基本とする1次ポアソン構造、2次ポアソン構造に対する変形量子化を応用できるような具体的体系を整え、その応用へ発展させている。保形形式で注目されているCohen-Rankin積は、不変量子化として考えられているが、この研究を研究分担者である宮崎(琢)の協力を求めて解明をはじめている。森吉は、トポロジーの立場から非可換多様体の不変量の構成、特に指数定理の研究を行なう。特に、非可換トーラス上のDirac作用素による指数定理の構成をめざし、佐々木多様体の指数定理を得ている。亀谷は4次元多様体の不変量として研究が進んでいるザイバーグ・ウィッテン不変量の研究、特に11/8予想についての研究を行って、成果を挙げている。これらの研究の展開のために、国外外研究者との討議等を行ない、海外研究者と研究交流のために、直接海外に赴き、以下の研究者と共同研究や研究討論を行ってきた。 Alan Weinstein : University of California Berkley, Prof., Poisson Geometry Albert Cattaneo : ETH, Zurich, Prof., Theoretical physics Alain Connes : IHES, Paris, Prof., Noncommutative geometry その成果をまとめるために、A.WeinsteinとA.Cattaneoが来年来日する。

  19. 楕円型作用素、並びに群作用に対する固有関数とその値分布の漸近挙動の研究

    楯 辰哉

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業 若手研究(B)

    Category: 若手研究(B)

    2003 - 2005

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    1.固有関数の分布関数の漸近挙動を理解するために研究を進めた。この問題は、一般的な解決は困難であるが、球面などの対称性の高い空間のラプラス作用素の標準的な固有関数や調和振動子などの良く知られた作用素の固有関数の分布関数について、その漸近挙動の様子を知ることを目標とした。球面の固有関数、つまり球面調和関数の分布関数を調べるために、固有関数を射影子の積分核の表示を用いて積分表示し、漸折挙動を調べるという当初の方針は妥当なものであろうと思われる。しかし積分の特異点の周りでの挙動を解析しきれず、思わしい結果を得ることが出来なかった。今後の課題として残されている。 2.前年度にZelditch氏との共同研究で得た結果を、前年度に引き続き報告した。これは、コンパクト群の既約表現のテンソル積表現内のウェイトの重複度の漸近挙動に関する結果であった。この研究においてテンソル積の次数は粒子の数に相当するが、これを一般の楕円型作用素を用いて考えるとき、ボゾンガスの状態数が自然と現れる。ボゾンガスの状態数は解析数論でしばしば現れる分割数のスペクトル論的な類似物でもある。本年度は分割数の漸近挙動についてのMeinardusの定理の手法を用いて、ボゾンガスの状態数の漸近挙動を書き下すことに成功した。現存論文を作成中である。また、この内容は幾つかの研究会で発表したが、特にESI Educational workshop on discrete probability (3/21, ESI, Vienna)で講演した。さらに日本数学会年会(3月28日)で特別講演を行う予定である。 3.コンパクト群のテンソル積表現のウェイトの重複度の漸近挙動の問題に関連して、ランダム行列理論に自然に現れるモーメント積分の、テンソル積のパラメータについての漸近挙動について、現在Stolz氏と共同研究中である。

  20. Study of Non-Commutative Geometry focusing on the Index theorem, and low-dimensional maniflod theory,

    MORIYOSHI Hitoshi, MAEDA Yoshiaki, KAMETANI Yukio, TATE Tatsuya, NATSUME Toshikazu, ONO Kaoru

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Keio University

    2002 - 2004

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    In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the basis of Low-dimensional Topology. Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. Let Γ be a discrete group and σ a 2-cocycle of Γ with values in U(1). We then twist the product on the group algebra C(Γ) in the following way : U_gU_h = σ(g,h)U_<gh> where U_g, U_h are the formal unitary elements corresponding to g, h ∈ Γ. Due to the cocycle condition we obtain an associative product on C(Γ). With respect to the operator norm on L^2(Γ) we take the C^*-closure of C(Γ) with product above. It is called group C^*-algebra twisted by a 2-cocycle σ and denoted by C^* (Γ,σ). There exists a Dirac operator whose index belongs to the K-group of C^* (Γ,σ). Let us denote the Dirac operator by D^▽. We then obtain the Index formula which express the trace τ(IndD^▽) of the index IndD^▽ in trems of characteristic classes A^^^(M/Γ) and a curvature R of the associated line bundle. As a corollary of the formula. we can prove the following result : Suppose that a closed symplectic manifold M is aspherical, then M does not admit a Riemanninan metric of positive scalar curvature. This yields a prtial solution to the Gromov-Lawson conjecture.

  21. Study of Poisson geometry and its application

    MAEDA Yoshiaki, ICHII Ippei, TANI Atsushi, KIKUCHI Norio, KAMETANI Yokio, MORIYOSHI Hitoshi

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Keio University

    2000 - 2002

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    Poisson Geometry, which is the extended notion of symplectic geometry, is now developing quite recently. However, this research fields is not common in Japan, while this research field is much developing in Europe and the United State. In our Study, we have a mission to develop this research fields in Japan, and to coorporate with the researchers who are working on this fields in Europe and the Unites States. The first our development is to study the convergence problems on deformation quantizations. For the case of formal deformations, M. Kontsevich has given a pretty result. It is important problem to study the convergence for the deformation quantizations. Through our research, we found the totally different feature for the convergence of the deformation quantizations, and propose a new geometric objects on gerbes. We will extend our research on the convergence of the deformation quantization as a future task. By this grant, we have two major international symposia in 2001 and 2002, which has noncommutative geometry and D-brane as main Topics. We could have many visitors from abroad and also from domestic research institutes. We have published a proceedings for our research developments. Maeda has been invited to the Poisson 2002 international conference at Lisbon and gave a talk on this problem, which was very interesting for the participants. We have also visited various meetings in Japan and outside of Japan to make strong activities. As a results, we could have a international research groups on noncommutative geometry, which is able to develop the research.

  22. ポアンカレ円板上のワイル型擬微分作用素とウィグナー変換

    楯 辰哉

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業

    Category: 奨励研究(A)

    Institution: 慶應義塾大学

    2001 - 2001

  23. Non-Commutative Geometry and the Spectral Flow Index Theorem

    MORIYOSHI Hitoshi, TATE Tatsuya, KAMETANI Yukio, MAEDA Yoshiaki, ONO Kaoru, NATSUME Toshikazu

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Keio University

    2000 - 2001

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    In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the bas1S of Low dimensional Topology. The objective of the project are the followings: 1. Establish the elaborated Index Theorem in the framework of Non-commutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invariants and the spectral flow; 2. Study the relationship between the elaborated Index Theorem and secondary characteristic classes including the Maslov class. Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. In July 2001, we are invited to give a talk with the title 'Analytic K-theory and the index theorem' at Topology Symposium of Japan, Akita. We are also working on the project are presented with the title 'Eta invariants, the Godbillon-Vey classes and the index theorem' in March 2001 at Workshop on Non-commutative Geometry and String Theory, Keio University. We also developed the research on Non-commutative Hopf invariants, and presented some results at Nagoya Technology Institute with the title 'Spectral flow, the Teoplitz index and the Hopf invariant' in March 2001, and with the title 'A index theorem of vector field and the Hopf invariant' at Kagoshima University in February 2002.

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