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YAMADA Yasuhiko
Graduate School of Science / Division of Mathematics
Professor

Researcher basic information

■ Research Keyword
  • integrable systems
  • special functions
  • mathematical physics
■ Research Areas
  • Natural sciences / Mathematical analysis

Research activity information

■ Award
  • 2017 Journal of Physics A, Journal of Physics A Highlights of 2017 collection, パンルヴェ方程式の幾何学的研究
    NOUMI MASATOSHI, YAMADA YASUHIKO
    Official journal

■ Paper
  • Hidetoshi Awata, Koji Hasegawa, Hiroaki Kanno, Ryo Ohkawa, Shamil Shakirov, Jun'ichi Shiraishi, Yasuhiko Yamada
    We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlevé VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions $\bigl(\mathcal{F}^{(1)},\mathcal{F}^{(2)}\bigr)$, which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.
    SIGMA (Symmetry, Integrability and Geometry: Methods and Application), Nov. 2023, Symmetry, Integrability and Geometry: Methods and Applications, 19, 47pp
    [Refereed]
    Scientific journal

  • Sanefumi Moriyama, Yasuhiko Yamada
    SIGMA (Symmetry, Integrability and Geometry: Methods and Application), Aug. 2021, Symmetry, Integrability and Geometry: Methods and Applications, 17, pp.24, English
    [Refereed]
    Scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Masatoshi Noumi, Simon Ruijsenaars, Yasuhiko Yamada
    SIGMA (Symmetry, Integrability and Geometry: Methods and Application), Jul. 2020, Symmetry, Integrability and Geometry: Methods and Applications, 16, pp.16, English
    [Refereed]
    Scientific journal

  • Hidehito Nagao, Yasuhiko Yamada
    Institute of Physics Publishing, Mar. 2018, Journal of Physics A: Mathematical and Theoretical, 51(13) (13), 135204(19pp), English
    [Refereed]
    Scientific journal

  • Hidehito Nagao, Yasuhiko Yamada
    KOBE UNIV, 2018, Funkcialaj Ekvacioj, 61(1) (1), 109 - 133, English
    [Refereed]
    Scientific journal

  • Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada
    Feb. 2017, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 50(7) (7), 073001 (164pp), English
    [Refereed]
    Scientific journal

  • Yasuhiko Yamada
    2017, SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 13(69) (69), 8pp, English
    [Refereed]
    Scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Christopher Michael Ormerod, Yasuhiko Yamada
    2015, SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 11(56) (56), 36 pages, English
    [Refereed]
    Scientific journal

  • A simple expression for discrete Painlevé equations
    YAMADA Yasuhiko
    Dec. 2014, RIMS講究録別冊, B47, 087 - 095, English
    [Refereed]
    Scientific journal

  • Hajime Nagoya, Yasuhiko Yamada
    Feb. 2014, ANNALES HENRI POINCARE, 15(2) (2), 313 - 344, English
    [Refereed]
    Scientific journal

  • Masatoshi Noumi, Satoshi Tsujimoto, Yasuhiko Yamada
    Springer New York LLC, 2013, Springer Proceedings in Mathematics and Statistics, 40(40) (40), 463 - 482, English
    [Refereed]
    International conference proceedings

  • Atsuo Kuniba, Masato Okado, Yasuhiko Yamada
    2013, SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 9(049) (049), 23pages, English
    [Refereed]
    Scientific journal

  • Localization with a surface operator, irregular conformal blocks and open topological string
    Hidetoshi Awata, Hiroyuki Fuji, Hiroaki Kanno, Masahide Manabe, Yasuhiko Yamada
    Jun. 2012, ADVANCES IN THEORETICAL AND MATHEMATICAL PHYSICS, 16(3) (3), 725 - 804, English
    [Refereed]
    Scientific journal

  • Yasuhiko Yamada
    2011, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (17) (17), 3823 - 3838, English
    [Refereed]
    Scientific journal

  • GENERALIZED ENERGIES AND INTEGRABLE D-n((1)) CELLULAR AUTOMATON
    Atsuo Kuniba, Reiho Sakamoto, Yasuhiko Yamada
    2011, NEW TRENDS IN QUANTUM INTEGRABLE SYSTEMS, 221 - 242, English
    [Refereed]
    International conference proceedings

  • A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
    YAMADA Yasuhiko
    Jan. 2011, J. Phys. A, Math. Theor., Vol 44. 055403, English
    [Refereed]
    Scientific journal

  • Five-Dimensional AGT Relation and the Deformed beta-Ensemble
    Hidetoshi Awata, Yasuhiko Yamada
    Aug. 2010, PROGRESS OF THEORETICAL PHYSICS, 124(2) (2), 227 - 262, English
    [Refereed]
    Scientific journal

  • Hidetoshi Awata, Yasuhiko Yamada
    Jan. 2010, JOURNAL OF HIGH ENERGY PHYSICS, Vol 01. 125(1) (1), English
    [Refereed]
    Scientific journal

  • Yasuhiko Yamada
    Apr. 2009, Funkcialaj Ekvacioj, 52(1) (1), 83 - 92, English
    [Refereed]
    Scientific journal

  • Yasuhiko Yamada
    2009, SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 5, English
    [Refereed]
    Scientific journal

  • Atsuo Kuniba, Reiho Sakamoto, Yasuhiko Yamada
    Dec. 2007, NUCLEAR PHYSICS B, 786(3) (3), 207 - 266, English
    [Refereed]
    Scientific journal

  • Symmetry and holomorphy of Painlevé type systems
    YAMADA Yasuhiko, Y. Sasano
    Res. Inst. Math. Sci., Mar. 2007, Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, RIMS Kokyuroku Bessatsu, B2, pp. 215-225, 73 - 88, English
    [Refereed]
    Scientific journal

  • Shun'ya Mizoguchi, Kenji Mohri, Yasuhiko Yamada
    May 2006, CLASSICAL AND QUANTUM GRAVITY, 23(9) (9), 3181 - 3193, English
    [Refereed]
    Scientific journal

  • A Kuniba, M Okado, R Sakamoto, T Takagi, Y Yamada
    Apr. 2006, NUCLEAR PHYSICS B, 740(3) (3), 299 - 327, English
    [Refereed]
    Scientific journal

  • Point configurations, Cremona transformations and the elliptic difference Painlevé equation
    NOUMI Masatoshi, KAJIWARA Kenji, MASUDA Tetsu, OHTA Yasuhiro, YAMADA Yasuhiko
    2006, Séminaires et Congrès, Vol 14, pp. 169 - 198, English
    [Refereed]
    Scientific journal

  • Construction of hypergeometric solutions to the q-Painleve equations
    K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada
    May 2005, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, vol 24, pp 1439～1463(24) (24), 1439 - 1463, English
    Scientific journal

  • $q$-Painlevé方程式の超幾何解 (可積分系数理の展望と応用)
    梶原 健司, 増田 哲, 野海 正俊, 太田 泰広, 山田 泰彦
    京都大学, Apr. 2005, 数理解析研究所講究録, 1422, 77 - 98, Japanese

  • Cubic Pencils and Painleve Hamiltonians
    K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada
    Apr. 2005, FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 48(1) (1), 147 - 160, English
    Scientific journal

  • Box-ball system with reflecting end
    A Kuniba, M Okado, Y Yamada
    2005, JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 12(4) (4), 475 - 507, English
    Scientific journal

  • Cremona変換と楕円差分Painleve方程式 : 高次元的な枠組みへの試論 (可積分系理論とその周辺 : 課題と展望を探る)
    梶原 健司, 増田 哲, 野海 正俊, 太田 泰広, 山田 泰彦
    京都大学, Oct. 2004, 数理解析研究所講究録, 1400, 197 - 263, Japanese

  • A Kuniba, M Okado, T Takagi, Y Yamada
    Mar. 2004, COMMUNICATIONS IN MATHEMATICAL PHYSICS, 245(3) (3), 491 - 517, English
    [Refereed]
    Scientific journal

  • Hypergeometric solutions to the q-painleve equations
    K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada
    2004, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (47) (47), 2497 - 2521, English
    [Refereed]
    Scientific journal

  • Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions.
    NOUMI Masatoshi, YAMADA, Y
    Math. Soc. Japan, 2004, Advanced Studies in Pure Mathematics., 40, 371-442., English
    [Refereed]
    Scientific journal

  • E-10(9) solution to the elliptic Painleve equation
    K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada
    May 2003, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 36(17) (17), L263 - L272, English
    Scientific journal

  • Geometric crystal and tropical R for D-n((1))
    A Kuniba, M Okado, T Takagi, Y Yamada
    2003, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2565-2620(48) (48), 2565 - 2620, English
    Scientific journal

  • Difference L operators related to q-characters
    A Kuniba, M Okado, J Suzuki, Y Yamada
    Feb. 2002, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 35(6) (6), 1415 - 1435, English
    [Refereed]
    Scientific journal

  • Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada
    A system of q-Painlev\'e type equations with multi-time variables t_1,...,t_M
    is obtained as a similarity reduction of the N-reduced q-KP hierarchy. This
    system has affine Weyl group symmetry of type A^{(1)}_{M-1} \times
    A^{(1)}_{N-1}. Its rational solutions are constructed in terms of q-Schur
    functions.
    Dec. 2001

  • Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada
    We give a birational realization of affine Weyl group of type $A^{(1)}_{m-1}
    \times A^{(1)}_{n-1}$. We apply this representation to construct some discrete
    integrable systems and discrete Painlev\'e equations. Our construction has a
    combinatorial counterpart through the ultra-discretization procedure.
    Jun. 2001

  • 野海 正俊, 山田 泰彦
    社団法人日本数学会, Jan. 2001, 数学, 53(1) (1), 62 - 75, Japanese

  • Tableau representation for Macdonald's ninth variation of Schur functions
    J Nakagawa, M Noumi, M Shirakawa, Y Yamada
    2001, PHYSICS AND COMBINATORICS, 180 - 195, English
    [Refereed]
    International conference proceedings

  • Birational Weyl group action arising from a nilpotent Poisson algebra
    M Noumi, Y Yamada
    2001, PHYSICS AND COMBINATORICS 1999, 287 - 319, English
    [Refereed]
    International conference proceedings

  • Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada
    A q-difference analogue of the fourth Painlev\'e equation is proposed. Its
    symmetry structure and some particular solutions are investigated.
    Dec. 2000

  • Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada
    Determinant formulas for the general solutions of the Toda and discrete Toda
    equations are presented. Application to the $\tau$ functions for the Painlev\'e
    equations is also discussed.
    Aug. 1999

  • Symmetries in the fourth Painleve equation and Okamoto polynomials
    M Noumi, Y Yamada
    Mar. 1999, NAGOYA MATHEMATICAL JOURNAL, 153, 53 - 86, English
    [Refereed]
    Scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Goro Hatayama, Anatol N. Kirillov, Atsuo Kuniba, Masato Okado, Taichiro Takagi, Yasuhiko Yamada
    Elsevier, Dec. 1998, Nuclear Physics B, 536(3) (3), 575 - 616, English
    Scientific journal

  • Yamada Yasuhiko
    約100年前にPainleveにより発見されたPainleve方程式は, 理論的にも応用面でも極めて重要なものであるが, その複雑さゆえに扱いはなかなか大変である. ところが, これらの方程式のもつ大きな対称性を積極的に用いると, その扱いが簡単化され, また, 方程式の一般化も可能となる. 可積分系との関係で重要なτ関数の理論と合わせて, 現在進行中の諸結果について簡潔に紹介したい.
    The Physical Society of Japan (JPS), Dec. 1998, Butsuri, 53(12) (12), 926 - 928, Japanese

  • Affine Weyl groups, discrete dynamical systems and Painleve equations
    M Noumi, Y Yamada
    Dec. 1998, COMMUNICATIONS IN MATHEMATICAL PHYSICS, 199(2) (2), 281 - 295, English
    [Refereed]
    Scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Umemura polynomials for the Painleve V equation
    M Noumi, Y Yamada
    Oct. 1998, PHYSICS LETTERS A, 247(1-2) (1-2), 65 - 69, English
    [Refereed]
    Scientific journal

  • Atsushi Nakayashiki, Yasuhiko Yamada
    1997, Selecta Math. New Ser., 3, 547 - 599, English
    [Refereed]
    Scientific journal

  • Atsushi Nakayashiki, Yasuhiko Yamada
    Springer New York, May 1996, Communications in Mathematical Physics, 178(1) (1), 179 - 200, English
    Scientific journal

  • On spinon character formulas
    Atsushi Nakayashiki, Yasuhiko Yamada
    1996, Frontiers in quantum field theory, World Scientific, 367 - 371, English
    [Refereed]
    International conference proceedings

  • Atsushi Nakayashiki, Yasuhiko Yamada
    1996, Int. J. of Mod. Phys. A, 11(2) (2), 395 - 408, English
    [Refereed]
    Scientific journal

  • Katsuhisa Mimachi, Yasuhiko Yamada
    Springer-Verlag, Dec. 1995, Communications in Mathematical Physics, 174(2) (2), 447 - 455, English
    [Refereed]
    Scientific journal

  • Hidetoshi Awata, Akihiro Tsuchiya, Yasuhiko Yamada
    Nov. 1991, Nuclear Physics, Section B, 365(3) (3), 680 - 696, English
    Scientific journal

  • Akihiro Tsuchiya, Kenji Ueno, Yasuhiko Yamada
    Mathematical Society of Japan, 1989, Advanced Studies in Pure Mathematics, 19, 459

■ MISC
  • Symmetry of factorized Lax matrices (Mathematical structures of integrable systems, its deepening and expansion)
    PARK Kanam, YAMADA Yasuhiko
    We study bi-rational Weyl group actions on certain matrix Lax operators given in factorized form. These actions generalize the W(A[m-1] x A[n-1]) symmetry considered before by Kajiwara et.al. Our study is motivated by two recent developments: one is on discrete isomonodromic systems and the other is on the bi-rational Weyl group actions arising from the quiver mutations. We also discuss further generalizations using the results by G. Frieden on the geometric crystals.
    Research Institute for Mathematical Sciences, Kyoto University, Aug. 2021, RIMS Kokyuroku Bessatsu, (87) (87), 135 - 147, Japanese

  • 量子座標環,PBW基底と3次元反射方程式 (組合せ論的表現論とその周辺)
    国場 敦夫, 尾角 正人, 山田 泰彦
    京都大学, Dec. 2013, 数理解析研究所講究録, 1870, 106 - 114, Japanese

  • 楕円差分Painleve方程式のLax形式 (可積分系数理とその応用)
    山田 泰彦
    京都大学, Jul. 2010, 数理解析研究所講究録, 1700, 179 - 201, Japanese

  • 27aXE-1 A new interpretation of the combinatorial mapping associated with Bethe ansatz
    Kuniba A., Okado M., Sakamoto R., Takagi T., Yamada Y.
    The Physical Society of Japan (JPS), 04 Mar. 2006, Meeting abstracts of the Physical Society of Japan, 61(1) (1), 253 - 253, Japanese

  • Tropical $R$ : 例と応用 (可積分系の組合せ論的側面)
    国場 敦夫, 尾角 正人, 高木 太一郎, 山田 泰彦
    京都大学, Apr. 2005, 数理解析研究所講究録, 1429, 57 - 69, Japanese

  • クリスタルから見た箱玉系 (可積分系数理の展望と応用)
    国場 敦夫, 尾角 正人, 山田 泰彦
    京都大学, Apr. 2005, 数理解析研究所講究録, 1422, 44 - 55, Japanese

  • KAJIWARA K, MASUDA Tetsu, NOUMI Masatoshi, OHTA Yasuhiro, YAMADA Yasuhiko
    神戸大学理学部数学科, 2005, Rokko Lectures in Mathematics 18 Elliptic integrable systems, 18, 43 - 48, English
    Introduction scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Bilinearization of the tropical R
    KUNIBA Atsuo, OKADO Masato, TAKAGI Taichiro, YAMADA Yasuhiko
    The Physical Society of Japan (JPS), 15 Aug. 2003, Meeting abstracts of the Physical Society of Japan, 58(2) (2), 259 - 259, Japanese

  • Vertex operators and partition functions for the box-ball system II
    KUNIBA Atsuo, OKADO Masato, TAKAGI Taichiro, YAMADA Yasuhiko
    The Physical Society of Japan (JPS), 15 Aug. 2003, Meeting abstracts of the Physical Society of Japan, 58(2) (2), 259 - 259, Japanese

  • Vertex operators and partition functions for the box-ball system
    KUNIBA Atsno, OKADO Masato, TAKAGI Taichiro, YAMADA Yasuhiko
    The Physical Society of Japan (JPS), 06 Mar. 2003, Meeting abstracts of the Physical Society of Japan, 58(1) (1), 301 - 301, Japanese

  • 箱玉系の頂点作用素と分配関数 (可積分系研究の新展開 : 連続・離散・超離散)
    国場 敦夫, 尾角 正人, 高木 太一郎, 山田 泰彦
    京都大学, Feb. 2003, 数理解析研究所講究録, 1302, 91 - 107, Japanese

  • Geometric crystals and tropical R
    KUNIBA Atsuo, OKADO Masato, TAKAGI Taichiro, YAMADA Yasuhiko
    The Physical Society of Japan (JPS), 13 Aug. 2002, Meeting abstracts of the Physical Society of Japan, 57(2) (2), 275 - 275, Japanese

  • S Mizoguchi, Y Yamada
    Jun. 2002, PHYSICS LETTERS B, 537(1-2) (1-2), 130 - 140, English

  • K Fukuda, Y Yamada, M Okado
    Apr. 2000, INTERNATIONAL JOURNAL OF MODERN PHYSICS A, 15(9) (9), 1379 - 1392, English

  • Painleve型hierarchyのaffine Weyl群対称性(Painleve系, 超幾何系, 漸近解析)
    野海 正俊, 山田 泰彦
    京都大学, Feb. 2000, 数理解析研究所講究録, 1133, 117 - 123, Japanese

  • 30p-P-1 Symmetry of the Painleve equations and discrete integrable systems
    Yamada Y.
    The Physical Society of Japan (JPS), 15 Mar. 1999, Meeting abstracts of the Physical Society of Japan, 54(1) (1), 773 - 773, Japanese

  • Combinatorial completeness of the Bethe's state
    HATAYAMA G., KUNIBA A., OKADO M., TAKAGI T., YAMADA Y.
    The Physical Society of Japan (JPS), 05 Sep. 1998, Meeting abstracts of the Physical Society of Japan, 53(2) (2), 784 - 784, Japanese

  • 6p-YJ-4 the Caluculation of the 1-dim Configuration Sum
    Hatayama G, Kirillov A.N., Kuniba A, Okado M, Yamada Y
    The Physical Society of Japan (JPS), 16 Sep. 1997, Meeting abstracts of the Physical Society of Japan, 52(2) (2), 761 - 761, Japanese

  • Kostka Polynomials and Crystals
    Yamada Yasuhiko
    Kyoto University, Aug. 1996, RIMS Kokyuroku, 962, 86 - 96, Japanese

  • Singular vectors of Virasoro algebra in terms of Jack symmetric polynomials(Various aspects of hypergeometric functions)
    Mimachi Katsuhisa, Yamada Yasuhiko
    Kyoto University, Aug. 1995, RIMS Kokyuroku, 919, 68 - 78, Japanese

■ Books And Other Publications
  • Padé Methods for Painlevé Equations
    Nagao Hidehito, Yamada Yasuhiko
    Joint work, Springer Briefs in Mathematical Physics, Sep. 2021, ISBN: 9789811629983

  • 「共形場理論入門」
    YAMADA Yasuhiko
    Single work, 培風館, 2006, Japanese
    General book

■ Lectures, oral presentations, etc.
  • Quantum q-P_{VI}, q-KZ equation and q-AGT relation
    Yasuhiko Yamada
    Elliptic Integrable Systems, Representation Theory and Hypergeometric Functions (MSJ-SI 2023), Aug. 2023, English
    [Invited]
    Invited oral presentation

  • Quantum representation of affine Weyl groups and mirror symmetry
    Yasuhiko Yamada
    Combinatorial Representation Theory and Connections with Related Fields, Oct. 2021, Japanese
    [Invited]
    Invited oral presentation

  • Quantum representation of Weyl group W(E^{(1)}_8)
    Yasuhiko Yamada
    Web-seminar on Painlev\'e Equations and related topics, May 2021, English
    [Invited]
    Invited oral presentation

  • On q-Garnier systems
    Yasuhiko YAMADA
    SIDE-13, Nov. 2018, English, Fukuoka Japan, International conference
    [Invited]
    Invited oral presentation

  • Geometry of isomonodromy deformations
    Yasuhiko YAMADA
    MS Seminar, Sep. 2018, English, IPMU (Kashiwa Japan), International conference
    [Invited]
    Invited oral presentation

  • Comments on q-Garnier systems
    Yasuhiko YAMADA
    Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, Apr. 2018, English, TSIMF Sanya, China, International conference
    [Invited]
    Invited oral presentation

  • Theory and applications of the elliptic Painlevé equation
    Yasuhiko YAMADA
    Partition Functions and Automorphic Forms, Feb. 2018, English, JINR (Dubna, Russia), International conference
    [Invited]
    Invited oral presentation

  • ゲージ理論とガルニエ系
    Yasuhiko YAMADA
    駒場研究会「弦・場・素粒子」, Nov. 2017, Japanese, 東京大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometric aspects of discrete Painlevé equations
    Yasuhiko YAMADA
    Exceptional and ubiquitous Painlevé equations for Physics, Aug. 2017, English, LPT-ENS (Paris, France), International conference
    [Invited]
    Invited oral presentation

  • パンルヴェ方程式の幾何と対称性
    Yasuhiko YAMADA
    Exceptional Groups as Symmetries of Nature '17, Jul. 2017, Japanese, KEK (つくば市), Domestic conference
    [Invited]
    Invited oral presentation

  • Geometric formulation of discrete Painlevé equations
    Yasuhiko YAMADA
    Geometric Correspondences of Gauge Theories, Jul. 2017, English, SISSA (Trieste, Italy), International conference
    [Invited]
    Invited oral presentation

  • q-差分ガルニエ系について, q-差分ガルニエ系からq 差分パンルヴェ系への簡約について
    Yamada Yasuhiko, NAGAO Hidehito
    日本数学会無限可積分系セッション一般講, Mar. 2017, Japanese, Tokyo Metroporitan University, Domestic conference
    Oral presentation

  • q-Garnier system and its autonomous limit
    Yamada Yasuhiko
    Workshop ”Elliptic Hypergeometric Functionsin Combinatorics, Integrable Systems and Physics”, Mar. 2017, English, Osaka City University, International conference
    [Invited]
    Invited oral presentation

  • Isomonodromy equations and gauge theories
    Yamada Yasuhiko
    Workshop ”Progress in Quantum Field Theoryand String Theory II”, Mar. 2017, English, Osaka City University, International conference
    [Invited]
    Invited oral presentation

  • Dual Lax pairs for discrete isomonodromy equations
    Yamada Yasuhiko
    Workshop ”Progress in Quantum FieldTheory and String Theory II”, Mar. 2017, English, Osaka City University, International conference
    [Invited]
    Invited oral presentation

  • A geometric formulation of the q-Garnier system
    Yamada Yasuhiko
    Workshop ”Geometry, Analysis and MathematicalPhysics”, Feb. 2017, English, Kyoto University, International conference
    [Invited]
    Invited oral presentation

  • On the q-Garnier system
    Yamada Yasuhiko
    Workshop on Integrable Systems, Dec. 2016, English, University of Sydney, International conference
    [Invited]
    Invited oral presentation

  • On q-Garnier systems
    Yamada Yasuhiko
    Workshop ”Conformal Field Theory, Isomonodromic tau-functions and Painlevé equations”, Nov. 2016, English, Kobe University, International conference
    [Invited]
    Invited oral presentation

  • q-ガルニエ系の種々のラックス形式について
    Yamada Yasuhiko
    RIMS 研究集会「可積分系数理の現状と展望」, Sep. 2016, Japanese, 京都大学益川ホール, Domestic conference
    [Invited]
    Invited oral presentation

  • パデ法とq差分ガルニエ系
    YAMADA Yasuhiko
    日本数学会無限可積分系セッション一般講, Mar. 2016, Japanese, 筑波大学, Domestic conference
    Oral presentation

  • q-ガルニエ系と超楕円QRT系
    YAMADA Yasuhiko
    有理函数近似が繋ぐ可積分系・直交多項式・パンルヴェ方程式, Jan. 2016, Japanese, 一橋大学, Domestic conference
    [Invited]
    Invited oral presentation

  • q-Garnier系とその自励化
    YAMADA Yasuhiko
    超幾何研究会2016, Jan. 2016, Japanese, 神戸大学, Domestic conference
    Oral presentation

  • モノドロミー保存変形と代数曲線
    YAMADA Yasuhiko
    琉球大数学科談話会, Nov. 2015, Japanese, 琉球大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometric introduction to discrete Painlevé equations
    YAMADA Yasuhiko
    Needs 2015, May 2015, English, Santa Margherita di Pula (Italy), International conference
    [Invited]
    Invited oral presentation

  • Conserved curves for autonomous (ultra-)discrete Painlevé equations
    YAMADA Yasuhiko
    The 9th IMACS Conference, Waves 2015, Apr. 2015, English, Georgia, Athens (USA), International conference
    [Invited]
    Invited oral presentation

  • SW curve and its quantization
    YAMADA Yasuhiko
    ミニワークショップ「数学・物理における可積分性の諸相」, Mar. 2015, Japanese, 大阪市立大学 理学部, Domestic conference
    [Invited]
    Invited oral presentation

  • Quantum curves associated with quantum Painlevé equations
    YAMADA Yasuhiko
    Workshop "Curves, Moduli and Integrable Systems", Feb. 2015, English, Tsuda College, International conference
    [Invited]
    Invited oral presentation

  • モノドロミー保存変形の幾何学と量子化
    YAMADA Yasuhiko
    大岡山談話会, Dec. 2014, Japanese, 東京工業大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometry of Painlevé equations
    YAMADA Yasuhiko
    Integrable Systems in Newcastle, Department of Mathematics and Information Sciences, Sep. 2014, English, Northumbria University, International conference
    [Invited]
    Invited oral presentation

  • Quantum Lax pairs for Painlevé systems and their solutionｓ
    YAMADA yasuhiko
    学振二国間交流事業共同研究ロシアとの共同研究(RFBR)「ゲージ理論と弦理論の双対性による可積分性の統合と進展」, Mar. 2014, Japanese, KKRびわこホテル, International conference
    [Invited]
    Invited oral presentation

  • パデ補間によるパンルヴェ型方程式の構成
    YAMADA yasuhiko
    「有理函数近似が繋ぐー可積分系・直交多項式・パンルヴェ方程式」, Feb. 2014, Japanese, 一橋大学, 一橋大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Lax pairs for quantum Painlevé equations and their solutions
    YAMADA yasuhiko
    Workshop on Integrable Systems, Dec. 2013, English, University of Sydney, Australia, International conference
    [Invited]
    Invited oral presentation

  • Box-ball system and soliton tau function
    YAMADA yasuhiko
    Around Sato's Theory on Soliton Equations, Dec. 2013, English, 津田塾大学, 東京, International conference
    [Invited]
    Invited oral presentation

  • A simple expression for discrete Painlevé equations
    YAMADA YASUHIKO
    New developments in nonlinear discrete integrable systems, Sep. 2013, Japanese, RIMS Kyoto Univ., Domestic conference
    [Invited]
    Invited oral presentation

  • PBW bases for Uq+ and quantum coordinate rings
    Okado Masato, Kuniba Atsuo, YAMADA YASUHIKO
    Mathematical Society of Japan, Sep. 2013, Japanese, Ehime Univ., Domestic conference
    Oral presentation

  • The q-Painlevé equations arising from the q-interpolation problems
    YAMADA YASUHIKO
    Follow-up meeting on Discrete Integrable Systems, Jul. 2013, English, Newton Institute, International conference
    [Invited]
    Invited oral presentation

  • A simple expression for the elliptic Painlevé equation and its Lax pair
    YAMADA YASUHIKO
    Elliptic Integrable Systems and Hypergeometric Functions, Jul. 2013, English, Lorentz Center, International conference
    [Invited]
    Invited oral presentation

  • 連続/離散パンルヴェ方程式のラックス形
    YAMADA yasuhiko
    立教大学数理物理セミナー, May 2013, Japanese, 立教大学, 立教大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Lax formalism for discrete Painlevé equations
    YAMADA YASUHIKO
    8th IMACS International Conference, Mar. 2013, English, Georgia Univ. Athens, International conference
    [Invited]
    Invited oral presentation

  • Symmetries of quantum Lax equations for Painlev\'e equations
    HAJIME NAGOYA, YAMADA YASUHIKO
    Mathematical Society of Japan, Sep. 2012, Japanese, Kyushu Univ., Domestic conference
    Oral presentation

  • An interpolation problem related with q-E^{(1)}_8 Painlevé equation
    YAMADA Yasuhiko
    Infinite Analysis 11, Jul. 2011, English, 東大数理, 東大数理, International conference
    [Invited]
    Invited oral presentation

  • 量子 Fuji-Suzuki-Tsuda 方程式とインスタントン分配関数
    YAMADA Yasuhiko
    日本数学会無限可積分系セッション, Mar. 2011, Japanese, 日本数学会, 早稲田大学, Domestic conference
    Oral presentation

  • Quantum isomonodromy deformation and N=2 gauge theory
    YAMADA Yasuhiko
    -物理セミナー, Feb. 2011, Japanese, 静岡大理学部, 静岡大理学部, Domestic conference
    [Invited]
    Invited oral presentation

  • Quantum isomonodromy deformation and N=2 gauge theory
    YAMADA Yasuhiko
    弦理論セミナー, Jan. 2011, Japanese, 名大多元数理・KMI, 名大多元数理・KMI, Domestic conference
    [Invited]
    Invited oral presentation

  • 共形場理論, モノドロミー保存変形と AGT 予想
    YAMADA Yasuhiko
    表現論セミナー, Dec. 2010, Japanese, 京大数理解析研究所, 京大数理解析研究所, Domestic conference
    [Invited]
    Invited oral presentation

  • AGT 予想が切り開いた新たな数学の世界
    YAMADA Yasuhiko
    「量子可積分系の新展開」研究会, Dec. 2010, Japanese, 富士教育研修所, Domestic conference
    [Invited]
    Invited oral presentation

  • モノドロミー保存変形とN=2 ゲージ理論
    YAMADA Yasuhiko
    第4回日露ワーキングセミナー, Nov. 2010, Japanese, 大阪市立大梅田文化交流センター, Domestic conference
    Invited oral presentation

  • A quantum isomonodromy equation and its application to gauge theories
    YAMADA Yasuhiko
    神戸可積分系セミナー, Nov. 2010, Japanese, 神戸大, 神戸大, Domestic conference
    Oral presentation

  • N=2ゲージ理論とモノドロミー保存変形
    YAMADA Yasuhiko
    「重力・幾何・素粒子」研究会, Sep. 2010, Japanese, 大阪市立大, 大阪市立大, Domestic conference
    [Invited]
    Invited oral presentation

  • E_n 型 q-Painlevé方程式のLax形式
    YAMADA Yasuhiko
    日本数学会無限可積分系セッション, Sep. 2010, Japanese, 名古屋大学, 名古屋大学, Domestic conference
    Oral presentation

  • CFT, モノドロミー保存変形, Nekrasov関数
    YAMADA Yasuhiko
    「BC系とAGT予想の周辺」研究会, Sep. 2010, Japanese, 東大数理, 東大数理, Domestic conference
    [Invited]
    Invited oral presentation

  • q-Virasoro 代数とインスタントン分配関数
    AWATA Hidetoshi, YAMADA Yasuhiko
    日本数学会無限可積分系セッション, Mar. 2010, Japanese, 日本数学会, 慶應大学, Domestic conference
    Oral presentation

  • Five-dimensional AGT conjecture and the deformed Virasoro algebra
    YAMADA Yasuhiko
    九州可積分系セミナー, Dec. 2009, Japanese, 九州大学, 九州大学, Domestic conference
    [Invited]
    Invited oral presentation

  • 離散Painlevé方程式のLax形式
    YAMADA Yasuhiko
    可積分系数理とその応用, Aug. 2009, Japanese, はこだて未来大学, はこだて未来大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Lax formalism for the elliptic difference Painlevé equation
    YAMADA Yasuhiko
    Geometric aspects of discrete and ultra-discrete integrable systems, Apr. 2009, English, Glasgow大学, Glasgow大学, International conference
    [Invited]
    Invited oral presentation

  • 離散Painlevé方程式の幾何学的Lax形式
    YAMADA Yasuhiko
    日本数学会無限可積分系セッション特別講演, Mar. 2009, Japanese, 日本数学会, 東京大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometry of elliptic Painlevé equation and its Lax formalism
    YAMADA Yasuhiko
    テータ関数と可積分系, Dec. 2008, Japanese, 九州大学, 九州大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometry of elliptic Painlevé equation and their hypergeometric solutions
    YAMADA Yasuhiko
    Elliptic integrable systems, isomonodromy problems, and hypergeometric functions, Jul. 2008, English, MPIM, Bonn, Germany, MPIM, Bonn, Germany, International conference
    [Invited]
    Invited oral presentation

  • Padé近似 と Painlevé方程式の特殊解
    YAMADA Yasuhiko
    日本数学会無限可積分系セッション一般講演, Mar. 2008, Japanese, 日本数学会, 近畿大学, Domestic conference
    Oral presentation

  • Padé近似 と Painlevé方程式
    YAMADA Yasuhiko
    早稲田大学代数解析セミナー, Feb. 2008, Japanese, 早稲田大学, 早稲田大学, Domestic conference
    Oral presentation

  • E型アメーバ図鑑 --その採取と観察--
    YAMADA Yasuhiko
    物理部会談話会, Nov. 2007, Japanese, 東京大学, 東京大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Geometric formulation of discrete Painlevé equation and its tropical ana logue
    YAMADA Yasuhiko
    研究会「トロピカル幾何学と関連分野」, Sep. 2007, Japanese, 北海道大学理学部, 北海道大学理学部, Domestic conference
    [Invited]
    Invited oral presentation

  • Knizhnik-Zamolodchikov, Schlesinger and reduction of ASDYM
    YAMADA Yasuhiko
    阪大素粒子論研究室セミナー, Jul. 2007, Japanese, 大阪大学, 大阪大学理学部, Domestic conference
    [Invited]
    Invited oral presentation

  • 可積分系と双有理変換
    YAMADA Yasuhiko
    日本数学会2005 年秋季総合分科会, Sep. 2005, Japanese, 日本数学会, 岡山大学, Domestic conference
    Invited oral presentation

  • 平面代数曲線とPainleve 方程式
    山田 泰彦
    筑波大学(数学) 談話会, Oct. 2004, Japanese, 筑波大学(数学), Domestic conference
    Oral presentation

  • Tropical affine Weyl group representation of type E.n.
    YAMADA Yasuhiko
    Tropical Algebraic Geometry and Tropical Combinatorics., Aug. 2004, English, 未記入, Kyoto, Domestic conference
    Invited oral presentation

  • Tropical R: Examples and applications
    KUNIBA, OKADO, TAKAGI, Y, YAMADA Yasuhiko
    Combinatorial Aspect of Integrable Systems., Jul. 2004, English, 未記入, Kyoto, Domestic conference
    Invited oral presentation

■ Affiliated Academic Society
  • 日本数学会

■ Research Themes
  • 量子曲線に基づく量子パンルヴェ方程式の構築と応用
    山田 泰彦, 太田 泰広, 高山 信毅
    日本学術振興会, 科学研究費助成事業, 基盤研究(B), 神戸大学, 01 Apr. 2022 - 31 Mar. 2027

  • Algebraic Geometry and Integrable Systems -- Moduli theory and Equations of Painleve type
    齋藤 政彦, 山田 泰彦, 岩木 耕平, 望月 拓郎, 吉岡 康太, Rossman W.F, 稲場 道明, 光明 新
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Kobe Gakuin University, 01 Apr. 2022 - 31 Mar. 2027

  • 太田 泰広
    科学研究費補助金／基盤研究(B), Apr. 2018 - Mar. 2023
    Competitive research funding

  • 齋藤 政彦
    科学研究費補助金／基盤研究(S), Apr. 2017 - Mar. 2022
    Competitive research funding

  • 齋藤 政彦
    科学研究費補助金／基盤研究(A), Apr. 2017 - Mar. 2022
    Competitive research funding

  • 野海 正俊
    科学研究費補助金／基盤研究(B), Apr. 2015 - Mar. 2020
    Competitive research funding

  • 山田 泰彦
    科学研究費補助金／基盤研究(B), Apr. 2014 - Mar. 2019, Principal investigator
    Competitive research funding

  • Bilinear method for multi-component coupled integrable systems
    Ohta Yasuhiro, YAMADA Yasuhiko, NOUMI Masatoshi
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 01 Apr. 2012 - 31 Mar. 2018
    For the integrable evolution equations, there are several methods to generalize them to the coupled systems of equations keeping the integrability, and those integrable coupled equations admit various types of solutions with many internal freedoms. It is important to study the multi-component coupled systems and their solutions systematically both in theoretical study and applications of integrable systems. Based on the theory of bilinear method in the classical integrable systems, the way of constructing new multi-component coupled integrable systems is proposed, and various types of interactions are described by using the internal freedoms of the solutions.

  • Pursuit of analogies between quantum difference isomonodromic systems, quantum Teichmuller theory, and solvable lattice models
    Hasegawa Koji, Yamada Yasuhiko, Kuroki Gen
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Tohoku University, 01 Apr. 2014 - 31 Mar. 2017
    This project is to pursuit the analogies between the quantum isomonodromic systems, quantum Teichmuller theory, and the solvable statistical lattice models in two dimension. It involves many structures nd aspects such as symmetries in quantum discrete Garnier systems and quantized tau function, as well as quantization of the so-called confluent procedure from the viewpoint of lattice models, and the quantized Teichmuller theory as the geometric counterpart of the solvable lattice theory. For all of these theme the proper understanding for the quantum discrete La x matrix is important. We have explored some of the relevant structures e.g. the appropriate root ordering problem in the formula of universal R matrix to obtain the appropriate Lax matrices. Still there are remaining problems, one of the main issue is to understand the infinitely many poles arising from the imaginary root factors in a proper way.

  • 太田 泰広
    学術研究助成基金助成金／挑戦的萌芽研究, Apr. 2014 - Mar. 2017
    Competitive research funding

  • Studies of the algebraic and combinatorial structures related to quantum integrable systems
    OKADO Masato, KUNIBA Atsuo, NAKANISHI Tomoki, YAMADA Yasuhiko, SCHILLING Anne
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), 01 Apr. 2011 - 31 Mar. 2016
    Principal investigator Okado constructed, with Schilling and Sakamoto, a bijection between highest weight paths and rigged configurations for type D in full generality, and thereby settled the X=M conjecture in a combinatorial way. Nakanishi studied, with his collaborators, the cluster algebra from the aspect of quantum integrable systems and clarified the periodicity, dilogarithm identities, the relation between exact WKB analysis and mutation, etc. Although it was not included in the purposes of this project in the beginning of the project period, Kuniba started the study of 3-dimensional quantum integrable systems. Later, together with Okado and other collaborators, he produced results, such as the relation between quantum coordinate rings and PBW bases, 2-dimensional reduction of the tetrahedron equation, applications to Markovian processes.

  • 齋藤 政彦
    科学研究費補助金／基盤研究(S), Jun. 2012 - Mar. 2016
    Competitive research funding

  • 齋藤 政彦
    科学研究費補助金／基盤研究(A), Apr. 2012 - Mar. 2016
    Competitive research funding

  • Integrable models and quantum cluster algebras
    KUNIBA ATSUO, YAMADA Yasuhiko, NAKANISHI Tomoki, OKADO Masato
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 01 Apr. 2012 - 31 Mar. 2015
    Several results illuminating novel mathematical structures of 3-dimensional integrable systems are obtained. (1) By invoking the representation theory of quantized coordinate ring of SP(4), two kinds of solutions are constructed for the 3-dimensional reflection equation proposed by Isaev and Kulish in 1997 for the first time. Its combinatorial limit, the classical analogue and a polynomial formula are obtained. (2) The transition coefficients of the PBW bases of the positive part of the quantized universal enveloping algebra Uq(g) are shown to coincide with the matrix elements of the intertwiner of the Soibelman representations of the quantized coordinate ring of g for all g of finite classical type. (3) For the two solutions known as R-operator and L-operator of the tetrahedron equation, 2-dimensional reduction is performed, and the results are identified with the quantum R matrices for generalized quantum groups including quantum superalgebras.

  • Study of vector bundles using the theory of complexes
    YOSHIOKA Kota, NOUMI Masatoshi, YAMADA Yasuhiko, SAITO Masahiko, NAKAJIMA Hiraku, ABE Takeshi
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 01 Apr. 2010 - 31 Mar. 2014
    I studied moduli of Bridgeland stable objects on an abelian or a K3 surface. In particular, I proved that moduli spaces are projective varieties, and studied birational properties of the spaces. I also apply these results to the classification of vector bundles on abelian surfaces. I also proved the Witten conjecture of Donaldson invariants for algebraic surfaces.

  • Approach to the polynomials related to representation theory from quantum integrable systems
    OKADO Masato, KUNIBA Atsuo, YAMADA Yasuhiko, SAKAMOTO Reiho, SCHILLING Anne
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Challenging Exploratory Research, 2011 - 2013
    The study of X=M conjecture, which originates in quantum integrable systems, equating the generating functions of highest weight elements of the tensor product of KR crystals and rigged configurations has advanced about 80% to the goal for type D. Research for the exceptional case E6 was also begun. However, the study of the relation to LLT polynomial remained to be incomplete. We also studied the relation between tetrahedron equation and quantum groups, namely, explicit formula for the solution to the 3D reflection equation, relation between matrix elements of the intertwiner of the quantum coordinate ring and PBW bases of the quantum enveloping algebra, coincidence of the 2D reduction and the intertwiner of the tensor product of q-oscillator representations of a quantum affine algebra.

  • Combinatorics and difference structure in Bethe ansatz
    KUNIBA Atsuo, YAMADA Yasuhiko, NAKANISHI Tomoki
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 2009 - 2011
    Structure of Markov matrix of multi-species ASEP is clarified. Generalizations, and solutions of periodicity/dilogarithm conjectures for T and Y-systems among transfer matrices in quantum integrable systems. Derivation of fermionic partition function of periodic box-ball system. Solution of higher-rank periodic box-ball system. Theory of generalized energies for D-type crystals is constructed. Derivation of quantum R-matrices for spin representations from 3D integrable system.

  • Theory of Painleve systems and its new development
    KAJIWARA Kenji, SHIRAI Tomoyuki, IWASAKI Katsunori, NOUMI Masatoshi, YAMADA Yasuhiko, SAKAI Hidetaka, MASUDA Tetsu, TSUDA Teruhisa
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyushu University, 2007 - 2010
    Theory of the Painleve systems, which are a certain family of second-order nonlinear integrable differential and difference equations, has been constructed by using the underlying affine Weyl group symmetries and algebraic geometric structures. Based on this framework, detaild studies on solutions have been carried out, such as determination of the sequence of hypergeometric functions arising as solutions. Also, generalizations of the theory of Painleve systems have been developed to higher-order and higher-dimensional systems. Moreover, based on the results obtained above, the theory has been extended to various areas, such as discrete soliton equations, discrete differential geometry, solvable chaotic systems, tropical geometry, complex dynamical systems, and random matrices.

  • 山田 泰彦
    科学研究費補助金／基盤研究(B), 2009, Principal investigator
    Competitive research funding

  • Ultradiscrete solitons and solvable lattice models
    KUNIBA Atsuo, OKADO Masato
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 2007 - 2008
    超離散ソリトン系の代表的なモデルである箱玉系について, 以下の結果を得た. 多状態かつ箱の容量が任意に非一様な無限系, 2状態で箱の容量が任意で一様な周期系のそれぞれについて, 初期値問題の解のアルゴリズムおよび明示式を得た, 特に明示式として, ソリトン理論や代数曲線の理論に登場するタウ関数やリーマンテータ関数の超離散類似を初めて導出した. この他, T-systemの周期性や多状態非対称排他過程のスペクトルについても結果を得た.

  • Integrable Systems and Combinatorial Representation Theory
    OKADO Masato, YAMADA Yasuhiko, KUNIBA Atsuo, NOBE Atsushi
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Osaka University, 2006 - 2007
    During the period of research project, we mainly obtained the following results. 1. [Affine geometric crystal] In collaboration with M. Kashiwara and T. Nakashima, we constructed geometric crystals associated to nonexceptional affine Lie algebras. We confirmed that the ultra-discrete limit of these geometric crystals coincide with the limit of previously known perfect crystals. Moreover, except type C, we obtained explicit formulas for birational maps, called tropical R maps, that satisfy the Yang-Baxter equation. 2. [Existence of crystal bases of the KR modules for nonexceptional types] There was a conjecture saying that any finite-dimensional representation of a quantum affirm algebra that has an integer multiple of a level 0 fundamental weight as highest weight (KR module) has a crystal base. We solved this conjecture for all affine Lie algebras of nonexceptional types. In collaboration with A. Schilling, we also proved that the crystals of type B^<(1)>_n, D^<(1)>_n, and A^<(2)>_<2n-1> are isomorphic to the combinatorial crystals recently constructed by Schilling. 3. [Construction of the coherent family of perfect crystals for exceptional types] In collaboration with M. Kashiwara, K.C. Misra and D. Yamada, we revealed the crystal structure of the perfect crystals associated to the exceptional affine lie algebra D^<(3)>_4 at any level.

  • 太田 泰広
    科学研究費補助金／基盤研究(B), 2007
    Competitive research funding

  • 齋藤 政彦
    科学研究費補助金／基盤研究(S), 2007
    Competitive research funding

  • Mathematical theory of dynamical systems of Painleve type
    KAJIWARA Kenji, NOUMI Masatoshi, YAMADA Yasuhiko, IWASAKI Katsunori
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), KYUSHU UNIVERCITY, 2003 - 2006
    1. We have extended the theory of symmetric form for q-Painleve IV equation with (A_2+A_1)^<(1)> affine Weyl group symmetry to formulate the q-KP hierarchy. By the similarity reduction we have constructed the hierarchy of discrete systems with (A_+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_+A_)^<(1)> affine Weyl group symmetry. 2. We have presented a formulation of the elliptic Painleve equation and its generalizations, the former of which is located on the top of all the Painleve systems of second order. Namely, we have formulated the time evolution and the Backlund transformations as Cremona transformations on the configuration space of generic points in the complex projective space, and given their realization as birational transformations parametrized by the theta functions on the level of the τ functions. We have also formulated the time evolution as the addition formula on the moving pencil of plane cubic curves, and clarified the geometric meaning of the Hamilitonians of the Painleve differential equations. 3. Applying the above formulation, we have constructed the simplest hypergeometric solutions to the elliptic Painleve equation and all the q-Painleve equations, and completed the list of coalscent diagram of hypergeometric functions, starting from the elliptic hypergeometric function _<10>E_9 to the q-Airy function. 4. Combining the algebro-geometric formulation of the Painleve VI equation and the ergodic theory of birational mapping on the algebraic surface by using the Riemann-Hilbert correspondence, we have shown that the nonlinear monodromy of the Painleve VI equation is chaotic along almost all the loops. 5. We have shown that the entries of Hankel determinant formula for the solutions of the Painleve differential equations arise as coefficients of asymptotic expansion of the ratio of solutions to the auxiliary linear problem, and that this phenomenon originates from the structure of the KP hierarchy. 6. Applying the above results we have discussed some new extentions or new solutions to the discrete and ultradiscrete Toda equation.

  • Geometry of integrable systems with infinite degrees of freedom and new development of moduli theory.
    UENO Kenji, KATO Fumiharu, KAWAGUCHI Shu, MOCHIZUKI Shinichi, TAKASAKI Kanehisa, KATSURA Toshiyuki
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (S), Kyoto University, 2002 - 2006
    From non-abelian conformal field theory (WSWN-model) and abelian conformal field Ueno and J.E. Andersen constructed modular functor and studied properties of the associated topological field theory. They also showed that S-matrices of non-abelian conformal field theory are determined by the genus 0 data. Moreover they showed that the Nielsen-Thurston classification of mapping class groups of 4-pointed spheres is determined by their quantum SU (n) representations. F. Kato and his collaborators have constructed the most general rigid geometry so that it can be applied to study moduli spaces by analytic method. S. Mochizuki has studied categorical aspect of moduli space of algebraic curves from different viewpoints. For example, he constructed theory of Frobenioids and that of etale theta functions, which give new direction of study of moduli spaces. Moreover, many interesting results on geometric and arithmetic properties of moduli spaces were obtained. For theory of integrable systems K. Takasaki and his collaborators found that moduli spaces play important roles in the soliton theory. Also interesting relationship between geometry of special algebraic curves and Painleve equations were found. Moreover several important properties of quantum cohomology and quantum K-theory of flag manifolds have been found. Our studies show that mathematical structure of moduli space is deeper than what we thought at the beginning and we need new mathematical tools for further investigations. Our results give a part of such tools and also show new directions to further investigations.

  • 吉岡 康太
    科学研究費補助金／基盤研究(B), 2006
    Competitive research funding

  • Study of Classical and Quantum Integrable Systems and their discretizations
    HASEGAWA Koji, KUROKI Gen, YAMADA Yasuhiko, IKEDA Takeshi
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Tohoku University, 2004 - 2005
    K.Hasegawa studied the quantization of the discretized Painleve equation and its symmetries (Backlund transformations) proposed by K.Kajiwara, M.Noumi and Y.Yamada. Especially he succeeded to construct the quatization of the discrete Painleve VIth equation of Jimbo and Sakai. G.Kuroki studied the quantization and discretization of the monodromy preserving deformation in general. He succeeded to reconstruct Hasegawa's quantization of Bcklund transformations from the viewpoint of dressing chains and geometric crystals. Also he tried to formulate the quantized theory as the deformation of the conformal field theory under the symmetry of W-algebras. Y.Yamada studied the tropicalization of the structures relevant in the two dimensional solvable lattice statistical models, yielding many combinatorial corresspondances. Also studied are the discrete Painleve systems. Hypergeometric solutions for the elliptic and/or degenerate discrete Painleve equations are obtained. The method was applied to the Hamiltonian sturucture for the differential case. T.Ikeda studied the reductions of solitonic equations. Heused the Fermionic Fock space to obtain a combinatorial formula for Schur's Q- functions. Also he succeded to identify Schur's Q-functions as some classes in equivariant cohomologies, suggesting that the study of special functions in the setting of torus action and its fixed points will be fruitful.

  • Parabolic Kostka polynomials, quiver varieties, crystal bases and tropicalcombinatorics
    KIRILLOV Anatol N., ARIKI Susumu, NAKAJIMA Hiraku, NOUMI Masatoshi, YAMADA Yasuhiko, MAENO Toshiaki
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyoto University, 2003 - 2005
    During 2003-2006 years in the course of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics", the members of the Project have published about 25 papers in the leading mathematical journals, have organized and attended several International and Domestic Conferences and Workshops, participated regularly in the joint discussions and collaboration. The main event of 2003 year term of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the "International Workshop on Quantum Cohomology" to be held during June 16-19, 2003 at RIMS, Kyoto University, and organized by A.N.Kirillov and M.Guest (Tokyo Metropolitan University). The Workshop collected several leading specialists in the field such as Profs. H.Nakajima (Kyodai), K.Saito (RIMS), B.Kim (S.Korea), A.-L.Mare (Canada), A.Buch (Sweden), as well as many (about 50) domestic participants. The main event of 2004 year term of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the International Workshop "Tropical algebraic geometry and tropical combinatorics" carried out August 8-22, RIMS, Kyoto University, and organized by A.N.Kirillov and M.Noumi. The Workshop collected several leading specialist in "Tropical Mathematics" such as Profs. A.Knutson (UC Berkeley, USA), E.Miller (Univ. of Minnesota, USA), G.Mikhalkin (Toronto Univ., Canada), D.Speyer (UC Berkeley, USA), O.Viro (Uppsala Univ., Sweden), M.Kashiwara (RIMS), M.Okado (Osaka), Y.Yamada (Kobe), as well as many (near 60) domestic participants. The both Workshops were recognized to be successful and gave rise to notable interest to Tropical Mathematics and Quantum Cohomology in Japan. In the body of the Project, A.N.Kirillov attended as invited speaker the International Workshop "Combinatorics, Special Functions and Physics, 2004" held August 2-4, Nankai Univ., China, and several domestic Conferences. Part of basic results about parabolic Kostka polynomials obtained by A.N.Kirillov in the body of project, has been published in the Publications of RIMS, vol. 40. In particular, this paper contains a proof of the so-called Generalized Saturation Conjecture, as well as proofs of several new interesting properties of parabolic Kostka polynomials, Schur functions and so on. Several important results about connections between Schubert Calculus and non-commutative differential calculus have been obtained and published by A.N.Kirillov and T.Maeno. In particular, we described the algebra generated by flat connections for some noncommutative algebraic varieties, and prove Monk formula for noncommutative B_n Schubert polynomials.

  • 山田 泰彦
    科学研究費補助金／基盤研究(B), 2005, Principal investigator
    Competitive research funding

  • 野海 正俊
    科学研究費補助金／基盤研究(A), 2005
    Competitive research funding

  • 齋藤 政彦
    科学研究費補助金／基盤研究(B), 2005
    Competitive research funding

  • 太田 泰広
    科学研究費補助金／基盤研究(C), 2005
    Competitive research funding

  • Defining manifolds and symmetries of nonlinear special functions in several variables
    TAKANO Kyochi, NOUMI Masatoshi, YAMADA Yasuhiko, SAITO Masa-hiko, IWASAKI Katsunori
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 2001 - 2004
    1.We have constructed the defining manifolds of the Garnier system and all its degenerate systems in two varibales. We have also solved the same problem for the Noumi-Yamada system of type A^<(1)>_4. 2.We have proved that the manifold defined by means of Backlund transformation group for each Painleve equation is isomorphic to the corresponding defining manifold constructed by K.Okamoto. 3.We have shown that there exists an hierarchy in the Backlund transformation groups for Painleve equations, namely, the Backlund transformation groups for all Painleve equations can be obtained successively from that for the sixth Painleve equation by the use of the usual confluence procedures. 4.We have characterized the Backlund transformation group for the sixth Painleve equation by means of Riemann-Hilbert correspondence, namely, the correspondence from the phase space(the space of initial conditions) of the sixth Painleve equation to the moduli space of the monodromy representation is just a covering mapping with the affine Weyl group of type D^<(1)>_4 as the covering transformation group.

  • 無限自由度の可積分系の数論幾何学的研究
    上野 健爾, 山田 泰彦, 齋藤 政彦, 加藤 文元, 神保 道夫, 齋藤 秀司
    日本学術振興会, 科学研究費助成事業, 基盤研究(A), 京都大学, 2001 - 2004
    上野はJ.Andersenとの共同研究で,曲線が退化する際のアーベル的共形場理論(bc系の理論)を構成した.この結果は,非アーベル的共形場理論からモジュラー函手を構成する際に,アーベル的共形場理論の分数ベキとのテンソル積を取ることが必要となり,点付き代数曲線のモジュライ空間の境界でのテンソル積の挙動を調べるために使われた.さらに,このモジュラー函手から構成される3次元多様体の不変量は,リー代数がsl(2,C)の時はReshetikhin-Turaevが構成した不変量と一致することがほぼ明らかになった.証明の詳細な詰めは次年度の研究で行う予定である.また,上野はアーベル的共形場理論を代数曲面の場合に拡張するための予備的な考察を行った. 齋藤政彦はパンルヴェ方程式の初期値空間の研究を行い,初期値空間として登場する岡本・パンルヴェ対が逆にパンルヴェ方程式を決定することを,岡本・パンルヴェ対に変形理論を適用することによって示した.山田は多変数のパンルヴェ方程式を対称性の観点から研究した.また,神保は量子場の相関関数とq直交多項式との関連を考察した. また,齋藤秀司は非特異代数多様体のChow群に関するBloch-Beilinsonフィルター付けについて考察した.加藤はMumford曲線に関する研究を行い,Mumford曲線を被覆として持つ非アルキメデス的オービフォールドの特徴付けを与え,またモジュライ空間でのMumford曲線のなす軌跡の性質について新しい知見を得た.またMumfordによる擬射影平面の志村多様体としての具体的な構成を与えた.

  • The study of vector bundles on an algebraic manifold with the trivial canonical bundle
    YOSHIOKA Kota, YAMADA Yasuhiko, SAITO Masahiko
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), KOBE UNIVERSITY, 2000 - 2002
    Yoshioka find the condition for the existence of stable sheaves on K3 and abelian surfaces and showed the connectedness of the moduli spaces. If the moduli space is compact, then he determined the albanese map and showed that the fiber is a hyperkaehler manifold. Moreover he showed that the deformation type of the manifold is determined by the dimension. By this result, the study of the topological type of the moduli space is reduced to the rank 1 case. In order to get these results, he used the Fourier-Mukai transform and the deformation of the underlying K3 surfaces. Hence he also studied the relation between the stability and the Fourier-Mukai transform. Moreover he studied the moduli of stable sheaves on an elliptic surface, surface components of the moduli of stable sheaves on a K3 surface and the Gromov-Witten invariants and got some results. Saito studied Gopakumar-Vafa conjecture on BPS states. He proposed a mathematical definition of the BPS invariants by using the moduli of purely 1-dimensional sheaves and checked the consistency for some cases. Yamada studied D-brane on a rational elliptic surface. Mathematical beautiful structures such as monodromy group, Mordell-Weil group, affine Weyl group are studied from the Physical point of view. Saito and yamada studied the Painleve equation in terms of the symmetry and the geometry. In particular, Saito showed that the Backlund transform is the flop in the birational geometry and the Painleve equation is derived from the deformation theory of a rational surface, which has an application of the classification of the Riccati solution.

  • Geometry on String Theory and Moduli spaces
    SAITO Masa-hiko, HOSONO Shinobu, YAMADA Yasuhiko, NOUMI Masatoshi, YOSHIOKA Kota, MUKAI Shigeru
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 2000 - 2002
    During the period of the project, we have investigated the following subjects and obtained the following results. (i) Gromov-Witten invariants and BPS invariants for Calabi-Yau manifolds-Gopakumar-Vafa conjecture, (ii) Homological Mirror Symmetry and Geometry of derived categories, (iii) Algebraic Geometry of spaces of initial conditions of Painleve equations, (iv) Lie theoretic studies for Painleve equations and its generalizations, (v) Symmetries of the moduli spaces of vector bundles, (vi) New development of invariant theory. As for (i), we proposed a mathematical definition of BPS invariants for Calabi-Yau 3-fold and verified their compatibility for the known calculation of Gromov-Witten invariants assuming Gopakumar-Vafa conjecture. In (iii), we introduce the notion of Okamoto-Painleve pairs and proved that one can derive Painleve equations through deformation theory of Okamoto-Painleve pairs. As for the studies of Painleve equations (iv), our group have been developing Lie theoretic approach and algebro-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.

  • Painleve equations and integrable systems
    YAMADA Yasuhiko, TAKANO Kyouichi, SAITO Masahiko, NOUMI Masatoshi, KAJIWARA Kenji, MASUDA Tetsu
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Faculty of Science, Kobe Univ., 1999 - 2002
    Noumi and Yamada gave a systematic generalization of Painleve-type differential equations from the point of view of affine Weyl group symmetry. This result is presented in the Noumi's book and activate the research of this area. A new Lax formalism for the sixth Painleve equation is also obtained. The universal structure with respect to the root systems was discovered on the birational representation of the affine Weyl group arising from Painleve equations. Lie theoretic background is also explained based on the gauss decomposition. The representation was lifted to the tau-functions. The tau functions are certain matrix elements of affine Lie algebras. This construction proved that the representation gives the symmetry of the Painleve type equations arising as the similarity reduction of the Drinfeld-Sokolov hierarchy. On the other hand, Kajiwara, Noumi, Yamada studied the q-Painleve IV equation and its generalization with Weyl group symmetry of type W (A^<(1)>_ × A^<(1)>_). This representation is "tropical" (=subtraction free) and has some combinatorial applications through the ultra-discretization. q-KP hierarchy and their polynomial solutions are obtained. Masuda gave the determinant formulas for the (q-)Painleve V and VI equations. Takano constructed the space of initial value based on the Backlund transformations. Saito gave the algebro-geometric characterization of the space of initial value. In summary, we obtained sufficient results for almost all the problems of the project.

  • Global Analysis of Painleve Equations
    TAKANO Kyoichi, YAMADA Yasuhiko, NOUMI Masatoshi, SASAKI Takeshi, TAKEI Yoshitsugu, IWASAKI Katsunori
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B)., Kobe University, 1998 - 2000
    1. Symmetries of Painleve equations : Theory of Backlund transformations (realization of affine Weyl groups) for Painleve equations has been constructed. The theory gives not only good perspective but also usefull tools to the study of Painleve equations. For example, we can easily obtain the form of each Backlund transformation as birational transformation and various kinds of special polynomials associated with Painleve equations. Similar theory is now being devepoled for discrete Painleve equations. 2. The spaces of initial conditions : (1) A relation between the spaces of initial conditions and Backlund transformations has been made clear, namely, the manifold obtained by patching affine charts via Backlund transformations are isomorphic to Okamoto's space of initial conditions. Fromx this fact, we can derive that the spaces of initial conditions whose papameters are equivalent under the affine Weyl group are isomorphic to each other. (2) Spaces of initial conditions for a higher order Painleve equation of type A^<(1)>_4 and degenerated Garnier systems of two variables have been obtained. 3. Exact WKB analysisi : (1) The connection problem for the second Painleve equation with a large paraneter has been solved by the use of exact WKB analysis. The connection formulas are given by compositions of those for the first Painleve equation with a large parameter. For this purpose, a reduction theorem to Birkhoff's normal form has been shown. The usual steepest descent method has been extended to one for third order linear differential equations. 4. Hypergeometric equations : A problem of studying Schwarz theory in the case where all parameters are pure imaginary numbers has been proposed. Some experiments were carried out.

  • Modern development of special functions - approach from the representation theory and the integrals
    MIMACHI Katsuhisa, YAMADA Yasuhiko, NOUMI Masatoshi, HANAMURA Masaki
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyushu University, 1997 - 1999
    The purpose of the present research was to settle the viewpoint to unify the theory of hypergeometic function associated with the root system and the theory of integrals. Concrete theme of this work was the following : 1. De Rham theory (Study of homology and cohomology associated with Selberg type integrals, which appear as the spherical functions of A type), 2. Relationship between the representations of several kinds of algebras (Hecke algebras and so on) and the integrals, 3. Application to Painleve equations (special polynomials such as Okamoto polynomials), 4. Application to mathematical physics (Calogero system, correlation functions in conformal field theory or solvable lattice models). The results of the head investigator were mainly about 1 and 2, those of Hanamura were about 2, those of Noumi and Yamada were about 3. Matsui's help was valuable in the study of 4, Ochiai's in 2 and 4, Wakayama's in 2, Kato's in 1. Anyway, we have obtained a lot of results through the period of the present research project. As an evidence, many of papers had appeared in the journal of excellent level.

  • Research on Periods of Algebraic Varieties and Hypergeometric Functions
    SAITO Masa-hiko, TAKANO Kyoichi, SASAKI Takeshi, NOUMI Masatoshi, YOSHIOKA Kota, TAKAYAMA Nobuki
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 1997 - 1999
    During the period of the project, we have investigated the following subjects and obtained the following results. (i)Mirror Symmetry Conjecture for Calabi-Yau manifolds, (ii)Counting Curves of higher genus in Rational Elliptic Surfaces, (iii)Lie theoretic aspects on Painleve equations, Algebro-geometric aspects on Painleve equations, GKZ hypergeometric systems and their Grobner deformations. As for (i), we have been studying Gromov-Witteninvariants for certain Calabi-Yau 3-folds and computed a part of A-model prepotentials by means of the theta function of the EィイD28ィエD2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series. As a result, we have checked MSC mathematically for those cases. Developing further, in (ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces. We propose the holomorphic anomaly equation(HAE), which the prepotential should satisfy. By using the Jacobi's triple product formula and the relative Lefschetz decomposition, we have checked the prepotential satisfies the HAE. As for the studies of Painleve equations ((iii)), our group have been developing Lie theoretic approach and algeblo-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.

  • Differential equations for periods and super string and susy gauge theories
    YAMADA Yasuhiko, YANG Sung-Kil, HOSONO Shinobu
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), KOBE UNIVERSITY, 1997 - 1998
    In this project, each of the three investigators carried out his research independently with in close co-operation. Inspiring with each other, we can obtain successful results more than our initial aim. 1. Y.Yamada studied the relation between the integrable structure in the two dimensional quantum gravity and singularity theories. He found deep connections between the KP equations, Drinfeld-Sokolov equations and Painleve equations, and obtained successful results on the theory of Painleve equations and their symmetries. This result is expected to be applied to other problems in mathematical physics. 2. S.K.Yang investigated the Seiberg-Witten curves (or more general geometries) in N = 2 susy gauge theories in systematic way with explicit examples. He clarify the profound relation with the singularity theories and determine the geometrical structures for exceptional gauge groups. 3. S.Hosono has continued the research on the mirror symmetry. He made important contribution for the evaluation of the Gromov-Witten invariants for elliptic- or K3-fibered Calabi-Yau manifolds. There is also some progress in counting the higher genus curves from theoretical and computing points of view.

  • Various aspects of hypergeometric functions
    YOSHIDA Masaaki, MATSUMOTO Keiji, WATANABE Fumihiko, HANAMURA Masaki, KANEKO Masanobu, KATO Fumiharu
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), KYUSHU UNIVERSITY, 1996 - 1998
    Hypergeometric integrals found by Euler was re-formulated in terms of a modern language by many authors : the dualﾟCpairing of twisted homologies and twisted cohomologies. Expected intersection theories were established by M.Kita and Yoshida for homologies, and by K.Cho and Matsumoto for cohomologies. Further developments are in progress, especially those for confluent case by Matsumoto. These can be considered to be twisted versions of Riemann's equality for period integrals. Twisted versions of Riemann's inequality were found, via twisted Hodge theory, by Hanamura and Yoshida. Modular interpretations of configuration spaces. Let X(k, n) be the configuration space of n-point-sets in the k-1-dimensional projective space. Several configuration spaces can be presented as quotient spaces of symmetric spaces under discontinuous groups ; the original one is X(2, 4) * H/GAMMA(2), where H is the upper half space and GAMMA(2) is an elliptic modular group. Yoshida found, with Matsumoto and T.Sasaki, a modular interpretation of the space X(3, 6) through hepergeometric function of type (3, 6)), which can be summerized as X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA, where GAMMA is an arithmetic subgroup acting on the hermitian symmetric domain of type IV.Yoshida wrote two books about this interpretation. Kaneko found, with D.Zagier, automorphic forms which connect hypergeometric functions and supersingular elliptic curves. Kaneko found a new arithmetic formulae for the Fourier coefficients of j(gamma). F.Watanabe established a new very transparent way to find Okamoto transformations for Painlv_ functions by using the Takano's construction of the phase spaces. F.Kato is ambitiously trying to find examples of algebraic varieties which are p-adically uniformized by Drinfeld symmetric spaces and the uniformizing differential equations ; he already found, with M.Ishida, new fake projective planes, and studied their uniformizations complex anlytically as well as p-adically.

  • Mathematics of Symmetry
    MIWA Tetsuji, KONNO Hitoshi, OKADO Masato, HASEGAWA Koji, SHIRAISHI Junichi, KUNIBA Atsuo
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), KYOTO UNIVERSITY, 1996 - 1998
    The main theme of this project is the symmetry approach to solvable lattice models. As for this problem, In the case of elliptic models, which had not been solved In the symmetric approach, a bosonization of the vertex operators were obtained by Shiraishi and Odake using the representation theory of quasl-Hopf algebra, in particular, the twist of quantum groups. By Miwa and Konno, in the trigonometric limit (with |q |=1) of this model, an integral formula is obtained for the difference analogue of the Knnizhnik-Zamolodchhikov equation. Hasegawa constructed Ruijsenaars' commuting difference operators by using the intertwining vectors In the elliptic model. The theory of crystal is a key In the connection between solvable lattice models and combinatorics. As for this, Miwa, Okado, Kuniba, Yamada (Yasuhiko) found that the set of inhomogeneous paths give the crystal of tensor products of integrable highest weight representations. Several interesting examples of non-perfect crystals and the corresponding paths are also studied. As for non-solvable models, Matsui showed that the matrix product states which represent the ground states correspond to the representations of the Kuntz algebra. The toroidal algebra is important because it governs the symmetry of the vertex operators. Mild found an automorphism of this algebra which connects two affine quantum algebras inside thereof. Solvable models In quantum field theory Is another main subject. Kawahigashi developed the method for calculating modular invariant quantities in the two-dimensional conformal field theory. It is also important to apply techniques developed in the two-dimensional solvable models to the problems in string theory and the four-dimensional gauge theory. As for this, Nakatsu showed that in the adiabatic limit the tau function of the Toda lattice gives the effective action of the N=2 super-symmetric Yang-Mills theory. Kanno and Yang obtained a generalization of the Donaldson-Witten Invariant for the four dimensional manifolds. Kato analysed the condition for a matrix model to be understood as quantum gravity theory In a curved space-time. The developments In the past three years Include (1) the symmetry method for the solvable lattice models of elliptic type ; (2) the method of representation theory for the mixed spin chains ; (3) the discovery of new links between lattice models and combinatorics ; (4) the quantization of several geometric invariants ; (5) the algebraic approach connecting the operator algebras and the conformal field theory.

  • 位相的場の理論とそのモジュライ
    山田 泰彦, 金子 昌信, 三町 勝久, 梶原 壌二, 柳川 堯, 小西 貞則
    日本学術振興会, 科学研究費助成事業, 一般研究(C), 九州大学, 1995 - 1995
    本研究では、位相的場の理論とそのモジュライに関して、関連する基礎研究をおこなった。この問題へは、適用する数学手法において、代数的(表現論、組合せ論)、幾何的(代数幾何、複素多様体)、解析的(非線形方程式、特殊関数)等の様々なアプローチが考えられる。これらは、位相的場の理論のモジュライという広い対象をいかなるカテゴリーにおいて捉えるかの見方の違いによる。このように、多様なアプローチが可能であることが、この問題の特質と言える。いずれの方法にも長所と短所があるので、それぞれの立場からの多角的研究が必要とされる。以下に、分担者ごとの研究の概要を述べる。 山田は、表現論的、組合せ論的側面を研究し、可解格子模型に関連した興味ある結果を得た。小西は、有効な予測モデルの構成の観点から、モデルの評価法の研究をした。梶原は、無限次元複素解析の立場から、極小曲面に近い場合のガウス写像の擬等角性を研究した。柳川は、多変量離散データ解析を研究し、疎な分割表解析、および非線形構造を持つ場合に成果を得た。三町は、球関数の立場から研究し、Macdonald多項式の積分表示について重要な結果を与えた。金子は、楕円曲線と保型形式の立場からの詳しい研究を行なった。 以上の各研究の結果、それぞれのアプローチの有効性、問題点をさらに明らかにできた。当面、代数的方法が重要となるであろうと予測される。

  • 共形場理論の可積分系への応用
    山田 泰彦
    日本学術振興会, 科学研究費助成事業, 重点領域研究, 九州大学, 1995 - 1995
    最近の研究により、共形場理論と関連する可解格子模型との関連が深く理解され、その様々な応用が現れてきている。なかでも、格子模型のスペクトルの粒子的構造を反映した、スピノン指標公式は、アフィン リー代数の表現論に新たな知見を与えるものとして興味深い。本研究では、中屋敷氏との共同で、このスピノン指標公式の一般化を行なった。 初めに、sl(2)の場合について、Bouwknegt Ludwing Schoutensにより予想されていた高いレベルのスピノン指標公式を証明した。もともとの共形場理論での定式化には、技術的に大きな困難が伴う。そのため我々は、粒子のなす代数のクリスタル理論的定式化を与え、これを用いることにより証明に成功した。 次に、上の結果をを対応するRSOS模型に拡張した。粒子の種類やその交換関係は、基本的に上と同じであることがわかり、我々のスピノン代数の有効性が示された。 最後に、sl(n)への拡張を試みた。最終的な証明には至っていないが、対応するスピノン指標公式の一般的予想を立てた。 関連して、格子模型の一次元状態和とKostka多項式との対応を明らかにし、アフィン リー代数の分岐関数に対するKirillowの予想を証明した。 得られたsl(n)の場合の予想の証明、および他のリー代数への拡張が今後の課題である。

  • ファノ多様体と超幾何函数の代数幾何的研究
    趙 康治, 若山 正人, 山田 泰彦, 吉田 正章, 塩浜 勝博, 梶原 壌二
    日本学術振興会, 科学研究費助成事業, 一般研究(C), 九州大学, 1994 - 1994
    (1)mixed Hodge structureと高次元におけるglobal residueを応用して、高次元の複素射影空間上のlocal systemに対する交点理論の構成の研究。 (2)層の複体のhypercohomologiesとLie環の表現、Verma moduleとの対応を調べることにより、超幾何微分方程式の量子化の導出のための指導原理を与える研究。 (3)Selberg型積分の組み合わせ論的側面を注目し、これらを統一的にあつかう研究。 (4)代数幾何で既知の各種の消滅定理とlocal systemsの(co)homology群の消滅定理間の対応関係を明確にすることにより、より一般的かつ有効な消滅定理の研究。 (5)量子群GL_q(n)上の定数係数微分作用素を導入し、それが持つ良い性質を示し、古典的な不変式論で重要な役割を果たしたCapelli恒等式の量子群版を得た。 (6)R.Howeによって提唱され、群の表現論、不変式論において重要な概念であるdual pairの理論を、最も簡単ではあるが非自明で重要な組(sl_2,on)に対し量子群類似を行った。あわせてこの組に付随したCapelli恒等式も得た。 (7)極小曲面のGauss Mapに関する藤本坦孝の結果を極小に近い曲面に量的に精密化する、笹倉の問題を解決し、その擬等角性を量的に表現し、平均曲率に関する評価式より擬等角性の評価式を導いた。 (8)自乗可積分な正則関数の作る関数空間の再生核としての核関数を、抽象Wiener空間の任意の領域に対して与え、Bergmanの核関数の無限次元化に成功した。

  • 位相的場の理論とミラー対称性
    山田 泰彦
    日本学術振興会, 科学研究費助成事業, 重点領域研究, 九州大学, 1994 - 1994
    N=2超共形場理論およびランダウ=ギンツブルグ模型に基づく位相的場の理論の立場からのカラビ=ヤオ多様体のミラー対称性が本研究の研究課題であった。 研究計画にあげた3つのテーマ、1.シグマ模型と可積分系、2.周期積分とミラー射像、3.BRSコホモロジーとA無限大代数、のうちでは、特に1.について大きな進展が得られた。 すなわち、シグマ模型として見た場合の標的空間の複素次元cが1未満の場合について、位相的場の理論の可積分構造を多い種数のリーマン面の寄与も含めて完全に解明することができた。具体的には、まず、多い種数のリーマン面への拡張を許すc<1の模型の分類を扱い、種数0で許される多くの解のうち、N=2超共形場理論による定式化を持つADE模型のみに制限されることをしめした。次に、幾何学的解釈をもつ模型、特に複素1次元(2次元)射影空間の場合に、高い種数における可積分構造を解明した。 テーマ2.については、複素1次元トーラスの場合の計算が進行中である。 本研究の結果、cが1以上の場合、位相的場の理論の可積分構造は高い種数で一般に任意性を持つことが明らかになった。これはマージナル作用素の存在に由来しており、 テーマ3.のBRSコホモロジーの構造を反映している。ミラー対称性では、 まさにこの任意性部分が本質的であり、この点の解明が将来の課題として残された。

  • Quantum Field Theories in Low Dimensions and Their Application
    INAMI Takeo, YAMADA Yasuhiko, ODAKE Satoru, ISHIBASHI Nobuyuki, DEGUCHI Tetsuo, FUJIKAWA Kazuo, MATSUO Yutaka, SASAKI Ryu, YANG Sung-kil
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Overseas Scientific Survey., Kyoto University, 1994 - 1994
    Many of the important physical phenomena and the methods to deal with them in particle physics and condensed matter physics are of nonperturbative nature and are difficult to solve. The same type of problems appears in physical systems in low dimensions. They are more easily accessible to rigorous methods. We intended to develop physical and mathematical methods in quantum field theories in low dimensions and to pursue their applications. Specifically we planed to investigate the following problems : 1) Construction of solvable lattice models/integrable field theories with boundaries and their application to condensed matter physics. 2) Construction of other new types of integrable field theories. 3) Non-perturbative methods for 2-dimensional quantum gravity. 4) Topological gravity. 5) Representation theories of W_* and W_N algebras and their application to condensed matter physics. 6) Application of the knot theory to statistical mechanics. 7) Nonperturbative methods in QCD and nonlinear sigma model in low dimensions. 8) Application of supersymmetry and anomalies to physics. We have made progress in the research of most of the problems written above. We give below only a few of them. Problems 1) We have constructed a few kinds of solvable lattice models and integrable field theories [8,17 of the references] and studied the stability problems in systems on a half line [18]. The Korean team (Nam and his collaborators) has a common interest with lnami and Sasaki on these problems and they have exchnged ideas. As an applicatication of the boundary CFT condensed matter physics, finite-size scaling spectrum in the Kondo problem has been derived [20]. Problems 5) : Progress has been made in the representation theoretic studies of W_* algebras [11-14], The Calogero-Sutherland type models has been studied from a representation theoretic viepoint of W_N algebras [15,16,19]. Nam has joined the discussion of the Japanese team (Odake, Matsuo). Problem 7) : This problem has been pursued mainly by the Korean team (Park and his collaborators) ; Inami has joined the discussion [7].

  • 共形場理論における相関関数の積分表示とその拡張
    山田 泰彦
    日本学術振興会, 科学研究費助成事業, 重点領域研究, 高エネルギー物理学研究所, 1993 - 1993
    1)位相的場の理論、および位相的重力理論は、リーマン面や各種のインスタントンのモジュライ空間の幾何を場の理論的に調べる手段を与える。これらの理論は、N=2超共形対称性、特異点理論、非線形可積分系等と結びつくことにより、様々な角度から、繊細な研究が可能となる。我々は、江口 徹、菅野 浩明、S.-K.Yang氏との共同研究で、これらの様々な理論の関係を調べてきた。主な成果は、次の二点である。 a)位相的場の理論を位相的重力理論に結合させた時に生ずる、重力的descendantsについて、その起源をN=2超共形対称性のBRST形式に基づいて明らかにした。 b)位相的重力理論の相関関数に対する、recursion relationと特異点理論のGauss-Manin方程式の関係を明らかにし、0点関数の周期積分公式を与えた。これらの成果により、少くとも種数0かつc<1に関する限り、上述のような様々な理論の関係が完全に明らかになった。 2)N=2超共形対称性とCalabi-Yau多様体の関係は、最近のmirror対称性の発見により、多くの研究者の注目するところとなった。これらの理論の具体的計算にあたっては、N=2Landau-Ginzburg模型が有効であることが知られている。元来N=2Landau-Ginzburg模型は、GrobalなN=2超対称性により特徴づけされており、LocalなN=2超共形対称性やCalabi-Yau多様体との関係は、必ずしも明確ではなかった。我々は、Wittenによる最近の研究に基づき、河合 俊哉、S.-K.Yang氏との共同研究でこの問題を考察した。すなわち、N=2超共形対称性と楕円的種数の一般的関係に基づき、N=2Landau-Ginzburg模型とCalabi-Yauシグマ模型の各々の楕円的種数の公式を与え、両者の比較をした。

  • 共形場理論における相関関数の積分表示とその拡張
    山田 泰彦, 石橋 延幸
    日本学術振興会, 科学研究費助成事業, 重点領域研究, 高エネルギー物理学研究所, 1992 - 1992
    q-変型アフィンリー代数U_q(sl_2^)のq-変型頂点演算子について、存在と一意性、2点関数、および交換関係について研究した。一昨年、q-変型アフィンリー代数に対応するq-変型頂点演算子がFrenkel-Reshetkhinによって導入され、その一般的性質が明らかにされた。我々は、q-変型アフィンリー代数U_q(sl_2^)に対して、このq-変型頂点演算子の存在と一意性を調べ、それらの2点関数、および交換関係を具体的に計算した。q-頂点演算子の存在と一意性については、q=1の場合と同様に、特にintegrable表現に対して,いくつかの結果が知られている。我々は、U_q(sl_2^)について知られている、singular vectorの具体的表示を用いることにより、integrable表現に限らず、一般のdegenerate表現について、q-変型頂点算子の存在条件を具体的に与えた。また、存在すれば一意的であることも示した。これは、q=1の場合に我々の結果の拡張である。我々はTsuchiya-Kanie理論にならって、U_q(sl_2^)で2点関数がq-超幾何関数に帰着される場合、すなわち、2つの外線spinのうちj=1/2である場合ついて、具体的に以下の計算を実行した。まずspin configuration(j1,j2,1/2,j4)に場合にq-KZ方程式を解き解をq-超幾何関数で具体的に与えた。次に、q-超幾何関数に接続公式を用いることにより、q-変型頂点演算子の交換関係を計算した。結果として交換関係に現れる行列が楕円的Yang-Baxter方程式の解(およびそのfusion)で与えられることを示した。q-KZ方程式の解は、積分表示を持つと一般に期待されている。U_q(sl_2^)の場合には、いくつかのグループにより、そのような表示が得られている。我々は、それらの一般化およびその接続問題への応用を試みている。

  • 超弦理論と基本相互作用
    小林 誠, 山田 泰彦, 九後 太一, 江口 徹, 東島 清, 米谷 民明
    日本学術振興会, 科学研究費助成事業, 重点領域研究, 高エネルギー物理学研究所, 1990 - 1990
    この研究では、10次元空間における超弦理論から我々の4次元時空の理論を導くことを目標にしている。残りの6次元部分がどのようなコンパクト空間であれば良いかについてはかなり明らかになってきたが、力学的にどのような機構でそのようなコンパクト化が起こるのか未だわかっていない。現在我々の持っている計算方法は、このような問題に答えるには不十分である事が明らかになってきた。特にコンパクト化などの非摂動効果の研究には、超弦理論を2次元場の理論と見たとき、2次元面のすべてのトポロジ-についてのたし上げが必要になるが、その技術が欠けている。一般の2次元場の理論の場合には難しいが、2次元面のトポロジ-だけで理論が決まっているような場合には足し上げられる可能性がある。そこで今年は主に2次元重力理論及びトポロジカル場の理論の研究を行った。特にどのような時に、非線形シグマ模型などの2次元場の理論がトポロジカルな理論になるかを調べた。 この研究は全国の研究者の協力によって成り立っている。今年度も平成2年12月18日から12月21日までの4日間にわたって、高エネルギ-物理学研究所において研究会を開催した。全国から百名を越える参加者があり活発な質議・討論がおこなわれた。 なお、この研究会の報告はKEK REPORTとして出版する予定である。

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