Updated on 2024/02/01

写真b

 
NOUMI Masatoshi
 
*Items subject to periodic update by Rikkyo University (The rest are reprinted from information registered on researchmap.)
Affiliation*
Graduate School of Science Master's Program in Mathematics
Graduate School of Science Doctoral Program in Mathematics
Title*
Specially Appointed Professor
Degree
理学修士 ( 上智大学 ) / 理学博士 ( 上智大学 )
Research Theme*
  • 可積分系や表現論を背景とする様々な特殊函数の代数解析的研究を行なっている。これまでの研究の主なテーマは、(a) 量子代数・アフィンヘッケ環の表現論とマクドナルド多項式、(b) パンルヴェ方程式の対称性とアフィンワイル群の双有理表現、(c) 楕円可積分系と楕円超幾何函数、などである。

  • Campus Career*
    • 4 2022 - Present 
      Graduate School of Science   Master's Program in Mathematics   Specially Appointed Professor
    • 4 2022 - Present 
      Graduate School of Science   Doctoral Program in Mathematics   Specially Appointed Professor
     

    Research Areas

    • Natural Science / Basic analysis

    Education

    • - 3 1983 
      上智大学 大学院理工学研究科数学専攻博士課程修了

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    Awards

    • 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

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      Award type:Honored in official journal of a scientific society, scientific journal  Country:United Kingdom

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    • 1 2013  
      The American Mathematical Society  The initial class of Fellows of the American Mathematical Society  -
       
      NOUMI MASATOSHI

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      Award type:International academic award (Japan or overseas)  Country:United States

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    Papers

    • Elliptic extension of Gustafson's q-integral of type G2 Peer-reviewed

      Masahiko Ito, Masatoshi Noumi

      Advances in Mathematics370   107211 - 107211   8 2020

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      Publishing type:Research paper (scientific journal)   Publisher:Elsevier BV  

      DOI: 10.1016/j.aim.2020.107211

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    • The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian

      Masatoshi Noumi, Simon Ruijsenaars, Yasuhiko Yamada

      Symmetry, Integrability and Geometry: Methods and Applications   8 7 2020

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      Publishing type:Research paper (scientific journal)   Publisher:SIGMA (Symmetry, Integrability and Geometry: Methods and Application)  

      DOI: 10.3842/sigma.2020.063

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    • A determinant formula associated with the elliptic hypergeometric integrals of type BCn Peer-reviewed

      Masahiko Ito, Masatoshi Noumi

      Journal of Mathematical Physics60 ( 7 ) 071705 - 071705   7 2019

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      Publishing type:Research paper (scientific journal)   Publisher:AIP Publishing  

      DOI: 10.1063/1.5094116

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    • Remarks on τ-functions for the difference Painlevé equations of type E8 Peer-reviewed

      Masatoshi NOUMI

      Advanced Studies inPure Mathematics Representation Theory, Special Functions and Painlevé Equations —RIMS 2015pp. 1–6576   2018

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      Language:English   Publishing type:Research paper (scientific journal)  

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    • Connection formula for the Jackson integral of type An and elliptic Lagrangeinterpolation Peer-reviewed

      M. Ito, Masatoshi NOUMI

      SIGMA14 ( No. 077, ) 42   2018

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      Language:English   Publishing type:Research paper (scientific journal)  

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    • EVALUATION OF THE BCn ELLIPTIC SELBERG INTEGRAL VIA THE FUNDAMENTAL INVARIANTS Peer-reviewed

      Masahiko Ito, Masatoshi Noumi

      PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY145 ( 2 ) 689 - 703   2 2017

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:AMER MATHEMATICAL SOC  

      We give a proof of the evaluation formula for the elliptic Selberg integral of type BCn as an application of the fundamental BCn-invariants.

      DOI: 10.1090/proc/13234

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    • Derivation of a BCn Elliptic Summation Formula via the Fundamental Invariants Peer-reviewed

      Masahiko Ito, Masatoshi Noumi

      CONSTRUCTIVE APPROXIMATION45 ( 1 ) 33 - 46   2 2017

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

      We give an alternative proof of an elliptic summation formula of type by applying the fundamental invariants to the study of Jackson integrals associated with the summation formula.

      DOI: 10.1007/s00365-016-9340-8

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    • Geometric aspects of Painleve equations Peer-reviewed

      Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada

      JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL50 ( 7 ) 073001 (164pp)   2 2017

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:IOP PUBLISHING LTD  

      In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painleve equations, with a particular emphasis on the discrete Painleve equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on. P-1 x P-1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

      DOI: 10.1088/1751-8121/50/7/073001

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    • Duality Transformation Formulas for Multiple Elliptic Hypergeometric Series of Type BC Peer-reviewed

      Yasushi Komori, Yasuho Masuda, Masatoshi Noumi

      CONSTRUCTIVE APPROXIMATION44 ( 3 ) 483 - 516   12 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

      New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type BC and of type C. Various transformation and summation formulas are derived as special cases to recover some previously known results.

      DOI: 10.1007/s00365-015-9316-0

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    • SOLUTIONS IN TERMS OF INTEGRALS OF MULTIVALUED FUNCTIONS FOR THE CLASSICAL HYPERGEOMETRIC EQUATIONS AND THE HYPERGEOMETRIC SYSTEM ON THE CONFIGURATION SPACE Peer-reviewed

      Katsuhisa Mimachi, Masatoshi Noumi

      KYUSHU JOURNAL OF MATHEMATICS70 ( 2 ) 315 - 342   9 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:KYUSHU UNIV, FAC MATHEMATICS  

      The systems of differential equations associated with the classical hypergeometric functions and the hypergeometric functions on the space of point configurations are investigated from the viewpoint of the twisted de Rham theory. In each case, it is proved that the integral of a certain multivalued function over an arbitrary twisted (or loaded) cycle satisfies the system of differential equations in question. The classical hypergeometric functions studied here include Appell's hypergeometric functions F-1, F-2, F-3, F-4, Lauricella's hypergeometric functions F-A, F-B, F-C, F-D, and the generalized hypergeometric function F-n+1(n).

      DOI: 10.2206/kyushujm.70.315

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    • A generalization of the Sears-Slater transformation and elliptic Lagrange interpolation of type BCn Peer-reviewed

      Masahiko Ito, Masatoshi Noumi

      ADVANCES IN MATHEMATICS299   361 - 380   8 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC ELSEVIER SCIENCE  

      The connection formula for the Jackson integral of type BCn is obtained in the form of a Sears-Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific bilateral multiple series. The coefficients of this expansion are expressed by certain elliptic Lagrange interpolation functions. Analyzing basic properties of the elliptic Lagrange interpolation functions, an explicit determinant formula is provided for a fundamental solution matrix of the associated system of q-difference equations. (C) 2016 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.aim.2016.05.016

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    • SOME TRANSFORMATION FORMULAS ASSOCIATED WITH ASKEY-WILSON POLYNOMIALS AND LASSALLE'S FORMULAS FOR MACDONALD-KOORNWINDER POLYNOMIALS Peer-reviewed

      A. Hoshino, M. Noumi, J. Shiraishi

      MOSCOW MATHEMATICAL JOURNAL15 ( 2 ) 293 - 318   4 2015

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:INDEPENDENT UNIV MOSCOW-IUM  

      We present a fourfold series expansion representing the Askey-Wilson polynomials. To obtain the result, a sequential use is made of several summation and transformation formulas for the basic hypergeometric series, including the Verma's q-extension of the Field and Wimp expansion, Andrews terminating q-analogue of Watson's F-3(2) sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type BC,, (n is an element of Z(>0)) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type B-n, C-n and D-n with one row diagram, thereby proving his conjectures.

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    • Tableau Formulas for One-Row Macdonald Polynomials of Types C-n and D-n Peer-reviewed

      Boris Feigin, Ayumu Hoshino, Masatoshi Noumi, Jun Shibahara, Jun'ichi Shiraishi

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS11 ( 100 ) 21 pages   2015

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:NATL ACAD SCI UKRAINE, INST MATH  

      We present explicit formulas for the Macdonald polynomials of types C-n and D-n in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced W-12(11) series. We remark that the correlation functions of the deformed W algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.

      DOI: 10.3842/SIGMA.2015.100

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    • Pade Interpolation and Hypergeometric Series Peer-reviewed

      Masatoshi Noumi

      ALGEBRAIC AND ANALYTIC ASPECTS OF INTEGRABLE SYSTEMS AND PAINLEVE EQUATIONS651   1 - 23   2015

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:AMER MATHEMATICAL SOC  

      We propose a class of Fade interpolation problems whose general solution is expressible in terms of determinants of hypergeometric series.

      DOI: 10.1090/conm/651/13034

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    • Padé interpolation for elliptic painlevé equation Peer-reviewed

      Masatoshi Noumi, Satoshi Tsujimoto, Yasuhiko Yamada

      Springer Proceedings in Mathematics and Statistics40 ( 40 ) 463 - 482   2013

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:Springer New York LLC  

      An interpolation problem related to the elliptic Painlevé equation is formulated and solved. A simple form of the elliptic Painlevé equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given. © Springer-Verlag London 2013.

      DOI: 10.1007/978-1-4471-4863-0_18

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    • Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications Peer-reviewed

      Yasushi Komori, Masatoshi Noumi, Jun'ichi Shiraishi

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS5   2009

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:NATL ACAD SCI UKRAINE, INST MATH  

      A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.

      DOI: 10.3842/SIGMA.2009.054

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    • Special functions arising from discrete Painleve equations: A survey Peer-reviewed

      Masatoshi Noumi

      JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS202 ( 1 ) 48 - 55   5 2007

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

      This article is a survey on recent studies on special solutions of the discrete Painleve equations, especially on hypergeometric solutions of the q-Painleve equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painleve equations, IMRN 2004 47 (2004) 2497-252], K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painleve equations, IMRN 2005 24 (2005) 1439-1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painleve equations for comparison, we give an overview of the present status of studies on difference (discrete) Painleve equations as a source of special functions. (c) 2006 Elsevier B.V. All rights reserved.

      DOI: 10.1016/j.cam.2006.01.034

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    • Point configurations, Cremona transformations and the elliptic difference Painlevé equation Peer-reviewed

      NOUMI Masatoshi, KAJIWARA Kenji, MASUDA Tetsu, OHTA Yasuhiro, YAMADA Yasuhiko

      Séminaires et CongrèsVol 14   pp. 169 - 198   2006

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    • Construction of hypergeometric solutions to the q-Painleve equations

      K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada

      INTERNATIONAL MATHEMATICS RESEARCH NOTICESvol 24, pp 1439~1463 ( 24 ) 1439 - 1463   5 2005

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:HINDAWI PUBLISHING CORPORATION  

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    • $q$-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

      数理解析研究所講究録1422   77 - 98   4 2005

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      Publisher:京都大学  

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    • Cubic Pencils and Painleve Hamiltonians

      K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada

      FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA48 ( 1 ) 147 - 160   4 2005

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:KOBE UNIV, DEPT MATHEMATICS  

      We present a simple heuristic method to derive the Painleve differential equations from the corresponding geometry of rational surfaces

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    • We present a simple heuristic method to derive the Painleve differential equations from the corresponding geometry of rational surfacesPainleve equation, Hamiltonian, Cubic pencil, Seiberg Witten curveCubic Pencils and Painlevé Hamiltonians Peer-reviewed

      K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada

      Funkcialaj Ekvacioj48 ( 1 ) 147 - 160   2005

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      Language:English   Publishing type:Research paper (scientific journal)  

      We present a simple heuristic method to derive the Painleve differential equations from the corresponding geometry of rational surfaces. © 2005, Division of Functional Equations, The Mathematical Society of Japan. All rights reserved.

      DOI: 10.1619/fesi.48.147

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    • 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

      数理解析研究所講究録1400   197 - 263   10 2004

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      Publisher:京都大学  

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    • Hypergeometric solutions to the q-painleve equations Peer-reviewed

      K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada

      INTERNATIONAL MATHEMATICS RESEARCH NOTICES ( 47 ) 2497 - 2521   2004

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:HINDAWI PUBLISHING CORPORATION  

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    • Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. Peer-reviewed

      NOUMI Masatoshi, YAMADA, Y

      Advanced Studies in Pure Mathematics.40, 371-442.   2004

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    • Askey-Wilson polynomials: An affine Hecke algebra approach Peer-reviewed

      M Noumi, JV Stokman

      LAREDO LECTURES ON ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONSpp.111-144   111 - 144   2004

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:NOVA SCIENCE PUBLISHERS, INC  

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    • Multiple elliptic hypergeometric series. An approach from the Cauchy determinant

      Y Kajihara, M Noumi

      INDAGATIONES MATHEMATICAE-NEW SERIES14 ( 3-4 ) 395 - 421   12 2003

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

      A multiple generalization of elliptic hypergeometric series is studied through the Cauchy determinant for the Weierstrass sigma function. In particular, a duality transformation for multiple hypergeometric series is proposed. As an application, two types of Bailey transformations for very well-poised multiple elliptic hypergeometric series are derived.

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    • Monodromy groups of hypergeometric functions satisfying algebraic equations

      M Kato, M Noumi

      TOHOKU MATHEMATICAL JOURNAL55 ( 2 ) 189 - 205   6 2003

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:TOHOKU UNIVERSITY  

      The solutions of the algebraic equation y(mn) + xy(mp) - 1 = 0 with n > p and m greater than or equal to 2 satisfy a generalized hypergeometric differential equation with imprimitive finite irreducible monodromy group. Thanks to this fact, we can determine the monodromy group and the Schwarz map of the differential equation.

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    • E-10(9) solution to the elliptic Painleve equation

      K Kajiwara, T Masuda, M Noumi, Y Ohta, Y Yamada

      JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL36 ( 17 ) L263 - L272   5 2003

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:IOP PUBLISHING LTD  

      A tau function formalism for Sakai's elliptic Painleve equation is presented. This establishes the equivalence between-the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painleve equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painleve equation expressed in terms of the elliptic hypergeometric function E-10(9).

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    • q-Painlev\'e systems arising from q-KP hierarchy

      Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada

          28 12 2001

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      A system of q-Painlev\&#039;e type equations with multi-time variables t_1,...,t_M<br />
      is obtained as a similarity reduction of the N-reduced q-KP hierarchy. This<br />
      system has affine Weyl group symmetry of type A^{(1)}_{M-1} \times<br />
      A^{(1)}_{N-1}. Its rational solutions are constructed in terms of q-Schur<br />
      functions.

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    • Discrete dynamical systems with $W(A^{(1)}_{m-1} \times A^{(1)}_{n-1})$ symmetry

      Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada

          19 6 2001

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      We give a birational realization of affine Weyl group of type $A^{(1)}_{m-1}<br />
      \times A^{(1)}_{n-1}$. We apply this representation to construct some discrete<br />
      integrable systems and discrete Painlev\&#039;e equations. Our construction has a<br />
      combinatorial counterpart through the ultra-discretization procedure.

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    • Painleve方程式の対称性

      野海 正俊, 山田 泰彦

      数学53 ( 1 ) 62 - 75   30 1 2001

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      Language:Japanese   Publisher:社団法人日本数学会  

      DOI: 10.11429/sugaku1947.53.62

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      Other Link: http://www.lib.kobe-u.ac.jp/handle_kernel/90003250

    • Tableau representation for Macdonald's ninth variation of Schur functions Peer-reviewed

      J Nakagawa, M Noumi, M Shirakawa, Y Yamada

      PHYSICS AND COMBINATORICS   180 - 195   2001

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:WORLD SCIENTIFIC PUBL CO PTE LTD  

      The ninth variation of Schur functions associated with a general N x N matrix is studied. In particular, two types of tableau representations for them are constructed by using decomposition of matrices and properties of minor determinants.

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    • Birational Weyl group action arising from a nilpotent Poisson algebra Peer-reviewed

      M Noumi, Y Yamada

      PHYSICS AND COMBINATORICS 1999   287 - 319   2001

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:WORLD SCIENTIFIC PUBL CO PTE LTD  

      We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra.

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    • A study on the fourth q-Painlev\'e equation

      Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada

          28 12 2000

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      A q-difference analogue of the fourth Painlev\&#039;e equation is proposed. Its<br />
      symmetry structure and some particular solutions are investigated.

      DOI: 10.1088/0305-4470/34/41/312

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    • Determinant Formulas for the Toda and Discrete Toda Equations

      Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada

          18 8 1999

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      Determinant formulas for the general solutions of the Toda and discrete Toda<br />
      equations are presented. Application to the $\tau$ functions for the Painlev\&#039;e<br />
      equations is also discussed.

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    • Symmetries in the fourth Painleve equation and Okamoto polynomials Peer-reviewed

      M Noumi, Y Yamada

      NAGOYA MATHEMATICAL JOURNAL153   53 - 86   3 1999

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:NAGOYA UNIV  

      The fourth Painleve equation P-IV is known to have symmetry of the affine Weyl group of type A(2)((1)) with respect to the Backlund transformations. We introduce a new representation of P-IV, called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of P-IV is given in terms of this representation. Through the symmetric form, it turns out that P-IV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions P-IV, called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.

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    • Affine Weyl groups, discrete dynamical systems and Painleve equations Peer-reviewed

      M Noumi, Y Yamada

      COMMUNICATIONS IN MATHEMATICAL PHYSICS199 ( 2 ) 281 - 295   12 1998

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER VERLAG  

      A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.

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    • Umemura polynomials for the Painleve V equation Peer-reviewed

      M Noumi, Y Yamada

      PHYSICS LETTERS A247 ( 1-2 ) 65 - 69   10 1998

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

      We give an explicit formula for the one-parameter family of special polynomials associated with the fifth Painleve equation (Pv) in terms of the Schur functions. Remarkably, our formulae coincide with those for the discrete Painleve II equation (dP(II)), where the roles of the independent variable and the parameter are exchanged. (C) 1998 Elsevier Science B.V.

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    • A reproducing kernel for nonsymmetric Macdonald polynomials Peer-reviewed

      K Mimachi, M Noumi

      DUKE MATHEMATICAL JOURNAL91 ( 3 ) 621 - 634   2 1998

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:DUKE UNIV PRESS  

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    • An integral representation of eigenfunctions for Macdonald's q-difference operators Peer-reviewed

      K Mimachi, M Noumi

      TOHOKU MATHEMATICAL JOURNAL49 ( 4 ) 517 - 525   12 1997

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:TOHOKU UNIVERSITY  

      We give the eigenfunctions for Macdonald's q-difference operators in terms of q-Selberg type integrals. Our result can be applied not only to the case of Macdonald symmetric polynomials but also to the cases of rational and meromorphic solutions.

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    • Finite dimensional representations of the quantum group GL_q(n;C) and the zonal spherical functions on U_q(n-1)\U_q(n)

      NOUMI Masatoshi, YAMADA Hirofumi, MIMACHI Katsuhisa

      Japanese journal of mathematics. New series19 ( 1 ) 31 - 80   6 1993

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      Language:English   Publisher:The Mathematical Society of Japan  

      DOI: 10.4099/math1924.19.31

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    • SPHERICAL-FUNCTIONS ON A FAMILY OF QUANTUM 3-SPHERES Peer-reviewed

      M NOUMI, K MIMACHI

      COMPOSITIO MATHEMATICA83 ( 1 ) 19 - 42   7 1992

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:KLUWER ACADEMIC PUBL  

      A quantum space M which can be regarded as a total space of a family of quantum 3-spheres is introduced. By the action of the quantum group SU(q)(2), its algebra of functions is decomposed into irreducible components. Its spherical functions are explicitly described in terms of the big q-Jacobi polynomials and the q-Hahn polynomials.

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    • ASKEY-WILSON POLYNOMIALS AS SPHERICAL-FUNCTIONS ON SUQ(2) Peer-reviewed

      M NOUMI, K MIMACHI

      LECTURE NOTES IN MATHEMATICS1510   97 - 103   1992

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

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    • ASKEY-WILSON POLYNOMIALS AS SPHERICAL-FUNCTIONS ON SUQ(2) Peer-reviewed

      M NOUMI, K MIMACHI

      QUANTUM GROUPS1510   97 - 103   1992

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:SPRINGER-VERLAG BERLIN  

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    • REPRESENTATIONS OF THE QUANTUM GROUP SUQ(2) AND THE LITTLE Q-JACOBI POLYNOMIALS Peer-reviewed

      T MASUDA, K MIMACHI, Y NAKAGAMI, M NOUMI, K UENO

      JOURNAL OF FUNCTIONAL ANALYSIS99 ( 2 ) 357 - 386   8 1991

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS  

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    • ROGERS Q-ULTRASPHERICAL POLYNOMIALS ON A QUANTUM 2-SPHERE Peer-reviewed

      M NOUMI, K MIMACHI

      DUKE MATHEMATICAL JOURNAL63 ( 1 ) 65 - 80   6 1991

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      Language:French   Publishing type:Research paper (scientific journal)   Publisher:DUKE UNIV PRESS  

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    • ASKEY-WILSON POLYNOMIALS AND THE QUANTUM GROUP SUQ(2) Peer-reviewed

      M NOUMI, K MIMACHI

      PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES66 ( 6 ) 146 - 149   6 1990

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:JAPAN ACAD  

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    • BIG Q-JACOBI POLYNOMIALS, Q-HAHN POLYNOMIALS, AND A FAMILY OF QUANTUM 3-SPHERES Peer-reviewed

      M NOUMI, K MIMACHI

      LETTERS IN MATHEMATICAL PHYSICS19 ( 4 ) 299 - 305   5 1990

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:KLUWER ACADEMIC PUBL  

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    • Unitary representations of the quantum group SUq(1, 1): II - Matrix elements of unitary representations and the basic hypergoemetric functions Peer-reviewed

      Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, Yutaka Saburi, Kimio Ueno

      Letters in Mathematical Physics19 ( 3 ) 195 - 204   4 1990

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:Kluwer Academic Publishers  

      Some series of unitary representations of the quantum group SUq(1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SUq(1, 1) and a q-analogue of some classical identities are discussed. © 1990 Kluwer Academic Publishers.

      DOI: 10.1007/BF01039312

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    • Unitary representations of the quantum group SUq(1,1): Structure of the dual space of Uq(sl(2)) Peer-reviewed

      Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, Yutaka Saburi, Kimio Ueno

      Letters in Mathematical Physics19 ( 3 ) 187 - 194   4 1990

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:Kluwer Academic Publishers  

      Real forms of the quantum universal enveloping algebra Uq(sl(2)) and a topological quantum group associated with this algebra are discussed. © 1990 Kluwer Academic Publishers.

      DOI: 10.1007/BF01039311

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    • QUANTUM 2-SPHERES AND BIG Q-JACOBI POLYNOMIALS Peer-reviewed

      M NOUMI, K MIMACHI

      COMMUNICATIONS IN MATHEMATICAL PHYSICS128 ( 3 ) 521 - 531   1990

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER VERLAG  

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    • MATRIX-ELEMENTS OF UNITARY REPRESENTATION OF THE QUANTUM GROUP SUQ (1,1) AND THE BASIC HYPERGEOMETRIC-FUNCTIONS Peer-reviewed

      K UENO, T MASUDA, K MIMACHI, Y NAKAGAMI, M NOUMI, Y SABURI

      DIFFERENTIAL GEOMETRIC METHODS IN THEORETICAL PHYSICS245   331 - 343   1990

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:PLENUM PRESS DIV PLENUM PUBLISHING CORP  

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    • ZONAL SPHERICAL-FUNCTIONS ON THE QUANTUM HOMOGENEOUS SPACE SUQ(N+1)/SUQ(N) Peer-reviewed

      M NOUMI, H YAMADA, K MIMACHI

      PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES65 ( 6 ) 169 - 171   6 1989

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    • REPRESENTATIONS OF QUANTUM GROUPS AND A Q-ANALOGUE OF ORTHOGONAL POLYNOMIALS Peer-reviewed

      T MASUDA, K MIMACHI, Y NAKAGAMI, M NOUMI, K UENO

      COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE307 ( 11 ) 559 - 564   10 1988

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:GAUTHIER-VILLARS  

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    • On the structure of cohomology groups attached to the integral of certain many-valued analytic functions

      KITA Michitake, NOUMI Masatoshi

      Japanese journal of mathematics. New series9 ( 1 ) 113 - 157   1983

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      Language:English   Publisher:The Mathematical Society of Japan  

      DOI: 10.4099/math1924.9.113

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    Misc.

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    Books and Other Publications

    • Special Issue on Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems

      N.~Joshi, E.~Koelink, T.~Koornwinder, M.~Noumi, S.~Ruijsenaars, F, Smirnov, J.~Stokman( Role: Joint editor)

      SIGMA  4 2012 

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      Language:English Book type:Other

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    • Elliptic Integrable Systems

      NOUMI Masatoshi, TAKASAKI K( Role: Joint author)

      Rokko Lectures in Mathematics  2005 

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      Language:English Book type:Scholarly book

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    • Painleve equations through symmetry

      NOUMI Masatoshi

      American Mathematical Society  2004 

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      Language:English Book type:Scholarly book

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    • Askey-Wilson polynomails: an affine Hecke algebra approach, in Laredo Lectures on Orthogonal Polynomials and Special Functions, R.Alvarez-Nodarse et al, Eds. Pp.111-144

      NOUMI Masatoshi, STOKMAN J V( Role: Joint author)

      Nova Science Publishers  2004 

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      Language:English Book type:Scholarly book

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    Presentations

    • Ruijsenaars 系の固有函数:現状と課題 Invited

      Masatoshi NOUMI

      表現論と特殊函数セミナー  2 2019 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:於琉球大学理学部  

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    • Elliptic hypergeometric integrals and associated determinant formulas Invited International conference

      Masatoshi NOUMI

      13th International Conference on Symmetry and Integrability of Difference Equations (SIDE 13)  11 2018 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Fukuoka, Japan  

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    • G2 型 Gustafson q-ベータ積分の楕円化とその無限積表示について

      伊藤雅彦, Masatoshi NOUMI

      日本数学会2018年秋季総合分科会  9 2018 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:於岡山大学, 津島キャンパス  

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    • Elliptic hypergeometric integrals Invited International conference

      Masatoshi NOUMI

      Summer Research Institute on q-Series  8 2018 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Nankai Univerisity,Tianjin, P.R. China  

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    • Elliptic hypergeometric functions and elliptic difference Painlevé equation Invited International conference

      Masatoshi NOUMI

      XIX International Congress on Mathmatical Physics  7 2018 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Montreal, Canada  

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    • 楕円ガンマ函数と楕円超幾何函数 Invited

      Masatoshi NOUMI

      多重三角函数とその一般化  3 2018 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:神戸大学  

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    • G₂型Weyl 群不変なq 超幾何積分の行列式公式

      伊藤雅彦, Aiko MIYANAGA, Masatoshi NOUMI

      日本数学会2018 年度年会  3 2018 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:東京大学駒場キャンパス  

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    • BCn 型補間函数の話題から Invited

      Masatoshi NOUMI

      琉球超幾何ワークショップ  2 2018 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:琉球大学理学部  

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    • Selberg 型楕円超幾何積分 Invited

      Masatoshi NOUMI

      複素微分方程式の楽しみ  11 2017 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:熊本大学理学部  

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    • On variations of Selberg integrals Invited International conference

      Masatoshi NOUMI

      KTH Theoretical Physics Seminar  11 2017 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Stockholm (Sweden)  

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    • 楕円超幾何積分に付随する行列式公式 Invited

      Masatoshi NOUMI

      琉球大学理学部数学教室談話会  9 2017 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:琉球大学  

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    • Selberg 型BCn 楕円超幾何積分の行列式公式

      伊藤雅彦, Masatoshi NOUMI

      日本数学会2016 年度秋季総合分科会  9 2017 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:山形大学  

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    • Elliptic difference Painlevé equation and elliptic hypergeometric integrals Invited International conference

      Masatoshi NOUMI

      XXVth International Conference on Integrable Systems and Quantum Symmetries  6 2017 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Czech Technical University(Prague, Czech Republic)  

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    • Discrete Painlevé equations and special functions Invited International conference

      Noumi Masatoshi

      ESI workshop: “Elliptic Hypergeometic Func-tions in Combinatorics, Integrable Systems and Physics”  3 2017  Erwin Schrödinger Institute

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Vienna, Austria  

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    • Elliptic hypergeometric integrals and elliptic interpolationfunctions Invited International conference

      Noumi Masatoshi

      BIRS workshop:“Painelevé Equations and Discrete Dynamics”  10 2016  Banff International Reseach Station

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Banff, Canada  

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    • 楕円超幾何積分と楕円補間函数 Invited

      Noumi Masatoshi

      RIMS研究集会「可積分系数理の現状と展望」  9 2016  RIMS

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:京都大学北部総合研究科棟・益川ホール  

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    • Gustafson-Rkha和公式の楕円化

      Noumi Masatoshi, Ito Masahiko

      日本数学会 2016 年度秋季総合分科会・一般講演  9 2016 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:Kansai Universuty  

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    • Macdonald 多項式 – アフィン Hecke 環からのアプローチ Invited

      Noumi Masatoshi

      超幾何学校2016  8 2016  Kobe University

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:Kobe University  

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    • A 型楕円 Lagrange 補間函数の構成法

      ITO Masahiko, NOUMI Masatoshi

      日本数学会2016年度年会・一般講演  3 2016 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:筑波大学  

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    • A型Jackson積分とRamanujan ${}_{1}\psi_{1}$ 変換公式の一般化

      ITO Masahiko, NOUMI Masatoshi

      日本数学会2016年度年会・一般講演  3 2016 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:筑波大学  

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    • q Selberg 積分と楕円 Selberg 積分 Invited

      NOUMI Masatoshi

      研究集会「微分方程式の総合的研究」  12 2015 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:東京大学  

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    • 微分方程式の双対性と双線型恒等式 Invited

      NOUMI Masatoshi

      研究会「不変性と双対性」  9 2015 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:鹿児島大学理学部  

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    • Sears-Slater の変換公式の一般化と$BC_n$ 型楕円 Lagrange 補間函数

      ITO Masahiko, NOUMI Masatoshi

      日本数学会2015年度秋季総合分科会  9 2015 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:京都産業大学  

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    • 一行型 $C_n$ 型 Macdonald 多項式のタブロー和表示

      NOUMI Masatoshi, HOSHINO Wataru, SHIBAHARA Atushi, SHIRAISHI Jyunichi

      日本数学会2015年度年会  3 2015 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:明治大学  

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    • Special functions arising from elliptic integrable systems Invited International conference

      NOUMI Masatoshi

      International Workshop "Representation Theory, Special Functions and Painlevé Equations"  3 2015 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:RIMS, Kyoto  

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    • BC型楕円多重和公式の基本不変式による導出

      ITO Masahiko, NOUMI Masatoshi

      日本数学会2015年度年会  3 2015 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:明治大学  

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    • BC型多重楕円超幾何級数に対する双対変換公式

      NOUMI Masatoshi, KOMORI Yasushi, MASUDA Yasuho

      日本数学会2015年度年会  3 2015 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:明治大学  

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    • Affine Hecke algebras and nonsymmetric kernel functions Invited International conference

      NOUMI Masatoshi

      International Workshop Integrable Systems and Representation Theory  3 2015 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Tokyo University of Marine Science and Technology  

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    • Elliptic difference Painlevé equation Invited International conference

      NOUMI Masatoshi

      Kobe-Leuven workshop on Geometry and Integrable Systems  7 2014 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:KU Leuven, Belgium.  

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    • Affine Weyl groups Invited International conference

      NOUMI masatoshi

      Painlevé equations and special functions: I, II, The 6th TIMS-OCAMIWASEDAJoint International Workshop on Integrable Systems and Mathematical Physics  3 2014  National Taiwan University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Taipei, Taiwan  

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    • Padé approximations and hypergemetric seris Invited International conference

      NOUMI masatoshi

      2014 Joint Mathematical Meeting, AMS SpecialSession: Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations  1 2014  Baltimore Convention Center

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Baltimore, U.S.A.  

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    • Discrete Painlevé equations and special functions Invited International conference

      NOUMI masatoshi

      57th Annual Meeting of the Australian Mathematical Society  10 2013  Universtiy of Sydney

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Sydney, Australia  

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    • Remarks on $BC_n$ interpolation functions Invited International conference

      NOUMI masatoshi

      JSPS-NWO Seminar: Analysis, Geometry and GroupRepresentations for Homogeneous Spaces  8 2013  Nagoya University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Nagoya, Japan  

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    • Elliptic Askey-Wilson functions and associated elliptic Schur functions Invited International conference

      NOUMI masatoshi

      Integrable Systems and Mathematical Physics Seminar  7 2013  University of Glasgow

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:U.K.  

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    • Elliptic Askey-Wilson functions and associated elliptic Schur functions Invited International conference

      NOUMI masatoshi

      Elliptic Integrable Systems and Hypergeometric Functions  7 2013  Lorentz Center

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Leiden, The Netherlands  

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    • An elliptic extension of Askey-Wilson polynomials and associated elliptic Schur functions Invited International conference

      NOUMI Masatoshi

      {em Bethe Ansatz, Quantum Groups and Beyond}  3 2013 

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:RIMS, Kyoto University, Kyoto, Japan  

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    • Ruijsenaars-Macdonald q 差分作用素の双スペクトル問題

      NOUMI MASATOSHI, SHIRAISHI ZYUNICHI

      日本数学会 2012年度秋季総合分科会  9 2012 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:九州大学  

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    • 超幾何函数の楕円化とその周辺 Invited

      NOUMI Masatoshi

      第10回岡シンポジウム  12 2011  奈良女子大学

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:奈良女子大学  

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    • Padé interpolation and hypergeometric series Invited International conference

      NOUMI Masatoshi

      Geometry Seminar  8 2011  University of Sydney

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Sydney  

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    • Elliptic Schur functions Invited International conference

      NOUMI Masatoshi

      Integrable Systems Seminar  8 2011  University of Sydney

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Sydney  

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    • Birational Weyl group action on the space of matrices and a discrete integrable system Invited International conference

      NOUMI Masatoshi

      Algebraic and geometric aspects of discrete integrable systems: integrable systems and cluster algebras  12 2010  Graduate School of Mathematical Sciences, University of Tokyo

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, Japan  

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    • Some remarks on elliptic Schur functions Invited International conference

      NOUMI Masatoshi

      Analysis, Geometry and Group Representations for Homogeneous Spaces  11 2010  Lorentz Center

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Lorentz Center, Leiden, The Netherlands  

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    • 楕円 Schur 函数 Invited

      NOUMI Masatoshi

      研究集会「BC 系と AGT 予想の周辺」  9 2010  東京大学院数理科学研究科

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:東京大学大学院数理科学研究科  

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    • BC_m 型 Ruijsenaars 差分作用素と積分変換から生じる固有函数 Invited

      NOUMI Masatoshi

      研究集会「BC 系と AGT 予想の周辺」  9 2010  東京大学院数理科学研究科

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:東京大学大学院数理科学研究科  

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    • Discrete Painlevé equations and special functions Invited International conference

      NOUMI Masatoshi

      ICM 2010 Satellite Conference "Integrable Systems and Geometry"  8 2010  Pondicherry University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Pondicherry University, Puducherry, India  

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    • Affine Weyl groups and Painlevé equations Invited International conference

      NOUMI Masatoshi

      Guest Lecture at Summer School on "Classical and Quantum Integrable Models''  7 2010  University of Kent

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:SMSAS, University of Kent, U.K.  

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    • Affine Hecke algebra approach to Koornwinder polynomials and van Diejen's q-difference operators Invited International conference

      NOUMI Masatoshi

      Alegebra, Geometry and Integrable Systems Colloquium  7 2010  School of Mathematics, University of Leeds

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:School of Mathematics, University of Leeds, U.K.  

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    • 可換差分作用素と核函数 Invited

      NOUMI Masatoshi

      日本数学会2010年度年会・特別講演  3 2010  日本数学会

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:慶應大学  

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    • トロピカル RSK 対応 Invited

      NOUMI Masatoshi

      トロピカル幾何と超離散系の展開  3 2010  京都大学・工学部

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:京都大学・工学部  

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    • Padé補間と超幾何級数 Invited

      NOUMI Masatoshi

      数理解析の諸問題とその展望 ― 宮川鉄朗先生を追悼して ―  2 2010 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:金沢大学サテライト・プラザ  

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    • Some remarks on $\tau$-functions for the elliptic difference Painlevé equation Invited International conference

      NOUMI Masatoshi

      China-Japan Joint Workshop on Integrable Systems on the occasion of Prof. Hirota's 77th Birthday  1 2010  Shiaoxing University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Shiaoxing University, Shiaoxing(紹興), China  

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    • Kernel functions for van Diejen's q-difference operators Invited International conference

      NOUMI Masatoshi

      CTQM/QGM Workshop: q-Representa-tion Theory  12 2009  Department of Mathematical Sciences, Aarhus University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Department of Mathematical Sciences, Aarhus University, Denmark  

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    • 微分環と差分環の話題から Invited

      NOUMI Masatoshi

      JST CREST Gröbner School  9 2009  神戸大学理学部数学教室

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:神戸大学理学部数学教室  

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    • The elliptic Painlevé equation, I, II, III International conference

      NOUMI Masatoshi

      Discrete Integrable Systems Seminar  6 2009  Isaac Newton Institute for Mathematical Sciences

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Isaac Newton Institute for Mathematical Sciences  

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    • Darboux transformations for elliptic Hirota-Miwa equations Invited International conference

      NOUMI Masatoshi, Jonathan Nimmo

      Integrable Systems and Mathematical Physics Seminar  6 2009  University of Glasgow

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Glasgow  

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    • Some determinantal identities for generalized hypergeometric functions Invited International conference

      NOUMI Masatoshi

      Workshop: Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems  4 2009  University of Glasgow

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Glasgow, Glasgow, UK  

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    • Mellin 変換と接続問題 Invited

      NOUMI Masatoshi

      Workshop on Accessory Parameters at Kumamoto  2 2009  熊本大学

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:熊本大学  

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    • (常)微分方程式の双対性と双線型恒等式 Invited

      NOUMI Masatoshi

      Workshop on Accessory Parameters at Kumamoto  2 2009  熊本大学

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:熊本大学  

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    • 常微分方程式の解の接続問題について Invited

      NOUMI Masatoshi, KISHIOKA Hiroyuki

      超幾何関数研究会 2009  1 2009  神戸大学

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:神戸大学  

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    • Multiple elliptic hypergeometric series and difference operators of Ruijsenaars type Invited International conference

      NOUMI Masatoshi

      Workshop: Elliptic integrable systems, isomonodromy problems and hypergeometric functions  7 2008  MPIM, Bonn, Germany

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:MPIM, Bonn, Germany  

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    • Symmetries of Painlevé equations and special polynomials Invited International conference

      NOUMI Masatoshi

      International Conference: From Painlev'e to Okamoto  6 2008  University of Tokyo

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      Venue:University of Tokyo, Japan  

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    • Some remarks on tau-functions for discrete Painlevé equations Invited International conference

      NOUMI Masatoshi

      8th International Conference on Symmetries and Integrability of Difference Equations (SIDE 8)  6 2008  Sainte-Adèle, Canada

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Sainte-Adèle, Canada  

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    • Koornwinder 多項式のq差分作用素の核函数とその応用

      NOUMI Masatoshi, 白石 潤一

      日本数学会2007年度秋季総合分科会  9 2007  日本数学会

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:東北大学  

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    • Kernel functions for Koornwinder's q-difference operators Invited International conference

      NOUMI Masatoshi

      Integrable Systems: Linear and Nonlinear Dynamics (ISLAND 3)  7 2007  Department of Mathematics, University of Glasgow, Isle of Islay,UK

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Machrie Hotel, Isle of Islay,UK  

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    • 楕円差分パンルベ方程式と楕円超幾何積分

      NOUMI Masatoshi

      東京無限可積分系セミナー  11 2005 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:東京大学大学院数理科学研究科  

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    • 楕円差分パンルベ方程式と楕円超幾何積分

      NOUMI Masatoshi

      京都大学理学研究科数学教室談話会  6 2005 

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      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:京都大学  

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    • Tropical Robinson-Schensted-Knuth Correspondence. International conference

      NOUMI Masatoshi

      International Workshop on :Tropical combinatorics and Tropical Algebraic Geometry, RIMS  8 2004  未記入

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:RIMS, Kyoto University  

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    • Elliptic difference Painleve equation and its hypergeometric solutions. International conference

      NOUMI Masatoshi

      International Conference on Asymptotic Theories and Painleve Equations.  6 2004  未記入

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Angers, France  

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    • Discrete Painleve equations with affine Weyl group symmetry of type E8. International conference

      NOUMI Masatoshi

      SIDE6(Symmetry and Integrability of Difference Equations, Ⅵ)  6 2004  未記入

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Helsinki, Finland  

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    Teaching Experience

    •  
      微分積分2【再履修者】 ( 神戸大学 )

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      微分積分2 ( 神戸大学 )

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      微分積分1【再履修者】 ( 神戸大学 )

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      微分積分1 ( 神戸大学 )

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      先端融合科学特論B(理学) ( 神戸大学 )

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      神戸大学の研究最前線A ( 神戸大学 )

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      関数方程式論Ⅰ ( 神戸大学 )

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      解析数理特論4 ( 神戸大学 )

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      解析数理特論Ⅳ ( 神戸大学 )

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      解析学2 ( 神戸大学 )

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      解析学Ⅱ ( 神戸大学 )

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      解析学VIII ( 神戸大学 )

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      Topics in Analysis Ⅳ ( Kobe University )

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      Topics in Analysis 4 ( Kobe University )

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      Functional Equations I ( Kobe University )

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      Forefront of Research at Kobe University A ( Kobe University )

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      Calculua B2【Re-entry】 ( Kobe University )

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      Calculua B2 ( Kobe University )

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      Calculua B1【Re-entry】 ( Kobe University )

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      Calculua B1 ( Kobe University )

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      Analysis Ⅱ ( Kobe University )

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      Analysis VIII ( Kobe University )

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      Analysis 2 ( Kobe University )

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    Research Projects

    • タウ函数の特異極限における新しいソリトン方程式系とその応用

      科学研究費補助金/基盤研究(B) 

      太田 泰広

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      4 2018 - 3 2023

      Grant type:Competitive

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    • 代数幾何と可積分系の融合 - 理論の深化と数学・数理物理学における新展開 -

      科学研究費補助金/基盤研究(S) 

      齋藤 政彦

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      4 2017 - 3 2022

      Grant type:Competitive

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    • 代数幾何と可積分系の融合 - 種々のモジュライ空間と数学・数理物理学の新展開 -

      科学研究費補助金/基盤研究(A) 

      齋藤 政彦

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      4 2017 - 3 2022

      Grant type:Competitive

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    • カロジェロ・モーザー・サザランド模型の可積分な拡張に関連する特殊函数の研究

      特別研究員奨励費 

      野海 正俊

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      4 2017 - 3 2020

      Authorship:Principal investigator  Grant type:Competitive

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    • 楕円差分の可積分系と特殊函数の研究

      科学研究費補助金/基盤研究(B) 

      野海 正俊

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      4 2015 - 3 2020

      Authorship:Principal investigator  Grant type:Competitive

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    • 量子モノドロミー保存変形とラックス形式

      科学研究費補助金/基盤研究(B) 

      山田 泰彦

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      4 2014 - 3 2019

      Grant type:Competitive

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    • Research on the difference systems associated with multivariable elliptic hypergeometric functions with Weyl group symmetry

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      ITO Masahiko, NOUMI Masatoshi, OKADA Soichi, KANEKO Jouichi, FORRESTER Peter J.

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      4 2013 - 3 2018

      Grant number:25400118

      Grant amount:\4940000 ( Direct Cost: \3800000 、 Indirect Cost:\1140000 )

      The multivariable elliptic hypergeometric functions associated with root systems were studied via their systems of difference equations and Weyl group symmetry. Masahiko Ito (the leader of this research, Univ. of the Ryukyus) and Masatoshi Noumi (cooperating researcher of this research, Kobe univ.) provided a definition of the family of ``interpolation functions'' for the multivariable elliptic hypergeometric functions of type BCn. This is the main result of this research. As applications of the interpolation functions, several summation and translation formulae for the multivariable elliptic hypergeometric functions of type BCn were proved.

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    • Bilinear method for multi-component coupled integrable systems

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      Ohta Yasuhiro, YAMADA Yasuhiko, NOUMI Masatoshi

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      4 2012 - 3 2018

      Grant number:24340029

      Grant amount:\8320000 ( Direct Cost: \6400000 、 Indirect Cost:\1920000 )

      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.

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    • Study on Renormalization as Invariant Theory under Infinite-dimensional Groups

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEDA Toru, NOUMI Masatoshi

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      4 2014 - 3 2017

      Grant number:26610022

      Grant amount:\2990000 ( Direct Cost: \2300000 、 Indirect Cost:\690000 )

      For the renormalization from the invariant-theoretic point of view, we studied some basic facts on the Capelli identities, Bell polynomials and differential operator of infinite variables, and universal enveloping algebras.These are necessary for the study of the infinite dimensional group of general coordinate transformation, and for its invariants. Their importance is that they are related both group theoretic and combinatorial points of view. In particular, the study on the Bell polynomials using the differential operators of infinite variables are similar to classical invariant theoy in the sense of the featuring "ground form". This will be useful in the calculation of non-commuting variables like in the universal enveloping algebras.

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    • Representation of elliptic quantum groups and elliptic Ruijsenaars model

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      Konno Hitoshi, JIMBO MICHIO, NOUMI MASATOSHI, MIMACHI KATSUHISA, OSHIMA KAZUYUKI

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      4 2014 - 3 2017

      Grant number:26400046

      Grant amount:\4420000 ( Direct Cost: \3400000 、 Indirect Cost:\1020000 )

      We have formulated the quantum determinant as well as the quantum minor determinants of the L-operators for the face type elliptic quantum group U_q,p(gl_N). We have also obtained a formula for the elliptic half currents in terms of the quantum minor determinants. We then have completed a proof of the isomorphism between U_q,p(gl_N) and the central extension E_q,p(gl_N) of Felder’s elliptic quantum group. In addition, we have obtained formal elliptic hypergeometric integral solutions to the face type elliptic q-KZ equation and associated elliptic weight functions by considering the traces of the product of the intertwining operators of the level-1 U_q,p(sl_N)-modules.

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    • 非線形発展方程式系におけるrogue wave解の一般的構成とその代数構造の研究

      学術研究助成基金助成金/挑戦的萌芽研究 

      太田 泰広

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      4 2014 - 3 2017

      Grant type:Competitive

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    • 代数幾何と可積分系の融合と進化

      科学研究費補助金/基盤研究(S) 

      齋藤 政彦

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      6 2012 - 3 2016

      Grant type:Competitive

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    • モジュライ理論による代数幾何と可積分系の新たな展開

      科学研究費補助金/基盤研究(A) 

      齋藤 政彦

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      4 2012 - 3 2016

      Grant type:Competitive

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    • Study of general hypergeometric functions and integrable systems coming from monodromy preserving deformation

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KIMURA Hironobu, HARAOKA Yoshishige, NOUMI Masatoshi, IWASAKI Katsunori, SAKAI Hidetaka, NAGOYA Hajime

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      4 2011 - 3 2016

      Grant number:23540247

      Grant amount:\4940000 ( Direct Cost: \3800000 、 Indirect Cost:\1140000 )

      Among special functions, which have good properties, we know the Guass hypergeometric function and Painleve functions which can be characterized by differential equations, integral representations, and contiguity relations. Our study is to generalize and describe them in a unified way. This viewpoint enables to understand why the good properties hold for these objects. The general hypergeometric systems (GHGS) and the general Schlesinger systems (GSS), which generalize Gauss hypergeometric equation and Painleve equations, respectively, are both defined on the Grassmannian manifold. We gave the explicit form of monodromy preserving deformation which gives GSS. We studied, by examining the results of Shah and Woodhouse, when GSS has solutions expressed by the solutions of GHGS and how these solutions can be expressed using solutions of GHGS. As a by-product, we found the relation between the theory of semi-classical orthogonal polynomials and the particular solutions of GSS.

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    • Representation theoretical aproach to multi-variate elliptic hypergeometric functions

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KONNO Hitoshi, JIMBO Michio, NOUMI Masatoshi, NAKAYASHIKI Atsushi, OSHIMA Kazuyuki

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      4 2010 - 3 2014

      Grant number:22540022

      Grant amount:\4290000 ( Direct Cost: \3300000 、 Indirect Cost:\990000 )

      We have reformulated the two elliptic algebras U_{q,p}(g) and E_{q,p}(g) as topological algebras over the ring of formal power series in p and shown that they are isomorphic for g=gl_n. We also have formulated the dynamical quantum Z-algebras associated with U_{q,p}(g), and shown that the irreducibility of the U_{q,p}(g)-modules is governed by the corresponding Z-modules. As for the multivariate elliptic hypergeometric functions, we have shown that the sl_n type found by Kajihara-Noumi and Rosengren is given by a certain tensor product of the symmetric fusions of the sl_n type vertex-face intertwining vectors and their duals. Furthermore, as an application of the representations of U_{q,p}(sl_2), we have derived and analyzed the structure factor of the massless XXZ model exactly.

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    • Study of vector bundles using the theory of complexes

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      YOSHIOKA Kota, NOUMI Masatoshi, YAMADA Yasuhiko, SAITO Masahiko, NAKAJIMA Hiraku, ABE Takeshi

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      4 2010 - 3 2014

      Grant number:22340010

      Grant amount:\11570000 ( Direct Cost: \8900000 、 Indirect Cost:\2670000 )

      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.

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    • Unified viewpoint for hypergeometric functions and the pentagonal number theorem based on representation theory

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEDA Toru, NOUMI Masatoshi, ITOH Minoru

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      2011 - 2013

      Grant number:23654050

      Grant amount:\2080000 ( Direct Cost: \1600000 、 Indirect Cost:\480000 )

      The famous pentagonal number theorem by Euler has been extended as the description of the anomaly, that is, the difference of the trace of two matrices of infinite size, which are almost conjugate. Notable examples, among them, are inversion formulas for q-hypergeometric series. We investigate the mechanism behind that phenomena from the invariant-theoretic and representation-theoretic points of view, so that some new horizon for the hypergeometric series is opened. Dual pair theory is the key to withdraw the symmetry, even if it is not apparent.

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    • Theory of Painleve systems and its new development

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KAJIWARA Kenji, SHIRAI Tomoyuki, IWASAKI Katsunori, NOUMI Masatoshi, YAMADA Yasuhiko, SAKAI Hidetaka, MASUDA Tetsu, TSUDA Teruhisa

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      2007 - 2010

      Grant number:19340039

      Grant amount:\9490000 ( Direct Cost: \7300000 、 Indirect Cost:\2190000 )

      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.

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    • Representations of elliptic quantum groups and their applications to elliptic special functions

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KONNO Hitoshi, JIMBO Michio, NOUMI Masatoshi

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      2007 - 2009

      Grant number:19540033

      Grant amount:\4420000 ( Direct Cost: \3400000 、 Indirect Cost:\1020000 )

      By adding a Hopf algebroid structure to the face type elliptic algebra U_<q,p>(AN^<(1)>), we have formulated it as an elliptic quantum group. Thanks to this structure, we have formulated the intertwining operators on the infinite-dimensional dynamical modules and obtained a consistent result to the previous one derived by using the quasi-Hopf algebra structure. We also have formulated the theta function analogue of the Drinfeld polynomials and shown that they specify the finite-dimensional irreducible dynamical representations uniquely. In the case N=2, we have derived the elliptic analogues of the Clebsch-Gordan coefficients for the tensor product of the finite-dimensional dynamical representations, and shown that they are expressed by using the elliptic hypergeometric series _<12>V_<11>.

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    • Studies on moduli of stable sheaves

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      YOSHIOKA Kota, SAITO Masahiko, YAMADA Yasuhiko, NOUMI Masatoshi, NAKAJIMA Hiraku, MASTUSHITA Daisuke, INABA Michiaki

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      2006 - 2009

      Grant number:18340010

      Grant amount:\17110000 ( Direct Cost: \13900000 、 Indirect Cost:\3210000 )

      We derived the wall crossing formula and the blow-up formula of the Donaldson invariants. We also formulated K-theoretic analogue of the Doinaldson invariants and got the wall crossing formula. We studied the betation of the Fourier-Mukai transform and the stability condition ang got a nice result. As an application, we also studied the moduli of stable sheaves on abelian surfaces.

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    • 離散可積分系における幾何的方法

      科学研究費補助金/基盤研究(B) 

      山田 泰彦

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      2009

      Grant type:Competitive

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    • A study of commutative rings of differential operators in completely integrable models and their eigenfunctions

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      DATE Etsuro, KAWANAKA Noriaki, MATSUMURA Akitaka, KOMATSU Gen, MIKI Kei, YAMANE Hiroyuki, OKADO Masato, NOUMI Masatoshi

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      2005 - 2008

      Grant number:17340046

      Grant amount:\8700000 ( Direct Cost: \7800000 、 Indirect Cost:\900000 )

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    • Representation Theory and Duality associated with Non-commutative Special Functions

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEDA Toru, NOUMI Masatoshi, WAKAYAMA Masato, OCHIAI Hiroyuki, MATSUZAWA Jun-ichi, ITOH Minoru

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      2004 - 2007

      Grant number:16340039

      Grant amount:\8770000 ( Direct Cost: \8200000 、 Indirect Cost:\570000 )

      Theory of special functions of non-commutative variables is a framework for us to understand deeply the dual pairs, which is a sort of revival of the classical invariant theory. The aim of the research is to link each other the three theories on special functions, invariants, and representations, to throw a new light to the mathematical world under the non-commutativity. In the center of our study, we have the Capelli identities, the equalities of the invariant differential operators, which arise from the representations of the centers of the universal enveloping algebras. Around the Capelli identities, we found many interesting phenomena caused from the transition from commutative theories to non-commutativeones. And even behind the usual commutative theories, we often found the dominating non-commutative variables.
      For the goal to obtain the ultimate Capelli type identities, we made a big progress through the new ideas which combine the non-commutative matrix elements, the generalization of the notion of transfer, symbolic method, and the method of generating functions. Though the program has not been accomplished, the results we had will be very useful for the understanding of the ultimate Capelli identities.
      An important example is in the treatment of Euler's pentagonal number theorem, which we understand a trace identity of matrices of infinite size. In there, we discover the fact that some identities generalizing the pentagonal number theorem are indeed sort of summation formula for q-hypergeometric series. This point of view will lead us to a new link between the representation theory and invariant theory through the infinite-dimensional spaces.
      Another important discovery is the algebra, which is a very useful toolfor the higher Capelli identities as the non-commutative formal variables. This is done by M. Itoh, an investigator of this research.

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    • 普遍指標の拡張と新しいソリトン方程式系

      科学研究費補助金/基盤研究(B) 

      太田 泰広

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      2007

      Grant type:Competitive

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    • 代数幾何と可積分系の融合と新しい展開

      科学研究費補助金/基盤研究(S) 

      齋藤 政彦

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      2007

      Grant type:Competitive

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    • 計算による数理科学の展開

      科学研究費補助金/基盤研究(A) 

      高山 信毅

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      2007

      Grant type:Competitive

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    • Mathematical theory of dynamical systems of Painleve type

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KAJIWARA Kenji, NOUMI Masatoshi, YAMADA Yasuhiko, IWASAKI Katsunori

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      2003 - 2006

      Grant number:15340057

      Grant amount:\8800000 ( Direct Cost: \8800000 )

      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_<m-1>+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_<m-1>+A_<n-1>)^<(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.

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    • Algebra, Geometry, Analysis in non-linear equations

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEMURA Hiroshi, FUJIWARA Kazuhiro, OKADA Soichi, OKAMOTO Kazuo, MUKAI Shigeru, NOUMI Masatoshi

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      2003 - 2006

      Grant number:15340004

      Grant amount:\13400000 ( Direct Cost: \13400000 )

      One of the central theme of this research is the differential Galois theory of infinite dimension. The head investigator presented in 1996 one such theory. Malgrange interested in this theory himself proposed a general differential Galois theory. Umemura's differential Galois theory is a Galois theory of differential field extension. Namely when a differential field extension L/K is given, we construct a kind of Galois closure of the extension, The Galois group p is th infinitesimal automorphism group of this Galois closure. On the other hand, Malgrange's Galois roupois is attached to a foliation on a variety. Namely when a foliation F on a variety is given, the Galois groupoid is the smallest algebraic Lie groupoid whose Lie algebra contains the tangent vectors to the foliation F.
      These two definitions are seemingly different but specialists observed that they coincide in Examples. In recent three years the development in this direction was remarkable. On can show in the absolute case L/K, by which we mean the base field K is a subfield in the constant field of L, these two definitions are equivalent. The proof is done through the universal
      On the other hand one of the investigator of this project, M. Noumi at Kobe University studied the most general Painleve equation or the Master equation called the elliptic Painleve equation. He showed among other things that as Riccati solutions to the elliptic Painleve equation, there appear hyperelliptic geometric functions. This result is one of the most remarkable results in this field of research.

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    • Parabolic Kostka polynomials, quiver varieties, crystal bases and tropicalcombinatorics

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KIRILLOV Anatol N., ARIKI Susumu, NAKAJIMA Hiraku, NOUMI Masatoshi, YAMADA Yasuhiko, MAENO Toshiaki

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      2003 - 2005

      Grant number:15340006

      Grant amount:\10200000 ( Direct Cost: \10200000 )

      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.

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    • Mathematical Studies on the Painleve equations

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      OKAMOTO Kazuo, KATSURA Toshiyuki, JIMBO Michio, SAKAI Hidetaka, SATSUMA Junkichi, NOUMI Masatoshi

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      2002 - 2005

      Grant number:14204012

      Grant amount:\48490000 ( Direct Cost: \37300000 、 Indirect Cost:\11190000 )

      The aim of present project is to study in synthetic manner the Painleve equations and the Garnier systems from many different points of view ; the latter is a generalization in the case of several variables of the former. In other words, we pursue studies on the Painleve equations with analytic, geometric and algebraic method and magnify, based on results obtained concerning Painleve equations, our research to investigation on general integrable systems, such as the Ganier system, discrete and q-difference Painleve equations. As for analytic behavior of solutions of Painleve equations, we have obtained new results by using Nevanlinna theory on meromorphic functions. The space of initial conditions of Painleve equations was constructed by K. OKAMOTO ; recently an analogous object has been considered for the Garnier systems and this gives us a geometric aspect on studies of the Garnier systems. From an algebraic view point of nonlinear integrable systems, we have studied birational canonical transformations of the Garnier systems by exteoding results on those of Painleve equations.
      At the beginning of pursuit of present project, we studied algebraic transformation of Painleve equations, called folding transformations of the space of initial conditions, we have arrived at the final result, which are published in the first articles cited in the list of publications. By means of geometrical researches on Painleve equations, done by H.SAKAI, the third Painleve equations has two special degenerate types and it is natural to consider the eight types of Painleve equations from a geometric point of view. We have completed studies on Painleve equations by considering the Hamiltonian structure of these two types of equations ; the space of initial conditions, the group of birational canonical transformations and special solutions are given in the second article of the list of the next page.

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    • 変分問題的な曲率条件を持つ曲面への可積分系の方法の応用

      科学研究費補助金/基盤研究(B) 

      ROSSMAN WAYNE, FREMONT

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      2005

      Grant type:Competitive

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    • 箱玉系と組合せ的Bethe仮説

      科学研究費補助金/基盤研究(B) 

      山田 泰彦

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      2005

      Grant type:Competitive

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    • 超幾何・パンルヴェ系の総合的研究

      科学研究費補助金/基盤研究(A) 

      野海 正俊

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      2005

      Authorship:Principal investigator  Grant type:Competitive

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    • モジュライ空間と可積分系の新しい展開

      科学研究費補助金/基盤研究(B) 

      齋藤 政彦

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      2005

      Grant type:Competitive

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    • 国際研究集会「離散可積分系理論の新展開(仮題)」に関する企画調査

      日本学術振興会  科学研究費助成事業 

      時弘 哲治, 薩摩 順吉, 岡本 和夫, 中村 佳正, 野海 正俊, 梶原 健司

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      2004 - 2004

      Grant number:16634003

      Grant amount:\2100000 ( Direct Cost: \2100000 )

      離散可積分系研究の現状に関して,以下の新しい展開があることがわかった.
      (1)離散パンルベ方程式系が生物・疫学の数理モデルに応用され始めている.
      (2)多成分シュレディンガー方程式系の離散化が成功し,そのソリトン解や超離散化についての研究が進展している.
      (3)QRT写像など離散可積分写像の楕円曲線等を用いた幾何学的な意味づけがなされ始めた.
      (4)パデ近似などの数値計算アルゴリズムと離散可積分系対応が明確になってきた.
      (5)逆超離散化の手法がほぼ確立された.
      (6)有限体上のτ函数の研究が進展した.
      (7)幾何クリスタルが組合せ論的R行列との対応の観点から研究されてきた.
      以上の新展開を考慮して,2006年秋に東京(東京大学・大学院数理科学研究科)において国際研究集会を開くことを決定した.
      国外からの招待講演者として,S.Carstea(INPE, Bucharest),C.Gilson(University of Glasgow),B.Grammaticos(Universite de Paris VII),J.Hietarinta(University of Turku),J.Nimmo(University of Glasgow),A.Ramani(Ecole Polytechnique),T.Tamizhmani(Kanchi Mamunivar, Pondicherry)を候補者と決定した.
      国内からは,太田泰広(神戸大),國場敦夫(東大),坂井秀隆(東大),高橋大輔(早稲田大),松木平淳太(龍谷大)を招待講演の候補者と決定した.
      また,国内からは数名の講演とポスターセッションとして十名程度の講演を募集することを決めた.

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    • Defining manifolds and symmetries of nonlinear special functions in several variables

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      TAKANO Kyochi, NOUMI Masatoshi, YAMADA Yasuhiko, SAITO Masa-hiko, IWASAKI Katsunori

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      2001 - 2004

      Grant number:13440054

      Grant amount:\10300000 ( Direct Cost: \10300000 )

      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.

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    • Geometry on String Theory and Moduli spaces

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      SAITO Masa-hiko, HOSONO Shinobu, YAMADA Yasuhiko, NOUMI Masatoshi, YOSHIOKA Kota, MUKAI Shigeru

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      2000 - 2002

      Grant number:12440008

      Grant amount:\16100000 ( Direct Cost: \16100000 )

      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.

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    • Painleve equations and integrable systems

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      YAMADA Yasuhiko, TAKANO Kyouichi, SAITO Masahiko, NOUMI Masatoshi, KAJIWARA Kenji, MASUDA Tetsu

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      1999 - 2002

      Grant number:11440047

      Grant amount:\12300000 ( Direct Cost: \12300000 )

      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)>_<m-1> × A^<(1)>_<n-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.

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    • Rational singularities, Young diagram, Painleve equation

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEMURA Hiroshi, NOUMI Masatoshi, OKAMOTO Kazuo, MUKAI Shigeru, OKADA Soichi

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      1999 - 2002

      Grant number:11440006

      Grant amount:\14500000 ( Direct Cost: \14500000 )

      1. Painleve equations and Special polynomial
      We disocvered that the Painleve equations generate special polynomials. If we consider the motivation of the discovery of the Painleve equtions, it is surprising that they have combinatorial aspects. We presented conjectures on the special polynomials and proved them.
      2. Generalization of the Painleve equations based on the symmetries
      Noumi considered that the conjectures should be solved in a natural frame work. To this end, he generalized the Painleve equations from the view point of theory of Lie algebra.
      3. Deformation of rational double points and Backhand tranhformations
      We proved that the Backhund transformations arise from deformation of rational double points.
      4. Infinite dimensional differential Galois theory and Painleve equations
      We applied our theory of infinite dimensional to the definition of the Painleve equations.

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    • Study of special functions based on representation and invariant theories

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      UMEDA Toru, NOUMI Masatoshi, MATSUZAWA Junichi, NOMURA Takaaki, OEHIAI Hiroyuki, WAKAYAMA Masato

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      1999 - 2001

      Grant number:11440043

      Grant amount:\9600000 ( Direct Cost: \9600000 )

      The main object of the research is to find the group theoretical background behind the world of special functions and to utilize the symonetious for the special functions. Among them the theory of "dual pairs" is the key to our study, which explains many phenomina from the view-point of representation theory and the theory of invariants. We have Capilli type identities, now commtative harnomic oscillatws as the typical investizations where and pains work very well as the griding principle. On the other hand, for the hyprogeinctic from Rons and Pain lene transcendents, we have claified the grop gymmetric behind them. The helps a lot for deeper investigations of these fnctions.
      As for the Capelli type identities, we got many interesting formlas including permanets and Pfuffians, not only for the determinants, Furthermore we found some Capelli type identity corresponding to the "group determinant". The invariant theoretic backgroud conneits these identities to some sphenicel functions. There are sort of unification of various objects.

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    • Global Analysis of Painleve Equations

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      TAKANO Kyoichi, YAMADA Yasuhiko, NOUMI Masatoshi, SASAKI Takeshi, TAKEI Yoshitsugu, IWASAKI Katsunori

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      1998 - 2000

      Grant number:10440058

      Grant amount:\9200000 ( Direct Cost: \9200000 )

      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.

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    • Modern development of special functions - approach from the representation theory and the integrals

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      MIMACHI Katsuhisa, YAMADA Yasuhiko, NOUMI Masatoshi, HANAMURA Masaki

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      1997 - 1999

      Grant number:09440020

      Grant amount:\9800000 ( Direct Cost: \9800000 )

      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.

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    • Research on Periods of Algebraic Varieties and Hypergeometric Functions

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      SAITO Masa-hiko, TAKANO Kyoichi, SASAKI Takeshi, NOUMI Masatoshi, YOSHIOKA Kota, TAKAYAMA Nobuki

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      1997 - 1999

      Grant number:09440015

      Grant amount:\12900000 ( Direct Cost: \12900000 )

      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.

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    • 幾何学における超幾何系の新展開

      日本学術振興会  科学研究費助成事業 

      佐々木 武, 山口 佳三, 大仁田 義裕, 宮岡 礼子, 野海 正俊, 高野 恭一

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      1998 - 1998

      Grant number:10894005

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      1. 調査研究の基本は神戸大学所属の分担者間での討議からはじめた。佐々木は幾何学に現れる可積分系のG.Darboux以来の歴史と日本とロシア・ドイツにおける研究の中心をまとめ、高野はパンルヴェ方程式の幾何構造の研究状況について、さらに野海は表現論と超幾何系の関連・パンルヴェ方程式の代数構造の問題・ヨーロッパにおける研究の現状について検討し、2つの調査研究会を開くこととした。1つは上記3者の主催する会議「超幾何系ワークショップin Kobe,98-幾何学・可積分系・パルヴェ系」(神戸大学滝川学術交流会館、1998年12月1日-4日)において、全国から90名の参加で超幾何系、パンルヴェ系、可積分系と幾何学について21個の報告を受けた。また、分担者宮岡と大仁田は「Hamiltonian Systems in Differential Geometry」(甲府市、1999年2月11日-13日)を開催し、平均曲率一定曲面、射影微分幾何における可積分系、周期的戸田方程式とループ群の量子コホモロジー、完備極小曲面のモジュライ空間上の微分幾何、等径超曲面の等質性にいたる可積分系の手法の解説等をおこなった。
      この会議に関連して、野海は「表現論ワークショップ」(アムステルダム)に講演及び研究視察に渡航する一方、ロシアの微分幾何と可積分系の研究状況に詳しいE.Ferapontov(Landau研究所)を招聘し(1999年1月30日-2月14日)、北海道大学での研究集会「Schwarz微分をめぐって」において連続講演を行い、上記甲府での研究集会において微分幾何と可積分系の諸問題について報告を行った。
      これらの幾何と可積分系をめぐる新しい動向は、超幾何系研究者・微分幾何の研究者に加え、表現論や代数の分野からも興味を呼び、2000年には第9回日本数学会国際研究集会を開催することとなり、この企画調査はその準備に有益な寄与をしたと考える。
      2. なお、研究課題に関連して代表者及び分担者の得た新しい結果の主要なものは次の通りである。
      ● 3次曲面のモジュライを4次元球の商空間として実現する線形微分方程式の具体形を求めたこと(佐々木・吉田)
      ● 等径超曲面の等質性についての最終的解決(宮岡)● アフィンWeyl群の表現に基づいてPainleve型差分系の系統的な構成法を与えたこと及びA型アフィンWeyl群がBaklund変換群として作用するような,高階Painleve微分方程式を構成したこと(野海)

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    • 国際共同研究「超弦理論と量子コホモロジー」を組織する為の企画調査

      日本学術振興会  科学研究費助成事業 

      齋藤 政彦, 清水 勇二, 細野 忍, 小野 薫, 深谷 賢治, 野海 正俊

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      1998 - 1998

      Grant number:10894003

      Grant amount:\2600000 ( Direct Cost: \2600000 )

      本企画調査の目的は「弦理論」と深く関わりある幾何学、特にシンプレクテック幾何学および代数幾何学等の諸分野において弦理論の新しい進展を取り入れながら21世紀の空間概念の創造の為の数学的基礎付けの為の研究を立ち上げる準備をする事であった。特に、深谷・小野のグロモフ・ウイッテン不変量および量子コホモロジー環のシンプレクテック幾何学的基礎付け、齋藤・細野・前野らのミラー対称性の研究や代数多様体の量子コホモロジーの具体的決定、上野・清水の共形場理論とヴイラソロ代数およびリーマン面のモジュライの幾何等の研究を踏まえて、国内外の研究者を巻き込んだ研究体制の構築の方法を調査研究した。この観点から平成11年度に京都大学数理解析研究所において「弦理論に関わる幾何学」という国際共同研究プロジェクトの採用が決定し、本企画調査の研究分担者が中心となり組織していく事が決定しており、その具体的企画調査を行なった。主な具体的企画調査については次の通り。
      1.細野が1998年7月26日から2週間、Canada,Kingstonに滞在し齋藤とクイーンズ大学の由井典子教授にその時点までの企画についてレビューを受け同時に国際的共同研究の可能性について情報収集および調査をした。
      2.1998年11月兵庫県城崎代数学シンポジュムにおいては、深谷および細野が研究発表を行なうとともに,数研プロジェクトの計画について議論した。
      3.1999年1月北海道大学にて、数研プロジェクトの企画会議および最新の研究情勢の報告会をおこなう。
      4.これ以外にも、齋藤、清水、細野、深谷、小野は不定期に企画会議および情報交換を行なった。この企画調査の結果として、数理研プロジェクトにおいて2件の国際研究集会および、3件の短期共同研究(ワークショップ)の企画が決定し現在その準備を行なっている。

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    • 数学研究支援のための自己学習型数式処理系の設計と開発

      日本学術振興会  科学研究費助成事業 

      野海 正俊, 高山 信毅

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      1997 - 1998

      Grant number:09874010

      Grant amount:\2200000 ( Direct Cost: \2200000 )

      本研究では,自己学習型の数式処理系の設計・開発を目標としつつ,数学研究の現状に照らし合わせて,その支援のための数式処理系のあり方についての検討を行った.その要点は,(1)無限個の変数や非可換な変数を含む数式処理の実際的研究と,(2)複数の数式処理システム間で数式データを相互にやりとりするためのプロトコルの問題,の2点である.
      研究代表者は,主に(1)を担当し,非線形の可積分な微分・差分方程式系と,Lie環とWeyl群の表現の研究における数式処理の実践を通じて,無限変数の問題の雛形として微分多項式のグレプナ基底について,また非可換変数の問題の雛形としてCoxeter群の数式処理について,実際的アルゴリスムの検討を行った.その成果についてはもう少し完成度を高めた上で公表したい.研究分担者は主に(2)を担当し,分散数式処理系のプロトコルである,openXMの設計と実装を行い,Risa/AsirとKan/smlの間で数式処理系のネットワークを実現した.当初の研究課題の実現には未だ遠いが,そのための第一歩を踏み出すことができたと思う.

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    • 新しい研究分野「超幾何系」を組織するための企画調査

      日本学術振興会  科学研究費助成事業 

      野海 正俊, 大島 利雄, 三町 勝久, 吉田 正章, 齋藤 政彦, 高野 恭一

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      1997 - 1997

      Grant number:09894001

      Grant amount:\2100000 ( Direct Cost: \2100000 )

      本企画調査の目的は、「超幾何系」の新分野を組織し、重点領域の領域申請のための企画調査を行うことであった。研究代表者は、「多変数超幾何函数に関連する諸問題を現在の観点から再編統合し、『超幾何系』の研究分野を組織して、21世紀の自然科学及び工学への新しい基盤を提供する」ことを目的として、各研究分担者とともにその方法についての具体的な検討を行い、平成9年11月に、「平成10年度発足特定領域」として「超幾何系の理論-数学新時代へのプラットフォーム」の領域申請を行った。これによって本企画調査の第一の目的は達成された。
      申請中の特定領域研究(B)「超幾何系の理論」は、総括班と4つの計画研究
      (ア)配置空間の幾何と超幾何函数による一意化
      (イ)代数多様体の周期と量子コホモロジー
      (ウ)超幾何積分のde Rham理論
      (エ)D加群と表現論の観点からの超幾何系
      からなり、基幹部分として配置空間と代数多様体の族に対する超幾何系のモジュライ理論の研究を、また基礎理論としてD加群と表現論からの代数適研究と超幾何積分の大域構造の研究を行い、これによって数理物理、数理工学、情報科学等への新しい応用を開くことを目指すものである。3つの拠点である神戸大学・九州大学・東京大学を「仮想超幾何系研究センター」として連携し、研究の機動的な体制をとることをその骨子とした。
      この間、特定領域の申請準備とともに、研究分担者間で、「超幾何系」の有機的な研究体制を整えるための検討を行った。また平成9年12月に神戸大学に於いて「超幾何系ワークショップinKobe」を開催し、超幾何系の研究分野の研究動向の確認と、関連分野の有機的な連携のための討論を行った。

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    • q完備空間の特徴付けについての研究

      日本学術振興会  科学研究費助成事業 

      渡邊 清, 名倉 利信, 池田 裕司, 野海 正俊

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      1996 - 1996

      Grant number:08640200

      Grant amount:\2100000 ( Direct Cost: \2100000 )

      正則函数の芽のなす層Oを係数とするコホモロジー群の消滅と Stein性の研究に関連して,まず次の問題を考えてみた。複素n次元空間内にある領域Dにおいて,Cousin-I 問題が解けることと1次元コホモロジー群H^1(D,O)=0となることは,同値であろうか?この問題は現在までも解かれていない。そこで、この問題に関連して,次の問題を考えてみた。ある領域Dにおける自明でない正則コサイクルα∈H^1(D,O)で,有理型コサイクルμ(α)と見てもH^1(D,m)内で自明でないものが存在するだろうか?
      本研究において、まず上の問題について,肯定的な解答を得た。すなわち,複素2次元空間より原点をぬいた領域をDとし,Dにおける正則コサイクルとして,α={exp(11/(ZW)}をとると,αは正則函数に依って分解されないばかりか,有理型函数に依っても分解されないことが分る。証明の要点は、次のことである。Dでは,Thullenに依れば,Cousin-II問題が常に解けること,さらに,係数の比較より定まるある無限次の連立方程式が自明な解しか持たないことを示すことである。
      次に,複素射影空間内のStein領域の在り様を明らかにする為に,次の様な問題を考えてみた。2次元複素射影空間より代数曲線をぬいたStein領域をDとするとき,代数曲線の次数が3でなければ,Dから複素2次元空間へのはめ込みが存在しないことが知られているが,3次のとこは,曲線が尖点をもつときは,はめ込みの存在が知られていた。ここでは,非特異3次曲線について,はめ込みの有無を研究した。その結果,4次元空間への埋蔵等いくつかの結果を得た。
      その他,関連研究として,フェルマ-曲線上のワイエルシュトラス点についての,いくつかの知見を得た。

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    • 保型形式に付随する特殊関数の研究

      日本学術振興会  科学研究費助成事業 

      山崎 正, 広森 勝久, 野海 正俊, 高山 信毅, 浜畑 芳紀

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      1996 - 1996

      Grant number:08640035

      Grant amount:\1600000 ( Direct Cost: \1600000 )

      保型形式の研究において、種々の形のフーリエ展開(周期)や保型的L関数との関連で、球等質空間の球関数の重要性が認識されてきた。本研究では、主に一般線型群の球部分群に対する球関数を中心に研究した。今後は、メタプレクティック群への拡張、整数論への応用などを研究したい。この問題を議論するために定期的にしたセミナーを開催した。さらに分担者と協力して、以下のような研究を行った。
      ・量子対称空間上の、有限次元表現に付随する、球関数を調べ多くの場合これらがルート系に対応するマクドナルド多項式を用いて記述されることを示した。またA型のルート系に付随するマクドナルド多項式の列型の昇降演算子(生成消滅演算子)を与えた。(野海)
      ・超幾何関数の研究において、ツイストホモロジーやツイストコホモロジーの交点数の計算が重要であるが、高階の場合にこれらの計算方法を開発した。グレブナ-基底・整数計画法の応用として超幾何級数に対する生成作用素を計算する方法を与えた。(高山)
      ・四元数体上のユニタリー群の主合同部分群に付随する2次の数論的(ジーゲル)多様体が、レベルが十分大のときに一般型になることが知られていたが、これをレベルを付けない場合に考え、四元数体の判別式が十分大のとき、数論的多様体が一般型であることを示した。また有限体上の2次特殊線形群のヒルベルト尖点形式の空間における表現を考え、正則レフシェッツ公式を用いて、ヘッケ-齊藤の結果を3変数の場合に拡張した。(浜畑)

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    • Special Functions in Many Variables

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      NOUMI Masatoshi, SEKIGUCHI Hideko, TAKAYAMA Nobuki, YAMAZAKI Tadashi, SASAKI Takeshi, TAKANO Kyoichi

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      1996 - 1996

      Grant number:08454030

      Grant amount:\5600000 ( Direct Cost: \5600000 )

      The main subject of the this research project has been to construct a new prototype of the theory of special functions, by originating a systematic study of hypergeometric special functions in many variables. In this repect, the following results have been obtained from the viewpoints of (1) quantum group symmetry of difference systems, (2) confluent hypergeometric functions and Hamiltonian systems, (3) geometry of configurations spaces, and (4) representation theory and integral transformations, respectively.
      (1) M.Noumi developed a theory of sperical functions on quantum symmetric spaces in relation to quantum group symmetry. In terms of quantum groups, he gave a representation-theoretic realization of commuting families of q-difference operators and of the q-hypergeometric orthogonal polynomials of Macdonald type.
      (2) K.Takano studied in detail the procedure of confluence for hypergeometric functions over the Grassmannians, namely the degeneration of a general regular singularity to a confluent singularity. He also clarified the structure of the spaces of initial values and the mechanism of degeneration in Hamiltonian systems of Painleve' type.
      (3) T.Sasaki investigated the spaces of configurations of one nondegenerate quadratic hypersurface and n hyperplanes in the projective space. He determined the differential system for the associated hypergeometric integrals and described the symmetry of them. For the configurations in the projective plane, in particular, he consturcted explicit power series solutions and independent cycles, and clarified the relationship with Appell's hypergeometric functions.
      (4) From the viewpoint of Penrose transformations in symmetric domains, H.Sekiguchi studied the generalization of hypergeometric integrals and hypergeometric differential equations to higher ranks. She also established the finite dimensionality of their solution spaces by means of the method of unitary representations.

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    • Research on Structures of Knots and Manifolds as Models

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      NAKANISHI Yasutaka, SAITO Masa-hiko, TAKAYAMA Nobuki, NOUMI Masatoshi, NAGURA Toshinobu, IKEDA Hiroshi

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      1996 - 1996

      Grant number:08454018

      Grant amount:\4000000 ( Direct Cost: \4000000 )

      According to the purpose and operating plan of the above project, the head investigator has obtained the following results.
      1.For two-bridge knots, necessary conditions on coefficients of Alexander polynomials are given.
      2.Shibuya's result that any union of two nontrivial knots without local knots is prime is proved in a general setting by tangle arguments. And the concrete example of exception are given.
      3.For every minimum-crossing knot-diagram among all unknotting-number-one two-bridge knots, the existence of crossings whose exchange yields the trivial knot is proved. This result is a part of answer of a conjecture on unknotting number.
      These results 1,2, and are published.
      4.The Delta-unknotting numbers for torus knots, positive closed 3-braids, and positive protzel knots are determined.
      5.The borromean rings and the 3-component trivial link cannot be deformable to each other by a finite sequence of link-homotopies and cancellations of consecutive 4-crossings.
      These results 4 and 5 are accepted to be published.
      He gave lectures based on the above results at international conferences : Conference/Workshop for The Fifth MSJ International Research Instutute Knot Theory (Waseda University, 1996.7.22-31), The fifth Japan-Korea Seminar on Knots and Links (Korean Advansed Institute for Science and Technology, 1997.2.17-20).
      Furthemore all investigators have obtained their results according to their roles.

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    • Systems of Holonomic Differential Equations

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      TAKANO Kyoichi, SASAKI Takeshi, HIGUCHI Yasunari, NOUMI Masatoshi, KABEYA Yoshitsugu, AIZAWA Sadakazu

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      1995 - 1996

      Grant number:07454028

      Grant amount:\5400000 ( Direct Cost: \5400000 )

      1.Hypergeometric differential equations : (1) We obtained the system of differential equations satisfied by integrals associated with a non-degenerated quadratic hypersurface and n hyperplanes in the (k-1)-dim. complex projective space and we studied certain sym-metries of the system.(2) We made clear geometrically the processes of confluence of general confluent hypergeometric functions on Grassmann manifolds.
      2.Painleve systems : (1) We found processes of conflunce among the spaces of initial con-ditions of Painleve systems. The processes are compatible with the well known ones.(2) We proved the irreducibility of the second and the fourth Painleve functions exept the known classical functions, by determining ivariant devisors of Hamiltonian vector fields associated with the Painleve systems.
      3.Quantum groups and q-functions : (1) We realized a family of quantum complex projective spaces as one of quantum homogeneous spaces associated with a family of coideals, and we expressed the zonal spherical functions in terms of Askey-Wilson polynomials. (2) We solved afflrmatively the integrality conjecture of Macdonald for the (q, t)-Kostka coefficients, by constructing raising operators for Macdonald's symmetric polynomials.
      4.Percolation problem : We obtained the order of the spectral gap in the case where + and - spins are randomly distributed on the boundary. The order is the same as that in the case where there are on spins no the boundary.
      5.We showed that the affine geometry of surfaces in the 3-dim. projective space works aiso in the case where some invariants are degenerated, and we classified projective homogeoenus surfaces.

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    • 無限自由度の可積分系および無限次元代数

      日本学術振興会  科学研究費助成事業 

      三輪 哲二, 梁 成吉, 土屋 昭博, 伊達 悦朗, 神保 道夫, 柏原 正樹, 浪川 幸彦, 大栗 博司, 野海 正俊, 上野 喜三雄

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      1992 - 1996

      Grant number:04245105

      Grant amount:\69800000 ( Direct Cost: \69800000 )

      三輪と神保は質量0のXXZ模型のボゾン化を、実直線上のボゾンを使うやり方で実現し、楕円的量子群の極限がレベル1の表現として実現していることを示した.伊達はオンサーガ-代数の商構造をイジング模型とそれを含む超可積分カイラル・ポッツ模型について調べた.増田はqの絶対値1でかつコンパクト型の量子群について調べ、それが2次元の非可換トーラスの量子対称性を記述することを明らかにした.野海はマクドナルド多項式に関係する昇降演算子を構成し、整性予想を解決した.上野はqの絶対値が1の場合のqの絶対値が1の場合のq変形された超幾何函数と多重ガンマ関数、多重サイン関数の関係を調べ、積分公式を得た.稲見は4次元の非線型シグマ模型の対称性を検討しトロイダル代数が対称性の代数になっていることを発見した.柏原は量子群の有限次元既約表現について考察し、ドリンフェルトによる結果を精密化して、任意の既約表現はその因子をスペクトルパラメタの大きさの順序に並べたテンソル積の中に最高ウェイトベクトルを含む部分加群として実現されることを示した.梁は戸田格子のスペクトル曲線を仲立ちとして、2次元位相的共形場理論、4次元N=2のサイバーグ・ウィッテン解、さらにN=2の超弦双対性が結びつくことを明らかにした.土屋は共形場理論のN点の行列要素の全体をリー環で不変な部分を法とした空間で考えたときに退化したアフィンヘッケ環の表現空間としてとらえられることを示し、その指標公式を自由分解する予想を提出した.

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    • 保型形式のL関数の解析的研究

      日本学術振興会  科学研究費助成事業 

      山崎 正, 広森 勝久, 野海 正俊, 高山 信毅, 浜畑 芳紀, 柳川 高明

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      1995 - 1995

      Grant number:07640043

      Grant amount:\2300000 ( Direct Cost: \2300000 )

      保型形式の研究の中で、種々の形のフーリエ展開(周期)や保型的L関数との関連で、球等質空間の球関数の重要性が認識されてきた。本研究では、主に一般線形群の球部分群に対する球関数を中心に研究した。これらの球関数はアフィンヘッケ環の構造論や表現論と深く結び付いているもので、保型的L関数と本質的関連があるものと期待される。この問題を議論するために定期的にしたセミナーを開催した。さらに以下のような研究を行った。
      ・量子対称空間上の、有限次元表現に付随する、球関数を調べ多くの場合これらがルート系に対応するMacdonald多項式を用いて記述されることを示した。また、BC_n型のAskey-Wilson多項式に対して、ヘッケ環を用いてこれらを同時固有関数にもつ可換なq差分作用素の族が構成できることを示した。(野海)
      ・超幾何関数の研究において、ツイストホモロジーやツイストコホモロジーの交点数の計算が重要であるが、高階の場合にこれらの計算方法を開発した。またq解析に現れるアフィンワイル群やアフィンヘッケ環などに対し、種々の実験を行うための数式処理系のための言語およびアルゴリズムを開発した。(高山)
      ・有理数体上階数1の数論的多様体の代数幾何的性質を調べた。不定符号四次元数体の極大orderから定まる離散的部分群の合同部分群による2次のSiegel上半平面の商は,四元数体の判別式が十分大きいとき、一般型であることを示した。また幾何種数が2以上のHilbert modular曲面を分類した。(浜畑)

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    • Topological invariants related to field theory

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KOHNO Toshitake, KATO Akishi, NOUMI Masatoshi, KATSURA Toshiyuki

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      1994 - 1995

      Grant number:06640111

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      Applying techniques of field theory in mathematical physics, we establishes a general framework to extract topological invariants of manifolds from infinite dimensional data.
      In late 80's Witten proposed a method to define topological invariants of 3-manifolds as the partition function of the Chern-Simons functional defined over the space of connections on the manifold. We clarified the relation between the Chern-Simons theory for 3-manifolds with boundary and the two-dimensional conformal field theory. We formulated the conformal field theory as the theory of connections on vector bundles over the moduli space of Riemann surfaces and expressed the Witten invariants by the holonomy of the connection. Moreover, from the above point of view we obtained lower estimates for classical invariants for knots and 3-manifolds.
      The critical points of the Chern-Simons functional are flat connections and it is known that the perturbative expansion at flat connections are described by Feynman diagrams. We investigated topological invariants arising from such perturbative expansion from the viewpoint of integral geometry-integral of Green forms on the configuration space. Motivated by Chern-Simons perturbative theory for 3-manifolds with boundary, we studied the space of chord diagrams on Riemann surfaces and its quantization, together with the symplectic geometry of the moduli space of flat connections. In particular, in the case of the torus, we investigated the holonomy of the elliptic KZ connection and defined Vassiliev type invariants for knots in the product of the torus and the unit inverval.

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    • The theory of transformation groups and its application

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      OKAMOTO Kazuo, YAMADA Michio, KIKUCHI Fumio, ORIHARA Akio, MIMURA Masayasu, KOHNO Toshitake

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      1994 - 1995

      Grant number:06452009

      Grant amount:\6500000 ( Direct Cost: \6500000 )

      The present project aimed studies on transformation groups, appearing in the various aspects of analysis. Investigating transformation groups of completely integrable systems defined in a complex domain, we obtaind a new point of view and various resultes particularly on this subject. The group of this project in the University of Tokyo concerned researchs on nonlinear integrable systems, in particular from the point of view of the algebraic theory. The main objects of our investigations is losted as follows :
      (1) Only special cases of partial differential equations admit fruitfull algebraic structure of their transformation groups. Since few of such equations, we attempt to construct the examples systematically.
      (2) We determine the transformation groups and their realization of nonlinear integrable system, in partcular these of the Garnier systems.
      (3) One of the most important points of studies on completely integrable systems is an application to theoretical physics. We try to establish the general method of determination of transformation groups of integrable systems.
      The investgators of this research project have continued their studies on transformation groups of integrable systems and announced their own results obtained during two years, 1994-95, in various occasins They have published some of results in journals.

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    • 高次元ソリトン方程式の解析

      日本学術振興会  科学研究費助成事業 

      薩摩 順吉, 木村 弘信, 野海 正俊, 岡本 和夫, 堀川 穎二, 谷島 賢二

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      1994 - 1994

      Grant number:06835003

      Grant amount:\1800000 ( Direct Cost: \1800000 )

      本研究の目的は、あるクラスの高次元ソリトン方程式について、解の挙動を解析的および数値的に調べることにより、その数学的な構造を明らかにし、ソリトン理論の拡張をはかることである。この目的に対して以下の成果を得た。
      1.磁場がかかったプラズマ中のイオン波の高次元挙動を調べ、その系がDavey-Stewartson方程式で記述されること、またあるパラメータ領域でドロミオン解を持つことを明らかにした。この結果はプラズマ中における高次元局在波の存在可能性を指摘したもので、応用上重要であるだけでなく、理工学の広い分野で同様の高次元波の存在を期待させる契機を与えるものである。
      2.2次元戸田方程式と関連して、適当なリダクションのもとでのその解が、離散型パンルヴェ方程式の解と密接に関係することを明らかにした。格子型の解は連続極限で特殊関数解に移行するのに対して、分子型の解は連続極限でその構造が壊れてしまうが、この結果は離散系におけるソリトン理論に新しい知見を与えるものである。
      3.2成分KPヒエラルキーに含まれる非線形シュレディンガー型方程式および微分型非線形シュレディンガー方程式に対して、その双線形構造を調べ、解空間に対する考察を行った。その結果、解空間の対称性がアフィン・リー環で明瞭に表されること、また物理的な解に要請される条件がアフィン・リー環の実形と密接に対応していることを示した。この結果はソリトン方程式および解の代数構造についてやはり新しい知見を与えるものである。

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    • 超幾何関数と量子群

      日本学術振興会  科学研究費助成事業 

      堀川 穎二, 木村 弘信, 野海 正俊, 小林 俊行, 岡本 和夫

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      1994 - 1994

      Grant number:06640017

      この研究では、多変数の超幾何関数の群論的な構造を研究の主題としている。ゲルファンドによる超幾何型のホロノミック偏微分方程式系の発見により、ガウスの超幾何関数はグラスマン多様体上の多変数の超幾何関数に拡張され、これらの関数の持つ変換公式と隣接関係が一般線形群の作用から自然に定義されることが判明した。その結果として、従来知られていたガウスの超幾何関数の場合にも新しい観点が生まれたと考えられる。その例としてI変数超幾何関数のq-類似関数について、その隣接関係を定めるq-差分作用素の計算が可能となり、それらの作用素全体がSL(4)の量子展開環をなすことが判明したのであった。さらにその発展として、ゲルファントの定義したグラスマン多様体上の超幾何関数の場合の隣接関係が一般線形群GL(n)の作用で説明できることに注目して、それを拡張した形でq-類似の場合にも隣接関係を与える差分作用素を定めることができた。これらの作用素の全体は対応する量子群の表現を与えるものである。
      もう一つの興味深い場合として、シンプレクチック群Sp(2n)に対応する場合が考えられる。この場合には交代形式のある2n次元ベクトル空間を考え、そのラグランジアン部分空間の作るグラスマン多様体の上で定義される関数で超幾何形のホロノミック系をみたすものを考えるわけである。qがつかない場合にはその隣接関係を与える微分作用素の計算はできていたのであるが、本年はそのq-類似の場合の計算を遂行してほぼ満足すべき結果を得た。ここでもC型のディンキン図形に対応する量子展開環が現れることがほぼ確実である。

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    • 離散型ソリトン方程式の解析

      日本学術振興会  科学研究費助成事業 

      薩摩 順吉, 木村 弘信, 野海 正俊, 岡本 和夫, 堀川 頴二, 谷島 賢二

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      1993 - 1993

      Grant number:05836008

      Grant amount:\2100000 ( Direct Cost: \2100000 )

      本研究の目的である離散型ソリトン方程式の解析に関して、以下の成果を得た。
      1)「特異性の閉じ込め」性に関連して現れる離散型パンルベ方程式のうち、2型のものについて、離散型特殊関数で表される新しい解を提案した。解は行方向と列方向で構造が異なる行列式で表されており、離散系特有の性質を持っている。また、解を構成する手法は他の離散型パンルベ方程式にも適用可能で、1型、3型について予備的な結果を得ている。とくに、3型の方程式では、q-離散系との関係が明らかになっている。
      2)代表的な離散型ソリトン方程式である戸田格子方程式を相対論的に拡張したものに対し、その解がカソラチ行列式で与えられることを示した。その解の特別な場合として、Nソリトン解が含まれている。この結果は、戸田方程式と関連したさまざまなソリトン方程式の相対論化に応用することが可能である。
      3)離散的なτ函数の理論における戸田方程式系列を利用して、非線形シュレディンガー方程式とSIT方程式が結合した系に対する厳密解を構成した。得られたソリトン解は非対称に振動するという特徴を持っており、実験との対応も期待される。またこの結果は、principal chiral field方程式などの解の構成にも応用することができる。
      4)行列の固有値問題におけるQR分解法とソリトン方程式の関連を調べ、その中から新しいタイプのLotoka-Volterra方程式を導出し、その解について検討を加えた。この方程式は、それぞれの種が最近接の種だけでなく、遠方の種とも相互作用している生態系を表している。また、有限系の場合の初期値問題は、QR分解法を用いて厳密に解くことができる。

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    • 微分方程式の変換群

      日本学術振興会  科学研究費助成事業 

      岡本 和夫, 河野 俊丈, 山本 昌宏, 木村 弘信, 野海 正俊, 金子 晃

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      1993 - 1993

      Grant number:05640154

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      本研究課題は,微分方程式とくに複素領域において定義された完全積分可能系の変換群を調べ,その代数的な構造をあきらかにすることが目標である。非線形完全積分可能系の中心的な役割を果たす2変数戸田方程式は,歴史的には古典的な曲面の変換論として19世紀末にダルブーにより見いだされたが,現代の数理物理学の観点からこの仕事を見直すことはなされていなかった。研究代表者は,この古典的曲面論の変換の理論,特にベックルント変換と梯子構造を不変にする変換の関係,という幾何学的研究を進め成果を得て,研究会などで発表した。現在論文を準備中であり,本年中に発表する予定である。
      また,線形方程式の変形理論の立場から,トーラス上の完全積分可能系についても研究を進め,いくつかの結果を得た。この成果は最近口頭発表したところであり,これについても論文を準備中である。
      完全積分可能系は広範な数学の分野と関連しており,研究協力者によりそれぞれの立場から研究が進められ成果が得られた。その主なものを報告する。量子群の表現論は,完全積分可能系のq-アナローグを追求するものであり,重要な研究課題である。この立場から研究を進めた。完全積分可能系の解析学を進めるためには偏微分方程式の解の構造を明らかにすることが必要となるが,この点について金子晃は熱方程式等について結果を得た。また。コンフィギュレイション空間上の超幾何関数の理論との関連を追求するという立場から,木村弘信は合流型超幾何関数について重要な結果を得た。

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    • 代数幾何学とその境界領域への応用

      日本学術振興会  科学研究費助成事業 

      堀川 穎二, 小林 俊行, 木村 弘信, 野海 正俊, 岡本 和夫, 清水 英男

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      1993 - 1993

      Grant number:05640014

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      本年度の研究では、代数多様体の構造の研究、および多変数の超幾何関数の群論的な構造の研究において進展を見た。
      まず、代数多様体の構造では6次曲面の大域変形を決定した論文を完成、投稿して出版した。引き続き、種数3のペンシルを持つ曲面の構造の研究の完成を目指している。
      後者の超幾何関数については、パラメーターを上げ下げした場合の隣接関係の観点から、これらの関数にはたらく対称性をリー群・リー環との関連でとらえることに興味を持ち、その立場からの研究を続けた。ゲルファントの定義したグラスマン多様体上の超幾何関数の場合の隣接関係が一般線形群GL(n)の作用で説明できることに注目して、それを拡張した形でq-類似の場合にも隣接関係を与える差分作用素を定めることができた。これらの作用素の全体は対応する量子群の表現を与えるものである。同様の関係がシンプレクチック群Sp(2n)の場合にも発見できることを予想していたが、本年の研究によって最終的な結果を得た。現在論文を作成中である。これはガウスの1変数超幾何関数でパラメーターが特別な関係を持つ場合を含みその点からも興味深いものである。
      さらにシンプレクチックの場合にもq-類似が存在することが期待されるので、このようなqつきべき級数の定義、それのみたす差分方程式系、およびそれらの隣接関係を研究し、Spの場合の量子展開環との関連を明らかにすることが次年度以降の研究で期待される。
      超幾何関数の研究は種々の専門が交差する分野にあたり、研究協力者からのサジェスションが有益であった。特に協力者の木村弘信は合流型の超幾何関数について類似の研究を行い、野海正俊はより抽象的に量子群の立場から隣接関係をとらえる研究を行った。

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    • MATHEMATICAL PHYSICS,TOPOLOGY AND RELATED ALGEBRAIC STRUCTURE

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KOHNO Toshitake, KIMURA Hironobu, NOUMI Masatoshi, KATSURA Toshiyuki

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      1991 - 1993

      Grant number:03640073

      Grant amount:\1800000 ( Direct Cost: \1800000 )

      1.Construction of 3-manifold invariants derived from conformal field theory and its applications
      Based on Chern-Simons gauge theory, Witten proposed topological invariants of 3-manifolds. Several works have been done afterwards from geometric or combinatorial viewpoints. We constructed 3-manifold invariants based on representations of mapping class groups appearing in conformal field theory and Heegaard splitting of 3-manifolds.
      As an application, using the unitarity of the monodromy of conformal field theory, we obtained lower estimates for classical invariants, such as Heegaard genus and tunnel numbers of knots etc. Investigating the symmetry derived from Dynkin diagram automorphisms, we refined Witten invariant and established the level-rank duality.
      2.Graph complex and differential forms on knot space
      The object of this research is differential forms on the space of all knots, which is an infinite dimensional space. We constructed a morphism from the graph complex to the de Rham complex on the knot space. This might be considered to be a generalization of the bar complex for the loop space. Especially, as the zero dimensional cohomology of the graph complex, the Vassiliev invariants can be represented by integrals appearing in Chern-Simons perturbation theory.
      Applying the de Rham homotopy theory to the pure braid group, we showed that the filtration derived from the Vassiliev invariants for pure braids coinsides with the lower central series. It turns out that the Vassiliev invariants are strong enough to distinguish any pure braid.

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    • スペクトル・散乱理論の研究

      日本学術振興会  科学研究費助成事業 

      谷島 賢二, 小林 俊行, 野海 正俊, 薩摩 順吉, 北田 均, 金子 晃

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      1992 - 1992

      Grant number:04640130

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      科学研究費補助金(一般研究C)「スペクトル・散乱理論の研究」によって得られた成果を箇条書きにする。
      1.多体Schroedinger作用素の散乱問題において波動作用素の完全性に関して北田均等によって短距離系に対する波動作用素の完全性の新たな証明が得られ,遠距離系に対する完全性の証明にも重要な進展があった。
      2.Schroedinger作用素の散乱・resonanceの問題に関して,resonanceのおよびtime-delayの準古典近似の問題が中村周等によって研究され準古典領域におけるresolvent等の振る舞い明確にされた
      3.Schroedinger作用素の波動作用素あるいはSchroedinger作用素の関数のSobolev空間あるいはBesov空間における写像としての性質が中村周,谷島賢二等によって研究され,種々の空間の間の有界性等多くの性質が明らかになった。これらの研究は非線形微分方程式のこれからの研究に大いに役立つものと期待される。
      4.スペクトルの逆問題に関連した研究が薩摩順吉等によって進められ,soliton方程式,非線形波動方程式のの解の構造,あるいは完全積分系に関する多くの新たな知見が得られた。
      5.スペクトルの数値解析に関連して有限要素法の数値解析の研究が菊地文雄,金子晃等によって進められ,多くの新たな知見が得られた。
      6.その他,分担者たちは多くの関連した研究を進展させた。主なものは野海正俊による量子群の研究,小林俊行による非可換調和解析の研究,山崎満によるBoltzmann方程式の研究,木村弘信による超幾何関数の研究及び太田啓史・古田幹雄による多様体の微分構造の研究等である。

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    • 特殊函数と代数構造

      日本学術振興会  科学研究費助成事業 

      野海 正俊, 山本 昌弘, 寺田 至, 木村 弘信, 岡本 和夫, 堀川 穎二

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      1991 - 1991

      Grant number:03640127

      Grant amount:\1900000 ( Direct Cost: \1900000 )

      本研究では、特殊函数と代数構造の結びつきに関し、いくつかの新たな側面を開拓した。本研究で得た成果の一部と、その現在の状況は下の通りである。
      1:野海は、特殊函数のqーanalogueの立場から、量子群上の等質空間の球函数に関する研究を行った。文献1は、Jacobi多項式の多様なqーanalogueと量子群SU_q(2)上の等質空間との本質的な関連を論じたものである。多変数の特殊函数に関連しても、等質空間GL(n)/SO(n),GL(2n)/Sp(2n)の量子群analogueを考察し、その帯球函数がMacdonald対称多項式の幾何学的実現を与えることを発見した(論文準備中)。
      2:また、量子群上の代数解析の基礎付けという観点から、量子一般線型群上で微分作用素の対応物を構成し、それに対するCapelli恒等式を得た(文献3)。Capelli恒等式は、Lie環の包絡環の中心元と不変微分作用素の関係を明示し、古典的不変式論と特殊函数論を結びつける重要な役割を演じてきたものである。
      3:堀川は、Grassmann多様体に付随するGelfand超幾何函数の隣接関係の対称性を研究した(文献4)。更にその立場から、Gaussの超幾何級数のqーanalogueを考察し、その隣接関係として量子群GL_q(4)の対称性が現れることを発見した(文献5)。堀川及び野海は現在、これを量子Grassmann多様体によって意味付けし、多変数に拡張する研究を行っている。
      4:木村は、多時間変数の確定特異点型Hamilton系の特異点の周りの標準形を考察し、Painleve超越函数の多変数への拡張であるGarnier方程式系の解に対し、特異点の周りでの挙動を調べることに成功した(文献6)。また、Gelfand超幾何函数に対しても、その合流に関する研究を現在行っている。

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    • Algebro-analytic study of partial differential equations

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      KANEKO Akira, KITADA Hitoshi, YAJIMA Kenji, HORIKAWA Eiji, KIKUCHI Fumio, NAMBA Kanji

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      1990 - 1991

      Grant number:02452006

      Grant amount:\5700000 ( Direct Cost: \5700000 )

      A. Kaneko studied the. solvability of convolution equation in the space of tempered distributions with a foreian research visitor, and got anew result for the necessity. Then in colaboration with graduate students developed the method of solution of convolution equations by finite difference method. K. Namba continued to study the Sato conjecture on the ancular distribution of the zeros of the zeta functions of algebraic carves, and recently found an interesting phenomenom concerning a kind of alaebraic curves which can be described in terms of a finite field analogue of the hypergeometric function. E. Horikawa rave a new viewpoint to the contiguity relation of the hypergeometric functions of AomotoGelfand, clarified relation with Lie algebras and made the theory transparent. K. Yajima proved that the solution of the Dirac equation for the Dirac particles in the electromagnetic field induced by finitely maly moving charges exists uniquely and the propagator conserves the Sobolev space of order 1. under a condition of bound for the velocities of the charges. He also treated the scattering theory of the one dimensional Stark Hamiltonian with long range potential, and showed that the existence and completeness of modified wave operator by means of modified propagator representing the classical trajectories asymptotically, thus clarifying the paradox between the classical scattering theory and the quantum one which has long been a theme of discussion. H. Kitada showed the asymptotic completeness of the wave operator for the Schroedinger operator with short range many particles employing only the microlocal asymptotic estimate of Enss. Further he gave a necessary and sufficient condition for the asymptotic completeness in the case of long range many particles. H. Ito gave a proof to the existence of an index map satisfying some arithmetic conditions in relation to subgroups mod 3 of SL(2, Z), and gave applications to the arithmetics of imaginary quadratic fields.

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    • 代数幾何学とその応用

      日本学術振興会  科学研究費助成事業 

      堀川 頴二, 黒田 成信, 山本 昌宏, 野海 正俊, 木村 弘信, 岡本 和夫

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      1990 - 1990

      Grant number:02640017

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      本年度の研究では、6次曲面の変形の決定、種数3の曲線ペンシルの構造の研究、グラスマン多様体上の超幾何関数の研究、および量子群と関連した特殊関数のqーアナロ-グについて研究を行ない多くの目覚ましい成果を得た。
      1.6次曲面の変形.射影平面内の非特異6次超曲面の変形として得られる可能性のある曲面の構造を完全に決定し、それらが実際6次曲面の変形であることを構成的に証明した。この結果このクラスの曲面の可微分多様体の上の複素構造の全体のモデュライ空間は既約であることが示された。これは一般型の曲面の例としては初めてである。この結果は曲面上のラインバンドルで標準バンドルの1/2であるものの定める有理写像を詳しく調べることによって得られる。このとき最も本質的かつ困難であるのは、この写像が種数3のペンシルになる場合である。この困難を克服することにより副産物として種数3のペンシルの構造の研究に対しても新しい対見が得られた。これを発展させて種数3のペンシルの一般的な構造の研究が近い将来に可能となるであろう。
      2.超幾何関数の研究。I.M.ゲルファンドと共同研究者によって精力的に研究されている一般化された超幾何関数についてリ-環的な見地からその変換法則と隣接関係を明らかにした。これらは従来の結果を非常に簡明で構造的な立場から解明したものである。
      3.研究分担者野海正俊は名古屋大学の三町勝久と共同で量子群を通して特殊関数のqーアナロ-グを研究してアスキ-・ウイルソン多項式と量子群の表現との関係を明らかにした。これは将来2.の超幾何関数との関連する研究に発展が期待されるものである。

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    • 量子群と可積分系

      日本学術振興会  科学研究費助成事業 

      野海 正俊

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      1989 - 1989

      Grant number:01740109

      Grant amount:\900000 ( Direct Cost: \900000 )

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    • 階別り一環の幾何学

      日本学術振興会  科学研究費助成事業 

      金行 壯二, 田原 秀敏, 野海 正俊, 横沼 健雄, 加藤 昌英, 長野 正

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      1989 - 1989

      Grant number:01540074

      Grant amount:\2000000 ( Direct Cost: \2000000 )

      1.シロフ境界の因果構造の研究(金行)。柱状対称有界領域D(次元>1)のシロフ境界Sに、或自己共役錐体をモデルとする因果構造を導入した。そしてその自己同型群を決定した。この群の作用はDの内部へのびてDの金正則変換郡を与へしかもそれで一杯であることを示した。2.擬ケ-ラ-アフィン対称空間の研究(金行)。Kε型の擬ケ-ラ-単純既約対称空間(7つある)を階別り一環を用いて統一的に構成しその正則埋込を作った。これによりこの種の対称空間のある稠密開集合は、拡張されたいみのジ-ゲル領域として実現できることを示した。前年度に研究したジョルダン代数に対するシルヴェスタ-の慣性律が証明に使われる。3.クラスL(=射影直線の近傍と正則同値な領域を含む)3次元複素曲面の構造の研究(加藤)。4.複素射影接続をもつ複素多様体にワイルの射影曲率テンソルを用いて新しい特性形式を導入した。そしてチャ-ン特性形式との関係を調べた(加藤)。5.トロイダルリ一環とヴァ-テックス表現の研究(横沼)。C[s,s^<-1>,t,t^<-1>]【cross product】〓(〓は複素単純り一環)の普遍中心拡大の、ヴァ-テックス作用素を用いた表現の構成(横沼)。6.特異点をもつ線型微分方程式に対してかなり一般的条件の下で局所可解性空理を示した(田原)。7.Briot-Bouguet型特異点をもつ非線型偏微分方程式のすべての解の決定(田原)。8.Gevrey級の発散形式解の研究とMaillet型定理の一般化(田原)。9.量子等質空間SU_q(n+1)/SU_q(n)上の帯球函数の研究(野海)。10.2次元量子球面と大gヤコビ多項式についての研究(野海)。

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    • 非線型微分方程式の代数的研究

      日本学術振興会  科学研究費助成事業 

      野海 正俊

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      1988 - 1988

      Grant number:63740103

      Grant amount:\1000000 ( Direct Cost: \1000000 )

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    • 非線型偏微分方程式の可積分系

      日本学術振興会  科学研究費助成事業 

      野海 正俊

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      1987 - 1987

      Grant number:62740106

      Grant amount:\900000 ( Direct Cost: \900000 )

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    • 非線型偏微分方程式の可積分系

      日本学術振興会  科学研究費助成事業 

      野海 正俊

      More details

      1986 - 1986

      Grant number:61740102

      Grant amount:\1000000 ( Direct Cost: \1000000 )

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