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2003 Fiscal Year Final Research Report Summary

Differential Equations and Reflection Groups(Polyhedral Harmonics, Hypergeometric Equations, and Painleve Equations)

Research Project

Project/Area Number 12440043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyushu University

Principal Investigator

IWASAKI Katsunori  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00176538)

Co-Investigator(Kenkyū-buntansha) INABA Michi-aki  Kyushu University, Faculty of Mathematics, Research Assistant, 大学院・数理学研究院, 助手 (80359934)
KAJIWARA Kenji  Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (40268115)
YOSHIDA Masaaki  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30030787)
上村 豊  東京海洋大学, 海洋科学部, 教授 (50134854)
SAITO Masa-hiko  Kobe University, Faculty of Science, Professor, 理学部, 教授 (80183044)
Project Period (FY) 2000 – 2003
KeywordsPainleve equations / hypergeometric equations / polyhedral harmonics / reflection groups / moduli spaces / twisted cohomology / modular groups / stable parabolic connections
Research Abstract

1.Holyhedral harminics : A general theory for polyhedral harmonics has been developed concerning the finite dimensionality of the space of polyhedral harmonic functions and those holonomic systems of partial differential equations which characterize the polyhedral harmonic functions. A survey article on the subject was written and the state of the art of the subject was addressed as a special lecture of the 2002 autumn meeting of the Mathematical Society of Japan.
2.Hypergeometric equation :' An intersection theory for twisted de Rham cohomology groups associated with isolated singularities has been established. By developing Kumano-go-Taniguchi-type pseudodifirential calculus for Witten's Laplacian, a version of Hodge-Kodaira decomposition and Poincare-Serre-type duality theorems have been proved. As an application, the intersection matrices of generalized Airy functions have been determined explicitly in terms of skew-Schur polynomials. A certain cohomology theory for systems of inhom … More ogeneous finite difference equations has been constructed. The theory was applied to contiguity relations of confluent hypergeometric systems to compute their Gevrey cohomology groups.
3.Painleve equations : The generating function for the rational solutions to the Painleve II equation has been determined explicitly in terms of the Airy function. The nonlinear monodromy of the Painleve VI equation has been realized as an action of the modular group on the four-parameter family of affine cubic surfaces. The phase spaces of the Painleve VI equation and the Gamier systems have been constructed algebro-geometrically as moduli spaces of stable parabolic connections. The affine-Weyl group symmetry of Backlund transformations has been constructed from the viewpoint of Riemann-Hilbert correspondence.
4.Inverse bifurcation problems : The solvability of singular Wiener-Hopf equations has been investigated and applied to the inverse problem for bifurcation phenomena as well as to some reaction-diffsion models in mathematical biology. Less

  • Research Products

    (82 results)

All Other

All Publications (82 results)

  • [Publications] Katsunori Iwasaki: "Recent progress in polyhedral harmonics"Acta Applicandae Mathematicae. 60(2). 179-197 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki: "Polytopes, invariants and harmonic functions"Advanced Studies in Pure Mathematics. 27. 145-156 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki, Keiji Matsumoto: "Intersection matrix of generalized Airy functions in terms of skew-Schur polynomials"Proceedings of the Japan Academy Series A. 76. 135-140 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masa-Hiko Saito, Hiroshi Umemura: "Painleve equations and deformations of rational surfaces rational double points"Proceedings of Nagoya International Workshop On Physics and Combinatorics, World Scientific. 320-365 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki, Yutaka Kamimura: "Inverse bifurcation problem, singular Wiener-Hopf equations and mathematical models in ecology"Journal of Mathematical Biology. 43(2). 101-143 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kyoichi Takano: "Confluence processes in defining manifolds for Painleve systems"Tohoku Mathematical Journal. 53(2). 319-335 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kyoichi Takano: "Defining manifolds for Painleve equations"Toward the Exact WKB Analysis of Differential Equations, Kyoto University Press. 261-269 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kazuo Okamoto, Kyoichi Takano: "The proof of the Painleve property by Masuo Hukuhara"Funkcialaj Ekvacioj. 44(2). 201-217 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada: "Determinant formula for the Toda and discrete Toda equations"Funkcialaj Ekvacioj. 44. 291-307 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara, M.Noumi, Y.Yamada: "A study on the fourth q-Painleve equation"Journal of Physics A : Mathematical and General. 34. 8563-8581 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Joe Kamimoto: "Nonanalytic Bergman and Szego kernels for weakly pseudoconves tube domains in C^2"Mathematische Zeitschrift. 236(3). 585-603 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Hosono, M.-H.Saito, A.Takahashi: "Relative Lefschetz action and BPS state counting"International Mathematics Research Notices. 2001:15. 783-816 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masa-Hiko Saito, Noriko Yui: "The modularity conjecture for rigid Calabi-Yau threefolds over Q"Journal of Mathematics, Kyoto University. 41(2). 403-419 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Noumi, K.Takano, Y.Yamada: "Backlund transformations and the manifolds of Painleve systems"Funkcialaj Ekvacioj. 45(2). 237-258 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Iwasaki, K.Kajiwara, T.Nakamura: "Generating function associated with the rational solutions of the Painleve II equation"Journal of Physics A : Mathematical and General. 35. L207-L211 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Iwasaki, A.Kenma, K.Matsumoto: "Polynomial invariants and harmonic functions related to exceptional regular polytopes"Experimental Mathematics. 11(2). 313-319 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki: "A modular group action on cubic surfaces and the monodromy of the Painleve VI equation"Proceedings of the Japan Academy Series A. 78(7). 131-135 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Masuda, Y.Ohta, K.Kajiwara: "A determinant formula for a class of rational solutions of Painleve V equation"Nagoya Mathematical Journal. 186. 1-25 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara, M.Noumi, Y.Yamada: "Discrete dynamical systems with W(A^{(1)}_{m-1}×A^{(1)}_{n-1}) symmetry"Letters in Mathematical Physics. 60. 211-219 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masa-Hiko Saito, T.Takebe: "Classifications of Olamoto-Painleve pairs"Kobe Journal of Mathematics. 19. 21-50 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Viet Nguen, Masa-Hiko Saito: "On Mordel-Weil lattices for non-hyperelliptic fibrations of Surfaces with zero geometric genus and irregularity"Izv.Ross.Acad.Nauk.Ser.Mat.English translation in Izv.Math.. 66. 789-805 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.-H.Saito, T.Takebe, H.Terajima: "Deformation of Okamoto-Painleve pairs and Painleve equations"Journal of Algebraic Geometry. 11. 311-362 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michi-aki Inaba: "On the moduli of stable sheaves on a reducible projective Scheme and examples of a reducible quadratic surface"Nagoya Mathematical Journal. 166. 135-181 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michi-aki Inaba: "Toward a definition of moduli of complexes of coherent sheaves on a projective scheme"Journal of Mathematics, Kyoto University. 42. 317-329 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki: "Isolated singularity, Witten's Laplacian and duality for Twisted de Rham cohomology"Communications in Partial Differential Equations. 28(1-2). 61-82 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki: "Cohomology groups for recurrence relations and Contiguity relations of hypergeometric systems"Journal of the Mathematical Society of Japan. 55(2). 289-321 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsunori Iwasaki: "An area-preserving action of the modular group on cubic surfaces and the Painleve VI equation"Communications in Mathematical Physics. 242(1-2). 185-219 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhisa Mimachi, Masaaki Yoshida: "Intersection numbers of twisted cycles and the correlation functions of the conformal field theory"Communications in Mathematical Physics. 243. 339-358 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takeshi Sasaki, Masaaki Yoshida: "Invariant theory for linear differential systems modeled after the Grassmannians G(n,2n)"Nagoya Mathematical Journal. 171. 163-186 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masaaki Yoshida: "A geometric interpretation of the Selberg integral"International Journal of Modern Physics A. 18. 4343-4359 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhisa Mimachi, Masaaki Yoshida: "The reciprocity relation of the Selberg function"Journal of Computational and Applied Mathematics. 160. 209-215 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masanobu Kaneko, Masaaki Yoshida: "The kappa function"International Journal of Mathematics. 14. 1003-1013 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Mimachi, H.Ochiai, M.Yoshida: "Intersection theory of loaded cycles IV --- resonant cases"Mathematische Nachrichten. 260. 66-77 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kenji Kajiwara, K.Kimura: "On a q-Painleve III equation. I. derivations, symmetry and Riccati type solutions"Journal of Nonlinear Mathematical Physics. 10. 86-102 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kenji Kajiwara: "On a q-Painleve III equation. II : rational solutions"Journal of Nonlinear Mathematical Physics. 10. 282-303 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada: "10E9 solution to the elliptic Painleve equation"Journal of Physics A : Mathematical and General. 36. L263-L272 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Inaba, K.Iwasaki, M.-H.Saito: "Backlund transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence"International Mathematics Research Notices. 2004:1. 1-30 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Joe Kamimoto: "Newton polyhedra and the Bergman kernel"Mathmatische Zeitschrift. 246(3). 405-440 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michi-aki Inaba: "On the moduli of stable sheaves on some nonreduced projective schemes"Journal of Algebraic Geometry. 13(1). 1-27 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Inaba, K.Iwasaki, M.-H.Saito: "Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of painleve equation of type VI"Kyushu University Preprint Series in Mathematics. 26. 1-71 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 上村 豊: "積分方程式---逆問題の視点から---"共立出版株式会社. 291 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Iwasaki: "Recent progress in polyhedral harmonics"Acta Applicandae Mathematicae. 60(2). 179-197 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki: "Polytopes, invariants and harmonic functions"Advanced Studies in Pure Mathematics. 27. 145-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki, K.Matsumoto: "Intersection matrix generalized Airy functions in terms of skew-Schur polynomials"Proceedings of the Japan Academy Series A. 76. 135-140 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.-H.Saito, H.Umemura: "Painleve equations and deformations of rational surfaces with rational double points"Proceedings of the Nagoya Int.Workshop Physics and combinatorics Nagoya University, 1999, World Scientific(Kirillov, A. et al.(eds.)). 320-365 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki, Y.Kamimura: "Inverse bifurcation problem, singular Wiener-Hopf equation and mathematical models in ecology"Journal of Mathematical Biology. 43(2). 101-143 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Takano: "Confluence processes in defining manifolds for Painleve systems"Tohoku Mathematical Journal. 53(2). 319-335 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Takano: "Defining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations, linear or non-linear(Kyoto,1998)(Kyoto Univ.Press, Kyoto). 204. 261-269 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Okamoto, K.Takano: "The proof of the Painleve property by Masuo Hukuhara"Funkcialaj Ekvacioj. 44(2). 201-217 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada: "Determinant formulas for the Toda and discrete Toda equations"Funkcialaj Ekvacioj. 44. 291-307 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara, M.Noumi, Y.Yamada: "A study on the fourth q-Painleve equation"Journal of Physics A : Mathematical and General. 34. 8563-8581 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Kamomito: "Non-analytic Bergman and Szego kernels for weakly pseudoconvex tube domains in C^2"Mathematische Zeitschrift. 236(3). 585-603 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Hosono, M.-H.Saito, A.Takahashi: "Relative Lefschetz action and BPS state counting"International Mathematics Re search Notices. {\bf2001:15}. 783-813 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.-H.Saito, N.Yui: "The modularity conjecture for rigid Calabi--Yauthreefolds over Q"Journal of Mathematics, Kyoto University. {\bf41}(2). 403-419 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Noumi, K.Takano, Y.Yamada: "Backlund transformations and the manifolds of Painleve systems"Funkcialaj Ekvacioj. 45(2). 237-258 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki, K.Kajiwara, T.Nakamura: "Generating function associated with the rational solutions of the Painleve II equation"Journal of Physics A : Mathematical and General. 35. L207-L211 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki, A.Kenma, K.Matsumoto: "Polynomial invariants and harmonic functions related to exceptional regular polytopes"Experimental Mathematics. 11(2). 313-319 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki: "A modular group action on cubic surfaces and the monodromy of the Painleve VT equation"Proceedings of the Japan Academy Series A. 78(7). 131-135 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Masuda, Y.Ohta, K.Kajiwara: "A determinant formula for a class of rational solutions of Painleve V equation"Nagoya Mathematical Journal. 168. 1-25 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara, M.Noumi, Y.Yamada: "Discrete dynamical systems with W(A^{(1)} {m-1}×A^{(1)}_{n-1}) symmetry"Letters in Mathematical Physics. 60. 211-219 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.-H.Saito, T.Takebe: "Classification of Okamoto-Painlev\'e pairs"Kobe Journal of Mathematics. 19. 21-50 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Viet, Nguen, M.-H.Saito: "On Mordell-Weil lattices for non-hyperelliptic fibrations of surfaces with zero geometric genus and irrgularity"Izv.Ross.Acad.Nauk.Ser.Mat.. 66,no.4. 137-154 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Viet, Nguen, M.-H.Saito: "On Mordell-Weil lattices for non-hyperelliptic fibrations of surfaces with zero geometric genus and irrgularity"translation in Izv.Math.. 66. 789-805 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.-H.Saito, T.Takebe, H.Terajima: "Deformation of Okamoto--Painleve pairs and Painleve equations"Journal of Algebraic Geometry. 11. 311-362 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Inaba: "On the moduli of stable sheaves on a reducible projective scheme and examples on a reducible quadric surface"Nagoya Mathematical Journal. 66. 135-181 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Inaba: "Toward a definition of moduli of complexes of coherent sheaves ona projective scbeme"Journal of Mathematics, Kyoto University. 42. 317-329 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki: "Isolated singularity, Witten's Laplacian and duality for twisted de Rham cohomology"Communications in Partial Differential Equations. 28(1-2). 61-82 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki: "Cohomology groups for recurrence relations and contiguity relations of hypergeometric systems"Journal of the Mathematical Society of Japan. 55(2). 289-321 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki: "An area-preserving action of the modular group on cubic surfaces and the Painleve VI equation"Communications in Mathematical. 242(1-2). 185-219 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Mimachi, M.Yoshida: "Intersection numbers of twisted cycles and the correlation functions of the conformal field theory"Communications in Mathematical Physics. 234(2). 339-358 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sasaki, M.Yoshida: "Invariant theory for linear differential systems modeled after the Grassmannian Gr(n,2n)"Nagoya Journal of Mathematics. 171. 163-186 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Yoshida: "A geometric interpretation of the Selberg integral"International Journal of Modern Physics A. 18. 4343-4359 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Mimachi, M.Yoshida: "The reciprocity relation of the Selberg function"Journal of Computational and Applied Mathematics. 160. 209-215 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kaneko, M.Yoshida: "The kappa function"International Journal of Mathematics. 14. 1003-1013 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Mimachi, H.Ochiai, M.Yoshida: "Intersection theory of loaded cycles IV --resonant cases"Mathematische Nachrichiten. 260. 66-77 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara, K.Kimura: "On a q-Painleve III equation. I : Derivations, symmetry and Riccati type solutions"Journal of Nonlinear Mathematical Physics. 10. 86-102 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara: "On a q-Painleve III equation. II : rational solutions"Journal of Nonlinear Mathematical Physics. 10. 282-303 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada: "10E9 solution to the elliptic Painleve equation"Journal of Physics, A : Mathematical and General. 36. L263-L272 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Inaba, K.Iwasaki, M.-H.Saito: "Backlund transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence"International Mathematics Research Notices. 2004:1. 1-30 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Kamimoto: "Newton polyhedra and the Bergman kernel"Mathematische Zeitschrift. 246(3). 405-440 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Inaba: "On the moduli of stable sheaves on some nonreduced projectiveschemes"Journal of Algebraic Geometry. 13(1). 1-27 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Inaba, K.Iwasaki, M.-H.Saito: "Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equations of type VI, Part I"Kyushu University Preprint Series in Mathematics. 2003-26. 1-71 (2003)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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