Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Profile and Settings

• Name (Japanese)

  Iwasaki

• Name (Kana)

  Katsunori

• Name

  201301068988103467

Alternate Names

http://kaken.nii.ac.jp/d/r/00176538.ja.html

Achievement

Research Interests

• hypergeometric group   K3 surfaces   超幾何関数   パンルヴェ方程式   リーマン・ヒルベルト対応   モジュライ空間   複素力学系   非線形モノドロミー   力学系   ガルニエ系   エアリー関数   三次曲面   逆分岐問題   ベックルント変換   カオス   差分方程式   パンルベ方程式   変分問題   ベックルント変換群   コホモロジー群   捩れコホモロジー群   特異積分方程式   多面体   エルゴード理論   指標多様体   モデュライ理論   安定放物型接続   多面体調和関数   ウィナー・ホップ方程式   安定放物接続   エントロピー   逆問題   

Research Areas

• Natural sciences / Mathematical analysis
• Natural sciences / Basic analysis
• Natural sciences / Geometry
• Natural sciences / Algebra

Research Experience

• 2011 - Today 北海道大学 理学(系)研究科(研究院) 教授

Committee Memberships

• 2022/07 - Today   Journal of the Mathematical Society of Japan   Editor-in-Chief

Published Papers

• K3 Surfaces, Picard Numbers and Siegel Disks

  Katsunori Iwasaki, Yuta Takada

  Journal of Pure and Applied Algebra 227 (3) 0022-4049 2023/03 [Refereed][Not invited]
  
• Hypergeometric groups and dynamics on K3 surfaces

  Katsunori Iwasaki, Yuta Takada

  Mathematische Zeitschrift 301 (1) 835 - 891 0025-5874 2022/01/03 [Refereed]
  
• Duality and reciprocity for hypergeometric series with a gamma product formula

  Katunori Iwasaki

  Kyushu Journal of Mathematics 73 (2) 251 - 294 2019/08 [Refereed][Not invited]
  
• Contiguous relations, Laplace's methods, and continued fractions for 3F2(1)

  Akihito Ebisu, Katsunori Iwasaki

  The Ramanujan Journal 49 (1) 159 - 213 2019 [Refereed][Not invited]
  
• Three term relations for 3F2(1)

  Akihito Ebisu, Katsunori Iwasaki

  Journal of Mathematical Analysis and Applications 463 (2) 593 - 610 2018 [Refereed][Not invited]
  
• Hypergeometric series with gamma product formula

  Katsunori Iwasaki

  INDAGATIONES MATHEMATICAE-NEW SERIES 28 (2) 463 - 493 0019-3577 2017/04 [Refereed][Not invited]
   

  We consider non-terminating Gauss hypergeometric series with one free parameter. Using various properties of hypergeometric functions we obtain some necessary conditions of arithmetic flavor for such series to admit gamma product formulas. (C) 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
• On an orbifold Hamiltonian structure for the first Painleve equation

  Katsunori Iwasaki, Shu Okada

  JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 68 (3) 961 - 974 0025-5645 2016/07 [Refereed][Not invited]
   

  For the first Painleve equation we establish an orbifold polynomial Hamiltonian structure on the fibration of Okamoto's spaces and show that this geometric structure uniquely recovers the original Painleve equation, thereby solving a problem posed by K. Takano.
• Around hypergeometric identities

  Katsunori Iwasaki, Akihito Ebisu

  Rokko Lectures in Mathematics 神戸大学理学部数学科 24 119 - 158 2016 [Not refereed][Invited]
  
• On algebraic solutions to Painleve VI

  Katsunori Iwasaki

  RIMS Kokyuroku Bessatsu 京都大学 B37 49 - 67 1881-6193 2013 [Refereed][Invited]
  
• Isolated Periodic solutions to Painleve VI equation

  Katsunori Iwasaki, Takato Uehara

  RIMS Kokyuroku Bessatsu 京都大学 B37 69 - 79 1881-6193 2013 [Refereed][Invited]
  
• Cubic harmonics and Bernoulli numbers

  Katsunori Iwasaki

  JOURNAL OF COMBINATORIAL THEORY SERIES A 119 (6) 1216 - 1234 0097-3165 2012/08 [Refereed][Not invited]
   

  The functions satisfying the mean value property for an n-dimensional cube are determined explicitly. This problem is related to invariant theory for a finite reflection group, especially to a system of invariant differential equations. Solving this problem is reduced to showing that a certain set of invariant polynomials forms an invariant basis. After establishing a certain summation formula over Young diagrams, the latter problem is settled by considering a recursion formula involving Bernoulli numbers. (C) 2012 Elsevier Inc. All rights reserved.
• 特殊関数の問題 : パンルヴェ性をめぐって (複素幾何学の諸問題)

  岩崎 克則

  数理解析研究所講究録 京都大学 1731 (0) 1 - 13 1880-2818 2011/03 [Not refereed][Not invited]
  
• Periodic points for area-preserving birational maps of surfaces

  Katsunori Iwasaki, Takato Uehara

  MATHEMATISCHE ZEITSCHRIFT 266 (2) 289 - 318 0025-5874 2010/10 [Refereed][Not invited]
   

  It is a basic problem to count the number of periodic points of a surface mapping, since the growth rate of this number as the period tends to infinity is an important dynamical invariant. However, this problem becomes difficult when the map admits curves of periodic points. In this situation we give a precise estimate of the number of isolated periodic points for an area-preserving birational map of a projective complex surface.
• On the Markoff-Painleve Transcendent (Integrated Research on Complex Dynamics and its Related Fields)

  Iwasaki Katsunori

  RIMS Kokyuroku 京都大学 1699 (0) 160 - 167 1880-2818 2010/07 [Not refereed][Not invited]
  
• Monodromy of Painleve VI Equation Around Classical Special Solutions (Monodromy of the differential equations and related problems)

  Iwasaki Katsunori

  RIMS Kokyuroku 京都大学 1662 (0) 136 - 147 1880-2818 2009/08 [Not refereed][Not invited]
  
• Finite branch solutions to Painleve VI around a fixed singular point

  Katsunori Iwasaki

  ADVANCES IN MATHEMATICS 217 (5) 1889 - 1934 0001-8708 2008/03 [Refereed][Not invited]
   

  Every finite branch local solution to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations. (c) 2007 Elsevier Inc. All rights reserved.
• Some dynamical aspects of Painlevé VI

  Katsunori Iwasaki

  Algebraic Analysis of Differential Equations: From Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai 143 - 156 2008 [Refereed][Not invited]
   

  We survey some results from our recent studies on the sixth Painlevé equation as a dynamical system. We discuss such topics as phase space and its compactification, Riemann-Hilbert correspondence, Poincaré section, bounded orbits, topological entropy and dynamical degree, and periodic solutions. © 2008 Springer Japan.
• An ergodic study of Painleve VI

  Katsunori Iwasaki, Takato Uehara

  MATHEMATISCHE ANNALEN 338 (2) 295 - 345 0025-5831 2007/06 [Refereed][Not invited]
   

  An ergodic study of Painleve VI is developed. The chaotic nature of its Poincare return map is established for almost all loops. The exponential growth of the numbers of periodic solutions is also shown. Principal ingredients of the arguments are a moduli-theoretical formulation of Painleve VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and the Lefschetz fixed point formula.
• Chaos in the sixth Painleve equation

  Katsunori Iwasaki, Takato Uehara

  RIMS Kokyuroku Bessatsu B2 73 - 88 2007 [Refereed][Invited]
  
• Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equation of type VI, Part I - Dedicated to Professor Kyoichi Takano on his 60th birthday

  Michi-aki Inaba, Katsunori Iwasaki, Masa-Hiko Saito

  PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 42 (4) 987 - 1089 0034-5318 2006/12 [Refereed][Not invited]
   

  In this paper, we will give a complete geometric background for the geometry of Painleve VI and Garnier equations. By geometric invariant theory, we will construct a smooth fine moduli space M-n(alpha)(t, lambda, L) of stable parabolic connections on P-1 with logarithmic poles at D(t) = t(1) +(...)+t(n) as well as its natural compactification. Moreover the moduli space R(P-n,P-t)(a) of Jordan equivalence classes of SL2 (C)-representations of the fundamental group pi(1) (P-1\D(t),(*)) are defined as the categorical quotient. We define the Riemann-Hilbert correspondence RH : M-n(alpha) (t, lambda, L) -> R (P-n.t)(a) and prove that RH is a bimeromorphic proper surjective analytic map. Painleve and Garnier equations can be derived from the isomonodromic flows and Painleve property of these equations are easily derived from the properties of RH. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and R(P-n,P-t)(a) is a symplectic resolution of singularities of from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Backlund transformations, special solutions of these equations.
• Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equations of type VI, Part II

  Michiaki Inaba, Katsunori Iwasaki, Masa-Hiko Saito

  Advanced Studies in Pure Mathematics 45 387 - 432 2006 [Refereed][Not invited]
  
• Dynamics of the sixth Painleve equation

  Michiaki Inaba, Katsunori Iwasaki, Masa-Hiko Saito

  Seminaires et Congres 14 103 - 167 2006 [Refereed][Invited]
  
• Sensitivity of minimaxity and admissibility in the estimation of a positive normal mean

  Y Maruyama, K Iwasaki

  ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 57 (1) 145 - 156 0020-3157 2005/03 [Refereed][Not invited]
   

  On the problem of estimating a positive normal mean with known variance, it is well known that one minimax admissible estimator is the generalized Bayes one with respect to the non-informative prior measure, the Lebesgue measure, restricted on the positive half-line. When the true variance is misspecified, however, it is shown that this estimator does not always retain minimaxity and admissibility. In particular, it is almost surely inadmissible in the misspecification case.
• Backlund transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

  M Inaba, K Iwasaki, MH Saito

  INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2004 (1) 1 - 30 1073-7928 2004 [Refereed][Not invited]
  
• An area-preserving action of the modular group on cubic surfaces and the Painleve VI equation

  K Iwasaki

  COMMUNICATIONS IN MATHEMATICAL PHYSICS 242 (1-2) 185 - 219 0010-3616 2003/11 [Refereed][Not invited]
   

  We construct an area-preserving action of the modular group on a general 4-parameter family of affine cubic surfaces. We present a geometrical background behind this construction, that is, a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2-dimensional sphere with four punctures, and a natural symplectic action upon it of the braid group on three strings. Studying this action as a discrete dynamical system will be important in discussing the monodromy of the Painleve VI equation.
• Cohomology groups for recurrence relations and contiguity relations of hypergeometric systems

  K Iwasaki

  JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 55 (2) 289 - 321 0025-5645 2003/04 [Refereed][Not invited]
   

  We develop the theory of cohomology groups for recurrence relations, based upon the asymptotic analysis of finite difference equations carried out in a previous paper. We apply it to compute-the Gevrey extension groups of the D-modules associated to some confluent hypergeometric systems. In those applications, recurrence relations appear as contiguity relations of hypergeometric systems.
• Isolated singularity, Witten's Laplacian, and duality for twisted de Rham cohomology

  K Iwasaki

  COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 28 (1-2) 61 - 82 0360-5302 2003 [Refereed][Not invited]
   

  In his famous paper (Witten E. Super-symmetry and Morse Theory. J Diff Geom 1982; 17:661-692), Witten used a twisted Laplacian, twisted by a Morse function, to develop his Morse theory as a super-symmetric quantum mechanics. Afterwards, in their studies on puits multiples en mecanique semi-classique, Helffer and Sjostrand (Helffer B, Sjostrand J. Puits Multiples en Mecanique Semi-classique IV, Etude du Complexe de Witten. Comm. Partial Differential Equations 1985; 10:245-340) made a detailed investigation of Witten's complexes. In this paper, we use a twisted Laplacian, twisted by a versal deformation of an isolated singularity (invariant under a finite. unitary reflection group), to construct a duality between a pair of polynomial twisted de Rham cohomology groups associated with the isolated singularity. The construction is based on a twisted version of Hodge-Kodaira decomposition derived through a pseudo-differential calculus of Witten's twisted Laplacian. Discussion is also made on the real structure and super-symmetry, compatible with the duality constructed, of the twisted de Rham cohomology.
• A modular group action on cubic surfaces and the monodromy of the Painleve VI equation

  K Iwasaki

  PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 78 (7) 131 - 135 0386-2194 2002/09 [Not refereed][Not invited]
   

  We construct an action of the modular group Gamma(2) on a general 4-parameter family of complex cubic surfaces and describe the nonlinear monodromy of the Painleve VI equation in terms of this action.
• Generating function associated with the rational solutions of the Painleve II equation

  K Iwasaki, K Kajiwara, T Nakamura

  JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 35 (16) L207 - L211 0305-4470 2002/04 [Refereed][Not invited]
   

  We consider the Hankel determinant representation for the rational solutions of the Painleve II equation. We give an explicit formula for the generating function of the entries in terms of logarithmic derivative of the Airy function.
• Polynomial invariants and harmonic functions related to exceptional regular polytopes

  K Iwasaki, A Kenma, K Matsumoto

  EXPERIMENTAL MATHEMATICS 11 (2) 313 - 319 1058-6458 2002 [Refereed][Not invited]
   

  We compute certain polynomial invariants for the finite reflection groups of the types H-3, H-4 and F-4. Using this result, we explicitly determine the solution space of functions satisfying a mean value property related to the exceptional regular polytopes, namely, the icosahedron and dodecahedron in three dimensions and the 24-cell, 600-cell, and 120-cell in four dimensions.
• Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

  K Iwasaki, Y Kamimura

  JOURNAL OF MATHEMATICAL BIOLOGY 43 (2) 101 - 143 0303-6812 2001/08 [Refereed][Not invited]
   

  A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equations.
• Inverse Bifurcation Problem in Mathematical Ecology and Related Wiener-Hopf Equations (Qualitative theory of functional equations and its application to mathematical science)

  Iwasaki Katsunori, Kamimura Yutaka

  RIMS Kokyuroku 京都大学 1216 (0) 115 - 127 1880-2818 2001/06 [Not refereed][Not invited]
  
• Witten Laplacian and Twisted de Rham Cohomology (Integral representations and twisted cohomology in the theory of differential equations)

  Iwasaki Katsunori

  RIMS Kokyuroku 京都大学 1212 (0) 1 - 17 1880-2818 2001/06 [Not refereed][Not invited]
  
• Intersection matrix of a generalized Airy function in terms of skew-Schur polynomials

  K Iwasaki, K Matsumoto

  PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 76 (9) 135 - 140 0386-2194 2000/11 [Refereed][Not invited]
   

  A duality is introduced between a pair of polynomial twisted de Rham cohomology groups associated with a generalized Airy function in several variables, Natural bases of the twisted de Rham groups are constructed in terms of Schur polynomials. Then the intersection matrix relative to these bases is calculated explicitly in terms of skew-Schur polynomials.
• Recent progress in polyhedral harmonics

  K Iwasaki

  ACTA APPLICANDAE MATHEMATICAE 60 (2) 179 - 197 0167-8019 2000/01 [Refereed][Not invited]
   

  Polyhedral harmonics is a subject which deals with the problem of characterizing the continuous functions satisfying the mean value property with respect to a polytope. The main feature of it is the finite dimensionality of the space of polyhedral harmonic functions. The theory involves not only analysis but also algebra and combinatorics, and has a rather different flavor from classical harmonic analysis. This paper aims at providing a survey on the subject, focusing on the author's recent results.
• Polytopes, invariants and harmonic functions

  Katsunori Iwasaki

  Advanced Studies in Pure Mathematics 27 145 - 156 2000 [Refereed][Not invited]
  
• Invariants of finite reflection groups and the mean value problem for polytopes

  K Iwasaki

  BULLETIN OF THE LONDON MATHEMATICAL SOCIETY 31 477 - 483 0024-6093 1999/07 [Refereed][Not invited]
   

  Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set H-P(k) of all continuous functions on R-n satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set H-G of all G-harmonic functions. Then a necessary and sufficient condition for the equality H-P(k) = H-G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.
• Gevrey Cohomology Groups for Confluent Hypergeometric Systems (Complex Analysis and Microlocal Analysis)

  Iwasaki Katsunori

  RIMS Kokyuroku 京都大学 1090 (0) 110 - 115 1880-2818 1999/04 [Not refereed][Not invited]
  
• Inverse Bifurcation Problem and Multiplicative Wiener-Hopf Equation (Methods and Applications for Functional Equations)

  Iwasaki Katsunori, kamimura Yutaka

  RIMS Kokyuroku 京都大学 1083 (0) 207 - 218 1880-2818 1999/02 [Not refereed][Not invited]
  
• A combinatorial formula of Leibniz type with application to Gegenbauer's polynomials

  K Iwasaki, H Kawamuko

  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 127 (1) 29 - 33 0002-9939 1999/01 [Refereed][Not invited]
   

  We establish a combinatorial formula of Leibniz type, which is an identity for a certain differential polynomial. The formula leads to new quadratic relations between Gegenbauer's orthogonal polynomials.
• Triangle mean value property

  Katsunori Iwasaki

  Aequationes Mathematicae 57 (2-3) 206 - 220 0001-9054 1999 [Refereed][Not invited]
   

  Given any triangle Δ, let Δ(d) be the d-skeleton of Δ for d = 0, 1, 2. The space HΔ(d) of all continuous functions in ℝ2 satisfying the mean value property with respect to Δ(d) is determined explicitly for each d = 0, 1, 2. We have dim H Δ(d) = 6 if the origin is the barycenter of Δ (resp. the incenter of Δ′) for d = 0, 2 (resp. for d = 1), where Δ′ is the reciprocal triangle of Δ otherwise dim HΔ(d) = 2. Moreover there exists a homogeneous polynomial Fd(x) such that HΔ(d) is generated by Fd(x) as a ℝ[∂]-module, F d(x) being determined explicitly. © Birkhäuser Verlag, Basel, 1999.
• Convolution calculus for a class of singular Volterra integral equations

  Katsunori Iwasaki, Yutaka Kamimura

  Journal of Integral Equations and Applications 11 (4) 461 - 499 0897-3962 1999 [Refereed][Not invited]
   

  For a class of singular Volterra integral equations we establish a necessary and sufficient condition for unique solvability in suitable function space settings. The discussion is based on the convolution calculus associated with the one-sided Mellin transform with weight 0. This study is motivated by some inverse nonlinear Sturm-Liouville problems, whose linearizations give rise to integral equations of our class. The method developed in this paper settles them in a unified manner. © 1999 Rocky Mountain Mathematics Consortium.
• Twisted homology of the configuration space of n-points with applications to hypergeometric functions

  Katsunori Iwasaki, Michitake Kita

  Kumamoto Journal of Mathematics 熊本大学 12 9 - 72 0914-675X 1999 [Refereed][Not invited]
  
• Asymptotic analysis for linear difference equations

  K Iwasaki

  TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 349 (10) 4107 - 4142 0002-9947 1997/10 [Refereed][Not invited]
   

  We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly, Finally we obtain Gevrey estimates fbr the solutions. In a forthcoming paper we will develop the theory of cohomology groups fcr recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.
• Basic invariants of finite reflection groups

  K Iwasaki

  JOURNAL OF ALGEBRA 195 (2) 538 - 547 0021-8693 1997/09 [Refereed][Not invited]
   

  Any finite reflection group G admits a distinguished basis of G-invariants canonically attached to a certain system of invariant differential equations. We determine. it explicitly for groups of types A, B, D, and I in a systematic way. (C) 1997 Academic Press.
• An inverse bifurcation problem and an integral equation of the Abel type

  K Iwasaki, Y Kamimura

  INVERSE PROBLEMS 13 (4) 1015 - 1031 0266-5611 1997/08 [Refereed][Not invited]
   

  This paper concerns an inverse problem for nonlinear Sturm-Liouville problems. Under some assumptions on the first eigenfunction of the linearized problem, we establish the existence of the nonlinearity realizing a given first bifurcating branch. The proof is based on solving an integral equation of the Abel type.
• Polytopes, Invariants, and PDEs

  IWASAKI KATSUNORI

  RIMS Kokyuroku 京都大学 983 (0) 14 - 21 1880-2818 1997/03 [Not refereed][Not invited]
  
• POLYHEDRAL HARMONICS(Structure of Functional Equations and Mathematical Methods)

  IWASAKI KATSUNORI

  RIMS Kokyuroku 京都大学 984 (0) 96 - 103 1880-2818 1997/03 [Not refereed][Not invited]
  
• Polytopes and the mean value property

  K Iwasaki

  DISCRETE & COMPUTATIONAL GEOMETRY 17 (2) 163 - 189 0179-5376 1997/03 [Refereed][Not invited]
   

  Let P be any (not necessarily convex nor connected) solid polytope in the n-dimensional Euclidean space R(n), and let P(k) be the k-skeleton of P. Let H-P(k) be the set of all continuous functions satisfying the mean value property with respect to P(k). For any k = 0, 1, ..., n, we show that H-P(k) is a finite-dimensional linear space of polynomials. This settles an open problem posed by Friedman and Littman [37] in 1962. Moreover, we show that if P admits ample symmetry, then H-P(k) is a finite-dimensional linear space of harmonic polynomials. Some interesting examples are also given.
• Regular simplices, symmetric polynomials and the mean value property

  K Iwasaki

  JOURNAL D ANALYSE MATHEMATIQUE 72 279 - 298 0021-7670 1997 [Refereed][Not invited]
   

  Let P be an n-dimensional regular simplex in R-n centered at the origin, and let P(k) be the k-skeleton of P for k = 0, 1,..., n. Then the set H-P(k) of all continuous functions in mn satisfying the mean value property with respect to P(k) forms a finite-dimensional linear space of harmonic polynomials. In this paper the function space H-P(k) is explicitly determined by group theoretic and combinatorial arguments for symmetric polynomials.
• Exterior power structure on the twisted de Rham cohomology of the complements of real veronese arrangements

  K Iwasaki, M Kita

  JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 75 (1) 69 - 84 0021-7824 1996 [Refereed][Not invited]
   

  Let S be the complex affine line minus real m points; then the n-th symmetric product T of S is the complement of a special arrangement of hyperplanes in the affine n-space, called the real Veronese arrangement. We show that under a simple condition, only the n-th twisted cohomology of T survives and it is isomorphic to the n-th exterior power of the first twisted cohomology of S. This isomorphism is explicitly written in terms of logarithmic forms.
• On a dynamical system on matrix algebra

  Katsunori Iwasaki

  Journal of Dynamics and Differential Equations 6 (4) 671 - 711 1040-7294 1994/10 [Refereed][Not invited]
   

  Let M(n) be the algebra of all n×n complex matrices. We consider a dynamical system on M(n) defined by the vector field V(X)=[[X*, X], X], (X ∃ M(n)). It arises as the gradient flow for two kinds of variational problems on M(n). Given any X0∃ M(n), let X(t) be the trajectory starting at X0. We study the global behavior of X(t) as t → ∞. We show that, if X0 is semisimple, then X(t) converges exponentially to a normal matrix. If X0 is not semisimple, then the behavior of X(t) is completely different and difficult to analyze. We give some results also in this case. Furthermore, we discuss about a center manifold approach to our dynamical system. © 1994 Plenum Publishing Corporation.
• A problem on the singulalities[singularities] of a real algebraic vector field(Singularities of Holomorphic Vector Fields and Related Topics)

  ISAWAKI KATSUNORI

  RIMS Kokyuroku 京都大学 878 (0) 75 - 78 1880-2818 1994/06 [Not refereed][Not invited]
  
• An intersection theory for hypergeometric functions(Singularities of Holomorphic Vector Fields and Related Topics)

  IWASAKI KATSUNORI

  RIMS Kokyuroku 京都大学 878 (0) 64 - 74 1880-2818 1994/06 [Not refereed][Not invited]
  
• Geometry of Fuchsian moduli spaces

  Katsunori Iwasaki

  Geometry, Topology and Field Theory, World Scientific Publishers 39 - 47 1994 [Refereed][Invited]
  
• FUCHSIAN MODULI ON A RIEMANN SURFACE - ITS POISSON STRUCTURE AND POINCARE-LEFSCHETZ DUALITY

  K IWASAKI

  PACIFIC JOURNAL OF MATHEMATICS 155 (2) 319 - 340 0030-8730 1992/10 [Refereed][Not invited]
   

  The moduli space of Fuchsian projective connections on a closed Riemann surface admits a Poisson structure. The moduli space of projective monodromy representations on the punctured Riemann surface also admits a Poisson structure which arises from the Poincare-Lefschetz duality for cohomology. We shall show that the former Poisson structure coincides with the pull-back of the latter by the projective monodromy map. This result explains intrinsically why a Hamiltonian structure arises in the monodromy preserving deformation.
• ON SOLUTIONS OF THE POINCARE EQUATION

  K IWASAKI

  PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 67 (6) 211 - 214 0386-2194 1991/06 [Refereed][Not invited]
  
• 微分方程式のモジュライと変形(複素解析幾何学とその周辺の研究:数理物理と複素解析幾何)

  岩崎 克則

  数理解析研究所講究録 京都大学 756 (0) 68 - 84 1880-2818 1991/06 [Not refereed][Not invited]
  
• Moduli and deformations for Fuchsian projective connections on a Riemann surface

  Katsunori Iwasaki

  Journal of Faculty of Sciences, University of Tokyo, Section IA, Mathematics 38 (3) 431 - 531 1991 [Refereed][Not invited]
  
• STRUCTURE OF THE MODULI SPACE OF SL-OPERATORS ON A RIEMANN SURFACE AND THE MONODROMY PRESERVING DEFORMATION

  IWASAKI Katsunori

  RIMS Kokyuroku 京都大学 683 (0) 9 - 31 1880-2818 1989/03 [Not refereed][Not invited]
  
• RIEMANN FUNCTION OF HARMONIC EQUATION AND APPELLS F4

  K IWASAKI

  SIAM JOURNAL ON MATHEMATICAL ANALYSIS 19 (4) 902 - 917 0036-1410 1988/07 [Refereed][Not invited]
  
• On the Riemann-Hilbert-Birkhoff problem for ordinary differential equations containing a parameter

  Katsunori Iwasaki

  Journal of Faculty of Science, University of Tokyo, Section IA, Mathematics Faculty of Science, The University of Tokyo 35 (2) 251 - 312 0040-8980 1988 [Refereed][Not invited]
  
• On the inverse Sturm-Liouville problem with spatial symmetry

  Katsunori Iwasaki

  Funkcialaj Ekvacioj 31 (1) 25 - 74 1988 [Refereed][Not invited]
  
• Scattering theory for fourth order differential operators II

  Katsunori Iwasaki

  Japanese Journal of Mathematics 一般社団法人 日本数学会 14 (1) 59 - 97 1988 [Refereed][Not invited]
  
• Scattering theory for fourth order differential operators I

  Katsunori Iwasaki

  Japanese Journal of Mathematics 一般社団法人 日本数学会 14 (1) 1 - 57 1988 [Refereed][Not invited]
  
• INVERSE PROBLEM FOR STURM-LIOUVILLE AND HILL EQUATIONS

  K IWASAKI

  ANNALI DI MATEMATICA PURA ED APPLICATA 149 (4) 185 - 206 0003-4622 1987 [Refereed][Not invited]
  

MISC

• F-3 正規分布の正の平均の推定におけるミニマクス性と許容性のロバストネス(決定理論)(2003年度統計関連学会連合大会記録(日本統計学会第71回大会))

  丸山 祐造, 岩崎 克則  日本統計学会誌  33-  (3)  412  -412  2003
• 25pXB-6 Generating function associated with the Yablonski-Vorob'ev polynomials

  Iwasaki K., Kajiwara K., Nakamura T.  Meeting abstracts of the Physical Society of Japan  57-  (1)  259  -259  2002/03/01
• A HAMILTONIAN SYSTEM ON THE FUCHSIAN MODULI SPACE(Algebraic Analysis and Number Theory)

  IWASAKI KATSUNORI  数理解析研究所講究録  810-  79  -84  1992/09

Presentations

• Around hypergeometric groups

  Katsunori Iwasaki

  アクセサリーパラメーター研究会 2022  2022/03
• Siegel disks on K3 surfaces and Picard numbers  [Not invited]

  Yuta Takada, Katsunori Iwasaki

  2021 年度日本数学会秋季総合分科会  2021/09
• Hypergeometric groups, K3 lattices, and root systems  [Not invited]

  Yuta Takada, Katsunori Iwasaki

  超幾何方程式研究会 2021  2021/01
• From hypergeometric groups to Siegel disks on K3 surfaces  [Not invited]

  Katsunori Iwasaki, Yuta Takada

  2020年度日本数学会秋季総合分科会  2020/09
• Hypergeometric groups and dynamics on K3 surfaces  [Invited]

  Katsunori Iwasaki

  Hokkaido University Mathematics Colloqium  2020/02
• Hypergeometric groups and K3 lattices  [Invited]

  Katsunori Iwasaki

  Differential Systems: from Theory to Computer Mathematics  2019/12
• On lattices generated by hypergeometric groups  [Invited]

  Katsunori Iwasaki

  Kumamoto University Mathematics Colloqium  2019/10
• Discrete Laplace method and hypergeometric continued fractions  [Invited]

  Katsunori Iwasaki

  15th International Conference on Orthogonal Polynomials, Special Functions and Applications  2019/07
• Discrete steepest descent method and Gauss continued fraction  [Not invited]

  Katsunori Iwasaki

  Workshop on accessary parameters  2019/03
• Discrete saddle point method for hypergeometric functions and its applications  [Invited]

  Katsunori Iwasaki

  Around differential equations and inverse problems  2019/03
• 超幾何連分数の漸近展開  [Invited]

  岩崎克則

  第１２回玉原特殊多様体研究集会、東京大学玉原国際セミナーハウス  2018/09
• 超幾何連分数の誤差評価 Error estimates for hypergeometric continued fraction  [Not invited]

  岩崎克則

  アクセサリ・パラメータ研究会，熊本大学理学部  2018/03
• 超幾何級数の離散鞍点法とその応用  [Not invited]

  岩崎克則

  超幾何方程式研究会 2018，神戸大学理学部  2018/01
• 代数曲面上の双有理写像の周期点とパンルヴェ方程式の周期解  [Invited]

  岩崎克則

  第11回玉原特殊多様体研究集会， 東京大学玉原国際セミナーハウス，群馬県沼田市  2017/09
• 超幾何級数の漸近挙動と離散鞍点法  [Not invited]

  岩崎克則

  超幾何学校 2017，小樽商科大学  2017/09
• 超幾何連分数について  [Invited]

  岩崎克則

  第10回玉原特殊多様体研究集会， 東京大学玉原国際セミナーハウス  2016/09
• 境界領域における超幾何関数のガンマ乗積表示について  [Not invited]

  日下部美奈, 岩崎克則

  2016年度函数方程式論サマーセミナー，石川県羽咋市  2016/08
• パンルヴェ方程式の幾何学  [Invited]

  岩崎克則

  幾何学コロキウム， 北海道大学  2016/06
• 超幾何関数の隣接関係式・鞍点法・連分数  [Not invited]

  岩崎克則, 蛭子彰仁

  アクセサリーパラメータ研究会，熊本大学  2016/03
• 超幾何恒等式と超幾何連分数について  [Invited]

  岩崎克則

  数論幾何・超幾何研究交流会，北海道大学  2016/03
• 超幾何恒等式をめぐって  [Invited]

  岩崎克則

  超幾何学校 2015， 神戸大学  2015/09
• 超幾何和の算術性について  [Invited]

  岩崎克則

  琉球超幾何セミナー, 琉球大学理学部, 沖縄県中頭郡  2015/02
• Hypergeometric series with gamma product formula  [Invited]

  Katsunori Iwasaki

  International Conference on Partial Differential Equations: General Theory and Variational Problems, Costabella Tropical Beach Hotel, Cebu, Philippines  2015/01
• Arithmetic conditions for HG sums = Gamma products,  [Not invited]

  岩崎克則

  超幾何方程式研究会 2015, 神戸大学  2015/01
• On some hypergeometric summations  [Invited]

  岩崎克則

  RIMS 研究集会「複素領域における微分方程式・その近年の発展」 京都大学数理解析研究所  2014/11
• 超幾何和の特殊値をめぐって  [Invited]

  岩崎克則

  微分方程式の展望, 熊本大学  2014/10
• パンルヴェ第Ｉ方程式の軌道体ハミルトン構造について  [Invited]

  岩崎克則, 岡田脩

  第5回ハミルトン系とその周辺、金沢大学サテライトプラザ  2014/05
• パンルヴェ第Ｉ方程式の軌道体ハミルトン構造  [Not invited]

  岡田脩, 岩崎克則

  超幾何方程式研究会 2014， 神戸大学  2014/01
• On an orbifold Hamiltonian structure for the first Painleve equation  [Not invited]

  Shu Okada, Katsunori Iwasaki

  International relation of young researchers in algebra and related fields, The 16th SNU-HU Joint Symposium, Seoul National University, Seoul, Korea  2013/12
• 超幾何和とサイン・サイン  [Not invited]

  岩崎克則

  超幾何方程式研究会 2013, 神戸大学  2013/01
• 超幾何和の超幾何性  [Invited]

  岩崎克則

  琉球超幾何セミナー, 琉球大学  2012/11
• 多面体と関数方程式  [Not invited]

  岩崎克則

  函数方程式サマーセミナー 2012，山形  2012/08
• 立方体とベルヌーイ数  [Not invited]

  岩崎克則

  超幾何方程式研究会 2012, 神戸大学  2012/01
• パンルヴェ性をめぐって  [Invited]

  岩崎克則

  アクセサリー・パラメーター研究会， 熊本大学  2011/03
• 複素曲面上の正則力学系について  [Invited]

  岩崎克則

  複素解析的ベクトル場・葉層構造とその周辺， 龍谷大学  2010/11
• パンルヴェ方程式の力学系  [Invited]

  岩崎克則

  Dynamics of Complex Systems セミナー，北海道大学  2010/11
• 複素曲面上の正則力学系について  [Invited]

  岩崎克則

  東北大学幾何セミナー， 東北大学  2010/10
• Algebraic analysis of the sixth Painleve equation  [Invited]

  Katsunori Iwasaki

  The 4th Workshop on Hamiltonian systems and related topics， Niigata University Satellite Campus ``Tokimeito"  2010/10
• 特殊関数の諸問題  [Invited]

  岩崎克則

  複素幾何学の諸問題，RIMS 共同研究， 京都大学数理解析研究所  2010/09
• Painleve VI 方程式のある特殊解について  [Invited]

  岩崎克則

  函数方程式論ワークショップ，東京大学  2010/07
• パンルヴェ方程式の代数解析と力学系  [Invited]

  岩崎克則

  北海道大学談話会，北海道大学  2010/05
• パンルヴェ方程式の代数解析と力学系  [Invited]

  岩崎克則

  日本数学会年会総合講演，慶應義塾大学  2010/03
• A note on the Markoff-Painleve transcendent  [Invited]

  岩崎克則

  複素力学系とその関連分野の総合的研究, RIMS 研究集会， 京都大学  2009/12
• Smale in Painleve around Gauss  [Invited]

  岩崎克則

  筑波大学解析セミナー, 筑波大学  2009/10
• Smale in Painleve around Gauss  [Not invited]

  岩崎克則

  2009 年度函数方程式サマーセミナー, 鳥羽  2009/08
• パンルヴェ方程式のモノドロミーとその力学系  [Invited]

  岩崎克則

  微分方程式のモノドロミーをめぐる諸問題，京都大学数理解析研究所  2009/02
• Periodic solutions to Painleve VI  [Invited]

  Katsunori Iwasaki

  Journees Franco-Japonaises en honneur de Kazuo Okamoto: autour de Equations de Painleve, IRMA, Universite Louis Pasteur, Strasbourg, France  2008/11
• パンルヴェ方程式の幾何学  [Invited]

  岩崎克則

  第 55 回幾何学シンポジウム, 弘前大学  2008/08
• パンルヴェ第 VI 方程式のの代数解について  [Not invited]

  岩崎克則

  函数方程式論サマーセミナー, 白樺ハイツ，富山市亀谷  2008/08
• 指標多様体上の有限軌道とパンルヴェ VI の代数解  [Invited]

  岩崎克則

  微分ガロア理論・モノドロミー保存変形・パンルヴェ方程式, 神戸大学  2008/07
• Painleve VI: from algebraic geometry to elementary geometry,  [Invited]

  Katsunori Iwasaki

  International Conference ``From Painleve to Okamoto", University of Tokyo  2008/06
• Finite orbits on character varieties and algebraic solutions to Painleve VI  [Invited]

  岩崎克則

  完全 WKB 解析と超局所解析, 京都大学数理解析研究所  2008/05
• Finite orbits on character varieties and algebraic solutions to Painleve VI  [Invited]

  Katsunori Iwasaki

  IRMAR, Universite de Rennes 1, France  2008/03
• パンルヴェ方程式とモデュライ空間上の力学系  [Invited]

  岩崎克則

  力学系と微分方程式, 広島大学  2007/11
• パンルヴェ方程式とモデュライ空間上の力学系  [Invited]

  岩崎克則

  京都大学談話会，京都大学  2007/10
• Area-preserving surface dynamics and S.Saito's fixed point formula  [Invited]

  Katsunori Iwasaki, Takato Uehara

  Differential Equations and Exact WKB Analysis, RIMS, Kyoto University  2007/10
• Geometry of Painleve equations  [Invited]

  Katsunori Iwasaki

  Geometry related to the theory of integrable systems, RIMS-OCAMI Project, Kyoto Universit  2007/09
• パンルヴェ第 VI 方程式の固定特異点のまわりの 有限分岐局所解について  [Not invited]

  岩崎克則

  日本数学会秋季総合分科会， 東北大学  2007/09
• Ergodic theory of Painleve foliation  [Invited]

  Katsunori Iwasaki

  Birational automorphisms of compact complex manifolds and dynamical systems, Nagoya University  2007/08
• パンルヴェ VI の固定特異点のまわりの有限分岐局所解  [Not invited]

  岩崎克則

  関数方程式論サマーセミナー，安曇野市  2007/08
• パンルヴェ方程式の力学系：可積分 vs カオス  [Invited]

  岩崎克則

  東北大学「２１世紀COE物質階層融合科学の構築」春の学校（続編)  2007/04
• Ergodic theory of Painleve foliation  [Invited]

  岩崎克則

  複素葉層構造研究会，龍谷大学  2007/02
• Algebraic analysis of the sixth Painleve equation,  [Invited]

  岩崎克則

  微分方程式の代数解析と完全 WKB 解析，京都大学数理解析研究所  2006/12
• Dynamics of the sixth Painleve equation  [Invited]

  岩崎克則

  複素力学系とその周辺，京都大学数理解析研究所  2006/10
• The sixth Painleve equation: a chaotic dynamical system  [Invited]

  Katsunori Iwasaki

  The Painleve Equations and Monodromy Problems: Recent Developments, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK  2006/09
• パンルヴェ方程式ののエルゴード力学系について  [Invited]

  上原崇人, 岩崎克則

  可積分系数理の眺望, 京都大学数理解析研究所  2006/08
• パンルヴェ方程式の幾何学  [Invited]

  岩崎克則

  福岡大学幾何セミナー，福岡大学  2006/07
• パンルヴェ方程式：モデュライ空間・特異点・力学系  [Invited]

  岩崎克則

  九州幾何セミナー，九州大学  2006/06
• An ergodic study of Painleve VI  [Invited]

  Katsunori Iwasaki, Takato Uehara

  Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painleve Hierarchies, Japanese-French Symposium, RIMS at Kyoto  2006/05
• Chaos in Painleve VI  [Not invited]

  岩崎克則, 上原崇人

  研究会「複素領域における微分方程式」 熊本大学  2006/03
• Painleve equations: moduli spaces, singularities, and dynamical systems  [Invited]

  岩崎克則

  代数・解析・幾何学セミナー, 鹿児島大学  2006/02
• Painleve VI: moduli spaces, singularities, and dynamical system  [Invited]

  岩崎克則

  可積分系極寒セミナー, 北見工業大学  2006/02
• Some dynamical aspects of Painleve VI  [Invited]

  岩崎克則

  ウインターセミナー「可積分系の理論」 新潟県南魚沼郡湯沢町  2006/01
• Poincare sections of the sixth Painleve dynamics  [Invited]

  岩崎克則

  Kobe Workshops on Integrable Systems and Painleve Systems, Kobe University  2005/11
• パンルヴェ方程式のダイナミクス  [Invited]

  岩崎克則

  Pathways Lecture Series in Mathematics, Keio, 慶応大学  2005/10
• Bounded trajectories in the sixth Painleve dynamics  [Invited]

  Katsunori Iwasaki

  International Conference on Algebraic Analysis of Differential Equations, RIMS and Clock Tower, Kyoto University  2005/07
• パンルヴェ方程式のダイナミクス  [Invited]

  岩崎克則

  日本数学会年会函数方程式分科会解析学賞受賞特別講演， 日本大学  2005/03
• Ｋ３曲面上の複素力学系  [Invited]

  岩崎克則

  第６回九州可積分系セミナー「Ｋ３曲面上の複素力学系」 九州大学  2005/03
• パンルヴェ方程式と特異点解消  [Invited]

  岩崎克則

  ウインターセミナー「可積分系の理論」 新潟県南魚沼郡湯沢町  2005/02
• パンルヴェ方程式と力学系理論  [Invited]

  岩崎克則

  慶応大学 21 世紀 COE 代数解析セミナー，慶応大学  2005/01
• パンルヴェ方程式の幾何学  [Invited]

  岩崎克則

  北海道大学談話会， 北海道大学  2004/12
• Ｋ３曲面上の複素力学系 (C.T. McMullen の仕事の紹介)  [Not invited]

  岩崎克則

  超幾何関数研究会, 神戸学生青年センター及び神戸大学  2004/11
• パンルヴェ方程式とリーマン・ヒルベルト対応  [Not invited]

  岩崎克則

  函数方程式サマーセミナー, 長野県下伊那郡  2004/08
• パンルヴェ方程式のダイナミクス  [Invited]

  岩崎克則

  多変数函数論サマーセミナー, 三重県菰野町  2004/08
• パンルヴェ方程式のダイナミクス  [Invited]

  岩崎克則

  別府夏の少人数セミナー「モジュライ空間と可積分系の新しい展開」 別府  2004/07
• Dynamics of the sixth Painleve equation  [Invited]

  Katsunori Iwasaki

  International Conference on Theories Asymptotiques et Equations de Painleve，Universite d'Angers，Angers，France  2004/06
• パンルヴェ方程式の幾何学  [Invited]

  岩崎克則

  志賀弘典先生還暦研究会， 北海道大学  2004/01
• パンルベ VI 型方程式およびガルニエ方程式の幾何学 "ベックルンド変換とτ関数を中心にして"  [Invited]

  稲場道明, 岩崎克則, 斎藤政彦

  複素領域の微分方程式， 神戸大学瀧川記念会館  2004/01
• パンルヴェ方程式のダイナミクス  [Invited]

  岩崎克則

  大岡山談話会，東京工業大学  2003/11
• リーマン・ヒルベルト対応とパンルヴェ第 VI 方程式  [Invited]

  岩崎克則

  複素領域における微分方程式の大域解析と漸近解析， 京都大学数理解析研究所  2003/10
• 極値問題とニュートン図形  [Invited]

  岩崎克則

  日本オペレーションズ・リサーチ学会 第 50 回シンポジウム「OR と数学」，九州大学国際学術研究交流プラザ  2003/09
• Geometry of the sixth Painleve equation  [Invited]

  岩崎克則

  New trends in microlocal analysis、RIMS at Kyoto  2003/08
• Backlund transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence  [Not invited]

  岩崎克則

  関数方程式サマーセミナー，香川県  2003/08
• パンルヴェ第 VI 方程式の幾何学  [Invited]

  岩崎克則

  微分方程式の総合的研究， 東京大学  2002/12
• パンルベ第 VI 方程式の幾何学  [Invited]

  岩崎克則

  ワークショップ「可積分系と パンルヴェ系」 神戸大学瀧川記念会館  2002/12
• モジュラー群の複素三次曲面への作用とパンルヴェ第六方程式の モノドロミー  [Not invited]

  岩崎克則

  日本数学会秋季総合分科会，島根大学  2002/09
• 多面体調和関数の数理 --- 多面体・不変式・偏微分方程式  [Invited]

  岩崎克則

  企画特別講演，日本数学会秋季総合分科会，島根大学  2002/09
• A modular group action on cubic surfaces and the monodromy of the sixth Painleve equation  [Not invited]

  岩崎克則

  関数方程式サマーセミナー， 慶應大学立科山荘  2002/08
• A modular group action on cubic surfaces and the monodromy of the sixth Painleve equation  [Invited]

  岩崎克則

  微分方程式の変形と漸近解析， 京都大学数理解析研究所  2002/06
• パンルベ第二方程式に付随する母関数  [Not invited]

  中村俊哉, 岩崎克則, 梶原健司

  日本数学会年会，明治大学  2002/03
• Yablonski-Vorobev 多項式の母関数  [Not invited]

  梶原健司, 岩崎克則, 中村俊哉

  日本物理学会，立命館大学  2002/03
• A modular group action on cubic surfaces and the monodromy of the sixth Painleve equation  [Invited]

  岩崎克則

  パンルベ方程式研究会、 九州大学  2002/02
• Generating function associated with the rational solutions of the Painleve II equation  [Invited]

  岩崎克則

  解析セミナー、熊本大学  2002/01
• Geometry of isomonodromic deformations  [Invited]

  Katsunori Iwasaki

  International Conference on Geometry of Moduli Spaces and Integrable Systems, RIMS at Kyoto University  2001/09
• Geometry of isomonodromic deformations  [Invited]

  岩崎克則

  大阪大学幾何セミナー， 大阪大学  2001/05
• Witten Laplacian and twisted de Rham cohomology  [Invited]

  岩崎克則

  微分方程式論における積分公式と twisted cohomology, 京都大学数理解析研究所  2001/01
• 一般のリーマン面におけるモノドロミー保存変形  [Invited]

  岩崎克則

  モノドロミー保存変形と可積分系，ピアザ淡海会議場，大津  2000/12
• 多面体調和関数の数理  [Invited]

  岩崎克則

  特殊函数をめぐって，慶應大学  2000/12
• Inverse problem in mathematical ecology and related Wiener-Hopf equation  [Invited]

  岩崎克則, 上村豊

  関数方程式の定性的理論と その現象解析への応用，京都大学数理解析研究所  2000/11
• Isolated singularity, Witten's Laplacian and twisted de Rham cohomology  [Invited]

  岩崎克則

  微分方程式の超局所解析・漸近解析，京都大学数理解析研究所  2000/10
• A certain inadmissible minimax estimator of a positive normal mean  [Not invited]

  丸山祐造, 岩崎克則

  日本数学会秋季総合分科会，統計数学分科会， 京都大学  2000/09
• 生態学と逆分岐問題  [Not invited]

  岩崎克則, 上村豊

  関数方程式サマーセミナー, 白馬  2000/07
• 多面体調和関数の数理  [Invited]

  岩崎克則

  微分方程式の総合的研究, 東京大学  1999/12
• Isolated singularities and harmonic analysis of twisted de Rham cohomology groups  [Invited]

  岩崎克則

  解析学火曜セミナー, 東京大学  1999/11
• 逆分岐問題と乗法的 Wiener-Hopf 方程式  [Not invited]

  岩崎克則, 上村豊

  日本数学会秋季総合分科会, 広島大学  1999/09
• 多面体調和関数の数理 -- ある問題の提示 --,  [Not invited]

  岩崎克則

  日本数学会 秋季総合分科会，広島大学  1999/09
• Airy integrals, Schur polynomials and harmonic theory for twisted de Rham cohomology groups  [Not invited]

  岩崎克則

  関数方程式サマーセミナー, 八ヶ岳  1999/08
• Asymptotic analysis for finite difference equations  [Invited]

  岩崎克則

  熊本大学談話会, 熊本大学  1999/07
• Twisted harmonic theory for de Rham cohomology groups  [Invited]

  岩崎克則

  Painleve 系、超幾何系、漸近展開，京都大学数理解析研究所  1999/06
• Inverse bifurcation problems and singular multiplicative Wiener-Hopf equations  [Invited]

  Katsunori Iwasaki

  Inverse and Direct Problems and Applications, Gargnano, Italy  1999/04
• Wiener-Hopf 方程式と逆分岐問題  [Invited]

  岩崎克則, 上村豊

  関数方程式の方法とその応用, 京都大学数理解析研究所  1998/11
• Intersection matrix of the generalized Airy function in terms of Schur polynomials  [Invited]

  岩崎克則

  Various aspects of special functions, Kumamoto University  1998/11
• Polyhedral harmonic functions and invariant theory  [Not invited]

  Katsunori Iwasaki

  ICM'98, Berlin, Germany  1998/08
• Introduction to Hodge theory  [Not invited]

  岩崎克則

  関数方程式サマーセミナー --- 微分方程式と特異点、 山形  1998/07
• Polytopes, invariants, and harmonic functions  [Invited]

  Katsunori Iwasaki

  Workshop on mathematics related to arrangements of hyperplanes, Tokyo Metropolytan University  1998/07
• 多面体調和関数と群調和関数  [Invited]

  岩崎克則

  日本数学会九州支部会特別講演、九州大学国際ホール  1998/02
• Contiguity relations and Gevrey cohomology groups for hypergeometric systems  [Invited]

  岩崎克則

  数論と特殊関数, 熊本大学  1998/01
• Cohomology groups for recurrence relations with applications to hypergeometric systems  [Invited]

  Katsunori Iwasaki

  Complex analysis and microlocal analysis, RIMS, Kyoto  1997/12
• 差分方程式のコホモロジー群と超幾何方程式 の隣接関係式  [Invited]

  岩崎克則

  Workshop on hypergeometric systems in Kobe '97, 神戸大学  1997/12
• 多面体調和関数と群調和関数 I,II  [Invited]

  岩崎克則

  表現論的組合わせ論と組合わせ論的表現論、 京都大学数理解析研究所  1997/11
• Invariant differential equations associated with finite reflection groups  [Not invited]

  岩崎克則

  関数方程式サマーセミナー、 九重国立大学研修センター  1997/07
• 多面体調和関数  [Invited]

  岩崎克則

  広島大学談話会  1997/07
• 多面体・不変式・ホロノミック系  [Invited]

  岩崎克則

  九州大学談話会  1997/07
• 多面体調和関数の数理 I, II  [Invited]

  岩崎克則

  北海道大学講演会  1997/06
• Polyhedral harmonics  [Invited]

  Katsunori Iwasaki

  Equadiff 9, Masaryk University, Brno, Czech  1996/11
• リーマン面・モノドロミー・微分方程式  [Invited]

  岩崎克則

  リーマン 面に関連した位相幾何学、北海道大学  1996/11
• Polytopes and the mean value property  [Invited]

  Katsunori Iwasaki

  Harmonic analysis session, ISAAC 97, Delaware University, U.S.A  1996/09
• Polytopes, Invariants and PDEs  [Invited]

  岩崎克則

  超局所解析に おける代数解析的方法、京都大学数理解析研究所  1996/07
• Polytopes, Invariants and PDEs  [Invited]

  岩崎克則

  解析学火曜セミナー、東京大学  1996/06
• Polytopes and the mean value property  [Invited]

  岩崎克則

  ポテンシャル論 Winter School, 岡山理科大学学術交流 センター  1996/02
• 多面体と平均値の性質  [Invited]

  岩崎克則

  広島大学談話会、広島大学  1996/02
• 多面体と平均値の性質  [Invited]

  岩崎克則

  熊本大学談話会、熊本大学  1996/01
• Gevrey cohomology 群に対する離散的熱方程式 の方法  [Invited]

  岩崎克則

  解析的微分方程式の解の構成、お茶ノ水女子大学  1995/12
• 多面体と平均値の性質  [Not invited]

  岩崎克則

  神戸大学談話会、神戸大学  1995/11
• 多面体遊び  [Not invited]

  岩崎克則

  超幾何関数の幾何，城崎  1995/10
• Gevrey cohomology 群に対する離散的熱方程式 の方法  [Invited]

  岩崎克則

  日本数学会秋季総合分科会特別講演，東北大学  1995/09
• An inverse bifurcation problem and an integral equation of the Abel type  [Not invited]

  岩崎克則, 上村豊

  日本数学会年会  1995/03
• Cohomology groups for recurrence relations  [Invited]

  Katsunori Iwasaki

  Colloquium at Universite Paris VI, Paris, France,  1995/01
• 超幾何関数に役立つ位相幾何  [Not invited]

  岩崎克則

  超幾何関数の 研究、神戸大学  1994/01
• Exterior power structure for the hypergeometric functions  [Invited]

  岩崎克則

  Symposium on Arithmetic Geometry, 東京都立大学、東京  1994/01
• 超幾何関数の交叉理論  [Not invited]

  岩崎克則

  関数方程式若手セミナー、乗鞍  1994/01
• A problem on the singularities of a real algebraic vector fields  [Invited]

  岩崎克則

  Symposium on singularities of holomorphic vector fields, RIMS, Kyoto  1993/11
• An intersection theory for hypergeometric functions  [Invited]

  岩崎克則

  Symposium on singularities of holomorphic vector fields, RIMS, Kyoto  1993/11
• Geometry of Fuchsian moduli spaces  [Invited]

  岩崎克則

  Symposium on geometry, topology and field theory, RIMS, Kyoto  1993
• Monodromy preserving deformation and Hamiltonian systems  [Invited]

  岩崎克則

  代数解析学と整数論、京都大学 数理解析研究所  1992/03
• Moduli and deformation for differential equations  [Invited]

  Katsunori Iwasaki

  Complex analytic geometry and related topics, RIMS, Kyoto,  1991/09
• Fuchsian moduli on Riemann surfaces  [Invited]

  Katsunori Iwasaki

  AMS conference on meromorphic differential equations, Portland State Univ., Portland, U.S.A.  1991/06
• Completely integrable Hamiltonian systems arising from monodromy preserving deformation  [Invited]

  Katsunori Iwasaki

  Norguet seminar on several complex variables, Universite Paris VII  1990/06
• Moduli and deformation for meromorphic differential equations  [Invited]

  Katsunori Iwasaki

  Phenomes de Stokes et resurgences, C.I.R.M., Universite de Marseille  1990/04
• Moduli space of Fuchsian projective connections on a Riemann surface and monodromy preserving deformation  [Invited]

  Katsunori Iwasaki

  日本数学会秋季総合分科会 特別講演  1989/09
• Local systems and dynamical sysytems on the configuration space of points  [Invited]

  Katsunori Iwasaki

  Colloquium at Universitat Ulm, Germany  1989/04
• Monodromy preserving deformation for Fuchsian projective connections on a Riemann surface  [Invited]

  Katsunori Iwasaki

  Die Gewohnlichen Differentialgleichungen und spetial Funktionen,  1989/04
• リーマン面上の SL 作用素の空間の構造と モノドロミー保存変形  [Not invited]

  岩崎克則

  、複素解析と大域解析 -- 微分 方程式の視点から、京都大学数理解析研究所  1988/11
• 微分方程式に対するひとつの見方  [Not invited]

  岩崎克則

  発展方程式若手セミナー、箱根  1988/08
• 助変数を含む微分方程式の Riemann-Hilbert- Birkhoff 問題  [Invited]

  岩崎克則

  微分方程式の総合的研究、日本大学  1987/12
• Asymptotic analysis for differential equations containing a large parameter  [Invited]

  Katsunori Iwasaki

  Taniguchi Symposium on differential equations in the complex domain  1987/08
• 調和方程式の Riemann 関数と Appell の F4  [Invited]

  岩崎克則

  東京大学関数方程式セミナー、東京大学  1986/11
• Sturm-Liouville 逆問題について  [Invited]

  岩崎克則

  Lie 環と 微分方程式、館山  1986/08
• 4 階微分作用素に対する逆散乱理論  [Invited]

  岩崎克則

  微分方程式の総合的研究、日本大学  1985/12
• 4 階微分作用素に対する逆散乱理論  [Invited]

  岩崎克則

  東京大学談話会, 東京大学  1985/05

Research Projects

• New developments of hypergeometric functions and hypergeometric groups

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

  Date (from‐to) : 2022/04 -2025/03 

  Author : 岩崎 克則
• Asymptotic and global analysis of hypergeometric functions

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

  Date (from‐to) : 2019/04 -2023/03 

  Author : 岩崎 克則

   

  本研究は、超幾何関数の漸近解析と大域解析を課題としている。本年度は、後者に係るテーマを主に研究した。超幾何関数の解の大域挙動はモノドロミー群により測られる。超幾何方程式のモノドロミー群をモデルとする行列群が超幾何群である。この群のクラスは、さまざまな有限群および無限群を含んでいるので興味深い。 超幾何群の概念は Beukers-Heckmanによるが、我々は最近の研究で超幾何群の理論に更なる基盤整備を与えた。また整数上定義される超幾何群に対して、超幾何格子の概念を展開した。さらに超幾何格子を用いてK3格子を実現することにより、K3曲面上のエントロピー正の正則自己同型を構成する方法を与えた。超幾何群とK3曲面上の力学系という、一見すると無関係に見える二つのテーマを結びつけるところに本研究の特色と意義がある。 本年度の第一の成果としては、研究協力者 高田 佑太との共著論文 Hypergeometric groups and dynamics on K3 surfaces が国際学術誌 Math. Z. にアクセプトされ、出版されたことが挙げられる。この論文では、超幾何群・超幾何格子の理論の基盤整備を行ったあと、超幾何格子を用いて非射影的 K3 曲面自己同型を構成する手法を確立した。 次に、Siegel円板をもつ正則自己同型を許容するK3曲面のPicard数を決定した。逆に、種々のPicard数に対して、上記のようなK3曲面自己同型を、超幾何群の方法を用いて構成する方法を与えた。なお、Siegel円板の存在のためには、K3曲面上の不変曲線と Siegel 円板が共存するような状況を設定することが必要である。そこでLefschetz型の正則不動点公式を整備するとともに、不変曲線上の孤立不動点の局所指数を表現する Grothendieck 留数を計算する手段を与えた。
• Hypergeometric functions and Painleve equations

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

  Date (from‐to) : 2016/04 -2019/03 

  Author : Iwasaki Katsunori, Ebisu Akihito

   

  For generalized hypergeometric functions 3F2(1) we developed a general theory of contiguous relations. We established the linear independence of contiguous functions, existence and uniquness of contiguous relations, and algorithms for calculating their coefficients, as well as their group symmetry. As an application we constructed an infinite number of 3F2(1) continued fractions and determined exactly the leading asymptotics of their truncation errors. To do so we developed a discrete analogue of saddle point method for obtaining the asymptotic behavior of hypergeometric series containing a large parameter. As for Painleve equations we summarized the results obtained so far and set up the direction in which the next study should take. We also obtained some conditions for hypergeometric functions to admit gamma product formulas.
• Developments in Interactions between Algebraic Geometry and Integrable Systems

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

  Date (from‐to) : 2012/05 -2017/03 

  Author : Saito Masa-Hiko, NORO Masayuki, KOIKE Tatsuya, INABA Michi-aki, MORI Shigefumi, MUKAI Shigeru, IWASAKI Katsunori, KANEKO Masanobu, HARAOKA Yoshishige, NAMIKAWA Yoshinori, ISHII Akira, FUJINO Osamu, HOSONO Shinobu, MATSUSHITA Daisuke, ABE Takeshi, IRITANI Hiroshi, TODA Yukinobu, NAKAJIMA Hiraku, NAKAMURA Iku, TANIGUCHI Takashi, ONO Kaoru, ROSSMAN Wayne, MITSUI Kentaro, SANO Taro

   

  We established the geometric Painleve property of nonlinear differential equations for isomonodromic deformations of connections with generic unramified irregular singularities and regular singularities with fixed spectral types. We also established theory of Mixed twister D-modules and developed several geometric theories for integrable systems. As for higher dimensional algebraic geometry, certain types of extremal contractions of 3-dimensinal terminal varieties were classified in detail. Fujino proved that canonical rings of compact Kahler manifolds are finitely generated. Several results for symplectic varieties, moduli theory were obtained in our research projects. Mathematical foundations of Quantum cohomology rings were developed by the group of Fukaya, Ono and others. Several developments of mirror symmetry, including the case of toric Calabi-Yau varieties, are obtained. We also obtained several important results on derived categories of sheaves on algebraic varieties.
• Painleve systems, hypergeometric systems and dynamical systems

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

  Date (from‐to) : 2013/04 -2016/03 

  Author : Iwasaki Katsunori, UEHARA Takato

   

  Hypergeometric equations are linear differential equations solved by an important class of functions called hypergeometric functions, while in certain sense Painleve equations may be thought of as nonlinear analogues of hypergeometric equations. Because of their nonlinearity, the study of Painleve equations requires various methods from dynamical systems. We constructed the phase space of a Painleve equation and gave a geometric characterization of it as an orbifold Hamiltonian dynamical system. We also discussed periodic solutions to another Painleve equation. As for hypergeometric functions we focused our attention on spacial-value formulas, especially on gamma product formulas, and obtained necessary conditions of arithmetic flavor for the existence of such formulas.
• 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

  Date (from‐to) : 2011/04 -2016/03 

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

   

  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.
• Geometric moduli theory and its theoretical applications

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

  Date (from‐to) : 2011/04 -2016/03 

  Author : Nakamura Iku, IWASAKI KATSUNORI, ONO KAORU, TERAO HIROAKI, WENG LIN, ASAKURA MASANORI, ISHII AKIRA, OOMOTO Toru, KATSURA TOSHIYUKI, KATSURADA HIDENORI, SAITO MASAHIKO, ABE NORIYUKI, TANABE KENICHIRO, NAKAMURA KENTARO, HARASHITA SHUSHI, YOSHINAGA MASAHIKO

   

  In this project we aimed at studying global structures of certain geometric spaces so that we may apply them to the related mathematical theories. The main results of our studies are 1) construction of the second compactifications of moduli spaces of abelian varieties, and study of the relation with the other important compactifications, 2) proof of Riemann hypothesis for some of zeta functions of the moduli spaces of semi-stable vector bundles over an algebraic curve, 3) a characterization of one of Painleve differential equations through the study of stable vector bundles of rank two, 4) proof of the isomorphism between the quantum cohomology ring and the Jacobi ring of a potential in mirror symmetry through the study of the moduli space of Lagrangian submanifolds of a toric manifold, 5) generalization and further study of Arrow's impossibility theorem in statistical economics in terms of hyperplane arrangement.
• Development of Integrable Geometry

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

  Date (from‐to) : 2011/04 -2015/03 

  Author : Miyaoka Reiko, KOTANI Motoko, NISHINOU Takeo, UEHARA Taketo, MATSUURA Nozomu, IWASAKI Katsunori, IRITANI Hiroshi, KAJIWARA Kenji, NAGATOMO Yasuyuki, NOMURA Takaaki, YAMADA Kotaro, ISHIKAWA Goo, UMEHARA Masaaki, GUEST Martin, SHODA Toshihiro, FUTAKI Akito, FUJIOKA Atsushi, RASSMAN Wayne, TAMARU Hiroshi

   

  Isoparametric hypersurfaces with 6 principal curvatures with multiplicity 2 are shown to be homogeneous, which solves one of Yau's problems. As for 4 principal curvature case, we gave a description by using the moment map of spin actions. Transnormal systems are investigated in details. We show the non-existence of L2 harmonic 1-form on a complete non-compact stable minimal Lagrangian submanifolds in a Kaheler manifold with positive Ricci curvature. Then the number of non-parabolic ends is less than two, and in the surface case, the genus should vanish. The Floer theory on the intersection of a Lagrangian submanifold with its Hamiltonian deformation is investigated. The Gauss images of isoparametric hypersurfaces in the sphere are Lagrangian submanifolds of complex hyperquadric, and in this case, we show that if the multiplicities of the principal curvatures are bigger than 1, then they are Hamiltonian non-displaceable,
• Dynamics on algebraic varieties and Painleve equations

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

  Date (from‐to) : 2008 -2012 

  Author : IWASAKI Katsunori, UEHARA Takato, KAJIWARA Kenji, KAMIMOTO Joe, TSUJII Masato, ISHII Yutaka, TSUDA Teruhisa

   

  We developed a dynamical study of the sixth Painleve equation on the algebro-geometrical and moduli theoretical foundations of the Painleve system. When the parameter lies on the walls of an affine Weyl group, we established the chaotic nature of the system and proved the exponential growth of the number of isolated periodic solutions. To obtain these results, we developed a general theory of periodic points for area-preserving birational maps on a projective surface. Constructing rational surface automorphisms of positive entropy has also been discussed.
• New developments and interaction between Algebraic Geometry and Integrable Systems

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

  Date (from‐to) : 2007 -2011 

  Author : SAITO Masa-hiko, NOUMI Masatoshi, YOSHIOKA Kota, YAMADA Yasuhiko, OHTA Yasuhiro, YAMAKAWA Daisuke, FUKAYA Kenji, INABA Michiaki, TAKASAKI Kanehisa, MORI Shigefumi, MUKAI Shigeru, IWASAKI Katsunori, KANEKO Masanobu, HARAOKA Yoshishige, NAMIKAWA Yoshinori, ISHII Akira, FUJINO Osamu, HOSONO Shinobu, MATSUSHITA Daisuke, YOSHINAGA Masahiko, KOIKE Tatsuya, MOCHIZUKI Takuro, IRITANI Hiroshi, HARASHITA Shushi, TODA Yukinobu

   

  We gave an algebro-geometric construction of the moduli spaces of stable parabolic connections over curves with unramified singularities, and showed the fundamental property of the Riemann-Hilbert correspondences. These results showed the geometric Painleve property of the nonlinear isomonodromic differential equations and established the geometry of isomonodromic deformations of connections, which enables us to investigate the phase space of differential equations deeply such as Okamoto's space of initial conditions for classical Painleve equations. Together with the progress in the field of higher dimensional birational geometry and the geometry related to mirror symmetry, these results reveal deep relations between algebraic geometry and integrable systems.
• Toward a unified understanding of general hypergeometric functions and general Schlesinger system by twistor theory

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

  Date (from‐to) : 2007 -2010 

  Author : KIMURA Hironobu, HARAOKA Yoshishige, TANABE Susumu, MISAWA Masashi, FURUSHIMA Mikio, OKAMOTO Kazuo, IWASAKI Katsunori, SHIMOMURA Shun, KAWAMUKO Hiroyuki

   

  We studied the theory of general hypergeometric functions(HGF) which generalize important special functions, like as Gauss hypergeometric functions, governed by linear differential equations to functions of several variables. We also studied nonlinear differential equations called general Schlesinger systems(GSS), which describe families of linear systems preserving monodromy data, from the point of view of twistor theory. For HGF, we determined the cohomology groups which are defined using the integrand of the integral representation of HGF. For GSS, we constructed its solutions expressed using HGF.
• Theory of Painleve systems and its new development

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

  Date (from‐to) : 2007 -2010 

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

   

  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.
• Fusion of geometry and the theory of integrable systems

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

  Date (from‐to) : 2007 -2010 

  Author : MIYAOKA Reiko, OHNITA Yoshihiro, MOTOKO Kotani, SASAKI Takeshi, IWASAKI Katsunori, OYSU Yukio, KAJIWARA Kenji, NAGATOMO Yasuyuki, NAKAYAAHIKI Atsushi, YAMADA Kotaro, FUTAKI Akito, MARTIN Guest, WAYNE Rossman, SHODA Toshihiro, IRITANI Hiroshi, ISHIKAWA Goo, UMEHARA Masaaki, KAWAKUBO Satoshi, TAMARU Hiroshi, FUJIOKA Atsushi, MATSUURA Nozomu, NISHINOU Takeo

   

  We classified almost all isoparametric hypersurface, and characterize them in terms of the moment map, which proves the evidence of a relation with integrable systems. A basic theory of surfaces with singularities, and a new method using the Legendre map have been established. Via the Riemann-Hilbert correspondence, the dynamical system of Painleve equations is investigated, and the view point of the chaos has been developed. The modularity of higher genus Gromov-Witten and the mirror symmetry are discussed. A surface with potential appeared in quantum cohomology is constructed, which contributes to the tt* geometry.
• Research on geometry related to theta functions

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

  Date (from‐to) : 2005 -2008 

  Author : MATSUMOTO Keiji, ONO Kaoru, NAKAMURA Iku, SHIMADA Ichiro, IWASAKI Katsunori, TERASOMA Tomohide, YOSHIDA Masaaki, ONO Kaoru, NAKAMURA Iku, IWASAKI Katsunori, SHIMADA Ichiro, TERASOMA Tomohide, YOSHIDA Masaaki

   

  テータ関数や超幾何関数のみたす関数等式を多数与えた。テータ関数のみたす関数等式によりWhitehead link と Borromean ringsの補空間に入る双曲構造を解明した。また、超幾何関数のみたす関数等式より、いくつかの多項版の算術幾何平均を定め、それらの値の表示公式を与えた。
• Geometry and global analysis of Pailneve equations

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

  Date (from‐to) : 2004 -2007 

  Author : IWASAKI Katsunori, KAJIWARA Kenji, KAMIMOTO Joe, SAITO Masa-hiko, INABA Mchiaki, HARAOKA Yoshishige

   

  Many results have been obtained for Painleve equations, especially for Painleve VI equation and its generalization, Gamier systems, from the viewpoint of algebraic geometry and dynamical system theory. They consist of the establishments of laws of Painleve dynamics mainly based on algebraic geometry, and the elucidations of the global phenomena of Painleve dynamics mainly based on dynamical system theory. More explicitly, the results on the laws of Painleve dynamics include the construction of the phase spaces of Painleve dynamics as the moduli space of stable parabolic connections, the establishment of Riemann-Hilbert correspondence, a characterization of Backlund transformations in terms of the Riemann-Hilbert correspondence, the discovery of an initimate relation between Riccati solutions and singularity theory, an intrinsic introduction of the Hamiltonian structure of Painleve equations, and so on. On the other hand, among the results on the phenomena of Painleve dynamics, it is most remarkable that we were able to show that the nonlinear monodromy of the Painleve flow is chaotic along almost all loops in the space of time variable. Namely, the proof of the positivity of the topological entropy, the construction of a maximal-entropy hyperbolic invariant probability measure of saddle type, the establishment of an algorithm of calculating entropy in terms of the reduced word of a given loop and the geometric representation of a universal Coxeter group. These results clearly show that the Painleve equation is in fact a chaotic dynamical system, although it has previously been studied from the viewpoint of integrable systems only. So it is expected that our results would stimulate people to change minds in the future direction of research in the field of Painleve equations. The above-mentioned achievements are the results of many cooperative researches, attendances at various conferences and exchanges of ideas, making the best use of this grant. By virtue of this grant, we were also able to announce or describe the details of our results in various conferences, workshops and other academic meetings, either domestic or overseas. The international conferences on Painleve equations in which the head investigator were invited to give a lecture include Theories asymptotiques et equations de Painleve, Universite d'Angers, France; The Painleve equations and monodromy problems, Isaac Newton Institute, Cambridge University. In summary, the original aims of this project, i.e., to develop an algebraic geometry in order to lay a sound foundation of Painleve equations and to explore the global phenomena of Painleve dynamics, have largely been achieved. A further advances along the line of this project can be expected based on these achievements.
• Development and relations between various geometries and integrable systems

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

  Date (from‐to) : 2004 -2007 

  Author : MIYAOKA Reiko, YAMADA Kotaro, IWASAKII Katunori, KAJIWARA Kenji, NAKAYASHIKI Atsushi, NAGATOMO Yasuyuki

   

  Miyaoka gives a new proof for the Dorfmeister Neher classification theorem on isoparametric hypersurfaces, and as applications of hypersurface geometry, clarifies the topological structure of the anti-self-dual bundle of complex projective plane and complete austere submanifolds, constructs Ricci flat metrics, special Lagrangian submanifolds. She also gets twister fibrations from the geometry of G2 orbits. Iwasaki connects the algebraic formulation of Painleve IV with the ergodic theory of birational maps of algebraic surfaces via Riemann-Hilbert correspondence, and shows the chaotic behavior of non-linear monodoromy. Kajiwara applies the theoretic formulation of the Painleve systems and constructs the determinant formula of the hypergeometric solutions of q-Painleve, and relates it with the solutions of the associate linear problems. Nakayashiki characterizes the coefficients of the series of sigma function by those of defining functions of the algebraic curves. Nagatomo obtains an essential relation between harmonic maps and the Yang-Mills connections, and generalizes Takahashi's theorem, de Carom-Wallach's theorem, and constructs harmonic maps from quaternion Kaehler manifold to Grassmannian manifolds. Yamada-Umehara-Rossman classify the behavior of the ends of complete flat fronts in the hyperbolic 3-space. Fujioka studies integrability and periodicity of the motion of curves in complex hyperbolics which depend on Burger's equation and have descritization. Ishikawa classifies singularities of inproper affine surfaces and surfaces with constant Gauss curvature, and their dual surfaces. He also clarifies moduli of the singularities, and obtains a relation between plane curves and their Legendle curves. Udagawa classifies compact isotropic submanifolds with parallel mean curvature vector wit the sectional curvature. Tamaru proves a fixed point theorem for cohomogeneity one action corresponding to homogeneous hypersurfaces in symmetric spaces of non-compact type. Matsuura studies a development of plane curves depending on KdV equation w..r.t. discrete time. Ikeda studies equi-energy surfaces of characteristic manifod of Whittaker abel group and full Kostant-Toda lattice via micro-local anaysis. Guest investigates harmonic maps, quantum cohomorogy and mirror symmetry, and writes an introductory book Futaki proves the existence of Sasaki-Einstein metrics on some toric Sasakian manifolds, in particular, the existence of compelete Ricci-flat metric on the canonical bundles of toric Fano manifolds.
• Inverse Problems in Nonlinear Phenomena

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

  Date (from‐to) : 2003 -2006 

  Author : KAMIMURA Yutaka, IWASAKI Katsunori, TSUBOI Kenji, NAKASHIMA Kimie

   

  This research was intended to make a scheme for determination of the nonlinearities and/or governed equations in nonlinear problems from a viewpoint of inverse problems. Main results are as follows : 1.An inverse problem to determine an unknown velocity in two-dimensional, time-independent advection-diffusion equation from data observed at a depth-level was discussed. A procedure by which the velocity is reconstructed from the observed data is established and, as a consequence, the uniqueness of the velocity realizing the prescribed data was proved. 2.Related with the problem in (1), an inverse scattering problem to recover the potentials of an energy dependent Schrodinger equation from the scattering data was discussed. A new inversion formula was developed, by which the potentials are recovered directly through the solution of a Marchenko equation. By means of this inverse formula, a necessary and sufficient condition for a given function to be the scattering data was obtained.
• General hypergeometric functions and geometry of the space of arrangements of points with infinitesimal neighborhoods

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

  Date (from‐to) : 2003 -2006 

  Author : KIMURA Hironobu, HARAOKA Yoshishige, TABABE Susumu, FURUSHIMA Mikio, MISAWA Masashi, IWASAKI Katsunori

   

  The general hypergeometric functions(GHF) and the structure of the twsted cohomology group. The conjugacy classes of the centralizers of regular elements of GL(N) are determined by partitions of N. GHF is a multi-valued function on the Grassmannian manifold Gr(n+1, N) defined as a Radon transform of a character of the universal covering group of the centralizer. For an integer q > 0, consider a partition (q, 1,...,1) of N. To clarify the structure of the solution space of general hypergeometric system, we computed the rank and a basis of the associated de Rham cohomology group. When GHF is given by n dimensional integral, we found that the k-th cohomology group vanishes for k different from n, and the rank of the n-th cohomology group is (N-2)!/n!(N-n-2)!. We gave a basis for this group explicitly using Schur functions. Schlesinger system and its generalizations. We started the research of giving this generalizations from the point of view of twistor theory. When one consider the generalized anti-self dual Yang-Mills equation(GASDYM) on the Grassmannian manifold Gr(2, N), its solution corresponds to a holomorphic vector bundle on the twistor space PN-1 via the Ward correspondence which is trivial when restricted to twistor lines. Let H be a maximal abelian subgroup of GL(N) as in 1) and consider its natural action on the twistor space PN-1. Moreover we assume that the action of H can be lifted to the holomorphic vector bundle corresponding to a solution to the GASYM equation. Then this action determines a flat connection on the bundle and when restricted to twistor lines, this flat connection describes a monodromy preserving deformation of ODEs. We gave the explicit form of the flat connection and by this explicit expression we made clear the analogy to the definition of GHF. We derived in a unified way the general Schlesinger systems from this point of view as the differential equations on Gr(2,N) which corresponds to the Painleve equations(including the degenerated ones). We also made clear that the Weyl group associated with H describes a group of symmetry of the general Schlesinger system. By this, we can give the group theoretic understanding for the fact that the number of parameters in the Painleve equations deceases after the degeneration. We could also construct the process of degeneration (confluence) for the general Schlesinger systems.
• Mathematical theory of dynamical systems of Painleve type

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

  Date (from‐to) : 2003 -2006 

  Author : KAJIWARA Kenji, NOUMI Masatoshi, YAMADA Yasuhiko, IWASAKI Katsunori

   

  1. We have extended the theory of symmetric form for q-Painleve IV equation with (A_2+A_1)^<(1)> affine Weyl group symmetry to formulate the q-KP hierarchy. By the similarity reduction we have constructed the hierarchy of discrete systems with (A_+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_+A_)^<(1)> affine Weyl group symmetry. 2. We have presented a formulation of the elliptic Painleve equation and its generalizations, the former of which is located on the top of all the Painleve systems of second order. Namely, we have formulated the time evolution and the Backlund transformations as Cremona transformations on the configuration space of generic points in the complex projective space, and given their realization as birational transformations parametrized by the theta functions on the level of the τ functions. We have also formulated the time evolution as the addition formula on the moving pencil of plane cubic curves, and clarified the geometric meaning of the Hamilitonians of the Painleve differential equations. 3. Applying the above formulation, we have constructed the simplest hypergeometric solutions to the elliptic Painleve equation and all the q-Painleve equations, and completed the list of coalscent diagram of hypergeometric functions, starting from the elliptic hypergeometric function _<10>E_9 to the q-Airy function. 4. Combining the algebro-geometric formulation of the Painleve VI equation and the ergodic theory of birational mapping on the algebraic surface by using the Riemann-Hilbert correspondence, we have shown that the nonlinear monodromy of the Painleve VI equation is chaotic along almost all the loops. 5. We have shown that the entries of Hankel determinant formula for the solutions of the Painleve differential equations arise as coefficients of asymptotic expansion of the ratio of solutions to the auxiliary linear problem, and that this phenomenon originates from the structure of the KP hierarchy. 6. Applying the above results we have discussed some new extentions or new solutions to the discrete and ultradiscrete Toda equation.
• パンルヴェ第VI走方程式の非線型モノドロミーと複素離散力学系

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2003 -2005 

  Author : 岩崎 克則, 石井 豊

   

  パンルヴェ第IV方程式の非線形モノドロミー,即ちポアンカレ回帰写像のなす複素力学系についての研究を行った。岩崎克則は,前年度までの成果である,パンリヴェ力学系の相空間,初期値空間のモジュライ理論的構成,リーマン・ヒルベルト対応による複素三次曲面上の離散力学系への共役写像の構成等の基盤整備に基づき,今年度は下記の研究を行った。 (1)ポアンカレ写像の周期点の個数の計算:パンルヴェ方程式のモジュライ理論的構成,リーマン・ヒルベルト対応の双正則性,複素三次曲面の幾何学,初期値空間上のポアンカレ写像に対応する三次曲面上の双有理写像の力学系の考察,特にその力学系のコホモロジー群への誘導線形写像の考察,力学次数の計算,レフシェッツの不動点公式の適用等の議論を経ることにより,ポッホハマー・ループに沿うポアンカレ写像の周期点の個数を計算した。この公式は,周期を与えたとき,その周期を持つ周期点の個数を明示的に表す具体的な公式であり,特にその個数が周期と共に指数関数的に増大することを示している。 (2)ポアンカレ写像のエルゴード理論的研究:パンルヴェ方程式の定義領域であるリーマン球面から3点を除いた領域上の,すべての非初等的な閉曲線に対して,その曲線に沿うポアンカレ写像がカオス的であることを示した。すなわち,位相的エントロピーが正であること,混合的であり鞍点型の双曲型不変測度が存在すること,その不変測度が最大エントロピー測度であること,その測度の台における双曲型不動点の稠密性等の発見である。その証明には,最近急速に発展している複素曲面上の双有理写像の力学系の成果が極めて有効に応用された。 これらの成果は,従来可積分系の立場から研究されることが殆どであったパンルヴェ方程式の分野に,カオス的な現象が実際に起こることを発見したものであり,その点で全く新しい成果であるといえる。 これらの研究を実行するに際して,研究代表者の岩崎克則は,分担者の石井豊と定期的に複素曲面上の力学系のエルゴード理論に関するセミナーを実施した。また,これまでの成果をまとめた概説論文を執筆した。また,石井豊は,双曲的な複素エノン写像の研究を行った。特に、拡大的1変数多項式の摂動としては決して得られないような双曲的エノン写像の構成に成功した。
• Geometric study of the hypergeometric function

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

  Date (from‐to) : 2002 -2005 

  Author : YOSHIDA Masaaki, SASAKI Takeshi, IWASAKI Katsunori, MIMACHI Katsuhisa, MATSUMO Keiji, CHO Koji

   

  I got the following results concerning the hypergeometric functions. 1)Studied the (co)homology groups attached to Selberg-tpe integrals, evaluated the intersection numbers, and discovered a combinatorial properties of the Selberg functions. 2)Presented co-variant function theory. Found the kappa function, and a 3-parameter families of hypergeometric polynomials, which are very different from the classical ones. 3)Found a new infinite-product formula for the elliptic modular function Lambda. 4)studied combinatorial-topologically the shape of the Schwarz triangles when the inner angles are general. 5)Studied the Whitehead-link-complement group, constructed automorphic functions for this group, and embedded the quotient space to a Euclidean space. 6)Studied the behavior of the solutions of the hypergeometric equation when the exponent-diffences are pure-imaginary, and studied the relation between the space of parameters and the Teichmuler space of genus 2 curves. 7)Invented the theory of hyperbolic Schwarz map. The target of the Schwarz map has been the sphere. Our hypergeometric one has the 3-dimensional hyperbolic space as its target. Group theoretically it is more natural 8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.
• Complex analysis of L holomorphic functions on pseudoconvex domains of finite type

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

  Date (from‐to) : 2002 -2005 

  Author : KAMIMOTO Joe, IWASAKI Katsunori, KAZAMA Hideaki, SATO Eiichi, TAKAGI Shunsuke, KIMURA Hironobu

   

  We studied many kinds of objects in the function theory of several complex variables from the viewpoints of the theory of singularities. In particular, we are interested in the boundary behavior of the holomorphic functions which are square integrabel. Concretely the Bergman kernel and Szegoe kernel are very important integral kernel and they have many important information of the boundary behavior of holomotphic functions. As is very well known, the case of strictly pseudoconvex domains has many strong results about the Bergman kernel and Szegoe kernel. For example, the asymptotic expansion due to C.Fefferman reveals completely their boundary behaviors. We are interested in the weakly pseudoconvex domains case. The general case is very difficult to analyze and so we restrict ourselves to the objects in the case of finite type in the sense of D'Angelo. From the definition of finite type, the argument from algebraic geometry and singularity theory are valuable. We introduced the concepts of"Newton polyhedra"into the analysis of the Bergman kernel and showed that its singularity can be expressed in terms of the topological information of the Newton polyhedra. Moreover, we analyzed the construction of peak functions of any finite type domains.
• Defining manifolds and symmetries of nonlinear special functions in several variables

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

  Date (from‐to) : 2001 -2004 

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

   

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

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

  Date (from‐to) : 2000 -2003 

  Author : IWASAKI Katsunori, INABA Michi-aki, KAJIWARA Kenji, YOSHIDA Masaaki, SAITO Masa-hiko

   

  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 inhomogeneous 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.
• 代数的可積分系のP-解析的手法による研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2001 -2002 

  Author : 岡本 和夫, 岩崎 克則, 桂 利行

   

  本研究課題の目的は、特別なタイプのp-線型微分方程式について、その解の構造をp-解析的手法で調べること、パンルベVI型方程式の代数解をp-解析的に調べること、超越的な解を持つ可積分系も有限体上では代数的可積分系となることがあるので、適当なモデルを作り数理実験を行うこと、p-解析の意味で代数的可積分系であるような可積分系を構成すること、線形の場合をモデルとして、p-線形微分方程式の一般論を目指すこと、の5点である。2年間の研究を通して、研究総括は研究代表者が行い、研究分担者は他分野の研究者との交流による情報と知見の収集、および共同研究の準備を行った。研究代表者は、パンルヴェI型方程式について以下のような結果を得た。パンルヴェI型方程式の一般解であるパンルヴェ超越関数は解析的には古典関数では表すことができない超越関数であるが、この事実をp-解析の対場から見直すと、標数が5の場合には代数的に可積分であることがわかる。具体的には、原点の周りの級数解は代数関数で表され、また無限遠点における形式級数解もp-解析の意味で収束し代数関数となる。同様な結果はパンルヴェII型方程式についても成り立つ。この場合の鍵になる標数は3である。さらに、他のタイプのパンルヴェ方程式の代数解について、共同研究により次の結果を得た。すなわち、パンルヴェ方程式はある2階線型常微分方程式のホロノミック変形で特徴付けられるが、代数解を代入すると、これらの線型方程式は古典関数で表され、そのモノドロミーを具体的に決定することができる。これらの結果は現在論文を準備中である。いずれの結果も解析的な研究が大いに参考になった。研究分担者の関連する研究は文献表に挙げた通りである。
• Integrated research of the general hypergeometric systems and nonlinear integrable systems

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

  Date (from‐to) : 1999 -2002 

  Author : KIMURA Hironobu, HARAOKA Yoshishige, KOHNO Mitsuhiko, YAMAKI Hiroyoshi, TAKANO Kyoichi, IWASAKI Katsunori

   

  The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometric functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities. For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function.
• A Geometric study of hypergeometric functions

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

  Date (from‐to) : 1999 -2001 

  Author : YOSHIDA Masaaki, MIMACHI Katsuhisa, IWASAKI Katsunori, SASAKI Takeshi, HANAMURA Masaki, MATSUMOTO Keiji

   

  We got the following results concerning hypergeometric functions. 0) Studied systems of linear partial differential equations modeled after grassmannians. 1) Investigated the Hodge structure of twisted cohomology groups and Got many integral formulae involving got twisted Riemann inequalities absolute values in the integrands. 2) Got the uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces. Proved that it is the restriction of the higher dimensional hypergeometric differential equation onto a d. 3) Developed the intersection theory for twisted cycles : got determinant formulae for not necessarily genetic hyperplane arrangements. Got partial results in the case that some quadratic hypersurfaces get into the arrangements. 4) Found a hyperlybolic structure on the real locus of the moduli space of marked cubic surfaces. Found that the corresponding group is the hyperbolic Coxeter group ; Constructed automorphic forms by the help of a modular embedding. 5) Made a geometric study of the hypergeometric function with Found that the monodromy groups turns out to be scottky imaginary exponents. Groups of genes 2. Constructed a modular ttu with rasp. To the monodromy group.
• Nonlinear Inverse Problems and Singular Integral Equations

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

  Date (from‐to) : 1998 -2001 

  Author : KAMIMURA Yutaka, HINO Yoshiyuki, SHIOTANI Nobuhira, TSUBOI Kenji, IWASAKI Katsunori, HARAOKA Yoshishige

   

  Several nonlinear inverse problems in mathematical sciences are discussed : a friction coefficient identification problem, a nonlinearity identification problem in nonlinear oscillations, a time-dependent thermal conductivity identification problem, inverse bifurcation problem in mathematical ecology (nonlinear reaction kinetics identification problem in a diffusional model of population dynamics from the relation between growth rates and certain central densities of population distributions). A certain class of singular integral equations plays a principal role in our mathematical analysis of the inverse problems ; some existence results of the solutions to the integral equations are established and applied to the inverse problems.
• Global Analysis of Painleve Equations

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

  Date (from‐to) : 1998 -2000 

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

   

  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.
• Studies on Differential and Difference Equations by Means of Geometric and Algebraic Methods

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

  Date (from‐to) : 1997 -1998 

  Author : IWASAKI Katsunori, KAMIMURA Yutaka, WATANABE Humihiko

   

  1. Polyhedral harmonics and invariant theory : K.Iwasaki settled a longstanding conjecture concerning the finite dimen- sionality of the space of polyhedral harmonic functions. He discovered new basic invariants of finite reflection groups in order to explicitly determine the polyhedral harmonic functions for polytopes with ample symmetry. In collaboration with A.Kenma and K.Matsumoto, he also made an explicit determination of polyhedral harmonic functions for the exceptional regular polytopes. He is planning to write a book on polyhedral harmonics and invariant theory. 2. Cohomology for reccurence relations and hypergeometric functions : K.Iwasaki obtained a sharp asymptotic formula for solutions to certain difference equations. Based on this formula, he is planning to develop a cohomology theory for recurrence relations. K.Iwasaki and M.Kita discovered an exterior power structure on the twisted de Rham cohomology groups associated to hypergeometric functions. K.Iwasaki and K.Matsumoto obtained a conjecture that the intersection matrix of the twisted cohomology groups associated to generalized Airy functions can be expressed in terms of skew-Schur polynomials. Attempts to prove this are now in progress. 3. Invese bifurcation problem and singular integral equations : K.Iwasaki and Y.Kamimura established the solvability of a class of singular integral equations. They applied this result to prove the existence of solutions to the inverse bifurcation problem for nonlinear Sturm-Liouville equations. Y.Kamimura is writing a book on integral equations which gives a detailed account of their results. 4. Painlev_ equations and combinatorics : K.Iwasaki and H.Kawamuko discovered a combinatorial formula of Leibniz type associated to the Hamiltonian structure of the fourth Painlev_ equation in several variables. As an application, they obtained a new quadratic relation among Gegenbauer's orthogonal polynomials. H.Watanabe found birational transformations of solutions to the sixth Painlev_ equation. He also determined classical solutions to that equation.
• STUDIES ON D-MODULES DERIVED FROM CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATIONS

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

  Date (from‐to) : 1997 -1998 

  Author : MAJIMA Hideyuki, IWASAKI Katsunori, ASAMOTO Nariko, YOSHIDA Hiroaki, TAKAYAMA Nobuki, KIMURA Hironobu

   

  1. Calculation of cohomology groups of solution complex of D-modules defined by confluent hypergeometric differential equations with values in the sheaves of germs of formal power-series ring and formal power-series ring with Gevrey order, by using projective resolutions of the D-modules similar to Koszul complex, which was invented by suggested information from 'KAN(a system' of computational algebraic analysis) made by Takayama 2. Asymptotic expansions of restrictions of generalized Airy functions by using relations between confluent hypergeometric functions with particular parameters and generalized Airy functions 3. Approximation formulas of coefficients of divergent solutions by using a vanishing theorem in asymptotic analysis in several variables 4. Constructionf of theory of intersection on the complex projective line for homology and cohomology groups defined by connections which are regular singular or not, and quadratic relations satisfied by confluent hypergeometric functions, as an analogue of period relations, by applying this theory.
• 微分方程式の幾何学的研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1996 -1996 

  Author : 岩崎 克則, 真島 秀行, 松尾 厚, 片岡 清臣, 岡本 和夫

   

  本年度特に国成果を挙げた研究は「多面体の組み合せ論,有限鏡映群の不変式論と偏微分方程式系」に関するものである。これに関しては、先ず1962年に応用数学者FriedmanとLittmanによって提出された、「多面体に関して平均値の性質を満たす関数(多面体調和関数)全体は有限次元線型空間をなすか?」という未解決問題を解決した。これは従来のFourier解析的手法と全く異なった,多面体の幾何と組み合せ論にホロノミックな偏微分方程式系の理論を組み合せる新しい手法に基づく。この結果は雑誌Geometryに掲載が決まっている。さらに高い対称性を持つ多面体に関する多面体調和関数全体を決定するために,有限鏡映群の不変式と不変偏微分方程式系について考察を進めた。そして一連の新しい不変式環の基底を発見した。この結果を用いて,一般次元での凸正多面体の大部分に対して,対応する多面体調和関数の空間を具体的に決定することができた。 他の研究主題に関しては,差分方程式の漸近解析とコホモロジー論,分岐理論の逆問題,多変数超幾何関数に付随する捩れコホモロジー論について研究を行い,成果を論文にまとめた。差分方程式の研究については,Gevreyコホモロジー群という新しい概念を導入し、研究が端緒についたばかりである。合流型超幾何関数への応用を含め,これからの大いなる発展を期していかねばならない。
• Analysis of the structure of solutions of nonlinear partial differential equations

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

  Date (from‐to) : 1995 -1996 

  Author : MATANO Hiroshi, IWASAKI Katunori, MATUMOTO Yukio, KUSUOKA Shigeo, OCHIAI Takushiro, TSUTSUMI Yoshio

   

  Westudied the structure of the infinite dimensional dynamical systems defined by nonlinear parabolic equations in one space dimension and showed that their global attractors are always finite dimensional manifolds. This result implies that the essential features of the long-time behavior of solutions can be described by a finite system of ordinary differential equations, and is therefore important from the point of view of qualitative theory. It should be noted that the so-called inertial manifold theory, which has been well-known since mid 1980's as a tool for studying the finite dimensionality of attractors, does not apply to the equation treated in our research. 2. We studied the behavior of solutions of degenerate diffusion equations and proved that any unstable equilibrium solution has an unstable manifold of infinite Hausdorff dimension. This result shows that there is essential difference between the dynamical structure of degenerate diffusion equations and that of nondegenerate equations. currently Matano is also studying the properties of traveling waves for nonlinear diffusion equations with spatially priodic coefficients and has obtained partial results. 3. We studied a mathematical model which combines Maxwell equation and Schrodinger equation. We obtained new results on the uniqueness and global existence of solutions. 4.Kusuoka, one of the investigaters of the research project, has been studying problems in mathematical finance with probabilistic method. He has obtained some interesting results.
• 微分方程式のモジュライ理論

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1993 -1993 

  Author : 岩崎 克則

   

  本年度の研究は二つの主題に分れる。第一の研究は多変数超幾何関数に関するものである。超幾何関数は数学の様々の分野や数理物理学に応用を持つ、極めて重要な特殊関数である。超幾何関数は、複素射影空間内の超平面配置の補集合上定義されたねじれホモロジー群とねじれコホモロジー群のカップリングとして定式化される。従って超幾何関数の性質を調べるためには、この(コ)ホモロジー群の構造を研究することが必要となる。これに関して金沢大学の喜多通武と共同研究を行なった。我々が発見したのは、上記(コ)ホモロジー群がある種の外積構造を持つということである。このことより特に、一般の超幾何関数を特別な超平面配置(ベロネ-ズ配置)に制限したものが、古典的によく知られたロ-リチェラの超幾何関数の外積として表わされる、という注目すべき結論が得られる。この結果は、まだ未知の部分が多い一般超幾何関数を、古典的なものと結びつけることができるという点で今後の研究に有用である。上記主題に関し、二篇の論文を喜多と共著で執筆し、現在数理科学研究科プレプリントシリーズとして公開中である。第二の研究では、行列代数上定義される、ある種の力学系を発見し、その軌道の挙動を詳細に調べた。この主題に関して一篇の論文を執筆し、やはりプレプリントシリーズとして公開中である。
• Microlocal Analysis of Differential Equations

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

  Date (from‐to) : 1991 -1992 

  Author : KOMATSU Hikosaburo, KAWAHIGASHI Yasuyuki, TSUTSUMI Yoshio, KATAOKA Kiyoomi, SUNADA Toshikazu, KOTANI Shinichi

   

  There are two ways of microlocal analysis, one by M. Sato et al. employs the theory of several complex variables and the cohomologies with coefficients in sheaves, and the other by L. Hormander et al. multiplication by cut-off functions and Fourier transforms. Komatsu established in between a third method of microlocal analysis employing Poisson integrals and their analytic continuations. This has the advantage of carrying out microlocalanalysis for various classes of generalized functions, including the Gevrey classes, between Sato's hyperfunctions and Schwartz' distributions at the same time. Komatsu extended, moreover, the theory of Laplace transforms of hyperfunctions to the case where hyperfunctions have values in a Banach space, and applied it in order to extend the Hille-Yosida theory of semigroups of linear operators to the case where semigroups are various classes of generalized functions. Kotani and Sunada investigated the spectra of Laplace operators and Schrodinger operators acting on the functions on Riemannian manifolds. In particular, Kotani gave a probabilistic proof to an estimate of the supremum of spectra in terms of curvatures. Sunada gave a sufficient condition for the spectrum has the band structure as a property of the C^*-group algebra of the discrete group acting on the manifold. Kataoka compared and distinguished many theories called the second microlocal analysis, and showed the importance of choozing a suitable theory in applying the second microlocal analysis to differential equations. Tsutsumi investigated the solvability of the initial value problem for the Zakharov equations describing the strong disturbance of Langmuin waves in plasmas. Kawahigashi gave rigorous formulations and their proofs to the so-called Ocneanu theory for the classification of subfactors in operator algebras for the first time. On this established foundation there will be fruitful applications of the theory.
• 非線型偏微分方程式の解の構造の解析

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1991 -1991 

  Author : 俣野 博, 岩崎 克則, 松本 幸夫, 小谷 真一, 落合 卓四郎, 堤 誉志雄

   

  1.研究代表者を中心として得られた知見(1)非線形熱方程式の爆発解の挙動について著しく解析が進展した。この研究には無限次元力学系の理論が役立った。(2)退化した拡散方程式の解のふるまいについて力学系の立場から考察し、ある場合にアトラクタ-の次元が無限大になることを示した(ロ-マ第2大学M.Pozioと共同研究)。(3)変分問題の立場から非線形偏微分方程式の解の形状を調べるのに有効な『リアレンジメント』の理論に関し、等可測連続変形の理論を提唱し、空間1次元の場合にその有効性を示した。これにより、これまで最小解に対して知られていた対称性や単調性などの性質が極小解に対しても成立することが明らかにな った(ハイデルベルク大学B.Kawohlとの共同研究)。 2.研究分担者を中心として得られた知見(1)非線形シュレディンガ-方程式の爆発解の興味ある挙動が明らかになった(提誉志雄)。爆発解の挙動は、非線形項が臨界指数をもつ場合は、シュレディンガ-方程式のそれはL^2ー凝縮と呼ばれるもので、非線形熱方程式の爆発解の挙動とは大きく様相を異にする。この差異を詳しく解析することは二つの方程式の構造の違いを深く理解することにつながり、当研究者と、研究代表者の間の研究討議は大変意義深いものであった。(2)リ-マン面土のフックス型微分方程式のなすモジュライ空間の構成をおこない、その空間のポアソン幾何的研究を行なった(岩崎克則)。この研究により、種々の完全積分ハ ミルトン方程式系が導出され、これら方程式系がハミルトン系である内在的理由が明らかにされた。(3)リ-マン面の1パラメ-タ族の退化曲面の写像類に関する研究(松本幸夫)
• 境界値問題及び混合問題の超局所解析

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1991 -1991 

  Author : 片岡 清臣, 岩崎 克則, 大島 利雄, 小松 彦三郎

   

  1.境界値問題や混合問題に限らず一般に線形偏微分方程式系の超局所解析において解の第2解析的特異性を調べることが重要であるが,種々の問題で柏原・河合による包合的多様体に沿う第2解析的特異スペクトラムの理論では説明し切れない現象があることがわかった。一方,仏のLebeauは既に,より微細な概念である陪特性帯に沿う第2特異スペクトラムを定義していたが佐藤超函数の枠内での意味は不明であった。研究代表者らはこのスペクトラムの定義函数による同値な表現を発見し,包合的な場合とのつながりも見い出した。特に,いわゆるPー解析性という,正則パラメ-タをもたないが一意接続性をもつマイクロ函数の性質を発見した。2.小松は調和函数とポアソン積分を用いる超局所解析の新しい基礎づけに対し,若干の補いを行った。またベクトル値ラプラス超函数の理論を整備した。3.岩崎はリ-マン面上のフックス型微分方程式のなすモジュライ空間の構成をおこない,その空間のポアソン幾何的研究をおこなった。更にモジュライを空間上にモノドロミ-保存葉層構造を定義し,それを記述する完全積分ハミルトン方程式系を導出した。更にこの方程式がハミルトン系なる内在的理由をコホモロジカルに説明した。4.片岡は微分方程式系からその導来系への自然な射をある種の分解を用いて具体的に表現することに成功した。これは混合問題の解析の際得られた,微分加群のHeaviside函数による切断操作に基づくものである。
• 非線型偏微分方程式の解の構造の解析およびその構造

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1990 -1990 

  Author : 俣野 博, 岩崎 克則, 小谷 真一, 松本 幸夫, 落合 卓四郎, 増田 久弥

   

  1.ある種の非線形拡散方程式の解は有限時間で消滅する(ある時刻以後は恒等的にゼロになる)ことが知られている。本研究では、以前知られていなかった、消滅時刻付近での解の詳しい挙動を解析することに成功した。この方程式は、例えばプラズマ中の熱の伝播(輻射のため熱エネルギ-は急速に失なわれる)や、多孔性媒質中の拡散現象(ただし蒸発等により総質量は急速に減少するような系)のモデルとして現れる。本研究で明らかにしたのは、解が消滅する際に、解の台(解が正の値をとる領域のこと;この外では解はゼロ)の各連結成分が収縮して、1点に縮まるという事実である(論文Finiteーpoint extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption)。これは、以前に行なわれていた数値実験でも結果がはっきりせず、結論がでていなかった。ただしこの研究の成果は空間1次元の場合に限られ、多次元の場合は今後の課題として残っている。上記の研究には、研究分担者の増田、岩崎(解析学)との討議が大いに役立った。また、拡散現象に対する確率論の立場からの示唆を小谷から受け、非常に参考になった。 2.非線形楕円型方程式の特異解の分類に大きな進展を見た(論文Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system)。これは量子力学におけるト-マス・フェルミ理論に現れるのと同種の方程式である。解析学の問題を、無限次元力学系の観点から定式化しなおし、解析学と幾何学の手法を併用することによって、大域的な研究を行ない得た。これに関し、分担者の落合、松本(幾何学)との研究討議が、解析学の問題に幾何学的視点を導入する上で非常に役立った。
• 境界値問題の超局所解析

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1989 -1989 

  Author : 片岡 清臣, 岩崎 克則, 戸瀬 信之, 山崎 昌男, 大島 利雄, 小松 彦三郎

   

  1.片岡は解析的線形微分方程式に対する混合問題に関する片岡-戸瀬による従来の理論をさらに発展させフランスのLebeauやSjo^¨strandらの回析波の伝播に関する重要な定理に幾何学的な別証を与えることに成功した。この証明によればさらに結果が退化した場合へも自然に拡張される。またそれと関連した第2超局所特異性理論の分野で片岡・戸瀬・岡田はより自然な第2マイクロ函数の理論を構成し,いくつかの基本的性質を導いた。小松はこれらの理論の基礎である佐藤の超函数とミクロ函数を調和函数を用いて新しく定義しなおし,ジュヴレイ族の正則性あるいは特異性をもつ函数及び超函数を超局所解析の立場から特徴付けた。大島はこれらの理論の表現論への応用として,関口と共同で半単純対称空間の普遍被覆空間に対するC-函数を具体的に計算することに成功し,線型とは限らない半単純リ-群の作用する半単純対称空間の場合にもPlancherelの公式が具体的に書けることを明らかにした。 2.岩崎は複素微分方程式に関する研究の中で,任意種数の閉リ-マン面上のフックス型射影接続のモジュライ空間を構成し,その解析空間あるいは複素多様体としての構造を研究した。更に,モジュライ空間上のモノドロミ-保存変形をポアソン幾何の観点から研究した。 3.山崎は超局所エネルギ-法を道具として分散を含む発展方程式の解の時間発展について研究した。まずポテンシャルが有界であるような2階の線型方程式について,初期値のある角領域での減衰から時間発展した後の解の超局所特異性が従うことを示した。また,非有界なポテンシャルを持つような方程式に上の結果を拡張するための準備として,楕円型でない発展作用素を持つ発展方程式がソボレフ空間上適切になるための十分条件を得た。