Researcher Database

Goo Ishikawa
Faculty of Science Mathematics Mathematics
Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

  • Professor

J-Global ID

Profile

  • Name: Goo Ishikawa

    Citizenship: Japanese.


    Degrees:

    B. S., Kyoto University (March, 1980)

    M. S., Kyoto University (March, 1982)

    Ph. D. (Doctor of Science), Kyoto University (March, 1985)


    Positions:

    Assistant, Nara Women's University (1985--87)

    Lecturer, Hokkaido University (1987--90)

    Associate Professor, Hokkaido University (1990-- )

    Associate Professor, Hokkaido University, Graduate School (1995--)

    Professor, Hokkaido University, Graduate School (2004--)

Research Interests

  • ケーレー8元体   パッチワーク   結び目の数え上げ   接触モデュライ空間   symplectic幾何   ルジャンドル曲線   擬直線配置   笠   接触同値   ヒルベルト第16問題   モンジュ・アンペール系   コイソトロピック写像   グルサ系   Web幾何   ケーリー8元数体   ガウス写像   非ホロノーム横断性定理   射影幾何   可展開面   可展面   特異ルジャンドル多様体   等径超曲面   サブリーマン幾何   ルジャンドル特異点   接触幾何   ラグランジュ特異点   単純特異点   接触構造   特異点論   モンジュ・アンペール方程式   

Research Areas

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

Academic & Professional Experience

  • 2006 - Today Hokkaido University Faculty of Science
  • 2004 - 2006 北海道大学 大学院理学研究科 教授
  • 1993 - 2004 北海道大学 大学院理学研究科 助教授
  • 1990 - 1993 Hokkaido University School of Science
  • 1987 - 1990 Hokkaido University School of Science
  • 1985 - 1987 Nara Women's University Faculty of Science

Association Memberships

  • THE MATHEMATICAL SOCIETY OF JAPAN   

Research Activities

Published Papers

  • G. Ishikawa, T. Yamashita
    Topology and its Applications 234 198 - 208 0166-8641 2018/02/01 [Refereed][Not invited]
     
    A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphism classification is completed for generic directed curves. Then it turns out that the swallowtails and open swallowtails appear generically for the classification on singularities of tangent surfaces.
  • Leibniz complexity of Nash functions on differentiations
    Goo Ishikawa, Tatsuya Yamashita
    Journal of the Mathematical Society of Japan 出版確定 2018 [Refereed][Not invited]
     
    arXiv:1509.08261 [math.AG]
  • Singularities of frontals
    Goo Ishikawa
    Advanced Studies in Pure Mathematics 78 55 - 106 2018 [Refereed][Not invited]
     
    http://www.math.sci.hokudai.ac.jp/~ishikawa/GooIshikawaProcKobeKyoto2015-HP.pdf
  • Residual Algebraic Restrictions of Differential Forms
    Goo Ishikawa, Janeczko
    Methods and Applications of Analysis 24 (1) 45 - 62 2017/03 [Refereed][Not invited]
  • Singularities of tangent surfaces to generic space curves
    Goo Ishikawa, Tatsuya Yamashita
    Journal of Geometry, 108 301 - 318 2017 [Refereed][Not invited]
  • Goo Ishikawa, Yoshinori Machida, Masatomo Takahashi
    ASIAN JOURNAL OF MATHEMATICS 20 (2) 353 - 382 1093-6106 2016/04 [Refereed][Not invited]
     
    In the split G(2)-geometry, we study the correspondence found by E. Cartan between the Cartan distribution and the contact distribution with Monge structure on spaces of five variables. Then the generic classification is given on singularities of tangent surfaces to Cartan curves and to Monge curves via the viewpoint of duality. The present paper completes the generic classification of singularitites for simple Lie algebras of rank 2, namely, for A(2), C-2 = B-2 and C-2.
  • D_n-geometry and singularities of tangent surfaces
    Goo Ishikawa, Yoshinori Machida, Masatomo Takahashi
    RIMS K\={o}ky\={u}roku Bessatsu B55 67 - 87 2016 [Refereed][Not invited]
  • Goo I. Shikawa, Yumiko Kitagawa, Wataru Yukuno
    Demonstratio Mathematica 48 (2) 193 - 216 2391-4661 2015/06/01 [Refereed][Not invited]
     
    Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. T hen it is shown in (15), that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.
  • Goo Ishikawa, Yumiko Kitagawa, Wataru Yukuno
    JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS 21 (2) 155 - 171 1079-2724 2015/04 [Refereed][Not invited]
     
    We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the viewpoint of geometric control theory. In fact, we consider the space of singular (or abnormal) paths on a given five-dimensional space endowed with a Cartan distribution, which form another five-dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover, we observe an asymmetry on this duality in terms of singular paths.
  • Goo Ishikawa, Yumiko Kitagawa, Wataru Yukuno
    JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS 21 (2) 155 - 171 1079-2724 2015/04 [Refereed][Not invited]
     
    We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the viewpoint of geometric control theory. In fact, we consider the space of singular (or abnormal) paths on a given five-dimensional space endowed with a Cartan distribution, which form another five-dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover, we observe an asymmetry on this duality in terms of singular paths.
  • Classification problems on singularities of mappings and their applications
    Goo Ishikawa
    Sugaku Exposition 28 (2) 189 - 214 2015 [Refereed][Not invited]
  • Goo Ishikawa, Yoshinori Machida
    SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 11 (081) 1815-0659 2015 [Refereed][Not invited]
     
    The classes of Monge-Ampere systems, decomposable and bi-decomposable Monge-Ampere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables >= 3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Ampere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Ampere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).
  • Goo Ishikawa, Yoshinori Machida, Masatomo Takahashi
    Journal of Singularities 12 27 - 52 1949-2006 2015 [Refereed][Not invited]
     
    It is well known that projective duality can be understood in the context of geometry of An-type. In this paper, as D4-geometry, we construct explicitly a ag manifold, its triple-fibration and différential systems which have D4-symmetry and conformal triality. Then we give the generic classification for singularities of the tangent surfaces to associated integral curves, which exhibits the triality. The classification is performed in terms of the classical theory on root systems combined with the singularity theory of mappings. The relations of D4-geometry with G2-geometry and B3-geometry are mentioned. The motivation of the tangent surface construction in D4-geometry is provided.
  • Goo Ishikawa
    Topics on Real and Complex Singularities, World Scientific 87 - 113 2014 [Refereed][Not invited]
     
    The algebraic notion of openings of a map-germ is introduced in this paper. An opening separates the self-intersections of the original map-germ, preserving its singularities. The notion of openings is different from the notion of unfoldings. Openings do not unfold the singularities. For example, the swallowtail is an opening of the Whitney cusp map-germ from plane to plane and the open swallowtail is a versal opening of them. Openings of map-germs appear as typical singularities in several problems of geometry and its applications. The notion of openings has close relations to isotropic map-germs in a symplectic space and integral map-germs in a contact space. We describe the openings of Morin singularities, namely, stable unfoldings of map-germs of corank one. The relation of unfoldings and openings are discussed. Moreover we provide a method to construct versal openings of map-germs and give versal openings of stable map-germs (R4, 0) → (R4, 0). Lastly the relation of lowerable vector fields and openings is discussed.
  • Tangent varieties and openings of map-germs
    Goo Ishikawa
    RIMS K\={o}ky\={u}roku Bessatsu B38 119 - 137 2013 [Refereed][Not invited]
  • Symplectic invariants of parametric singularities
    Goo Ishikawa, S. Janeczko
    Advances in Geometric Analysis, Advanced Lectures in Mathematics series 21 259 - 280 2012/07 [Refereed][Not invited]
  • Goo Ishikawa
    TOPOLOGY AND ITS APPLICATIONS 159 (2) 492 - 500 0166-8641 2012/02 [Refereed][Not invited]
     
    The generic singularities and bifurcations are classified for one-parameter families of curves with frames in the space forms En+1, Sn+1, Hn+1. Two kinds of frames are considered: adapted frames and osculating frames. In particular, we give the classification results on the singularities of envelopes associated to framed curves. The associated envelopes and their singularities are classified. characterised in term of geometric invariants of framed curves. We apply to the global problem of framed curves and to the extension problem of surfaces with boundaries in three space. generalising the results obtained in Ishikawa (2010) [12]. (C) 2011 Elsevier B.V. All rights reserved.
  • Classification problems on singularities of mappings and their applications
    Goo Ishikawa
    Sugaku 64 (1) 75 - 96 2012 [Refereed][Not invited]
  • Goo Ishikawa
    Journal of Singularities 6 54 - 83 1949-2006 2012 [Refereed][Not invited]
     
    It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable algebra. It is provided also the generic diffeomorphism classification of singularities on tangent varieties to contact-integral curves in the standard contact projective space. Moreover we give basic results on the classification of singularities of tangent varieties to generic surfaces and Legendre surfaces.
  • Ishikawa Goo
    RIMS Kokyuroku 京都大学 1764 149 - 164 1880-2818 2011/09 [Not refereed][Not invited]
  • Goo Ishikawa, Yoshinori Machida, Masatomo Takahashi
    Journal of Singularities 3 126 - 143 1949-2006 2011/01 [Refereed][Not invited]
     
    We give the generic classification on singularities of tangent surfaces to Legendre curves and to null curves by using the contact-cone duality between the contact 3-sphere and the Lagrange-Grassmannian with cone structure of a symplectic 4-space. As a consequence, we observe that the symmetry on the lists of such singularities is breaking for the contact-cone duality, compared with the ordinary projective duality.
  • Goo Ishikawa
    DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 28 (3) 341 - 354 0926-2245 2010/06 [Refereed][Not invited]
     
    We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreover we give formulae related to remote singularities of the boundary-envelope. (C) 2010 Elsevier B.V. All rights reserved.
  • Goo Ishikawa, Stanislaw Janeczko
    ANNALES POLONICI MATHEMATICI 99 (3) 263 - 284 0066-2216 2010 [Refereed][Not invited]
     
    Based on the discovery that the delta-invariant is the symplectic codimension of a parametric plane curve singularity, we classify the simple and uni-modal singularities of parametric plane curves under symplectic equivalence. A new symplectic deformation theory of curve singularities is established, and the corresponding cyclic symplectic moduli spaces are reconstructed as canonical ambient spaces for the diffeomorphism moduli spaces which are no longer Hausdorff spaces.
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 1610 38 - 50 1880-2818 2008/08 [Not refereed][Not invited]
  • A. A. Davydov, G. Ishikawa, S. Izumiya, W. -Z. Sun
    JAPANESE JOURNAL OF MATHEMATICS 3 (1) 93 - 119 0289-2316 2008 [Refereed][Not invited]
     
    For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.
  • Goo Ishikawa
    Contemporary Mathematics 459 51 - 72 2008 [Refereed][Not invited]
     
    We consider the global symplectic classification problem of plane curves.
    First we give the exact classification result under symplectomorphisms, for the
    case of generic plane curves, namely immersions with transverse
    self-intersections. Then the set of symplectic classes form the symplectic
    moduli space which we completely describe by its global topological term. For
    the general plane curves with singularities, the difference between
    symplectomorphism and diffeomorphism classifications is clearly described by
    local symplectic moduli spaces of singularities and a global topological term.
    We...
  • Bifurcations in symplectic space
    Goo Ishikawa, S. Janeczko
    Banach Center Publ. 82 111 - 124 2008 [Refereed][Not invited]
  • Jiro Adachi, Go-o Ishikawa
    NONLINEARITY 20 (8) 1907 - 1925 0951-7715 2007/08 [Refereed][Not invited]
     
    Motivated by the importance and universal character of phase singularities which were clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by Whitney and Thom. The classification of phase singularities is reduced to the classification of planar curves by radial transformations due to the theory of du Plessis, Gaffney and Wilson. Then fold singularities are classified into hyperbolic and elliptic singularities. We show that the elliptic singularities are never realized by any Helmholtz waves, while the hyperbolic singularities are realized in fact. Moreover, the classification and realizability of Whitney's cusp, as well as its bifurcation problem, are considered in order to explain the three point bifurcation of phase singularities. In this paper, we treat the dislocation of linear waves mainly, developing the basic and universal method, the method of jets and transversality, which is applicable also to nonlinear waves.
  • ISHIKAWA Go-o
    RIMS Kokyuroku 京都大学 1540 79 - 90 1880-2818 2007/04 [Not refereed][Not invited]
  • Go-o Ishikawa
    Real and Complex Singularities 56 - 84 2007 [Refereed][Not invited]
     
    In this survey article we show several recent results on the local classification of irreducible plane curve singularities, plane branches, based on Oscar Zariski's lectures [17], and that of Legendre curve singularities. We compare exact classifications, the classification of plane branches and the classifications of Legendre curves, in the cases of simple and uni-modal singularities, and, as a consequence, we observe remarkable difference and mysterious similarity of these classification results. Also the comparison is achieved for a bi-modal case of characteristic (6, 7) which Zariski treated.
  • ISHIKAWA Goo, MACHIDA Yoshinori
    RIMS Kokyuroku 京都大学 1502 41 - 53 1880-2818 2006/07 [Not refereed][Not invited]
  • GO Ishikawa, Y Machida
    INTERNATIONAL JOURNAL OF MATHEMATICS 17 (3) 269 - 293 0129-167X 2006/03 [Refereed][Not invited]
     
    We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvature and for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.
  • G. Ishikawa
    Asian Journal of Mathematics 9 (1) 133 - 166 2005/03 [Refereed][Not invited]
     
    We give the characterization of Arnol'd-Mather type for stable singular
    Legendre immersions. The most important building block of the theory is
    providing a module structure on the space of infinitesimal integral
    deformations by means of the notion of natural liftings of differential systems
    and of contact Hamiltonian vector fields.
  • GO Ishikawa
    JOURNAL OF GEOMETRY AND PHYSICS 52 (2) 113 - 126 0393-0440 2004/10 [Refereed][Not invited]
     
    We give basic results for classifying singular Legendre curves in the contact 3-space from the viewpoint of singularity theory of differentiable mappings. In particular, we introduce the notion of codimension, discuss the finite determinacy, and give an alternative proof to M. Zhitomirskii's theorem stating that locally diffeomorphic singular Legendre curves are locally contactomorphic, in the complex analytic category. (C) 2004 Elsevier B.V. All rights reserved.
  • Ishikawa Go-o
    RIMS Kokyuroku 京都大学 1374 1 - 14 1880-2818 2004/05 [Not refereed][Not invited]
  • Perturbations of Caustics and Fronts
    Goo Ishikawa
    Banach Center Publications 62 101 - 116 2004 [Refereed][Not invited]
  • Symplectic singularities of isotropic mappings
    Goo Ishikawa, S. Janeczko
    Banach Center Publications 65 85 - 106 2004 [Refereed][Not invited]
  • Ishikawa Go-o
    RIMS Kokyuroku 京都大学 1328 126 - 143 1880-2818 2003/06 [Not refereed][Not invited]
  • G Ishikawa, S Janeczko
    QUARTERLY JOURNAL OF MATHEMATICS 54 73 - 102 0033-5606 2003/03 [Refereed][Not invited]
     
    We study the classification of varieties in the Marsden-Weinstein reduction and their liftability. In particular the complete symplectic classification of the Bruce-Gaffney plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings.
  • Lagrange mappings of the first open Whitney umbrella
    IA Bogaevski, G Ishikawa
    PACIFIC JOURNAL OF MATHEMATICS 203 (1) 115 - 138 0030-8730 2002/03 [Refereed][Not invited]
     
    In this paper we give a classification of simple stable singularities of Lagrange projections of the first open Whitney umbrella, the simplest singularity of Lagrange varieties. Our classification extends the ADE-classification, due to Arnold, of simple stable singularities of Lagrange projections of smooth Lagrange submanifolds. We also prove a criterion of equivalence of stable Lagrange projections of open Whitney umbrellas which is analogous to Mather's fundamental theorem on stable map-germs.
  • Submanifolds with degenerate Gauss mappings in spheres
    Goo Ishikawa, M. Kimura, R. Miyaoka
    Advanced Study in Pure Mathematics 37 115 - 149 2002 [Refereed][Not invited]
  • Go-O Ishikawa, Tohru Morimoto
    Differential Geometry and its Application 14 (2) 113 - 124 0926-2245 2001/03 [Refereed][Not invited]
     
    In this paper we examine the singularities of solution surfaces of Monge-Ampère equations and study their global and local effects on the solutions for certain kinds of equations in the framework of contact geometry. In particular, as a byproduct, we give a simple proof to the classical Hartman-Nirenberg's theorem by using the notion of projective duality and provide a new example of compact developable hypersurfaces in the real projective space RP4 © 2001 Elsevier Science B.V.
  • Topological classification of the tangent developables of space curves
    G Ishikawa
    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 62 (2) 583 - 598 0024-6107 2000/10 [Refereed][Not invited]
     
    The tangent developables of space curves are classified locally and topologically. It turns out that only seven different tangent developables arise, except for very degenerate cases. An extension of the known classification due to Cleave, Shcherbak and Mend is given.
  • Ishikawa Go-o
    RIMS Kokyuroku 京都大学 1122 35 - 48 1880-2818 2000/01 [Not refereed][Not invited]
  • Ishikawa Go-o
    RIMS Kokyuroku 京都大学 1069 92 - 104 1880-2818 1998/11 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 952 131 - 146 1880-2818 1996/05 [Not refereed][Not invited]
  • Ishikawa G, Ohmoto T
    RIMS Kokyuroku 京都大学 926 144 - 148 1880-2818 1995/11 [Not refereed][Not invited]
  • 石川 剛郎
    数理解析研究所講究録 京都大学 926 59 - 72 1880-2818 1995/11 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 845 53 - 63 1880-2818 1993/06 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 815 78 - 88 1880-2818 1992/12 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 690 9 - 36 1880-2818 1989/05 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 619 1 - 11 1880-2818 1987/04 [Not refereed][Not invited]
  • ISHIKAWA Goo
    RIMS Kokyuroku 京都大学 595 39 - 58 1880-2818 1986/07 [Not refereed][Not invited]
  • Ishikawa Goo
    RIMS Kokyuroku 京都大学 493 101 - 133 1880-2818 1983/05 [Not refereed][Not invited]
  • ISHIKAWA GOO
    RIMS Kokyuroku 京都大学 403 88 - 95 1880-2818 1980/11 [Not refereed][Not invited]

Books etc

  • 論理・集合・数学語
    石川剛郎 (Single work)
    共立出版 2015/12
  • Singularities of Curves and Surfaces in Various Geometric Problems
    Goo Ishikawa (Single work)
    CAS Lecture Notes 10, Exact Sciences 2015
  • よろず数学問答
    石川剛郎 (Single work)
    日本評論社 2008
  • 愛ではじまる微積分
    石川剛郎 
    プレアデス出版 2008
  • 応用特異点論
    石川剛郎 (Joint work)
    1998
  • 代数曲線と特異点
    石川剛郎 (Joint work)

MISC

Educational Activities

Teaching Experience

  • Geometry
    開講年度 : 2018
    課程区分 : 修士課程
    開講学部 : 理学院
    キーワード : 多様体,非退化臨界点,リーマン幾何,測地線,リー群,対称空間,ホモトピー群,ボット周期定理
  • Advanced Geometry
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 多様体,非退化臨界点,リーマン幾何,測地線,リー群,対称空間,ホモトピー群,ボット周期定理
  • Linear Algebra I
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 行列, 連立1次方程式, 基本変形, 階数, 行列式, 逆行列
  • Linear Algebra II
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : ベクトル空間, 線形写像, 線形独立, 基底, 固有値, 固有ベクトル, 対角化
  • Exercises on Basic Mathematics A
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 基礎数学Aに準ずる.
  • The World of Science and Technology
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 回転数,メビウスの帯; フーリエ級数; 数列,力学系;3,4次方程式の解の公式、代数学の基本定理


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