Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Profile and Settings

• Name (Japanese)

  Kobayashi

• Name (Kana)

  Shimpei

• Name

  200901083779087841

Alternate Names

https://sites.google.com/site/kobayashishimpeisite/

Achievement

Research Interests

• 幾何学   微分幾何   ループ群   可積分系   線型常微分方程式   平均曲率一定曲面   可視化   コンピュータ実験   Discretization   

Research Areas

• Natural sciences / Geometry / Differential Geometry

Research Experience

• 2024/04 - Today Hokkaido University Faculty of Science Department of Mathematics Professor
• 2013/09 - 2024/03 Hokkaido University Faculty of Science Department of Mathematics Associate Professor
• 2011/04 - 2013/08 Hirosaki University Graduate School of Science and Technology
• 2008/04 - 2011/03 Hirosaki University Graduate School of Science and Technology
• 2005/04 - 2008/03 Tokyo Denki University School of Information Environment

Published Papers

• Half-dimensional immersions into the para-complex projective space and Ruh-Vilms type theorems

  Josef F. Dorfmeister, Roland Hildebrand, Shimpei Kobayashi

  2024/05/20 

  In this paper we study isometric immersions $f:M^n \to {\mathbb {C}^{\prime } }\!P^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb {C}^{\prime } }\!P^n$. We study the immersion $f$ by means of a lift $\mathfrak f$ of $f$ into a quadric hypersurface in ${S^{2n+1}_{n+1 } }$. We find the frame equations and compatibility conditions. We specialize these results to dimension $n = 2$ and a definite metric on $M^2$ in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.
• The complex landslide flow and the method of integrable systems

  Shimpei Kobayashi

  2024/04/22 

  We investigate a connection between the complex landslide flow, defined on a pair of Teichm\"uller spaces, and the integrable system method's approach to harmonic maps into a symmetric space. We will prove that the holonomy of the complex landslide flow can be derived from the holonomy of the family of flat connections determined by a harmonic map into the hyperbolic two-space.
• A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space

  Junichi Inoguchi, Shimpei Kobayashi

  2024/04/12 

  Weakly complete constant Gaussian curvature $-1-1$ but $K \neq 0$ by using harmonicities of Lagrangian and Legendrian Gauss maps. Then we will show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into $\mathbb H^2$ for $-1< K<0$ or $\mathbb S^2$ for $K>0$, respectively.
• Geodesics of multivariate normal distributions and a Toda lattice type Lax pair

  Shimpei Kobayashi

  Physica Scripta 98 (11) 115241 - 115241 0031-8949 2023/10/17 [Refereed]
   

  Abstract We study geodesics of multivariate normal distributions with respect to the Fisher metric. First it will be shown that a computational formula for geodesics can be understood using the block Cholesky decomposition and a natural Riemannian submersion. Next a mid point algorithm for geodesics will be obtained. And finally a new Toda lattice type Lax pair will be derived from the geodesic and the block Cholesky decomposition.
• Minimal cylinders in the three-dimensional Heisenberg group

  Shimpei Kobayashi

  Mathematische Annalen 388 (3) 3299 - 3317 0025-5831 2023/03/24 [Refereed]
  
• Discrete constant mean curvature surfaces on general graphs

  Tim Hoffmann, Shimpei Kobayashi, Zi Ye

  Geometriae Dedicata 216 (6) 0046-5755 2022/12 [Refereed][Not invited]
   

  Abstract The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.
• Minimal surfaces with non-trivial geometry in the three-dimensional Heisenberg group

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Complex Manifolds 9 (1) 285 - 336 2022/11/15 [Refereed]
   

  Abstract We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil₃ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil₃ with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso_◦(Nil₃) of Nil₃.
• Timelike Minimal Surfaces in the Three-Dimensional Heisenberg Group

  Hirotaka Kiyohara, Shimpei Kobayashi

  The Journal of Geometric Analysis 32 (8) 1050-6926 2022/08 [Refereed]
  
• On a constant curvature statistical manifold

  Shimpei Kobayashi, Yu Ohno

  Information Geometry 2511-2481 2022/02/15 [Refereed]
  
• A characterization of the alpha-connections on the statistical manifold of normal distributions

  Hitoshi Furuhata, Jun-ichi Inoguchi, Shimpei Kobayashi

  Information Geometry 4 (1) 177 - 188 2511-2481 2021/07 [Refereed]
  
• Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

  Josef F. Dorfmeister, Shimpei Kobayashi, Hui Ma

  Mathematische Zeitschrift 296 (3-4) 1751 - 1775 0025-5874 2020/12 [Refereed][Not invited]
  
• The Gauss maps of Demoulin surfaces with conformal coordinates

  Jun-ichi Inoguchi, Shimpei Kobayashi

  Science China Mathematics 64 (7) 1479 - 1492 1674-7283 2020/09/21 [Refereed][Not invited]
  
• Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

  Josef F. Dorfmeister, Shimpei Kobayashi

  Annali di Matematica Pura ed Applicata (1923 -) 200 (2) 521 - 546 0373-3114 2020/06/05 [Refereed][Not invited]
  
• Representation formula for discrete indefinite affine spheres

  Shimpei Kobayashi, Nozomu Matsuura

  Differential Geometry and its Applications 69 101592 - 101592 0926-2245 2020/04 [Refereed][Not invited]
  
• Survey on real forms of the complex A2(2)-Toda equation and surface theory

  Josef F. Dorfmeister, Walter Freyn, Shimpei Kobayashi, Erxiao Wang

  Complex Manifolds 6 (1) 194 - 227 2019/01/01 [Refereed][Not invited]
   

  The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8]. In this survey we will show that to each of the five different types of real forms for a loop group of A₂⁽²⁾ there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ², minimal Lagrangian surfaces in ℂℍ², timelike minimal Lagrangian surfaces in ℂℍ₁², proper definite affine spheres in ℝ³ and proper indefinite affine spheres in ℝ³, respectively.
• Nonlinear d'Alembert formula for discrete pseudospherical surfaces

  Shimpei Kobayashi

  JOURNAL OF GEOMETRY AND PHYSICS 119 208 - 223 0393-0440 2017/09 [Refereed][Not invited]
   

  On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail. (C) 2017 Elsevier B.V. All rights reserved.
• A loop group method for affine harmonic maps into Lie groups

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Advances in Mathematics 298 207 - 253 0001-8708 2016/08 [Refereed]
  
• A construction method for discrete constant negative Gaussian curvature surfaces

  Shimpei Kobayashi

  Mathematical Progress in Expressive Image Synthesis III, Extended and Selected Results from the Symposium MEIS2015 21 - 33 2016/04 [Not refereed][Not invited]
   

  This article is an application of the author's paper about a construction
  method for discrete constant negative Gaussian curvature surfaces, the
  nonlinear d'Alembert formula. The heart of this formula is the Birkhoff
  decomposition, and we give a simple algorithm for the Birkhoff decomposition.
  As an application, we draw figures of discrete constant negative Gaussian
  curvature surfaces given by this method.
• On the Bernstein Problem in the Three-dimensional Heisenberg Group

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Canadian Mathematical Bulletin 59 (01) 50 - 61 0008-4395 2016/03 [Refereed]
   

  Abstract In this note we present a simple alternative proof for the Bernstein problem in the threedimensional Heisenberg group Nil₃ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch- Rosenberg diòerential.
• A loop group method for minimal surfaces in the three-dimensional Heisenberg group

  Josef F. Dorfmeister, Jun-Ichi Inoguchi, Shimpei Kobayashi

  Asian Journal of Mathematics 20 (3) 409 - 448 1093-6106 2016 [Refereed][Not invited]
  
• A loop group method for Demoulin surfaces in the 3-dimensional real projective space

  Shimpei Kobayashi

  Differential Geometry and its Applications 40 57 - 66 0926-2245 2015/06 [Refereed]
  
• Constant Gaussian curvature surfaces in the 3-sphere via loop groups

  David Brander, Jun-ichi Inoguchi, Shimpei Kobayashi

  Pacific Journal of Mathematics 269 (2) 281 - 303 0030-8730 2014/07/26 [Refereed]
  
• Constant mean curvature surfaces in hyperbolic 3-space via loop groups

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Journal für die reine und angewandte Mathematik (Crelles Journal) 2014 (686) 1 - 36 0075-4102 2014/01/01 [Refereed]
  
• Discretization of integrable systems via dressing actions.

  KOBAYASHI Shimpei

  RIMS Kôkyûroku Bessatsu 京都大学 B41 161 - 171 1881-6193 2013 [Refereed][Not invited]
  
• Real forms of complex surfaces of constant mean curvature

  Shimpei Kobayashi

  Transactions of the American Mathematical Society 363 (04) 1765 - 1765 0002-9947 2011/04/01 [Refereed][Not invited]
  
• TOTALLY SYMMETRIC SURFACES OF CONSTANT MEAN CURVATURE IN HYPERBOLIC 3-SPACE

  SHIMPEI KOBAYASHI

  Bulletin of the Australian Mathematical Society 82 (2) 240 - 253 0004-9727 2010/06/18 

  Abstract We detail a construction of totally symmetric surfaces of constant mean curvature 0≤H<1 in hyperbolic 3-space of sectional curvature −1 via the generalized Weierstrass type representation.
• Complex surfaces of constant mean curvature fibered by minimal surfaces

  Josef Dorfmeister, Shimpei Kobayashi, Franz Pedit

  Hokkaido Mathematical Journal 39 (1) 1 - 55 0385-4035 2010/02 [Refereed][Not invited]
   

  We define complex constant mean curvature immersions in complex three space using a natural extension of constant mean curvature immersions in Euclidean three space via loop group techniques. We then discuss the fundamental properties of these complex constant mean curvature immersions. In particular, we prove that these immersions are doubly ruled by holomorphic null curves. We present a construction of minimal immersions from constant mean curvature immersions in Euclidean three space via the associated complex constant mean curvature immersions.
• Note on equivariant harmonic maps in complex projective spaces

  Shimpei Kobayashi

  Annals of Global Analysis and Geometry 36 (4) 375 - 380 0232-704X 2009/05/10 [Refereed]
  
• Asymptotics of ends of constant mean curvature surfaces with bubbletons

  Shimpei Kobayashi

  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136 (4) 1433 - 1443 0002-9939 2008 [Refereed][Not invited]
   

  A constant mean curvature surface with bubbletons is defined by the loop group action on the set of extended framings for constant mean curvature surfaces by simple factors. Classically such surfaces were obtained by the transformation of tangential line congruences, the so- called Bianchi-Backlund transformations. In this paper, we consider constant mean curvature surfaces with Delaunay ends in three- dimensional space forms R-3, S-3 and H-3 and their surfaces with bubbletons for which the topology is preserved. We show that the ends of such surfaces are again asymptotic to Delaunay surfaces.
• Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

  N. Schmitt, M. Kilian, S.-P. Kobayashi, W. Rossman

  JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 75 563 - 581 0024-6107 2007/06 [Refereed][Not invited]
   

  A theorem on the unitarizability of loop group valued monodromy representations is presented and applied to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply connected 3-dimensional space forms R-3, S-3 and H-3. Additionally, the extended frame for any associated family of Delaunay surfaces is computed.
• Coarse classification of constant mean curvature cylinders

  J. Dorfmeister, S.-P. Kobayashi

  Transactions of the American Mathematical Society 359 (6) 2483 - 2500 0002-9947 2007/01/04 

  We give a coarse classification of constant mean curvature (CMC) immersions of cylinders into via the loop group method. Particularly for this purpose, we consider double loop groups and a new type of “potentials” which are meromorphic 1-forms on Riemann surfaces.

• CONSTANT MEAN CURVATURE SURFACES OF ANY POSITIVE GENUS

  M. KILIAN, S.-P. KOBAYASHI, W. ROSSMAN, N. SCHMITT

  Journal of the London Mathematical Society 72 (01) 258 - 272 0024-6107 2005/07/20
• CHARACTERIZATIONS OF BIANCHI–BÄCKLUND TRANSFORMATIONS OF CONSTANT MEAN CURVATURE SURFACES

  SHIMPEI KOBAYASHI, JUN-ICHI INOGUCHI

  International Journal of Mathematics 16 (02) 101 - 110 0129-167X 2005/02 [Refereed]
   

  We show that Bianchi–Bäcklund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.
• Loop Group Methods for Constant Mean Curvature Surfaces

  Shimpei KOBAYASHI

  Rokko Lectures in Mathematics 2005
• Bubbletons in 3-dimensional space forms

  KOBAYASHI Shimpei

  Balkan Journal of Geometry and its Applications 9 (1) 44 - 68 2004 [Refereed][Not invited]
  

Books etc

• 曲面とベクトル解析

  小林, 真平 
  日本評論社 2016/12 (ISBN: 9784535806375) vi, 216p
• 部分多様体の微分幾何学の深化

  京都大学数理解析研究所, 小林, 真平 
  京都大学数理解析研究所 2014/04 ii, 189p
• 曲面の微分幾何学とソリトン方程式 : 可積分幾何入門

  井ノ口, 順一, 小林, 真平, 松浦, 望, 立教大学 
  [立教大学] 2005/02 ii, 117, 28, 56p
• Loop group methods for constant mean curvature surfaces

  Fujimori, Shoichi, 小林, 真平, Rossman, Wayne 
  Department of Mahtematics, Faculty of Science, Kobe University 2005 (ISBN: 4907719175) v, 118 p.

Research Projects

• Comprehensive study of various geometries with harmonic maps to symmetric spaces as a core

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

  Date (from‐to) : 2022/04 -2027/03 

  Author : 小林 真平
• Fusion of discrete and smooth integrable geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2018/04 -2022/03 

  Author : 小林 真平

   

  本年度は，まず離散平均曲率一定曲面の一般化について，ワイエルシュトラス型の表現公式を用いて研究した（ミュンヘン工科大学のHoffmann氏とYe氏との共同研究）．ワイエルシュトラス型の表現公式に付随する複比のシステムをaddtive-rational戸田系と対応づけること及び，ループ群の分解定理を用いることにより，自然に離散平均曲率一定曲面の離散化が得られる．現在，研究結果を纏めている．
  また，A_2^(2)型のアフィン・カッツ-ムーディリー代数の実形の分類を用いて，新しい可積分曲面の類を見つけた（ミュンヘン工科大学のDorfmeister氏との共同研究）．これは，これまでに見つかっていなかった可積分曲面の類であり，アフィン・カッツ-ムーディリー代数の実形の分類が，可積分曲面の研究に非常に重要であることの証左である．その研究結果を現在纏めている．また，これに関連して，A_2^(2)型のアフィン・カッツ-ムーディリー代数の実形（5つ存在する）と可積分曲面の完全な対応に関してのサーベイ論文を執筆した（Dorfmeister氏，Freyn氏，Wang氏との共著）．
  さらに，３次元ハイゼンベルグ群の極小曲面の大域的な性質について研究した（Dorfmeister氏と筑波大学の井ノ口氏との共同研究）．極小曲面がいつ非自明な位相を持つかの特徴づけなどを得ることができ，極小回転面の構成を具体的に与えた．また，今後の研究の基礎となる部分も一緒に纏め現在投稿中である．
• A Weierstrass type representation for surfaces via loop group method and its applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2014/04 -2018/03 

  Author : Kobayashi Shimpei

   

  Surfaces whose structure equation can be given by an integrable system are often called integrable surfaces. Here the integrable systems is a generic term used to refer to solvable (partial) differential equations. In particular many integrable surfaces have a Weierstrass type representation in terms of loop groups and holomorphic functions. In this research we studied integrable surfaces by using the Weierstrass type representation. Concretely, we studied affine harmonic maps, constant Gaussian curvature surfaces in 3-dimensional hyperbolic space, discrete affine spheres, affine plane curves and maximal surfaces in 3-dimensional Anti-de Sitter space.
• Systematic development and application of methods in differential geometry and integrable systems motivated by quantum cohomology

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

  Date (from‐to) : 2013/10 -2018/03 

  Author : Guest Martin, OHNITA Yoshihiro, MAEDA Yoshiaki, SERGEI V Ketov, SAKAI Takashi, OTOFUJI Takashi, AKAHO Manabu, KOBAYASHI Shimpei, IRITANI Hiroshi, HOSONO Shinobu

   

  A series of methods for solving the tt*-Toda equations were developed during the course of this project. These methods used p.d.e. theory, integrable systems theory, and Lie theory. Our main results were achieved for the tt*-Toda equations of type A_n. Here we give a complete treatment of the solutions and their asymptotic data and monodromy data. A more abstract approach was used in the case n=1, in order to describe the moduli space of solutions. These results were motivated in part by the special solutions corresponding to quantum cohomology rings of Kaehler manifolds. In order to promote research in this area, a number of conferences, workshops, and seminars by specialists were organised.
• Exploitation of new relations between differential geometry and quantum cohomology in the context of integrable systems

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

  Date (from‐to) : 2009 -2012 

  Author : GUEST Martin, KAMISHIMA Yoshinobu, TOKUNAGA Hiroo, MAEDA Yoshiaki, MIYAOKA Reiko, KOHNO Toshitake, OHNITA Yoshihiro, SAKAI Takashi, SERGEI V Ketov, AKAHO Manabu, OTOFUJI Takashi, KOBAYASHI Shinpei, KUROSU Sanae

   

  We have made progress with some key examples, which demonstrate interesting and nontrivial phenomena. In "Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa" (M. Guest and C.-S. Lin, J. reine angew. Math., 2012, in press) the existence of a family of smooth solutions of the tt*-Toda equation was established. This was a technical breakthrough: p.d.e. methods are well suited to the noncompact case, where standard loop group methods fail. In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (M. Guest, A. Its, and C.-S. Lin, arXiv:1209.2045), a second technical breakthrough was made, by relating the global smoothness of the solutions to the monodromy data (Stokes data) of an associated linear equation. This Stokes data was computed explicitly for all globally smooth solutions of the tt*-Toda equation. We expect that these techniques will be applicable to other problems in differential geometry.
• Construction of surfaces via complexifications of loop groups

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

  Date (from‐to) : 2011 -2011 

  Author : KOBAYASHI Shimpei

   

  When the structure equations (nonlinear partial differential equations) of a surface is an integrable system, the surface is called "integrable surface". In the research, we gave constructions and characterizations of integrable surfaces. In particular, using loop group structures of integrable surfaces, we gave a construction of minimal surfaces in the three-dimensional Heisenberg group, constant Gaussian curvature surfaces in the three-sphere and a characterization of Demoulin surfaces in the three-dimensional real projective space. Moreover, we gave a new method obtaining the discrete mKdV equation using a loop group action.
• Asymptotic behavior of surfaces with special properties via transformation theory and conserved quantities and integrable systems techniques

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

  Date (from‐to) : 2008 -2011 

  Author : W.F Rossma, NORO Masayuki, KOIKE Tatsuya

   

  The purpose of this research was to expand our understanding of discretizations in surface theory, in a way that would preserve the geometric mathematical structure (such as the notion of isothermicity, and various transformations like the Christoffel and Calapso and Darboux and Baecklund transformations) that exists for smooth surfaces.
• A construction of surfaces in spaces of constant curvature via integrable system method

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

  Date (from‐to) : 2008 -2010 

  Author : KOBAYASHI Sinpei

   

  We defined complex constant mean curvature surfaces by a natural generalization of constant mean curvature surfaces in Euclidean three space and classified real form surfaces, such as constant mean curvature or constant Gauss curvature surfaces in spaces of constant curvature, for a complex constant mean curvature surface. We also characterized equivariant harmonic maps in complex projective spaces via potentials, which are matrix valued 1-forms. Moreover, a construction method of equivariant harmonic maps in complex projective spaces has been obtained from the potentials.
• Differential geometric researches on surfaces in a space of constant curvature and their singularities

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

  Date (from‐to) : 2006 -2009 

  Author : KOKUBU Masatoshi, IRIE Hiroshi, KOBAYASHI Shimpei, ROSSMAN Wayne

   

  We studied surfaces in a three-dimensional manifold of constant negative curvature, called the hyperbolic space, requiring them to have good properties from the differential-geometric viewpoint. (Note that the hyperbolic space has interesting features beyond our common sense, e.g., a single hyperbolic line has infinitely many parallels.) We clarified the asymptotic behavior of ends of flat surfaces admitting singularities. Concerning linear Weingarten surfaces, we had a global representation formula, criterion for the shape of singularities, the orientability and co-orientability, and so on.
• Research on quantum cohomology, Frobenius manifolds, and harmonic maps related to integrable systems

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

  Date (from‐to) : 2006 -2008 

  Author : MARTIN Guest, SERGEI V. Ketov

   

  この研究は可積分系(大きな群対称性を持つ微分方程式系)に関連した現代幾何学の諸問題に関わる研究である。これらの問題は(曲面論を含む)古典的な微分幾何学および量子論と弦理論の幾何学に端を発する。 ループ群や無限次元グラスマン多様体の理論をはじめ、無限次元の手法が用いられる。主要な結果の1つとして、D加群による量子コホモロジーの理論への新しいアプローチが挙げられる。このプロジェクトの大きな特徴は、この研究領域を発展させるために、この分野をリードする国内外の研究者達と共同で研究活動を行うことである。