Braid groups, iterated integrals and geometric structures of configuration spaces

• Project/Area Number: 20340010
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: KOHNO Toshitake The University of Tokyo, 大学院・数理科学研究科, 教授 (80144111)
• Co-Investigator(Kenkyū-buntansha): 寺杣 友秀 東京大学, 大学院・数理科学研究科, 教授 (50192654) ; 森田 茂之 東京大学, 大学院・数理科学研究科, 教授 (70011674)
• Co-Investigator(Renkei-kenkyūsha): 寺尾 宏明 北海道大学, 大学院・理科学研究院, 教授 (90119058) ; 斎藤 恭司 東京大学, 数物連携宇宙研究機構, 主任研究員 (20012445)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥18,070,000 (Direct Cost: ¥13,900,000、Indirect Cost: ¥4,170,000); Fiscal Year 2010: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2009: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2008: ¥6,630,000 (Direct Cost: ¥5,100,000、Indirect Cost: ¥1,530,000)
• Keywords: 位相幾何 / 組みひも群 / 写像類群 / 反復積分 / 配置空間 / ループ空間 / 共形場理論 / モース理論 / 共同形場理論 / ドーラム理論 / 有限不変量

Research Abstract

We determined the structure of the image of the monodromy representations of mapping class groups in conformal field theory. We proved that the Novikov cohomology for the complement of a hyperplane arrangement vanishes except in the middle dimension for a large class of abelian representations of the fundamental group.

Research Products

• [Journal Article] KZ equation - structure of monodromy representations and their applications to invariants of knots (2011)
  • Author(s): T.Kohno
  • Journal Title: Theory of Hypergeometric Functions Pages: 283-301
  • Peer Reviewed
• [Journal Article] Bar complex, configuration spaces and finite type invariants for braids (2010)
  • Author(s): T.Kohno
  • Journal Title: Topology and Its Applications 157 Pages: 2-9
• [Journal Article] Limit Elements in the Configuration Algebra for a Cancellative Monoid (2010)
  • Author(s): K.Saito
  • Journal Title: Publ.RIMS Kyoto Univ. 46 Pages: 37-113
• [Journal Article] Algebraic correspondences between genus three curves and certain Calabi-Yau varieties (2010)
  • Author(s): T.Terasoma
  • Journal Title: Amer.J.Math. no.1 Pages: 181-200
• [Journal Article] A primitive derivation and logarithmic differential forms of Coxeter arrangements. (2010)
  • Author(s): A.Abe, H.Terao
  • Journal Title: Math.Z. 264 Pages: 813-828
  • NAID: 120002834671
• [Journal Article] Bar complex, configuration spaces and finite type invariants for braids (2010)
  • Author(s): T.Kohno
  • Journal Title: Topology and Its Applications Volume: 157 Pages: 2-9
  • Peer Reviewed
• [Journal Article] Algebraic correspondences between genus three curves and certain Calabi-Yau varieties (2010)
  • Author(s): T.Terasoma
  • Journal Title: American Journal of Matematics Volume: 132 Pages: 181-200
  • Peer Reviewed
• [Journal Article] Orbit configuration spaces associated to discrete subgroups of PSL (2, R) (2009)
  • Author(s): F.R.Cohen, T.Kohno, M.A.Xicontencatl
  • Journal Title: Journal of Pure and Applied Algebra 213 Pages: 2289-2300
• [Journal Article] Totally free arrangements (2009)
  • Author(s): T.Abe, H.Terao, M.Yoshinaga
  • Journal Title: Proc.Amer.Math.Soc. 137 Pages: 1405-1410
• [Journal Article] On orbit configuration spaces associated to groups acting on the upper half plane (2009)
  • Author(s): F.R.Cohen, T.Kohno, M.A.Xicontencatl
  • Journal Title: Journal of Pure and Applied Algebra 213 Pages: 2289-2300
  • Peer Reviewed
• [Journal Article] Orbit configuration spaces associated to discrete subgroups of PSL(2, R) (2009)
  • Author(s): F.R.Cohen, T.Kohno and M,A.Xicontencatl
  • Journal Title: Journal of Pure and Applied Algebra
  • Peer Reviewed
• [Journal Article] Bar complex, configuration spaces and finite type invariants for braids (2009)
  • Author(s): T.Kohno
  • Journal Title: Topology and Its Applications
  • Peer Reviewed
• [Presentation] Braid groups, configuration spaces and iterated integrals (2011)
  • Author(s): 河野俊丈
  • Organizer: CIMPA-Vietnam School and Workshop on Braids in Algebra, Geometry and Topology
  • Place of Presentation: Institute of Mathematics, Hanoi, Vietnam
  • Year and Date: 2011-01-18
• [Presentation] Quantum representations of mapping class groups (2010)
  • Author(s): T.Kohno
  • Organizer: International Conference Japan-Mexico on Topology and its Applications
  • Place of Presentation: Colima University, Mexico
  • Year and Date: 2010-09-28
• [Presentation] Quantum representations of mapping class groups (2010)
  • Author(s): 河野俊丈
  • Organizer: International Conference Japan-Mexico on Topology and its Applications
  • Place of Presentation: ColimaUniversity, Mexico
  • Year and Date: 2010-09-28
• [Presentation] Gauss-Manin connections for the space of conformal blocks (2010)
  • Author(s): 河野俊丈
  • Organizer: 研究集会「多重ゼータ値の諸相」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-09-07
• [Presentation] Quantum representations of mapping class groups and their images (2010)
  • Author(s): T.Kohno
  • Organizer: Workshop on "Algebra and Geometry of Configuration Spaces and Related Structures"
  • Place of Presentation: Centro de Ricerca Matematica Ennio De Giorgi, Pisa, Italy
  • Year and Date: 2010-06-24
• [Presentation] Quantum representations of mapping class groups and their images (2010)
  • Author(s): T.Kohno
  • Organizer: Algebra and Geometry of Configuration Spaces and Related Structures
  • Place of Presentation: Centro de Ricerca Matematica Ennio De Giorgi, Pisa, Italy
  • Year and Date: 2010-06-24
• [Presentation] Highest weight representation of Elliptic Lie Algebras (2010)
  • Author(s): K.Saito
  • Organizer: 4^<th> International Conference on Representation Thoery
  • Place of Presentation: Xian, China.
• [Presentation] Limit partition functions associated with Cancellative monoids (2010)
  • Author(s): K.Saito
  • Organizer: Kinosaki symposium of Algebraic geometry
  • Place of Presentation: Kinosaki, Japan
• [Presentation] The 2nd MSJ-SI "Arrangements of Hyperplanes", survey talk (2009)
  • Author(s): 河野俊丈
  • Organizer: The 2nd MSJ-SI "Arrangements of Hyperplanes"
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-05
• [Presentation] Local systems on configuration spaces and the space of conformal blocks (2009)
  • Author(s): 河野俊丈
  • Organizer: Conference Solstice
  • Place of Presentation: パリ第七大学パリ,フランス
  • Year and Date: 2009-06-17
• [Presentation] Homology of local systems on configuration spaces and conformal field theory (2009)
  • Author(s): T.Kohno
  • Organizer: Beijing Program on Algebraic Topology
  • Place of Presentation: Chinese Academy of Sciences, Beijing(中国・北京)
  • Year and Date: 2009-05-27
• [Presentation] Bar comPlex, configuration spaces and finite type invariants for braids (2009)
  • Author(s): T.Kohno
  • Organizer: East Asian School of Knots and Related Topics
  • Place of Presentation: 韓国,慶州文化会館
  • Year and Date: 2009-01-12
• [Presentation] Topology of the complements of hyperplane arrangements, local system homology and iterated integrals (2009)
  • Author(s): T.Kohno
  • Organizer: The 2nd MSJ-SI "Arrangements of Hyperplanes"
  • Place of Presentation: survey talk,Hokkaido University
• [Presentation] Bar Complex of Orlik-Solomon algebra and rational universal holonomy maps (2008)
  • Author(s): T.Kohno
  • Organizer: Conference in honour of Peter Orlik
  • Place of Presentation: Fields Institute, Toronto, Canada
  • Year and Date: 2008-08-21
• [Presentation] Braids, Drinfel'd associator and rational homotopy (2008)
  • Author(s): T.Kohno
  • Organizer: Braids, Knots and Applications
  • Place of Presentation: Univ.de Montpellier, France
  • Year and Date: 2008-06-10
• [Book] 反復積分の幾何学 (2009)
  • Author(s): 河野俊丈
  • Total Pages: 295
  • Publisher: シュプリンガー・ジャパン
• [Book] 新版組みひもの数理 (2009)
  • Author(s): 河野俊丈
  • Total Pages: 206
  • Publisher: 遊星社
• [Book] 場の理論とトポロジー (2008)
  • Author(s): 河野俊丈
  • Total Pages: 144
  • Publisher: 岩波書店
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/'kohno/