Visualization in geometry and construction of 3-dimensional mathematical models

• Project/Area Number: 23654022
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: KOHNO Toshitake 東京大学, 数理(科)学研究科(研究院), 教授 (80144111)
• Co-Investigator(Renkei-kenkyūsha): TSUBOI Takashi 東京大学, 大学院・数理科学研究科, 教授 (40114566) ; AHARA Kazushi 明治大学, 理工学部, 准教授 (80247147)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 離散データ / 代数曲面 / 定曲率曲面 / ガウス曲率 / 平均曲率 / 三次曲面 / 極小曲面 / 複素解析関数 / 離散幾何学 / 極限集合 / 可積分系

Research Abstract

Geometric objects of present research are given in various forms such as algebraic equations, complex analytic functions and differential equations. Starting from discrete data of surfaces we investigated a method to compute optimal values of invariants in differential geometry such as Gauss curvature and mean curvature. In the process of constructing geometric models it turns out that it is efficient to include information about normal direction at each point and we established techniques to construct models based on such enhanced data. We succeeded in constructing models of 27 lines on a cubic surface and a surface of constant negative curvature made of aluminum with high accuracy. We obtained a satisfactory procedure to establish an interface for constructing models from discrete data.

Research Products

• [Journal Article] On Burau representations at roots of unity (2014)
  • Author(s): L. Funar, T. Kohno
  • Journal Title: Geometriae Dedicata Volume: Volume169, Issue1 Pages: 145-163
• [Journal Article] Novikov homology, jump loci and Massey products (2014)
  • Author(s): T. Kohno and A. Pajitnov
  • Journal Title: Cent. Eur. J. Math. Volume: 12 Issue: 9 Pages: 1285-1304
  • DOI: 10.2478/s11533-014-0413-2
  • Peer Reviewed
• [Journal Article] Free subgroups within the images of quantum representations (2014)
  • Author(s): L. Funar and T. Kohno
  • Journal Title: Forum Mathematicum Volume: 印刷中 Issue: 2 Pages: 337-355
  • DOI: 10.1515/form.2011.162
  • Peer Reviewed
• [Journal Article] On Burau representations at roots of unity (2014)
  • Author(s): L. Funar and T. Kohno
  • Journal Title: Geometriae Dedicata Volume: 169 Issue: 1 Pages: 145-163
  • DOI: 10.1007/s10711-013-9847-0
  • Peer Reviewed
• [Journal Article] Novikov homology, jump loci and Massey products (2014)
  • Author(s): T. Kohno and A. Pajitnov
  • Journal Title: Cent. Eur. J. Math. Volume: 12 Pages: 1285-1304
  • Peer Reviewed
• [Journal Article] Circle-valued Morse theory for complex hyperplane arrangements (2013)
  • Author(s): T. Kohno, A. Pajitnov
  • Journal Title: Forum Mathematicum Volume: (Published on line) Issue: 4 Pages: 2113-2128
  • DOI: 10.1515/forum-2013-0032
  • Peer Reviewed
• [Journal Article] Homeomorphism groups of commutator width one (2013)
  • Author(s): Takashi TSUBOI
  • Journal Title: Proceedings Amer. Math. Soc Volume: 141 Issue: 5 Pages: 1839-1847
  • DOI: 10.1090/s0002-9939-2012-11595-3
  • Peer Reviewed
• [Journal Article] On Burau representations at roots of unity (2013)
  • Author(s): Louis Funar and Toshitake Kohno
  • Journal Title: Geometriae Dedicata Volume: --
  • Peer Reviewed
• [Journal Article] Circle-valued Morse theory for complex hyperplane arrangements (2013)
  • Author(s): Toshitake Kohno and Andrei Pajitnov
  • Journal Title: Forum Mathematicum Volume: --
  • Peer Reviewed
• [Journal Article] Homological representations of braid groups and KZ connections (2012)
  • Author(s): T. Kohno
  • Journal Title: Journal of Singularities Volume: 5 Pages: 94-108
• [Journal Article] Quantum and homological representations of braid groups (2012)
  • Author(s): T. Kohno
  • Journal Title: Edizioni della Normale Pages: 355-372
• [Journal Article] Hyperplane arrangements, local system homology and iterated integrals (2012)
  • Author(s): T. Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 62 Pages: 157-174
• [Journal Article] On the uniform perfectness of the groups of diffeomorphisms ofeven-dimensional manifolds (2012)
  • Author(s): Takashi Tsuboi
  • Journal Title: CommentariiMathematici Helvetici Volume: 87 Issue: 1 Pages: 141-185
  • DOI: 10.4171/cmh/251
  • Peer Reviewed
• [Journal Article] Homological representations of braid groups and KZ connections (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Journal of Singularities Volume: 5 Pages: 94-108
  • Peer Reviewed
• [Journal Article] Quantum and homological representations of braid groups (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Edizioni della Normale Volume: 14 Pages: 355-372
  • Peer Reviewed
• [Journal Article] Hyperplane arrangements, local system homology and iterated integrals (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 62
  • Peer Reviewed
• [Journal Article] Free subgroups within the images of quantum representation (2011)
  • Author(s): Louis Funar, Toshitake Kohno
  • Journal Title: Forum Mathematicum Volume: 20
  • Peer Reviewed
• [Presentation] Conformal block bundles and Gauss-Manin connections (2014)
  • Author(s): T. Kohno
  • Organizer: 第9回代数・解析・幾何学セミナー
  • Place of Presentation: 鹿児島大学
• [Presentation] Conformal block bundles and Gauss-Manin connections (2014)
  • Author(s): T. Kohno
  • Organizer: 第９回代数・解析・幾何学セミナー
  • Place of Presentation: 鹿児島大学
  • Invited
• [Presentation] Several problems on groups of diffeomorphisms (2013)
  • Author(s): T. Tsuboi
  • Organizer: Geometry and Foliations 2013
  • Place of Presentation: Komaba, Tokyo
  • Year and Date: 2013-09-11
• [Presentation] Discriminantal arrangements and fusion rules in WZW model, Hyperplane arrangements and characteristic classes (2013)
  • Author(s): T. Kohno
  • Organizer: RIMS
  • Place of Presentation: Kyoto University
• [Presentation] Braids, quantum symmetry and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Geometric representation theory of braid groups, related quantum groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: Geometry and Dynamics
  • Place of Presentation: The University of Tokyo
• [Presentation] Monodromy groups of conformal field theory, Session in Mathematical Physics (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: Sapporo
• [Presentation] Quantum symmetries in homological representations of braid groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: Sapporo
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2013)
  • Author(s): T. Kohno
  • Organizer: Joint International Meeting of American Mathematical Society and Romanian Mathematical Society
  • Place of Presentation: Alba-Iulia, Romania
• [Presentation] Quantum symmetries in homological representations of braid groups (2013)
  • Author(s): T. Kohno
  • Organizer: homological representations of braid groups, "Quantization of Teichmuller spaces"
  • Place of Presentation: Erwin Schrodinger Institute, Vienna
• [Presentation] Quantum symmetries in homological representations of braid groups (2013)
  • Author(s): T. Kohno
  • Organizer: Quantization of Teichmuller spaces
  • Place of Presentation: Erwin Schrodinger Institute, Vienna
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2013)
  • Author(s): T. Kohno
  • Organizer: Quantum Topology and Hyperbolic Geometry
  • Place of Presentation: Nha Trang, Vietnam
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2013)
  • Author(s): T. Kohno
  • Organizer: Joint International Meeting of American Mathematical Society and Romanian Mathematical Society
  • Place of Presentation: Alba-Iulia, Romania
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: 札幌コンベンションセンター
  • Invited
• [Presentation] Monodromy groups of conformal field theory (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: 札幌コンベンションセンター
  • Invited
• [Presentation] Geometric representation theory of braid groups related quantum groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: Geometry and Dynamics
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Presentation] Braids, quantum symmetry and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: 日本数学会秋季総合分科会，特別講演
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] Discriminantal arrangements and fusion rules in WZW model (2013)
  • Author(s): T. Kohno
  • Organizer: Hyperplane arrangements and characteristic classes
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Homeomorphism groups of commutator width one, Geometry in Dynamics - Satellite Thematic Session (2012)
  • Author(s): T. Tsuboi
  • Organizer: 6th European Congress of Mathematics
  • Place of Presentation: Krakow
  • Year and Date: 2012-07-01
• [Presentation] Homological representations of braid groups and KZ connections, "Braid groups and configuration spaces" (2012)
  • Author(s): T. Kohno
  • Organizer: 6th European Congress of Mathematics
  • Place of Presentation: Krakow, Poland
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2012)
  • Author(s): T. Kohno
  • Organizer: Geometric Group Theory
  • Place of Presentation: RIMS Kyoto University
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): T. Kohno
  • Organizer: Low Dimensional Topology and Number Theory IV
  • Place of Presentation: Kyushu University
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): T. Kohno
  • Organizer: East Asian School of Knots and Links
  • Place of Presentation: KAIST, Daejon
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2012)
  • Author(s): Toshitake Kohno
  • Organizer: Geometric Group Theory
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Homological representations of braid groups and KZ connections (2012)
  • Author(s): Toshitake Kohno
  • Organizer: 6th European Congress of Mathematics
  • Place of Presentation: Krakow, Poland
  • Invited
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): 河野俊丈
  • Organizer: East Asian School of Knots and Related Topics(招待講演)(招待講演)
  • Place of Presentation: KAIST (韓国)
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): 河野俊丈
  • Organizer: Low Dimensional Topology and Number Theory (招待講演)(招待講演)
  • Place of Presentation: 九州大学 (福岡県)
• [Presentation] Homeomorphism groups of commutator width one (2011)
  • Author(s): T. Tsuboi
  • Organizer: Poster at Geometry and Dynamics, Todai Forum
  • Place of Presentation: Ecole Normale Superieure de Lyon
  • Year and Date: 2011-10-17
• [Presentation] Morse-Novikov theory for hyperplane arrangements (2011)
  • Author(s): T. Kohno
  • Organizer: "Hyperplane arrangements and applications", PIMS
  • Place of Presentation: University of British Columbia
• [Presentation] Monodromy groups of conformal field theory, "Singularity theory and its applications" (2011)
  • Author(s): T. Kohno
  • Place of Presentation: Chinese University of Science and Technology, Hefei
• [Presentation] Monodromy groups of conformal field theory (2011)
  • Author(s): 河野俊丈
  • Organizer: Singularity theory and its applications (招待講演)(招待講演)
  • Place of Presentation: 中国科学技術大学
• [Presentation] Morse-Novikov theory for hyperplane arrangements (2011)
  • Author(s): 河野俊丈
  • Organizer: Hyperplane arrangements and applications (招待講演)(招待講演)
  • Place of Presentation: PIMS （カナダ）
• [Presentation] Homological representations of braid groups and KZ equations (2011)
  • Author(s): 河野俊丈
  • Organizer: 仙台シンポジウム(招待講演)(招待講演)
  • Place of Presentation: 東北大学（宮城県）
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks] Toshitake Kohno's Home Page
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks] Toshitake Kohno's Home Page
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/