Discrete geometry and creation of 3 dimensional geometric models

• Project/Area Number: 15K13434
• 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, 大学院数理科学研究科, 教授 (80144111)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 離散幾何学 / 定曲率曲面 / 極小曲面 / 代数曲面 / 特異点理論 / 可積分系 / 極限集合 / 離散データ / 単体分割 / モーデル・ヴェイユ格子 / 曲率

Outline of Final Research Achievements

In this research we developed a method to create extremely accurate 3 dimensional models of surfaces appearing in geometry and topology by curving metals. This method realized models much more accurate than those created by 3D printers. We also developed techniques realizing limit sets of discrete group and objects in singularity theory, foliation theory and integrable systems, which were difficult to create by previous methods. By adding data of normal direction to discrete point data sets and mesh, we realized accuracy of less than 1/100mm. Our research established a new interface for creating accurate 3 dimensional models.

Research Products

• [Journal Article] Quantum representations of braid groups and holonomy Lie algebras (2017)
  • Author(s): Toshitake Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 72 Pages: 117-144
  • Peer Reviewed
• [Journal Article] Homological Representations of Braid Groups and the Space of Conformal Blocks (2017)
  • Author(s): Toshitake Kohno
  • Journal Title: Springer INdAM Series Volume: 19 Pages: 427-442
  • DOI: 10.1007/978-3-319-58971-8_16
  • ISBN: 9783319589701, 9783319589718
  • Peer Reviewed
• [Journal Article] Higher holonomy maps of formal homology connections and braid cobordisms (2016)
  • Author(s): Toshitake Kohno
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 26 Issue: 12 Pages: 1642007-1642007
  • DOI: 10.1142/s0218216516420074
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Higher holonomy maps of formal homology connections and braid cobordisms (2016)
  • Author(s): T. Kohno
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [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
• [Presentation] Local systems on configuration spaces, KZ connections and conformal blocks (2018)
  • Author(s): Toshitake Kohno
  • Organizer: Topology of Arrangements and Representation Stability, Oberwolfach
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of holonomy maps and representations of braid cobordims (2018)
  • Author(s): Toshitake Kohno
  • Organizer: The 13th Algebra-Analysis-Geometry Seminar
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of holonomy mapsfor hyperplane arrangements (2017)
  • Author(s): Toshitake Kohno
  • Organizer: Arrangements and beyond: Combinatorics, geometry, topology and applications
  • Int'l Joint Research / Invited
• [Presentation] Representations of the category of braid cobordisms (2017)
  • Author(s): Toshitake Kohno
  • Organizer: Self-distributive system and quandle (co)homology theory in algebra and low-dimensional topology
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of holonomy maps for hyperplane arrangements (2016)
  • Author(s): Toshitake Kohno
  • Organizer: Summer Conference on Hyperplane Arrangements (SCHA) in Sapporo
  • Place of Presentation: Hokkaido University
  • Year and Date: 2016-08-08
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of the holonomy maps of KZ connections (2016)
  • Author(s): Toshitake Kohno
  • Organizer: Workshop on Grothendieck-Teichmuller Theories
  • Place of Presentation: Chern Institute of Mathematics, Nankai University, Tianjin, China
  • Year and Date: 2016-07-24
  • Int'l Joint Research / Invited
• [Presentation] Holonomy of braids and its 2-category extension (2015)
  • Author(s): T. Kohno
  • Organizer: Braids, Configuration Spaces and Quantum Topology
  • Place of Presentation: 東京大学
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] Holonomy of braids and its 2-category extension (2015)
  • Author(s): T. Kohno
  • Organizer: Integrability in Algebra, Geometry and Physics: New Trends
  • Place of Presentation: Congressi Stefano Franscini, Switzerland
  • Year and Date: 2015-07-13
  • Int'l Joint Research / Invited
• [Book] 結晶群 (2015)
  • Author(s): 河野俊丈
  • Total Pages: 200
  • Publisher: 共立出版
• [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/
• [Remarks] Collection of Mathematical Models
  • URL: http://faculty.ms.u-tokyo.ac.jp/~topology/models/index.html
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/teacher/kohno.html
• [Funded Workshop] Braids, Configuration Spaces and Quantum Topology (2015)
  • Place of Presentation: 東京大学大学院数理科学研究科大講義室
  • Year and Date: 2015-09-07