Infinite series of algebraic varieties

• Project/Area Number: 23540050
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: Takahashi Nobuyoshi 広島大学, 理学(系)研究科(研究院), 准教授 (60301298)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: カンドル多様体 / 母関数 / モチーフ的ゼータ関数 / 数論的カンドル / カンドル / 代数的整数環 / 遠アーベル幾何 / 代数多様体 / リー山口代数 / 対称空間 / 等質空間 / 彩色 / 彩色空間 / 代数群 / 有理性 / 巾級数 / Chow 多様体

Outline of Final Research Achievements

(1) I showed a theorem on rationality of multivariate series in a joint work with Shun-ichi Kimura(Hiroshima University) and Shigeru Kuroda. As an application, we showed that a motivic Euler-Chow series is not necessarily rational.

(2) I defined a quandle variety as an algebraic variety endowed with a quandle operation, and studied its structure. I also associated a quandle to each arithmetic curve, and studied how to reconstruct the arithmetic curve from the associated quandle.

Research Products

• [Journal Article] Quandle varieties, generalized symmetric spaces and \varphi-spaces (2016)
  • Author(s): Takahashi, Nobuyoshi
  • Journal Title: Transformation Groups Volume: 21 Issue: 2 Pages: 555-576
  • DOI: 10.1007/s00031-015-9351-8
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] The closed cone of a rational series is rational polyhedral (2014)
  • Author(s): Kimura, Shunichi; Kuroda, Shigeru; Takahashi, Nobuyoshi
  • Journal Title: Journal of Algebra Volume: 405 Pages: 243-258
  • DOI: 10.1016/j.jalgebra.2014.02.007
  • Peer Reviewed
• [Journal Article] Nonstandard Point Counting for Algebraic Varieties (2013)
  • Author(s): Takahashi, Nobuyoshi
  • Journal Title: Communications in Algebra Volume: 41 Issue: 3 Pages: 971-988
  • DOI: 10.1080/00927872.2012.737074
  • Peer Reviewed
• [Journal Article] Motivic zeta functions in additive monoidal categories (2012)
  • Author(s): Kimura, Kimura, Takahashi
  • Journal Title: Journal of K-theory Volume: (印刷中) Issue: 3 Pages: 459-473
  • DOI: 10.1017/is011011006jkt174
  • Peer Reviewed
• [Presentation] Quandles associated to arithmetic schemes (2016)
  • Author(s): 高橋 宣能
  • Organizer: Low dimensional topology and number theory VIII
  • Place of Presentation: 九州大学産学官連携イノベーションプラザ
  • Year and Date: 2016-03-25
  • Int'l Joint Research / Invited
• [Presentation] Quandles in algebraic and arithmetic geometry (2015)
  • Author(s): 高橋 宣能
  • Organizer: The 10-th Anniversary Tokyo-Seoul Conference in Mathematics - Algebraic/Arithmetic/Complex Geometry
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2015-12-05
  • Int'l Joint Research / Invited