Interface between singularity theory of mappings and knot theory

• Project/Area Number: 15K04880
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Seikei University
• Principal Investigator: Takase Masamichi 成蹊大学, 理工学部, 教授 (30447718)
• Co-Investigator(Kenkyū-buntansha): 渡邉 忠之 島根大学, 学術研究院理工学系, 講師 (70467447)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 中次元トポロジー / 幾何学 / トポロジー / 微分トポロジー / 多様体 / 写像 / 特異点

Outline of Final Research Achievements

I have been pursuing a new field of study lying between high dimensional topology and low dimensional topology and between differential topology and algebraic topology.
First, I studied the independence among the seven types of roseman moves in surface knot theory and published a joint paper with Dr. Kokoro Tanaka (Tokyo Gakugei University). Second, I studied the behaviour of complex tangents of smooth manifolds embedded in complex space and published two joint papers with Dr. Naohiko Kasuya (Kyoto Sangyo University). In addition, several projects, including a joint project with Dr. Tadayuki Watanabe (Shimane University), are underway.

Academic Significance and Societal Importance of the Research Achievements

これまでも「高次元トポロジー vs 低次元トポロジー」、「代数的トポロジー vs 微分トポロジー」、「PL トポロジー vs 微分トポロジー」という具合に分野横断的な研究を行ってきたが、本研究課題における成果はトポロジーをはみ出して複素微分幾何的な領域にまで広がってきた。このことは多様体のトポロジー、幾何学、ひいては数学の他分野への応用の幅を広げることに資する。

Research Products

• [Journal Article] 特異点と結び目 (2020)
  • Author(s): 高瀬将道
  • Journal Title: 数理科学, サイエンス社 Volume: 4月号 Pages: 14-20
• [Journal Article] Generic immersions and totally real embeddings (2018)
  • Author(s): Kasuya Naohiko、Takase Masamichi
  • Journal Title: International Journal of Mathematics Volume: 29 Issue: 11 Pages: 1850073-1850073
  • DOI: 10.1142/s0129167x18500738
  • Peer Reviewed
• [Journal Article] Knots and links of complex tangents (2017)
  • Author(s): Kasuya Naohiko、Takase Masamichi
  • Journal Title: Transactions of the American Mathematical Society Volume: 370 Issue: 3 Pages: 2023-2038
  • DOI: 10.1090/tran/7164
• [Journal Article] Regular-equivalence of $2$-knot diagrams and sphere eversions (2016)
  • Author(s): Masamichi Takase, Kokoro Tanaka
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 161 (No. 2) Issue: 2 Pages: 237-246
  • DOI: 10.1017/s0305004116000244
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Complex tangents and knots (2016)
  • Author(s): 高瀬将道
  • Organizer: トポロジー金曜セミナー、九州大学
  • Place of Presentation: 九州大学 伊都キャンパス
  • Year and Date: 2016-02-01
• [Presentation] Knots and links of complex tangents (2016)
  • Author(s): 粕谷直彦
  • Organizer: 可微分写像の特異点論とその応用
  • Place of Presentation: 京都大学数理解析研究所