Microlocal analysis of topological Radon transforms and their applications to singularity theory

• Project/Area Number: 15K17564
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kindai University
• Principal Investigator: MATSUI Yutaka 近畿大学, 理工学部, 准教授 (10510026)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: ラドン変換 / 構成可能関数 / 定義可能関数

Outline of Final Research Achievements

In this research, we study integral transforms of constructible functions and definable functions, whose integral theory is based on the topological Euler characteristics. We obtained the following results: the injectivity theorem of topological Radon transforms of constructible functions on compact Grassmann manifolds, several properties of topological Radon transforms of globally constructible functions on affine Grassmann manifolds and several properties of topological Radon transforms of definable functions.

Academic Significance and Societal Importance of the Research Achievements

本研究では位相的ラドン変換と呼ばれる積分変換の反転公式や像の特徴づけを中心としたさまざまな性質を研究している．それは切断面のオイラー数から元の集合を復元するという幾何学的CTスキャンの原理におけるその切断面の様子や元の集合の復元可能性についての研究であるといえる．今回の研究成果は，先行研究とは数学的に異なるさまざまな設定の下で，位相的ラドン変換の反転公式や像の特徴づけについて考察を行い，類似の結果が得られることを明らかにしたものである．

Research Products

• [Journal Article] Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets (2018)
  • Author(s): Y.Ike, Y.Matsui, K.Takeuchi
  • Journal Title: International Mathematics Research Notices Volume: 2018 Pages: 4852-4898
  • Peer Reviewed
• [Journal Article] Monodromy at infinity of polynomial maps and A-hypergeometric functions (2016)
  • Author(s): Kiyoshi TAKEUCHI and Yutaka MATSUI
  • Journal Title: Sugaku Volume: 29
  • NAID: 10031177295
  • Peer Reviewed
• [Journal Article] Ａ ｒａｎｇｅ ｃｈａｒａｃｔｅｒｉｚａｔｉｏｎ ｏｆ ｔｏｐｏｌｏｇｉｃａｌ Ｒａｄｏｎ ｔｒａｎｓｆｏｒｍｓ ｏｎ Ｇｒａｓｓｍａｎｎ ｍａｎｉｆｏｌｄｓ (2015)
  • Author(s): Ｙｕｔａｋａ ＭＡＴＳＵＩ
  • Journal Title: Ｒｅｖ． Ｒｏｕｍａｉｎｅ Ｍａｔｈ． ＰｕｒｅｓＡｐｐｌ． Volume: 60 Pages: 423-440
  • Peer Reviewed / Open Access
• [Journal Article] Ｏｎ ｔｈｅ ｓｉｚｅｓ ｏｆ ｔｈｅ Ｊｏｒｄａｎ ｂｌｏｃｋｓ ｏｆ ｍｏｎｏｄｒｏｍｉｅｓ ａｔ ｉｎｆｉｎｉｔｙ (2015)
  • Author(s): Ｙｕｔａｋａ ＭＡＴＳＵＩ， Ｋｉｙｏｓｈｉ ＴＡＫＥＵＣＨＩ
  • Journal Title: Ｈｏｋｋａｉｄｏ Ｍａｔｈ． Ｊ． Volume: 44 Pages: 1-14
  • Peer Reviewed / Open Access
• [Presentation] Topological Tomography (2018)
  • Author(s): Y. Matsui
  • Organizer: 近畿大学研究コア交流会
• [Presentation] Topological Radon transforms on various Grassmann manifolds (2018)
  • Author(s): Y. Matsui
  • Organizer: RIMS研究集会代数解析の諸問題
  • Int'l Joint Research / Invited
• [Presentation] Topological Radon transforms on Grassmann manifolds (2017)
  • Author(s): Yutaka MATSUI
  • Organizer: Australian-Japanese workshop on real and complex singularities
  • Int'l Joint Research / Invited
• [Presentation] Euler obstructionの応用について (2016)
  • Author(s): Yutaka MATSUI
  • Organizer: 若者のための現代幾何入門
  • Place of Presentation: 立教大学
  • Year and Date: 2016-12-10
  • Invited
• [Presentation] Topological Radon transforms and their applications (2016)
  • Author(s): Yutaka MATSUI
  • Organizer: New development of microlocal analysis and singular perturbation theory
  • Place of Presentation: 京都大学
  • Year and Date: 2016-10-04
  • Int'l Joint Research / Invited
• [Presentation] Euler obstructions and related topics I, II (2016)
  • Author(s): Yutaka MATSUI
  • Organizer: Research on global properties of singularities
  • Place of Presentation: 兵庫教育大学
  • Year and Date: 2016-06-23
  • Invited