New development of inverse spectral problems for singular spaces

• Project/Area Number: 24654010
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kyoto University (2014); University of Tsukuba (2012-2013)
• Principal Investigator: YAMAGUCHI Takao 京都大学, 理学(系)研究科(研究院), 教授 (00182444)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: スペクトル逆問題 / 特異空間 / 熱核

Outline of Final Research Achievements

We have obtained the following mathematical findings as results of this reserach project. (1) By a joint work with Ayato Mitsuishi, we found local Lipschitz homotopy structure of Alexandrov spaces.(2) We observed that there is a possibility of establishing Tataru's unique continuation theorem for waves on the GH-limit spaces of Riemannian manifolds whose Ricci curvatures are uniformly bounded below. (3)We found that considering spectral limits of Schrodinger operators on collapsed surfaces whose curvatures explode to -infinity should be usuful in the study of waves of some quantum graphs.

Research Products

• [Journal Article] Collaspsing three-dimensional closed Alexandrov spaces. (2015)
  • Author(s): A. Mitsuishi and T. Yamaguchi
  • Journal Title: Transactions of AMS. Volume: 367 Issue: 4 Pages: 2339-2410
  • DOI: 10.1090/s0002-9947-2014-06091-1
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stability of strongly Lipschitz contractible balls in Alexandrov spaces. (2014)
  • Author(s): A. Mitsuishi and T. Yamaguchi
  • Journal Title: Math. Zeitschrift Volume: 277 Issue: 3-4 Pages: 995-1009
  • DOI: 10.1007/s00209-014-1289-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Locally strongly Lipschitz contractibility of Alexandrov spaces (2014)
  • Author(s): A. Mitsuishi and T. Yamaguchi
  • Journal Title: Pacific J. Volume: 270 Issue: 1 Pages: 393-421
  • DOI: 10.1007/s10711-014-0017-9
  • Peer Reviewed / Open Access
• [Presentation] 境界つき３次元アレクサンドロフ空間の崩壊 (2015)
  • Author(s): 三石史人, 山口孝男
  • Organizer: 日本数学会・2015年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Lipschitz homotopy structure of Alexandrov spaces (2014)
  • Author(s): T. Yamaguchi
  • Organizer: Trends in Modern Geometry,
  • Place of Presentation: University of Tokyo
  • Year and Date: 2014-07-08
  • Invited
• [Presentation] アレクサンドロフ空間のリプシッツ・ホモトピー構造 (2014)
  • Author(s): 山口孝男
  • Organizer: 京都大学数学談話会
  • Place of Presentation: 京都大学
  • Year and Date: 2014-06-25
  • Invited
• [Presentation] アレクサンドロフ空間の局所リプシッツ可縮性とその応用 (2012)
  • Author(s): 三石史人、山口孝男
  • Organizer: 日本数学会 2012 年度秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] Inverse spectral problems for collapsed manifolds
  • Author(s): 山口孝男
  • Organizer: Probability theory and Geometry
  • Place of Presentation: Kyoto University
• [Presentation] 崩壊とスペクトル逆問題
  • Author(s): 山口孝男
  • Organizer: 第６０回幾何学シンポジウム
  • Place of Presentation: 東京工業大学