Topological structure of a compactification and its boundary on coarse geometry

• Project/Area Number: 25800035
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kanagawa University (2016); The University of Tokyo (2013-2015)
• Principal Investigator: MINE Kotaro 神奈川大学, 工学部, 助教 (90512525)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: Higson境界 / 粗空間 / 大スケール幾何学 / 漸近次元 / 一様空間 / 擬等長 / コンパクト化 / 位相空間 / 粗幾何 / Higsonコンパクト化 / Smirnovコンパクト化 / coarse空間 / 不動点

Outline of Final Research Achievements

We studied properties of large scale geometry from a view point of the topology of a boundary at infinity which is called the Higson corona. (1) We investigated a condition of a coarse map on the underlying space which induces a fixed point free homeomorphism on the corona. (2) We discussed a generalization of C_0 structures for uniform spaces and a relation between these Higson compactifications and Smirnov compactifications. (3) We also studied the Higson corona of the product of coarse spaces.

Research Products

• [Journal Article] Metric compactifications and coarse structures (2015)
  • Author(s): K. Mine and A. Yamashita
  • Journal Title: Canadian Journal of Mathematics Volume: 67 Issue: 5 Pages: 1091-1108
  • DOI: 10.4153/cjm-2015-029-8
  • Peer Reviewed
• [Journal Article] C0 coarse structures on uniform spaces (2015)
  • Author(s): K. Mine, A. Yamashita and T. Yamauchi
  • Journal Title: Houston Journal of Mathematics Volume: 41 Pages: 1351-1358
  • Peer Reviewed
• [Journal Article] Higsonコンパクト化のUniversalityについて (2015)
  • Author(s): 嶺幸太郎
  • Journal Title: 数理解析研究所講究録 Volume: 1932 Pages: 7-11
• [Journal Article] C0 coarse structures on uniform spaces (2014)
  • Author(s): 嶺幸太郎・山下温・山内貴光
  • Journal Title: 数理解析研究所講究録 Volume: 1884 Pages: 13-17
• [Journal Article] Higson 境界上の同相変換の不動点集合 (2014)
  • Author(s): 嶺幸太郎
  • Journal Title: 数理解析研究所講究録 Volume: 1884 Pages: 40-47
• [Journal Article] Detecting topological groups which are (locally)homeomorphic to LF-spaces (2013)
  • Author(s): T. Banakh, K. Mine, D. Repovs, K. Sakai, T. Yagasaki
  • Journal Title: Topology and its Applications Volume: 160 Issue: 18 Pages: 2272-2284
  • DOI: 10.1016/j.topol.2013.07.023
  • Peer Reviewed
• [Presentation] 微分積分学と位相空間論 (2017)
  • Author(s): 嶺 幸太郎
  • Organizer: 横浜セミナー
  • Place of Presentation: 横浜国立大学
• [Presentation] Coarse空間の正規性 (2016)
  • Author(s): 嶺 幸太郎
  • Organizer: 横浜セミナー
  • Place of Presentation: 横浜国立大学
• [Presentation] 粗空間の積とその境界 (2015)
  • Author(s): 嶺幸太郎
  • Organizer: 集合論的位相幾何学および幾何学的トポロジーの最近の動向と展望
  • Place of Presentation: 京都大学 数理解析研究所
  • Year and Date: 2015-11-17
• [Presentation] Product of coarse spaces and its boundary (2015)
  • Author(s): Kotaro Mine
  • Organizer: 葉層構造と微分同相群2015研究集会
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2015-10-30
• [Presentation] 位相空間論からの話題 (2015)
  • Author(s): 嶺幸太郎
  • Organizer: 無限群と幾何学の周辺
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2015-07-12
• [Presentation] Higson corona and fixed points (2015)
  • Author(s): Kotaro Mine
  • Organizer: Dubrovnik VIII - Geometric Topology, Geometric Group Theory & Dynamical Systems
  • Place of Presentation: Dubrovnik大学
  • Year and Date: 2015-06-25
  • Int'l Joint Research
• [Presentation] Coronae of coarse spaces and uniform spaces (2015)
  • Author(s): Kotaro Mine
  • Organizer: International Conference on Preserver Problems and Related Topics
  • Place of Presentation: 新潟大学
  • Year and Date: 2015-05-24
• [Presentation] Higsonコンパクト化の普遍性について (2015)
  • Author(s): 嶺幸太郎
  • Organizer: 第38回トポロジーセミナー
  • Place of Presentation: 山喜旅館
  • Year and Date: 2015-03-18
• [Presentation] Higson境界と不動点集合 (2015)
  • Author(s): 嶺幸太郎
  • Organizer: つくばセミナー
  • Place of Presentation: 筑波大学
  • Year and Date: 2015-03-06
• [Presentation] Higsonコンパクト化のUniversalityについて (2014)
  • Author(s): 嶺幸太郎
  • Organizer: 集合論的・幾何学的トポロジーと種々の分野の交流
  • Place of Presentation: 京都大学 数理解析研究所
  • Year and Date: 2014-10-22
• [Presentation] 位相空間論入門 (2014)
  • Author(s): 嶺幸太郎
  • Organizer: 第2回つくばフレッシュマンセミナー
  • Place of Presentation: 筑波大学
  • Year and Date: 2014-07-21
• [Presentation] コンパクト化…空間にふたをする (2014)
  • Author(s): 嶺幸太郎
  • Organizer: 無限群と幾何学の新展開のアウトリーチについての研究会, , 2014年6月6日.
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2014-06-06
• [Presentation] Coarse幾何学と位相空間論
  • Author(s): 嶺幸太郎
  • Organizer: 第１回つくばフレッシュマンセミナー
  • Place of Presentation: 筑波大学
• [Presentation] Metric compactifications and coarse structure
  • Author(s): K. Mine
  • Organizer: 28th Summer Conference on Topology and its Applications
  • Place of Presentation: Nipissin大学, North Bay, Canada
• [Presentation] Subdivisions of simplicial complexes preserving the metric topology
  • Author(s): K. Mine
  • Organizer: International Conference on Topology and Geometry 2013
  • Place of Presentation: 島根大学
• [Presentation] Approximation Theorems for compactifications
  • Author(s): K. Mine
  • Organizer: 5th Waseda Geometric topology meeting
  • Place of Presentation: 早稲田大学
• [Presentation] Homeomorphism of a Higson corona and its fixed points
  • Author(s): 嶺幸太郎
  • Organizer: 研究集会「群と幾何学の展望」
  • Place of Presentation: 東京大学玉原国際セミナーハウス
• [Presentation] Higson境界上の同相変換の不動点集合
  • Author(s): 嶺幸太郎
  • Organizer: 集合論的及び幾何学的トポロジーの現状とその展望
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Higson境界上の同相変換と不動点集合
  • Author(s): 嶺幸太郎
  • Organizer: 第37回トポロジーセミナー
  • Place of Presentation: いこいの村たてやま
• [Presentation] Graph-like連続体について
  • Author(s): 嶺幸太郎
  • Organizer: 第6回早稲田幾何学的トポロジー研究集会
  • Place of Presentation: 早稲田大学
• [Remarks] 嶺幸太郎研究室ホームページ
  • URL: http://www.math.kanagawa-u.ac.jp/mine/