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Applications of index theory to geometry and physics

Research Project

Project/Area Number 19K14544
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionShinshu University (2020-2022)
Institute of Physical and Chemical Research (2019)

Principal Investigator

Kubota Yosuke  信州大学, 学術研究院理学系, 講師 (30804075)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords非可換幾何学 / 高階指数理論 / トポロジカル相 / 作用素環論 / 指数理論 / 非可換幾何 / 作用素環 / K理論 / 正スカラー曲率計量 / トポロジカル絶縁体 / KK理論
Outline of Research at the Start

指数理論とはAtiyah-Singerの指数定理に始まる一連の研究を指す.本研究では,その中でもC*-環を中心的に用いる抽象理論(非可換幾何学)を用いて幾何学や物理学の問題に挑戦し,従来当該分野で考えられていたものとは異なる視点を導入することで問題解決することを目標としている.より具体的には,特異性のある多様体の正スカラー曲率計量の存在問題,強欠陥系のトポロジカル相とバルク・境界対応,散乱理論を用いた非コンパクト多様体の指数理論などを考えている.

Outline of Final Research Achievements

The main result of this research is an observation about the behavior of operators in higher index theory and some consequences obtained by applying it. The basic observation is a that operators with finite propagation on a metric space of a certain shape naturally lift to their covering space. It answers to several questions that has been asked in higher index theory, more specifically (1) index theory of codimensional 2 submanifolds, (2) obstractions to PSC metrics due to the infinite KO-bandwidth, and (3) a mathematical proof of the bulk-dislocation correspondence in 3-dimensional topological matter with a screw dislocation.

Academic Significance and Societal Importance of the Research Achievements

高階指数理論は,作用素のなす空間のトポロジーを扱う抽象理論で,これまでに高度に非自明な理論的枠組を構築することに成功している.一方それに比べると,その抽象論がどのような問題に適用されうるかについての知見はまだ不足している.本研究では,分野が培ってきた理論がどのようなことを証明する能力を持っているかについて,具体的な事例の研究をもって理解を推し進めることができた.このような方向性の研究は,分野の知見を数学全体の中に根付かせるために大切である.

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (26 results)

All 2023 2022 2021 2020 2019

All Journal Article (11 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 11 results) Presentation (14 results) (of which Int'l Joint Research: 1 results,  Invited: 2 results) Book (1 results)

  • [Journal Article] Codimension 2 transfer of higher index invariants2023

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Mathematische Annalen

      Volume: online Issue: 3 Pages: 1-59

    • DOI

      10.1007/s00208-023-02598-7

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Delocalized Spectra of Landau Operators on Helical Surfaces2022

    • Author(s)
      Kubota Yosuke、Ludewig Matthias、Thiang Guo Chuan
    • Journal Title

      Communications in Mathematical Physics

      Volume: 395 Issue: 3 Pages: 1211-1242

    • DOI

      10.1007/s00220-022-04452-4

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Band Width and the Rosenberg Index2022

    • Author(s)
      Kubota Yosuke
    • Journal Title

      International Mathematics Research Notices

      Volume: rnac124 Issue: 11 Pages: 1-17

    • DOI

      10.1093/imrn/rnac124

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Index Theorem of Lattice Wilson?Dirac Operators via Higher Index Theory2022

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Annales Henri Poincar?

      Volume: 23 Issue: 4 Pages: 1297-1319

    • DOI

      10.1007/s00023-022-01159-z

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The relative Mishchenko-Fomenko higher index and almost flat bundles II: Almost flat index pairing2022

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Journal of Noncommutative Geometry

      Volume: 16 Issue: 1 Pages: 215-264

    • DOI

      10.4171/jncg/432

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Twisted crystallographic T-duality via the Baum-Connes isomorphism2021

    • Author(s)
      Gomi Kiyonori、Kubota Yosuke、Thiang Guo Chuan
    • Journal Title

      International Journal of Mathematics

      Volume: 32 Issue: 10 Pages: 2150078-2150078

    • DOI

      10.1142/s0129167x21500786

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The bulk-dislocation correspondence for weak topological insulators on screw?dislocated lattices2021

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 54 Issue: 36 Pages: 364001-364001

    • DOI

      10.1088/1751-8121/ac190c

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Gromov?Lawson codimension 2 obstructionto positive scalar curvature and the C??index2021

    • Author(s)
      Kubota Yosuke、Schick Thomas
    • Journal Title

      Geometry & Topology

      Volume: 25 Issue: 2 Pages: 949-960

    • DOI

      10.2140/gt.2021.25.949

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Reconstructing the Bost--Connes semigroup actions from K-theory2020

    • Author(s)
      Kubota Yosuke, Takeishi Takuya
    • Journal Title

      Advances in Mathematics

      Volume: 366 Pages: 107070-107070

    • DOI

      10.1016/j.aim.2020.107070

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Almost flat relative vector bundles and the almost monodromy correspondence2020

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Journal of Topology and Analysis

      Volume: - Issue: 02 Pages: 1-30

    • DOI

      10.1142/s1793525320500545

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] The relative Mishchenko-Fomenko higher index and almost flat bundles. I. The relative Mishchenko-Fomenko index2020

    • Author(s)
      Kubota Yosuke
    • Journal Title

      Journal of Noncommutative Geometry

      Volume: 14 Issue: 3 Pages: 1209-1244

    • DOI

      10.4171/jncg/391

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Lifting finite propagation operators --- Applications to geometry and physics2023

    • Author(s)
      Yosuke Kubota
    • Organizer
      Japan-Netherlands Joint Seminar: Index Theory and Operator Algebras in Topological Physics
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] The bulk-dislocation correspondence: an operator-algebraic approach2021

    • Author(s)
      Yosuke Kubota
    • Organizer
      Topological phases of matter
    • Related Report
      2021 Research-status Report
  • [Presentation] The bulk-dislocation correspondence: an operator-algebraic approach2021

    • Author(s)
      Yosuke Kubota
    • Organizer
      The 21st International Conference on Discrete Geometric Analysis for Materials Design
    • Related Report
      2021 Research-status Report
  • [Presentation] Higher index theory in geometry and physics2021

    • Author(s)
      窪田陽介
    • Organizer
      日本数学会2021年度年会特別講演
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Operator algebra K-theory and topological phases2020

    • Author(s)
      Yosuke Kubota
    • Organizer
      CREST チュートリアル・ワークショップ 「物質のトポロジカル相の理論的探究」
    • Related Report
      2019 Research-status Report
  • [Presentation] On some structural properties of self-similar groupoids2020

    • Author(s)
      Yosuke Kubota
    • Organizer
      名古屋大学量子解析セミナー
    • Related Report
      2019 Research-status Report
  • [Presentation] Crystallographic T-duality via the Baum-Connes isomorphism2020

    • Author(s)
      Yosuke Kubota
    • Organizer
      研究会「トポロジカル表面状態,ソリトンとブレーン,指数定理」
    • Related Report
      2019 Research-status Report
  • [Presentation] Almost flat vector bundles on manifolds with boundary2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      RIMS workshop "Recent Developments in Ooperator Algebras"
    • Related Report
      2019 Research-status Report
  • [Presentation] Codimension 2 index obstruction to positive scalar curvature metrics2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] Operator algebra K-theory and topological phases2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      Rigorous Statistical Mechanics and Related Topics
    • Related Report
      2019 Research-status Report
  • [Presentation] Crystallogrphic T-duality as the Baum-Connes isomorphism2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      Materials Research Meetings 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Relative Khomology group of C algebras and almost flat vector bundle on manifolds with boundary2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      東京大学作用素環セミナー
    • Related Report
      2019 Research-status Report
  • [Presentation] A codimension 2 index obstruction to positive scalar curvature and the Calkin algebra2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      京都大学微分トポロジーセミナー
    • Related Report
      2019 Research-status Report
  • [Presentation] A codimension 2 index obstruction to positive scalar curvature and the Calkin algebra2019

    • Author(s)
      Yosuke Kubota
    • Organizer
      首都大学東京幾何セミナー
    • Related Report
      2019 Research-status Report
  • [Book] 物性物理とトポロジー2023

    • Author(s)
      窪田 陽介
    • Total Pages
      224
    • Publisher
      サイエンス社
    • ISBN
      9784781915715
    • Related Report
      2022 Annual Research Report

URL: 

Published: 2019-04-18   Modified: 2024-01-30  

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