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指数定理の展開とトポロジカル表面状態

Planned

Project AreaDiscrete Geometric Analysis for Materials Design
Project/Area Number 17H06461
Research Category

Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)

Allocation TypeSingle-year Grants
Review Section Science and Engineering
Research InstitutionThe University of Tokyo

Principal Investigator

古田 幹雄  東京大学, 大学院数理科学研究科, 教授 (50181459)

Co-Investigator(Kenkyū-buntansha) 五味 清紀  東京工業大学, 理学院, 教授 (00543109)
加藤 毅  京都大学, 理学研究科, 教授 (20273427)
Project Period (FY) 2017-06-30 – 2022-03-31
Project Status Granted (Fiscal Year 2021)
Budget Amount *help
¥52,130,000 (Direct Cost: ¥40,100,000、Indirect Cost: ¥12,030,000)
Fiscal Year 2021: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Fiscal Year 2020: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Fiscal Year 2019: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Fiscal Year 2018: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Fiscal Year 2017: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
Keywordsトポロジー / Bott周期性 / K群 / スペクトル切断 / 格子指数定理 / Wilson-Dirac作用素 / 指数定理 / ドメインウォール / ゲージ理論
Outline of Annual Research Achievements

Atiyah-Singerの指数定理は閉多様体上の線形偏微分方程式の解空間の次元から決まる「指数」を、ある条件の下でトポロジカルに与える公式である。 本研究のひとつの課題は、ある設定の下で、指数定理を(1)非線形偏微分作用素に対して、また(2)閉ではない多様体、に対して拡張し、ある種の非線形偏微 分作用素に対して解の散乱に相当する現象をトポロジカルに把握することである。ここで(1)としては3,4多様体上のゲージ理論の方程式、(2)としては 端のモデルが閉多様体の被覆空間である開多様体が対象である。
・昨年度の主要な成果は格子ゲージ理論におけるWilson-Dirac作用素の指数と連続理論におけるDirac作用素の指数を結び付けたことであったが、今年度は、前者をK理論的に定式化する記述方法を整備した。
・今年度の主要な成果として、K理論において基本的なBott周期性の、スペクトル切断による新しい別証明を与えた。数年前に、物性物理のバルクエッジ対応の考察にヒントを得てBott周期性のいわば"量子力学的証明"を得ていた。今回は、その証明から無限次元的考察を排除し、完全に有限次元の線形代数の範疇で証明を与えた。S^1上の微分作用素に対して、スペクトル切断の明示的な構成を行い、有限次元近似構成の基本的手段として用いる。この構成は、複素K群の周期性のみならず、実のKO群、あるいはKR理論の非自明な懸垂についての周期性に直接拡張可能である。
・上のスペクトル切断の構成の動機は上記(1)での有限次元近似への応用である。しかしこれについては今年度は明確な進展はなかった。
・物理学者の深谷英則・大野木哲也・山口哲の3氏および数学者の松尾信一郎・山下真由子の2氏との共同研究を進め。Atiyah-Patodi-Singer境界条件におけるmod 2 指数のドメインウォールを用いた具体的な記述を与えた。

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

●今年度は予定していた国際研究集会の計画をコロナの影響で取りやめた。それに限らず、3人以上のメンバーが一堂に会して広い黒板を前にして対面で議論を行う機会が持てなかった。代表者はオンラインでの講義の準備等に予想以上の労力が必要であった。代表者は本研究を進めるために「K理論的な定式化の記述方法の整備」のプロジェクトは、大学院向けの講義を用いた発表と軌を一にして行ったが、十分には行えなかった。
●昨年度に引き続く物理学者と数学者との共同研究は、準定期的なオンラインミーティングによって行った。対面とは質の異なる議論となり、長時間連続して行うことは難しかった。
●本研究の非線形作用素とかかわる側面については、新しい可能性のアイディアは得たが(スペクトル切断を"effective"に構成することによる可能性)結果的には、その実行を試みるまでには至らなかった。

Strategy for Future Research Activity

●2020年度に開催できなかった国際研究集会の代替の集会を2021年度に開催したい。オンラインが適切か否かを検討する。
●これまで若手の協力研究員を適宜追加し、若手人材の育成を行ってきた。概ねこの方針は有効に働いてきたと思われる。この方針を地道に堅持する。

Report

(4 results)
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report

Research Products

(64 results)

All 2021 2020 2019 2018 2017 Other

All Journal Article (24 results) (of which Int'l Joint Research: 15 results,  Peer Reviewed: 24 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (29 results) (of which Int'l Joint Research: 20 results,  Invited: 29 results) Remarks (6 results) Funded Workshop (5 results)

  • [Journal Article] Homological bulk-edge correspondence for Weyl semimetal2021

    • Author(s)
      Kiyonori Gomi
    • Journal Title

      Progress of Theoretical and Experimental Physics

      Volume: 0 Pages: 0-0

    • DOI

      10.1093/ptep/ptab035

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] The simple type conjecture for mod 2 Seiberg-Witten invariants2021

    • Author(s)
      Kato Tsuyoshi、 Nakamura Nobuhiro、 Yasui Koichi
    • Journal Title

      Journal of the European Mathematical Society

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Freed-Moore K-theory2021

    • Author(s)
      Kiyonori Gomi
    • Journal Title

      Communications in Analysis and Geometry

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A note on the Nielsen realization problem for K3 surfaces2021

    • Author(s)
      David Baraglia, Hokuto Konno
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: -

    • DOI

      10.1090/proc/15544

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Kitaev’s Stabilizer Code and Chain Complex Theory of Bicommutative Hopf Algebras2021

    • Author(s)
      Kim Minkyu
    • Journal Title

      Communications in Mathematical Physics

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Atiyah-Patodi-Singer Index and Domain-Wall Fermion Dirac Operators2020

    • Author(s)
      H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita
    • Journal Title

      Communications in Mathematical Physics

      Volume: 380 Pages: 1295-1311

    • DOI

      10.1007/s00220-020-03806-0

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Smoothness filtration of the magnitude complex2020

    • Author(s)
      Gomi Kiyonori
    • Journal Title

      Forum Mathematicum

      Volume: 32 Pages: 625-639

    • DOI

      10.1515/forum-2019-0091

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Eigenvalue crossings in Floquet topological systems2020

    • Author(s)
      Gomi Kiyonori、Tauber Clement
    • Journal Title

      Letters in Mathematical Physics

      Volume: 110 Pages: 465-500

    • DOI

      10.1007/s11005-019-01225-4

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Geometry of (non-Abelian) Landau levels2020

    • Author(s)
      Giuseppe De Nittis, Kiyonori Gomi, Massimo Moscolari
    • Journal Title

      Journal of Geometry and Physics

      Volume: 152 Pages: 103649-103649

    • DOI

      10.1016/j.geomphys.2020.103649

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends2020

    • Author(s)
      Hokuto Konno, Masaki Taniguchi
    • Journal Title

      Inventiones mathematicae

      Volume: 222 Pages: 833-880

    • DOI

      10.1007/s00222-020-00979-2

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the K-theoretic classification of dynamically stable systems2019

    • Author(s)
      Giuseppe De Nittis, Kiyonori Gomi
    • Journal Title

      Reviews in Mathematical Physics

      Volume: 31 Pages: 1950003-1950003

    • DOI

      10.1142/s0129055x1950003x

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices2019

    • Author(s)
      Kiyonori Gomi,Guo Chuan Thiang
    • Journal Title

      Letters in Mathematical Physics

      Volume: 109 Pages: 857-904

    • DOI

      10.1007/s11005-018-1129-1

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Crystallographic T-duality2019

    • Author(s)
      Kiyonori Gomi,Guo Chuan Thiang
    • Journal Title

      Journal of Geometry and Physics

      Volume: 139 Pages: 50-77

    • DOI

      10.1016/j.geomphys.2019.01.002

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Second-order topological phases protected by chiral symmetry2019

    • Author(s)
      Okugawa Ryo、Hayashi Shin、Nakanishi Takeshi
    • Journal Title

      Physical Review B

      Volume: 100 Pages: 205302-205302

    • DOI

      10.1103/physrevb.100.235302

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toeplitz operators on concave corners and topologically protected corner states2019

    • Author(s)
      Hayashi Shin
    • Journal Title

      Letters in Mathematical Physics

      Volume: 109 Pages: 2223-2254

    • DOI

      10.1007/s11005-019-01184-w

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Chiral vector bundles2018

    • Author(s)
      Giuseppe De Nittis, Kiyonori Gomi
    • Journal Title

      Mathematische Zeitschrift

      Volume: 290 Pages: 775-830

    • DOI

      10.1007/s00209-018-2041-1

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice2018

    • Author(s)
      Hayashi Shin
    • Journal Title

      Communications in Mathematical Physics

      Volume: 364 Pages: 343-356

    • DOI

      10.1007/s00220-018-3229-2

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries2018

    • Author(s)
      Ken Shiozaki, Hassan Shapourian, Kiyonori Gomi, Shinsei Ryu
    • Journal Title

      Physical Review B

      Volume: 98

    • DOI

      10.1103/physrevb.98.035151

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The FKMM-invariant in low dimension2018

    • Author(s)
      De Nittis Giuseppe、Gomi Kiyonori
    • Journal Title

      Letters in Mathematical Physics

      Volume: 108 Pages: 1225-1277

    • DOI

      10.1007/s11005-017-1029-9

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The cohomological nature of the Fu-Kane-Mele invariant2018

    • Author(s)
      Guseppe De Nittis, Kiyonori Gomi
    • Journal Title

      Journal of Geometry and Physics

      Volume: 124 Pages: 124-164

    • DOI

      10.1016/j.geomphys.2017.10.007

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Fu-Kane-Mele monopoles in semimetals2017

    • Author(s)
      Guo Chuan Thiang, Koji Sato and Kiyonori Gomi
    • Journal Title

      Nuclear Physics B

      Volume: 923 Pages: 107-125

    • DOI

      10.1016/j.nuclphysb.2017.07.018

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Twists on the torus equivariant under the 2-dimensional crystallographic point groups2017

    • Author(s)
      Kiyonori Gomi
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 13 Pages: 014-051

    • DOI

      10.3842/sigma.2017.014

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III2017

    • Author(s)
      Yoshiko Ogata
    • Journal Title

      Communications in Mathematical Physics

      Volume: 352 Pages: 1205-1263

    • DOI

      10.1007/s00220-016-2810-9

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Bulk-edge correspondence and the cobordism invariance of the index2017

    • Author(s)
      Hayashi Shin
    • Journal Title

      Reviews in Mathematical Physics

      Volume: 29 Pages: 1750033-1750033

    • DOI

      10.1142/s0129055x17500337

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 同変K理論によるバンド絶縁体の分類2021

    • Author(s)
      五味 清紀
    • Organizer
      オンライン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
      2020 Annual Research Report
    • Invited
  • [Presentation] Topological invariants and corner states for Hamiltonians on a lattice2021

    • Author(s)
      林 晋
    • Organizer
      第27回大阪市立大学国際学術シンポジウム 可視化の数理と対称性およびモジュライの深化
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] トーラス上のDirac作用素の正方格子による有限次元近似2020

    • Author(s)
      古田幹雄
    • Organizer
      Summer School 数理物理 2020
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] L^2 harmonic theory and Seiberg-Witten・Bauer-Furuta theory on non-compact complete Riemannian 4-manifolds2020

    • Author(s)
      Kato Tsuyoshi
    • Organizer
      The Eighth Pacific Rim Conference in Mathematics, Session on Differential Geometry
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] L^2 harmonic theory and Seiberg-Witten・Bauer-Furuta theory on non-compact complete Riemannian 4-manifolds2020

    • Author(s)
      Kato Tsuyoshi
    • Organizer
      SouthamptonKyoto Workshop 2
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A mathematical approach to higher-order topological insulators2020

    • Author(s)
      林 晋
    • Organizer
      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
      2020 Annual Research Report
    • Invited
  • [Presentation] Toeplitz作用素の指数理論と(高次)トポロジカル絶縁体のトポロジーについて2020

    • Author(s)
      林 晋
    • Organizer
      神戸幾何学セミナー, 神戸大学
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Index of the Wilson-Dirac operator revisited: a discrete version of Dirac operator on a finite lattice2020

    • Author(s)
      古田 幹雄
    • Organizer
      理研 iTHEMS Math Seminar
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls2019

    • Author(s)
      Furuta Mikio
    • Organizer
      MATERIALS 56 RESEARCH MEETING 2019 Materials Innovation for Sustainable Development Goals, YOKOHAMA SYMPOSIA
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls2019

    • Author(s)
      Furuta Mikio
    • Organizer
      International Molecule-type Workshop Frontiers in Lattice QCD and related topics, Yukawa Institute for Theoretical Physics, Kyoto University
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Seiberg-Witten 理論周辺の応用と発展2019

    • Author(s)
      加藤毅
    • Organizer
      第66回幾何学シンポジウム
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] A rigidity theorem of the Z_2-valued Seiberg-Witten invariants for spin families2019

    • Author(s)
      Kato Tsuyoshi
    • Organizer
      Index Theory, Duality and Related FieldsIn honour of Varghese Mathai’s 60th birthday
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted Donaldson invariant2019

    • Author(s)
      Kato Tsuyoshi
    • Organizer
      K-theory and Noncommutative Geometry
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological Phases of Interacting Quantum Systems2019

    • Author(s)
      Gomi Kiyonori
    • Organizer
      Casa Matematica Oaxaca, Mexico
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological phase and generalized (co)homology2019

    • Author(s)
      Gomi Kiyonori
    • Organizer
      Rigorous Statistical Mechanics and Related Topics, Kyoto University
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological invariants and corner states for Hamiltonians on a lattice2019

    • Author(s)
      林 晋
    • Organizer
      第66回トポロジーシンポジウム
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] ある種の角に対するテープリッツ作用素の指数理論とその応用2019

    • Author(s)
      林 晋
    • Organizer
      第66回幾何学シンポジウム
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Topological phases of interacting quantum systems2019

    • Author(s)
      Hayashi Shin
    • Organizer
      Casa Matematica Oaxaca, Mexico
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls2018

    • Author(s)
      古田幹雄
    • Organizer
      BEC2018X Variety and universality of bulk-edge correspondence in topological phases: From solid state physics to transdisciplinary concepts, Tsukuba University
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological invariants andcorner states for Hamiltonians on a three dimensional lattice2018

    • Author(s)
      林晋
    • Organizer
      Topology and computer 2018, Nara Women’s University
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted Donaldson invariant2018

    • Author(s)
      加藤毅
    • Organizer
      Gauge Theory and Applications, Universität Regensburg
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic volume growth of the instanton moduli spaces over the four-sphere2018

    • Author(s)
      松尾信一郎
    • Organizer
      Gauge theory and applications at Universität Regensburg
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Band topology and submanifolds of matrices2018

    • Author(s)
      五味清紀
    • Organizer
      Recent progress in mathematics of topological insulators, ETH Zürich, Switzerland
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted Donaldson invariant2018

    • Author(s)
      加藤毅
    • Organizer
      Gauge Theory in Fukuoka
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Relative higher index and almost flat bundles2018

    • Author(s)
      窪田陽介
    • Organizer
      Gauge Theory in Fukuoka
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A (3+1)-dimensional TQFT with TRS-like symmetry2018

    • Author(s)
      古田幹雄
    • Organizer
      Variety and universality of bulk-edge correspondence in topological phases, University of Tsukuba
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological crystalline insulators and K-theory2018

    • Author(s)
      五味清紀
    • Organizer
      Topology & Condensed Matter , National Tsing Hua University, Taiwan, March 22-23
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Pin(2)-Seiberg-Witten Floer functor on homology 3-spheres2017

    • Author(s)
      古田幹雄
    • Organizer
      Mathematical Aspects of Topological Phases of Matter and Quantum Computing, AIMR
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A class of asymmetric gapped Hamiltonians on quantum spin2017

    • Author(s)
      緒方芳子
    • Organizer
      "Operator algebras and Quantum Information Theory", IHP Paris
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] A01「トポロジカル物質」

    • URL

      https://www.math-materials.jp/outline/#a01

    • Related Report
      2020 Annual Research Report
  • [Remarks] 研究成果報告書

    • URL

      https://www.ms.u-tokyo.ac.jp/activity/annualreport.html

    • Related Report
      2020 Annual Research Report
  • [Remarks] 研究成果報告書 令和元年度 東京大学大学院数理科学研究科

    • URL

      https://www.ms.u-tokyo.ac.jp/activity/documents/AnnualReport2019.pdf

    • Related Report
      2019 Annual Research Report
  • [Remarks] 東大数理: 研究成果報告書平成30年度

    • URL

      https://www.ms.u-tokyo.ac.jp/activity/documents/AnnualReport2018.pdf

    • Related Report
      2018 Annual Research Report
  • [Remarks] 京都大学 教育研究活動データベース

    • URL

      https://kyouindb.iimc.kyoto-u.ac.jp/j/cA2iC

    • Related Report
      2018 Annual Research Report
  • [Remarks] Kiyonori Gomi's Website

    • URL

      http://www.math.titech.ac.jp/~kgomi/

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] PROGRESS IN THE MATHEMATICS OF TOPOLOGICAL STATES OF MATTER, AIMR2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Topology and Computer 2018, Oct. 12 (Fri) - 14 (Sun) around midday Venue: Nara Women’s University, Collaboration Center Z3062018

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      2018 Annual Research Report
  • [Funded Workshop] Symposium on Transformation Groups, Kumamoto City International Center,December 6(Thu) - December 8(Sat), 20182018

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  • [Funded Workshop] East Asian Conference on Gauge theory and Related Topics, Kyoto university, September 11- September 15, 20182018

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  • [Funded Workshop] Gauge Theory in Fukuoka2018

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Published: 2017-07-04   Modified: 2021-12-27  

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