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Universal polynomials for multi-singularities of maps and quantum Schubert calculus

Research Project

Project/Area Number 17H02838
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionWaseda University (2021-2022)
Hokkaido University (2017-2020)

Principal Investigator

Ohmoto Toru  早稲田大学, 理工学術院, 教授 (20264400)

Co-Investigator(Kenkyū-buntansha) 諏訪 立雄  北海道大学, 理学研究院, 名誉教授 (40109418)
池田 岳  早稲田大学, 理工学術院, 教授 (40309539)
成瀬 弘  山梨大学, 大学院総合研究部, 教授 (20172596)
Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥17,420,000 (Direct Cost: ¥13,400,000、Indirect Cost: ¥4,020,000)
Fiscal Year 2021: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2020: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2019: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2018: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2017: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Keywords写像の特異点論 / 特性類理論 / ヒルベルト・スキーム / 代数的コボルディズム / 応用代数幾何 / 特異点論 / 数え上げ幾何学 / シューベルト・カルキュラス / 特性類 / 普遍多項式 / トム多項式 / 相対ヒルベルト・スキーム / チャーン・シュワルツ・マクファーソン類 / 多重特異点 / 数え上げ幾何 / モティヴィック・チャーン類 / 量子シューベルト算法 / コボルディズム / リーマン・ロッホ定理
Outline of Final Research Achievements

This research project aims to give a firm foundation of Enumerative Geometry from Classics to Modern. In particular, the existence of universal polynomials for multi-singularities of proper maps (Multi-singularity Thom polynomials) has long been an important unsolved problem so far. We could partly give an afirmative solution to this problem by our novel approach using Hilbert schemes, algebraic cobordism and algebraic cohomology operations.

Academic Significance and Societal Importance of the Research Achievements

古くはアポロニウスの円の問題から現代の弦理論にまで関わる《数え上げ幾何学》は,未だその厳密な基礎付けが完成されていないと言える.とりわけ,「多重特異点跡を表す普遍多項式(トム多項式)の存在問題」は重要な未解決問題であって,本研究において,この肯定的解決を部分的に完成させた.これは数学のみならず,数理物理や工学分野への波及効果が期待される.

Report

(6 results)
  • 2022 Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • Research Products

    (70 results)

All 2023 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (15 results) Journal Article (20 results) (of which Int'l Joint Research: 8 results,  Peer Reviewed: 19 results,  Open Access: 6 results) Presentation (30 results) (of which Int'l Joint Research: 17 results,  Invited: 21 results) Book (2 results) Funded Workshop (3 results)

  • [Int'l Joint Research] ハレ大学/キール大学(ドイツ)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] サンパウロ大学(ブラジル)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] フランス国立科学研究センター(フランス)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] ミナスジェライス大学(ブラジル)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] フィレンツェ大学/パルマ大学(イタリア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] リュミニ数学研究所(フランス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] フィレンツェ大学/パルマ大学(イタリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] マルセイユ・リュミニ数学研究所(フランス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] オハイオ州立大学/ヴァージニア工科大学(米国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] ローマ大学/フィレンツェ大学/パルマ大学(イタリア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] リュミニ数学研究所(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] ヴァージニア工科大学/オハイオ州立大学(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] 中山大学(中国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] ICMC-USP(ブラジル)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] ワルシャワ工科大学/ワルシャワ大学(ポーランド)

    • Related Report
      2017 Annual Research Report
  • [Journal Article] Sato hyperfunctions via relative Dolbeault cohomology2023

    • Author(s)
      Naofumi Honda, Takeshi Izawa and Tatsuo Suwa
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 75 Issue: 1 Pages: 229-290

    • DOI

      10.2969/jmsj/87668766

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The dually flat structure for singular models2021

    • Author(s)
      N. Nakajiama and T. Ohmoto
    • Journal Title

      Information Geometry

      Volume: 4 Issue: 1 Pages: 31-64

    • DOI

      10.1007/s41884-021-00044-8

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local and global coincidence homology classes2021

    • Author(s)
      Jean-Paul Brasselet and Tatsuo Suwa
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 23 Issue: 2

    • DOI

      10.1007/s11784-021-00857-1

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Geometric algebra and singularities of ruled and developable surfaces2020

    • Author(s)
      J. Tanaka, To. Ohmoto
    • Journal Title

      Journal of Singularities

      Volume: 21 Pages: 249-267

    • DOI

      10.5427/jsing.2020.21o

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Localization of Bott-Chern classes and Hermitian residues2020

    • Author(s)
      Mauricio Correa Jr and Tatsuo Suwa
    • Journal Title

      J. London Math. Soc.

      Volume: 101 (2) Issue: 1 Pages: 349-372

    • DOI

      10.1112/jlms.12273

    • NAID

      120006803960

    • Related Report
      2020 Annual Research Report 2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms2020

    • Author(s)
      Daniele Angella, Tatsuo Suwa, Nicoletta Tardini and Adriano Tomassini
    • Journal Title

      Complex Manifolds

      Volume: 7 Issue: 1 Pages: 194-214

    • DOI

      10.1515/coma-2020-0103

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Double Grothendieck polynomials for Symplectic and Odd Orthogonal Grassmannians2020

    • Author(s)
      Hudson Thomas、Ikeda Takeshi、Matsumura Tomoo、Naruse Hiroshi
    • Journal Title

      Journal of Algebra

      Volume: 546 Pages: 294-314

    • DOI

      10.1016/j.jalgebra.2019.11.002

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Skew hook formula for d-complete posets via equivariant K-theory2019

    • Author(s)
      Naruse Hiroshi、Okada Soichi
    • Journal Title

      Algebraic Combinatorics

      Volume: 2-4 Issue: 4 Pages: 541-571

    • DOI

      10.5802/alco.54

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Multiplicites of Schubert varieties in the symplectic flag variety2019

    • Author(s)
      D. Anderson, T. Ikeda, M. Jeon, R. Kawago
    • Journal Title

      Seminaire Lotharingien de Combinatoire

      Volume: 82B

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Peterson isomorphism in K-theory and relativistic Toda lattice2018

    • Author(s)
      Takeshi Ikeda, Shinsuke Iwao and Toshiaki Maeno
    • Journal Title

      International Mathematics Research Notices

      Volume: - Issue: 19 Pages: 6421-6462

    • DOI

      10.1093/imrn/rny051

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Elementary proof and application of the generating functions for generalized Hall-Littlewood functions2018

    • Author(s)
      Naruse Hiroshi
    • Journal Title

      Journal of Algebra

      Volume: 516 Pages: 197-209

    • DOI

      10.1016/j.jalgebra.2018.09.010

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Universal Gysin formulas for the universal Hall-Littlewood functions2018

    • Author(s)
      M. Nakagawa and H. Naruse
    • Journal Title

      Contemporary Mathematics

      Volume: 708

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Thom polynomials in A-classification I: counting singular projections of a surface2018

    • Author(s)
      T. Sasajima and T. Ohmoto
    • Journal Title

      EMS Series of Congress Reports

      Volume: - Pages: 237-260

    • DOI

      10.4171/182-1/11

    • ISBN
      9783037191828, 9783037196823
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pieri rule for the factorial Schur P-functions2018

    • Author(s)
      T. Ikeda, S. Cho
    • Journal Title

      EMS Series, IMPANGA15, Schubert Varieties, Equivariant Cohomology and Characteristic Classes

      Volume: - Pages: 25-48

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Binary differential equations at parabolic and umbilical points for 2-parameter families of surfaces2018

    • Author(s)
      Deolindo-Silva J.L.、Kabata Y.、Ohmoto T.
    • Journal Title

      Topology and its Applications

      Volume: 234 Pages: 457-473

    • DOI

      10.1016/j.topol.2017.11.014

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Relative cohomology for the sections of a complex of fine sheaves2018

    • Author(s)
      Tatsuo Suwa
    • Journal Title

      代数幾何学シンポジウム記録

      Volume: -

    • NAID

      120006394752

    • Related Report
      2017 Annual Research Report
    • Open Access
  • [Journal Article] C^1-triangulation of semialgebraic sets2017

    • Author(s)
      T. Ohmoto and M. Shiota
    • Journal Title

      Journal of Topology

      Volume: 10 Issue: 3 Pages: 765-775

    • DOI

      10.1112/topo.12024

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Thom polynomials in A-classification I: counting singular projections of a surface2017

    • Author(s)
      T. Sasajima and T. Ohmoto
    • Journal Title

      Eroupean Math. Soc. Series of Congress Reports, IMPANGA Lecture Notes `Vector bundles, Schubert varieties, and equivariant cohomology'

      Volume: -

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Classical formulae on projective characters of surfaces and 3-folds, revisited2017

    • Author(s)
      T. Sasajima and T. Ohmoto
    • Journal Title

      Saitama J. Math.

      Volume: 31

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Degeneracy loci classes in K -theory ? determinantal and Pfaffian formula2017

    • Author(s)
      Hudson Thomas、Ikeda Takeshi、Matsumura Tomoo、Naruse Hiroshi
    • Journal Title

      Advances in Mathematics

      Volume: 320 Pages: 115-156

    • DOI

      10.1016/j.aim.2017.08.038

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 多重特異点とヒルベルト第15問題2022

    • Author(s)
      大本 亨
    • Organizer
      研究集会「Singularities of Differentiable Maps and its Applications」
    • Related Report
      2021 Annual Research Report
  • [Presentation] Thom polynomials since 19572022

    • Author(s)
      大本 亨
    • Organizer
      研究集会「可微分写像の特異点論とその応用」
    • Related Report
      2021 Annual Research Report
  • [Presentation] Global Theory of Singularities of Differentiable Maps - Osamu Saeki’s works and others2022

    • Author(s)
      T. Ohmoto
    • Organizer
      Workshop on Real and Complex Singularities, Univ. Sao Paulo, San Carlos, Brazil.
    • Related Report
      2021 Annual Research Report
  • [Presentation] Persistent characteristic classes and Riemann-Roch2021

    • Author(s)
      T. Ohmoto
    • Organizer
      Asia Pacific Seminar on Applied Topology and Geometry (Online)
    • Related Report
      2021 Annual Research Report
  • [Presentation] Applied Singularity Theory -- an updated new catastrophe theory2021

    • Author(s)
      T. Ohmoto
    • Organizer
      International Symposium on Big-Data, Cybersecurity and IoT (Online)
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Information Geometry from Singularity Theory Viewpoint2021

    • Author(s)
      大本 亨・中島 直道
    • Organizer
      研究集会「接触構造、特異点、微分方程式及びその周辺」(オンライン)
    • Related Report
      2020 Annual Research Report
  • [Presentation] 情報幾何への特異点論的アプローチ2020

    • Author(s)
      大本 亨
    • Organizer
      ワークショップ「統計多様体の幾何学とその周辺」(オンライン)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Information Geometry from Singularity Theory Viewpoint2020

    • Author(s)
      N. Nakajima and T. Ohmoto
    • Organizer
      16th International Workshop on Real and Complex Singularities
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Coxeter groups and hook formulas2019

    • Author(s)
      H. Naruse
    • Organizer
      Flags, Galleries and Reflection Groups
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Generalized colored hook formula2019

    • Author(s)
      H. Naruse
    • Organizer
      New interactions between Geometry and Combinatorics
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Multiplicites of Schubert varieties in the symplectic flag variety2019

    • Author(s)
      D. Anderson, T. Ikeda, M. Jeon, R. Kawago
    • Organizer
      Formai Power Series and Algebraic Combinatorics 2019 Ljublijana
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Multiple-point formulas, revisited2019

    • Author(s)
      T. Ohmoto
    • Organizer
      The 14th Algebra-Analysis-Geometry Seminar, Kagoshima
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some combinatorial properties of dual factorial Schur functions2019

    • Author(s)
      H. Naruse
    • Organizer
      Workshop on Algebraic and Enumerative Combinatorics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Hook formula and equivariant K-theory2019

    • Author(s)
      H. Naruse
    • Organizer
      Crystals and Their Generalizations
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 逆平面分割の母関数と同変K-理論2019

    • Author(s)
      成瀬 弘、岡田聡一
    • Organizer
      日本数学会年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Relative Dolbeault cohomology and hyperfunctions2019

    • Author(s)
      T. Suwa
    • Organizer
      The 14th Algebra-Analysis-Geometry Seminar, Kagoshima
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Multiple-point formulas, revisited2018

    • Author(s)
      T. Ohmoto
    • Organizer
      Characteristic Classes and Singularities/RR50, Univ. Geneve
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Representation of Sato hyperfunctions by relative Dolbeault classes2018

    • Author(s)
      T. Suwa
    • Organizer
      Hayama Symposium on Complex Analysis in Several Variables XX & Pacific Rim Complex-Symplectic Geometry Conference
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 相対 Dolbeault コホモロジーとその佐藤超関数論への応用2018

    • Author(s)
      諏訪立雄
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] K 理論的 Peterson 同型2018

    • Author(s)
      池田岳
    • Organizer
      日本数学会秋季総合分科会・無限可積分系セッション特別講演
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] K-theory Schubert calculus of the maximal orthogonal Grassmannian and set-valued decomposition tableaux2018

    • Author(s)
      T. Ikeda
    • Organizer
      Geometry, Combinatorics and Integral Systems Seminar(Ohio State University)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Relativistic Toda lattice and K-theoretic Peterson isomorphism2018

    • Author(s)
      T. Ikeda
    • Organizer
      Quantum K-theory and related topics (Korea Institute for Advanced Study)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] シューベルト・カルキュラスの視点からの Hall-Littlewood函数の一般化・母函数表示と応用2018

    • Author(s)
      成瀬 弘
    • Organizer
      日本数学会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Geometric Algebrai and Singularities arising in Differential Line Geometry2017

    • Author(s)
      大本 亨
    • Organizer
      可微分写像の特異点論の局所的研究と大域的研究
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] C^1-triangulations and semialgebraic de Rham homotopy theory2017

    • Author(s)
      T. Ohmoto
    • Organizer
      Singularity Conference, Shanghai
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Relative de Rham, relative Dolbeault cohomologies and their applications I-IV2017

    • Author(s)
      Tatsuo Suwa
    • Organizer
      Hokkaido University Summer Institute, Complex Analytic Geometry -- Residue and Dynamics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Relative Bott-Chern cohomology2017

    • Author(s)
      Tatsuo Suwa
    • Organizer
      Residues, Dynamics and Hyperfuntions
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Relative Dolbeault cohomology and Sato hyperfunctions2017

    • Author(s)
      諏訪立雄
    • Organizer
      超局所解析と漸近解析
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Relative Dolbeault cohomology and Sato hyperfunctions2017

    • Author(s)
      Tatsuo Suwa
    • Organizer
      Kinosaki Algebraic Geometry Symposium 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Relative Dolbeault cohomology and Sato hyperfunctions2017

    • Author(s)
      Tatsuo Suwa
    • Organizer
      The 5-th Franco-Japanese-Vietnamese Symposium on Singularities
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] Cisneros-Molina, J.L., Du;ng Trang, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities III.2022

    • Author(s)
      T. Suwa
    • Total Pages
      802
    • Publisher
      Springer, Cham
    • Related Report
      2021 Annual Research Report
  • [Book] 数え上げ幾何学講義2018

    • Author(s)
      池田 岳
    • Total Pages
      280
    • Publisher
      東京大学出版会
    • ISBN
      9784130613125
    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] New interactions between Geometry and Combinatorics2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] The 5th Franco-Japanese-Vietnamese Symposium on Singularities2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Geometric and Algebraic Singularity Theory2017

    • Related Report
      2017 Annual Research Report

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Published: 2017-04-28   Modified: 2024-01-30  

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