Homological mirror symmetry and tropical geometry

• Project/Area Number: 16H03930
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: Ueda Kazushi 東京大学, 大学院数理科学研究科, 准教授 (60432465)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥17,810,000 (Direct Cost: ¥13,700,000、Indirect Cost: ¥4,110,000); Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2019: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2018: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2017: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2016: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
• Keywords: ミラー対称性 / 非可換代数幾何学 / ホモロジー的ミラー対称性 / 保型形式 / トロピカル幾何学 / シンプレクティック幾何 / 代数幾何

Outline of Final Research Achievements

Formulation of a conjecture on homological mirror symmetry for Milnor fibers of invertible polynomials and its proof in special cases, the relation between the Grothendieck ring of algebraic varieties and the derived categories of coherent sheaves, classification of noncommutative Hirzebruch surfaces, a definition and a construction　of noncommutative del Pezzo surfaces, mirror symmetry for Grassmannians, reconstruction of K3 surfaces from their Gromov-Hausdorff limits, structures of rings of automorphic forms on some symmetric domains of type IV.

Academic Significance and Societal Importance of the Research Achievements

ミラー対称性は超弦理論に由来する数学的現象で、ある空間の複素幾何と、そのミラーと呼ばれる別の空間のシンプレクティック幾何の間に不思議な関係があることを指す。ミラー対称性に属する現象には様々なものがあるが、ホモロジー的ミラー対称性はその中でも最も強いものの一つであり、A-infinity圏を空間と見てその幾何学を研究するという新しい視点を提供する。研究期間内に実施した研究によって、ホモロジー的ミラー対称性と、それに深く関連する非可換代数幾何学の理解が大きく進展し、これまでになかった定理の定式化と証明が可能になった。

Research Products

• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] 韓国高等科学院(韓国)
• [Int'l Joint Research] King's College London(英国)
• [Int'l Joint Research] University of California, Riverside(米国)
• [Int'l Joint Research] King's College London(英国)
• [Int'l Joint Research] 高等科学院(韓国)
• [Journal Article] Spherical 2-Designs as Stationary Points of Many-Body Systems (2020)
  • Author(s): Miura Makoto、Ueda Kazushi
  • Journal Title: Graphs and Combinatorics Volume: 37 Issue: 2 Pages: 485-492
  • DOI: 10.1007/s00373-020-02259-5
  • Peer Reviewed
• [Journal Article] 3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians (2020)
  • Author(s): Ueda Kazushi、Yoshida Yutaka
  • Journal Title: Journal of High Energy Physics Volume: 2020 Issue: 8 Pages: 1-43
  • DOI: 10.1007/jhep08(2020)157
  • Peer Reviewed / Open Access
• [Journal Article] Potential functions on Grassmannians of planes and cluster transformations (2020)
  • Author(s): Yuichi Nohara and Kazushi Ueda
  • Journal Title: J. Symplectic Geom. Volume: 18 Issue: 2 Pages: 559-612
  • DOI: 10.4310/jsg.2020.v18.n2.a6
  • Peer Reviewed
• [Journal Article] Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties (2020)
  • Author(s): Ito Atsushi、Miura Makoto、Okawa Shinnosuke、Ueda Kazushi
  • Journal Title: Selecta Mathematica Volume: 26 Issue: 3 Pages: 1-27
  • DOI: 10.1007/s00029-020-00561-x
  • Peer Reviewed
• [Journal Article] $$G_2$$-Grassmannians and derived equivalences (2019)
  • Author(s): Ueda Kazushi
  • Journal Title: manuscripta mathematica Volume: 159 Issue: 3-4 Pages: 549-559
  • DOI: 10.1007/s00229-018-1090-4
  • Peer Reviewed
• [Journal Article] Reconstruction of general elliptic K3 surfaces from their Gromov?Hausdorff limits (2019)
  • Author(s): Hashimoto Kenji、Ueda Kazushi
  • Journal Title: Proceedings of the American Mathematical Society Volume: 147 Issue: 5 Pages: 1963-1969
  • DOI: 10.1090/proc/14428
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The class of the affine line is a zero divisor in the Grothendieck ring: via G2-Grassmannians (2019)
  • Author(s): Atsushi Ito, Makoto Miura, Shinnosuke Okawa, Kazushi Ueda
  • Journal Title: Journal of Algebraic Geometry Volume: 28 Issue: 2 Pages: 245-250
  • DOI: 10.1090/jag/731
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Equivariant A-twisted GLSM and Gromov-Witten invariants of CY 3-folds in Grassmannians (2017)
  • Author(s): Kazushi Ueda and Yutaka Yoshida
  • Journal Title: Journal of High Energy Physics Volume: 9 Issue: 9 Pages: 1-19
  • DOI: 10.1007/jhep09(2017)128
  • Peer Reviewed / Open Access
• [Journal Article] On a certain generalization of triangle singularities (2017)
  • Author(s): Hashimoto Kenji、Lee Hwayoung、Ueda Kazushi
  • Journal Title: manuscripta mathematica Volume: 153 Issue: 1-2 Pages: 35-51
  • DOI: 10.1007/s00229-016-0876-5
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Homological mirror symmetry for affine K3 surfaces (2021)
  • Author(s): Kazushi Ueda
  • Organizer: The 6th Workshop "Complex Geometry and Lie Groups"
  • Int'l Joint Research / Invited
• [Presentation] Noncommutative del Pezzo surfaces (2020)
  • Author(s): Kazushi Ueda
  • Organizer: ZAG seminar
  • Int'l Joint Research / Invited
• [Presentation] Noncommutative del Pezzo surfaces (2020)
  • Author(s): Kazushi Ueda
  • Organizer: Freemath seminar
  • Int'l Joint Research / Invited
• [Presentation] Matrix factorizations and mirror symmetry (2020)
  • Author(s): Kazushi Ueda
  • Organizer: University of Melbourne Moduli Spaces seminar
  • Int'l Joint Research / Invited
• [Presentation] 3d N=2 Chern-Simons matter theory and K-theoretic quasimap invariants of Grassmannians (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Mirror Symmetry and Related Topics, 2019
  • Int'l Joint Research / Invited
• [Presentation] Periods and mirrors of noncommutative deformations of simple singularities (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry Ⅶ
  • Int'l Joint Research / Invited
• [Presentation] Mirror symmetry for Grassmannians and cluster transformations (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Conference on Fukaya Category and Homological Mirror Symmetry
  • Int'l Joint Research / Invited
• [Presentation] Mirror symmetry for Grassmannians and cluster transformations (2019)
  • Author(s): Kazushi Ueda
  • Organizer: International Workshop on Verlinde Algebras and Grassmannians
  • Int'l Joint Research / Invited
• [Presentation] Homological mirror symmetry for Milnor fibers of invertible polynomials (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Who is Who in Mirror Symmetry
  • Int'l Joint Research / Invited
• [Presentation] Homological mirror symmetry for Milnor fibers of invertible polynomials (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Special Session on SYZ Mirror Symmetry and Enumerative Geometry, AMS Sectional Meeting
  • Int'l Joint Research / Invited
• [Presentation] Homological mirror symmetry for Milnor fibers of invertible polynomials (2019)
  • Author(s): Kazushi Ueda
  • Organizer: Mirror Symmetry and related stuff, Sanya
  • Int'l Joint Research / Invited
• [Presentation] Gamma conjecture for Brieskorn-Pham singularities (2018)
  • Author(s): Kazushi Ueda
  • Organizer: Hypergeometric functions and mirror symmetry, the University of Tokyo
  • Int'l Joint Research / Invited
• [Presentation] Reconstruction of K3 surfaces from their Gromov-Hausdorff limits (2018)
  • Author(s): Kazushi Ueda
  • Organizer: 2018 Seoul-Tokyo Conference
  • Int'l Joint Research / Invited
• [Presentation] Homological mirror symmetry for Milnor fibers of invertible polynomials (2018)
  • Author(s): Kazushi Ueda
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry VI, Hokkaido Univeristy
  • Int'l Joint Research / Invited
• [Presentation] A-infinity algebras and singularities (2018)
  • Author(s): Kazushi Ueda
  • Organizer: The 6th Franco - Japanese - Vietnamese symposium on singularites, Nha Trang
  • Int'l Joint Research / Invited
• [Presentation] Moduli of K3 as moduli of A-infinity structures (2018)
  • Author(s): Kazushi Ueda
  • Organizer: Japanese-European symposium on symplectic varieties and moduli spaces, Tokyo University of Science
  • Int'l Joint Research / Invited
• [Presentation] Moduli of K3 surfaces as moduli of A-infinity structures (2018)
  • Author(s): Kazushi Ueda
  • Organizer: Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory
  • Int'l Joint Research / Invited
• [Presentation] Mirror symmetry and Grassmannians (2017)
  • Author(s): 植田 一石
  • Organizer: Singularities, symmetries and submanifolds
  • Place of Presentation: University College London
  • Year and Date: 2017-01-04
  • Int'l Joint Research / Invited
• [Presentation] Moduli of K3 surfaces as moduli of A-infinity structures (2017)
  • Author(s): Kazushi Ueda
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry V
  • Int'l Joint Research / Invited
• [Presentation] Gamma conjecture for Brieskorn-Pham singularities (2017)
  • Author(s): Kazushi Ueda
  • Organizer: The 5th Franco-Japanese-Vietnamese Symposium on Singularities
  • Int'l Joint Research / Invited
• [Presentation] Dimer models and homological mirror symmetry (2017)
  • Author(s): Kazushi Ueda
  • Organizer: Berkeley-Tokyo Summer School "Geometry, Representation Theory, and Mathematical Physics"
  • Int'l Joint Research / Invited
• [Presentation] An update on tropical coamoebas (2017)
  • Author(s): Kazushi Ueda
  • Organizer: Inaugural conference for the Laboratory of Mirror Symmetry and Automorphic forms
  • Int'l Joint Research / Invited
• [Presentation] Gamma conjecture for Brieskorn-Pham singularities (2017)
  • Author(s): Kazushi Ueda
  • Organizer: Hodge Theory, Stokes Phenomenon and Applications
  • Int'l Joint Research / Invited
• [Presentation] Calabi-Yau 3-folds in Grassmannians of exceptional types (2016)
  • Author(s): 植田 一石
  • Organizer: Workshop on mirror symmetry and related topics
  • Place of Presentation: 京都大学
  • Year and Date: 2016-12-16
  • Int'l Joint Research / Invited
• [Presentation] Compact moduli of K3 surfaces (2016)
  • Author(s): 植田 一石
  • Organizer: BICMR & IBS-CGP Joint Symplectic Geometry Workshop
  • Place of Presentation: 済州島, 大韓民国
  • Year and Date: 2016-11-03
  • Int'l Joint Research / Invited
• [Presentation] Residue mirror symmetry for Grassmannians (2016)
  • Author(s): 植田 一石
  • Organizer: Modern Interactions between Algebra, Geometry and Physics
  • Place of Presentation: 東北大学
  • Year and Date: 2016-06-27
  • Int'l Joint Research / Invited
• [Presentation] Residue mirror symmetry for Grassmannians (2016)
  • Author(s): 植田 一石
  • Organizer: Workshop on Symplectic Geometry and Mathematical Physics
  • Place of Presentation: 北京大学
  • Year and Date: 2016-05-24
  • Int'l Joint Research / Invited
• [Remarks]
  • URL: https://www.ms.u-tokyo.ac.jp/~kazushi/index.html
• [Remarks]
  • URL: https://www.ms.u-tokyo.ac.jp/~kazushi/
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/~kazushi/index.html
• [Funded Workshop] Hypergeometric functions and mirror symmetry, the University of Tokyo (2018)