Homological Mirror Symmetry of Isolated Hypersurface Singularities With Group Actions

2010 Fiscal Year Self-evaluation Report

• Project/Area Number: 20684003
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: TAKAHASHI Atsushi Osaka University, 大学院・理学研究科, 准教授 (50314290)
• Project Period (FY): 2008 – 2011
• Keywords: 幾何学 / 代数学 / 数理物理学 / ミラー対称性

Research Abstract

ミラー対称性とは,リーマン面から多様体への正則写像の数え上げに対応するA模型と複素多様体の変形理論に対応するB模型という,2つの位相的弦理論の等価性のことである.ミラー対称性は多様体の定性的および定量的な性質に関するさまざまな予想を与え,これまで個別に研究されてきた数学を結び付けるので極めて興味深い.
本研究の目的は,離散群・特異点・ルート系・リー環・有限次元代数の間にある不思議な関係を解明するため,ホモロジー的ミラー対称性の観点から,群作用付孤立超曲面特異点に対して,「群作用付き行列因子化の圏およびその安定性条件の空間の構造解明」および「ミルナーファイバーの消滅ラグランジアン輪体の行列因子化による構成」という課題を解決することである.

Research Products

• [Journal Article] Weighted projective lines associated to regular systems of weights of dual type (2010)
  • Author(s): A.Takahashi
  • Journal Title: Advanced Studies in Pure Mathematics 59 Pages: 371-388
  • Peer Reviewed
• [Journal Article] Categories of Graded Matrix Factorizations for Regular Systems of Weights with e=-1 (2009)
  • Author(s): H.Kajiura, K.Saito , A.Takahashi
  • Journal Title: Advances in Mathematics 220 Pages: 1602-1654
  • Peer Reviewed
• [Journal Article] From Primitive Forms to Frobenius Manifolds (2008)
  • Author(s): K.Saito , A.Takahashi
  • Journal Title: Proceedings of Symposia in Pure Mathematics 78 Pages: 31-48
  • Peer Reviewed
• [Journal Article] Strange duality of weighted homogeneous polynomials
  • Author(s): W.Ebeling , A.Takahashi
  • Journal Title: to appear in Compositio Math 2011
  • Peer Reviewed
• [Presentation] Mirror symmetry for weighted homogeneous polynomials (2011)
  • Author(s): A.Takahashi
  • Organizer: Derived Category Tbkyo 2011
  • Place of Presentation: 東京大学
  • Year and Date: 2011-01-28
• [Presentation] Strange duality of weighted homogeneous polynomials (2010)
  • Author(s): A.Takahashi
  • Organizer: Workshop on Geometry and Physics of Landau-Ginzburg model
  • Place of Presentation: Institut Fourier (Grenoble, France)
  • Year and Date: 2010-06-01
• [Presentation] Triangulated categories for isolated hvpersurface singularitie and mirror symmetry (2009)
  • Author(s): A.Takahashi
  • Organizer: 可換環論シンポジウム
  • Place of Presentation: ホテルアウィーナ大阪
  • Year and Date: 2009-11-27
• [Presentation] Homological Mirror Symmetry for Cusp Singularities (2009)
  • Author(s): A.Takahashi
  • Organizer: Workshop on Homological Mirror Symmetry and Related Topics
  • Place of Presentation: University of Miami (Miami, USA)
  • Year and Date: 2009-01-23
• [Presentation] Homological Mirror Symmetry for Isolated Hypersurface Similarities (2008)
  • Author(s): A.Takahashi
  • Organizer: Workshop on Homological Mirror Symmetry and Symplectic Geometry
  • Place of Presentation: MIT (Cambridge.USA)
  • Year and Date: 2008-05-27
• [Book] ミラー対称性入門 (2009)
  • Author(s): 江口徹, 深谷賢治, 高橋篤史, 並河良典, 細野忍, 今野宏, マーティンゲスト, 藤隆史, 中島啓
  • Total Pages: 33-45,60-70,137-148
  • Publisher: 日本評論社