Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.

可接受的拉格朗日量、深谷范畴和同调镜像对称性。

基本信息

  • 批准号:
    1702049
  • 负责人:
  • 金额:
    $ 44.14万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2017
  • 资助国家:
    美国
  • 起止时间:
    2017-07-01 至 2019-07-31
  • 项目状态:
    已结题

项目摘要

The idea of noncommutative geometry, where one tries to think of spaces in nonlocal terms and concepts such as position no longer make sense, has emerged as a promising language to unify diverse areas of mathematics. In symplectic geometry -- the geometry of phase spaces of classical mechanics -- points are "too small" to be relevant, and it is more natural to consider a class of half-dimensional subspaces called Lagrangian submanifolds. Considering the geometry of these submanifolds, rather than points, naturally gives rise to a noncommutative space: the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry," asserts that many Fukaya categories are in fact equivalent to conventional commutative spaces such as those studied in algebraic geometry. The main goal of this research project is to expand the range of settings to which mirror symmetry is applicable. Specifically, the goal is to establish homological mirror symmetry in a broad enough setting to exhibit all (commutative) algebraic spaces defined by systems of polynomial equations as instances of Fukaya categories. A key step in this program is to study the non-commutative geometry that arises from a symplectic manifold equipped with one or more (commuting) functions. More specifically, the main goal of this project is to prove Kontsevich's homological mirror symmetry conjecture for all complete intersections in (possibly noncompact) toric varieties. The mirror spaces in this setting are so-called toric Landau-Ginzburg models (i.e., noncompact toric Calabi-Yau varieties equipped with regular functions). A key ingredient in the study of their Fukaya categories is the concept of simultaneous admissibility with respect to a collection of toric monomials. This gives a new approach to the Floer-theoretic calculations needed to prove homological mirror symmetry. The expected outcome will be a framework for understanding mirror symmetry that places varieties of general type (including noncompact ones) on the same footing as the more classical Calabi-Yau and Fano cases. This project will also investigate some apparently new structural features of Fukaya categories for Landau-Ginzburg models and for affine varieties, which suggest previously unnoticed functoriality properties of homological mirror symmetry, as well as a new approach to computations for affine varieties.
非对易几何的概念,即人们试图用非局部术语和概念(如位置)来思考空间不再有意义,已经成为一种有希望的语言,可以统一数学的不同领域。在辛几何中--经典力学中相空间的几何学--点“太小”而不相关,考虑一类称为拉格朗日子流形的半维子空间更为自然。考虑到这些子流形的几何,而不是点,自然会产生一个非交换空间:Fukaya范畴。在理论物理思想的启发下,一个引人注目的数学猜想“同调镜像对称性”断言,许多Fukaya范畴实际上等价于传统的交换空间,例如那些在代数几何中研究的空间。本研究项目的主要目的是扩大镜面对称性的适用范围。具体地说,目标是在足够广泛的背景下建立同调镜像对称性,以将由多项式方程组定义的所有(交换)代数空间显示为Fukaya范畴的实例。本程序的一个关键步骤是研究具有一个或多个(交换)函数的辛流形所产生的非对易几何。更具体地说,这个项目的主要目的是证明Kontsevich的同调镜像对称猜想适用于(可能非紧的)环簇中的所有完全交集。在此背景下的镜像空间是所谓的Toric Landau-Ginzburg模型(即,配备正则函数的非紧致Toric Calabi-Yau变种)。研究它们的Fukaya范畴的一个关键因素是关于环面单项集合的同时可容许性的概念。这为证明同系镜像对称性所需的Floer理论计算提供了一种新的途径。预期的结果将是一个理解镜像对称的框架,它将各种一般类型(包括非紧凑类型)置于与更经典的Calabi-Yau和Fano情形相同的基础上。这个项目还将研究Landau-Ginzburg模型和仿射簇的Fukaya范畴的一些明显的新结构特征,这些特征暗示了以前未被注意到的同调镜像对称性的函数性质,以及计算仿射簇的一种新方法。

项目成果

期刊论文数量(2)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties
单项允许的 Fukaya-Seidel 类别的单项性与复曲面簇的镜像
  • DOI:
    10.1016/j.aim.2019.04.056
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Hanlon, Andrew
  • 通讯作者:
    Hanlon, Andrew
Categorical mirror symmetry on cohomology for a complex genus 2 curve
复属 2 曲线上同调的分类镜像对称性
  • DOI:
    10.1016/j.aim.2020.107392
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Cannizzo, Catherine
  • 通讯作者:
    Cannizzo, Catherine
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Denis Auroux其他文献

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves
  • DOI:
    10.1007/s00029-024-00988-6
  • 发表时间:
    2024-10-22
  • 期刊:
  • 影响因子:
    1.200
  • 作者:
    Denis Auroux;Alexander I. Efimov;Ludmil Katzarkov
  • 通讯作者:
    Ludmil Katzarkov
Khovanov–Seidel quiver algebras and bordered Floer homology
  • DOI:
    10.1007/s00029-012-0106-2
  • 发表时间:
    2012-10-11
  • 期刊:
  • 影响因子:
    1.200
  • 作者:
    Denis Auroux;J. Elisenda Grigsby;Stephan M. Wehrli
  • 通讯作者:
    Stephan M. Wehrli
Infinitely many monotone Lagrangian tori in $$\mathbb {R}^6$$
  • DOI:
    10.1007/s00222-014-0561-9
  • 发表时间:
    2014-11-13
  • 期刊:
  • 影响因子:
    3.600
  • 作者:
    Denis Auroux
  • 通讯作者:
    Denis Auroux

Denis Auroux的其他文献

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{{ truncateString('Denis Auroux', 18)}}的其他基金

Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry
镜像对称中的部分包裹深谷范畴和函子性
  • 批准号:
    2202984
  • 财政年份:
    2022
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
Conference: Current Developments in Mathematics
会议:数学的当前发展
  • 批准号:
    1933415
  • 财政年份:
    2019
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.
可接受的拉格朗日量、深谷范畴和同调镜像对称性。
  • 批准号:
    1937869
  • 财政年份:
    2019
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
Lagrangian Floer homology and the geometry of homological mirror symmetry
拉格朗日弗洛尔同调和同调镜像对称的几何
  • 批准号:
    1406274
  • 财政年份:
    2014
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
FRG: Collaborative Research: Wall-crossings in Geometry and Physics
FRG:合作研究:几何和物理的跨越
  • 批准号:
    1264662
  • 财政年份:
    2013
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Standard Grant
Floer homology, low-dimensional topology, and mirror symmetry
Florer 同调、低维拓扑和镜像对称
  • 批准号:
    1007177
  • 财政年份:
    2010
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
FRG Collaborative Research: Homological Mirror Symmetry and its applications
FRG合作研究:同调镜像对称及其应用
  • 批准号:
    0652630
  • 财政年份:
    2007
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Standard Grant
Geometric and Algebraic Structures in the Group of Hamiltonian Diffeomorphisms
哈密​​顿微分同胚群中的几何和代数结构
  • 批准号:
    0706976
  • 财政年份:
    2007
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Standard Grant
Lefschetz fibrations in symplectic topology and applications to mirror symmetry
辛拓扑中的莱夫谢茨纤维及其在镜像对称中的应用
  • 批准号:
    0600148
  • 财政年份:
    2006
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant
Approximately holomorphic techniques and monodromy invariants in symplectic topology
辛拓扑中的近似全纯技术和单向不变量
  • 批准号:
    0244844
  • 财政年份:
    2003
  • 资助金额:
    $ 44.14万
  • 项目类别:
    Continuing Grant

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代数和组合学中的拉格朗日量
  • 批准号:
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Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.
可接受的拉格朗日量、深谷范畴和同调镜像对称性。
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  • 财政年份:
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    $ 44.14万
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