Derived Categories and Mirror Symmetry
派生范畴和镜像对称
基本信息
- 批准号:RGPIN-2015-04596
- 负责人:
- 金额:$ 1.6万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2018
- 资助国家:加拿大
- 起止时间:2018-01-01 至 2019-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Mirror symmetry began as a duality between superconformal field theories. In 1991, Maxim Kontsevich re-interpreted this concept from physics as an incredibly deep and ubiquitous mathematical duality now known as homological mirror symmetry (HMS). In his famous Fields Medal address, he proposed a new type of geometry based on derived categories, creating a frenzy of activity in the mathematical community which lead to a remarkable synergy of diverse mathematical disciplines: symplectic geometry, algebraic geometry, and category theory. HMS is now the cornerstone of an immense field of active mathematical research which utilizes ever-advancing state-of-the-art techniques from string theory.***This proposal constitutes a 5-year plan to study derived categories in algebraic geometry and more specifically, their relationship to constructions in string theory and mirror symmetry. Based on my previous work, I expect applications to range from partial solutions of the Hodge Conjecture (a "Millennium Problem") to a deep conjecture of Kawamata on the relationship between birational geometry and derived categories. I intend to fully solve the latter problem. In doing so, I will be aided by an international network of collaborators which includes Kontsevich himself. ***The central mathematical objectives of this proposal are:***1) Study algebraic varieties with group actions via stratifications with the intent to use these stratifications to extend the relationship between geometric invariant theory and derived categories.***2) Unify, develop, and expand mirror symmetry constructions with gauge groups in the setting of derived categories and apply the theory above towards classical algebro-geometric invariants: Noether-Lefschetz loci, algebraic cycles (and the Hodge conjecture), Griffiths groups.***The first objective is in many ways a fundamental study of group actions in algebraic geometry. On the other hand, it has enormous potential for furthering the specific interests of my field. Indeed, through Objective 1, I envision a new understanding of semi-orthogonal decompositions and equivalences of derived categories which I expect to re-write and unify the current literature. This is highly evidenced by my previous work on derived categories and geometric invariant theory which recovers many of the most exciting theorems in derived categories and algebraic geometry.***The second objective has an outward focus on the greater field of mirror symmetry and algebraic geometry. Much of my previous work has been building towards this objective. Developing mirror symmetry constructions with gauge groups is natural to the existing literature and is of great importance, as it expands on the highly influential role of toric mirror symmetry. In addition, applications towards classical algebro-geometric invariants are of great interest to the larger algebraic geometry community. *** *** *** *** **
镜像对称开始于超共形场论之间的对偶。1991年,马克西姆·孔采维奇(Maxim Kontsevich)将这个概念从物理学中重新解释为一种令人难以置信的深刻和普遍存在的数学对偶性,现在被称为同调镜像对称(HMS)。在他著名的菲尔兹奖的地址,他提出了一种新的几何类型的基础上派生类别,创造了狂热的活动,在数学界导致了显着的协同作用,不同的数学学科:辛几何,代数几何和范畴理论。HMS现在是一个巨大的活跃的数学研究领域的基石,它利用了弦理论中不断进步的最先进的技术。这个建议构成了一个5年计划,研究代数几何中的导出范畴,更具体地说,它们与弦理论和镜像对称中的结构的关系。 基于我以前的工作,我希望应用范围从霍奇猜想(“千年问题”)的部分解决方案,以深川俣双有理几何和派生类别之间的关系的猜想。 我打算彻底解决后一个问题。 在此过程中,我将得到包括孔采维奇本人在内的国际合作者网络的帮助。 * 这个提议的中心数学目标是:*1)通过分层研究具有群作用的代数簇,目的是使用这些分层来扩展几何不变理论和导出范畴之间的关系。2)统一、发展和扩展镜像对称结构与规范群在导出范畴的设置,并将上述理论应用于经典的代数几何不变量:诺特-莱夫谢茨轨迹,代数循环(和霍奇猜想),格里菲斯群。第一个目标是在许多方面的基本研究群作用的代数几何。 另一方面,它有巨大的潜力,促进我的领域的具体利益。 事实上,通过目标1,我设想一个新的理解半正交分解和等价的派生类别,我希望重写和统一目前的文献。 这是高度证明了我以前的工作导出范畴和几何不变理论恢复了许多最令人兴奋的定理在导出范畴和代数几何。第二个目标有一个外在的重点,对镜像对称和代数几何更大的领域。 我以前的大部分工作都是朝着这个目标发展的。发展规范群的镜像对称结构对现有文献来说是很自然的,并且非常重要,因为它扩展了复曲面镜像对称的高度影响力。 此外,对经典代数几何不变量的应用是更大的代数几何社区的极大兴趣。
项目成果
期刊论文数量(0)
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会议论文数量(0)
专利数量(0)
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Favero, David其他文献
Variation of geometric invariant theory quotients and derived categories
- DOI:
10.1515/crelle-2015-0096 - 发表时间:
2019-01-01 - 期刊:
- 影响因子:1.5
- 作者:
Ballard, Matthew;Favero, David;Katzarkov, Ludmil - 通讯作者:
Katzarkov, Ludmil
Derived categories of BHK mirrors
- DOI:
10.1016/j.aim.2019.06.013 - 发表时间:
2019-08-20 - 期刊:
- 影响因子:1.7
- 作者:
Favero, David;Kelly, Tyler L. - 通讯作者:
Kelly, Tyler L.
Favero, David的其他文献
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{{ truncateString('Favero, David', 18)}}的其他基金
Windows and Mirror Symmetry
窗口和镜像对称
- 批准号:
RGPIN-2022-03400 - 财政年份:2022
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2021
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2020
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2019
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2017
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
相似海外基金
Postdoctoral Fellowship: MPS-Ascend: Understanding Fukaya categories through Homological Mirror Symmetry
博士后奖学金:MPS-Ascend:通过同调镜像对称理解深谷范畴
- 批准号:
2316538 - 财政年份:2023
- 资助金额:
$ 1.6万 - 项目类别:
Fellowship Award
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:
$ 1.6万 - 项目类别:
Continuing Grant
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2021
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2020
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2019
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
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:
$ 1.6万 - 项目类别:
Continuing Grant
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2017
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
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:
1702049 - 财政年份:2017
- 资助金额:
$ 1.6万 - 项目类别:
Continuing Grant
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2016
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual
Derived Categories and Mirror Symmetry
派生范畴和镜像对称
- 批准号:
RGPIN-2015-04596 - 财政年份:2015
- 资助金额:
$ 1.6万 - 项目类别:
Discovery Grants Program - Individual