Collaborative Research: Derived Categories in Birational Geometry, Enumerative Geometry, and Non-commutative Algebra
合作研究:双有理几何、枚举几何和非交换代数中的派生范畴
基本信息
- 批准号:2302262
- 负责人:
- 金额:$ 28万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2023
- 资助国家:美国
- 起止时间:2023-09-01 至 2026-08-31
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
Solutions to polynomial equations form geometric shapes (for example lines, planes, curves). Thus, we can solve equations using geometry (by finding where lines, planes, curves intersect). These ideas have evolved into algebraic geometry, a field of modern mathematics with applications ranging from physics to cybersecurity. In this collaborative project we are particularly interested in the connections to physics. These connections are made through “derived categories”, a complex system of mathematical data associated to a geometric shape. While derived categories have experienced explosive development in recent decades, many questions about them remain unsolved. This project attempts to solve some of the central and most recent questions about derived categories using techniques developed by the principal investigators and many others over the last decade. This award will also support undergraduate and graduate students. This project focuses on three specific questions about derived categories. Together, these questions incorporate numerous areas of mathematics including birational geometry, enumerative geometry, and non-commutative algebra. Specifically, our first question studies connections between derived categories and birational geometry (and the possibly concealed connection between flips/flops and derived partial compactifications). We ask whether K-equivalent varieties have equivalent derived categories, a central conjecture about derived categories due to Bondal-Orlov and Kawamata. Our second question asks how decompositions of quantum cohomology are related to semi-orthogonal decompositions of derived categories. This connection is conjecturally made through mirror symmetry as proposed by Kontsevich and Kuznetsov. Finally, our third question asks how derived categories shed light on resolutions of singularities (namely as moduli spaces coming from non-commutative algebra). Here, we aim to construct non-commutative crepant resolutions using ideas from homological mirror symmetry. The existence of these resolutions has been conjectured by Van den Bergh. As a whole, this project aims to interpolate between these three questions/conjectures using techniques from geometric invariant theory, non-commutative algebra, derived algebraic geometry, and mirror symmetry. The central theme is the explicit use and construction of Fourier-Mukai kernels whose geometry provides a foothold into understanding these problems. Combining the past work of the principal investigators on the construction of Fourier-Mukai kernels, wall crossing for derived categories, and virtual fundamental cycles, we expect to advance our understanding of these fundamental questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
多项式方程的解形成几何形状(例如直线、平面、曲线)。 因此,我们可以用几何学来解方程(通过寻找直线、平面、曲线相交的地方)。 这些想法已经演变成代数几何,这是现代数学的一个领域,其应用范围从物理到网络安全。 在这个合作项目中,我们对物理学的联系特别感兴趣。 这些联系是通过“派生类别”,一个复杂的系统的数学数据相关联的几何形状。 虽然派生范畴在近几十年来经历了爆炸性的发展,但关于它们的许多问题仍然没有解决。 这个项目试图使用主要研究者和许多其他人在过去十年中开发的技术来解决一些关于派生类别的核心和最新问题。 该奖项还将支持本科生和研究生。这个项目集中在三个关于派生范畴的具体问题上。 总之,这些问题包含了数学的许多领域,包括双有理几何,枚举几何和非交换代数。 具体地说,我们的第一个问题研究导出范畴和双有理几何之间的联系(以及翻转/触发器和导出部分紧化之间可能隐藏的联系)。 我们问是否K-等价品种有等价的派生范畴,一个中央猜想派生范畴由于Bondal-Orlov和Kawamata。 我们的第二个问题是问量子上同调的分解与导出范畴的半正交分解有什么关系。 这种联系是通过孔采维奇和库兹涅佐夫提出的镜像对称来实现的。 最后,我们的第三个问题是问导出范畴如何阐明奇点的分解(即作为来自非交换代数的模空间)。 在这里,我们的目标是使用同调镜像对称的思想来构建非交换的crepant决议。 货车登贝格证实了这些决议的存在。 作为一个整体,这个项目的目的是使用几何不变理论,非交换代数,导出代数几何和镜像对称的技术在这三个问题之间进行插值。中心主题是明确使用和建设的傅立叶-向井内核的几何提供了一个立足点到理解这些问题。 结合主要研究人员过去在构建Fourier-Mukai内核、衍生类别的跨壁和虚拟基本周期方面的工作,我们希望推进我们对这些基本问题的理解。该奖项反映了NSF的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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David Favero其他文献
Kernels for Grassmann flops
Grassmann 失败的内核
- DOI:
10.1016/j.matpur.2021.01.005 - 发表时间:
2019 - 期刊:
- 影响因子:0
- 作者:
Matthew R. Ballard;N. Chidambaram;David Favero;P. McFaddin;Robert Vandermolen - 通讯作者:
Robert Vandermolen
A category of kernels for equivariant factorizations, II: Further implications
等变分解的核的类别,II:进一步的含义
- DOI:
10.1016/j.matpur.2014.02.004 - 发表时间:
2013 - 期刊:
- 影响因子:0
- 作者:
Matthew R. Ballard;David Favero;L. Katzarkov - 通讯作者:
L. Katzarkov
An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity
幻影、奥尔洛夫谱和克诺勒周期的轨道构建
- DOI:
- 发表时间:
2014 - 期刊:
- 影响因子:0
- 作者:
David Favero;F. Haiden;L. Katzarkov - 通讯作者:
L. Katzarkov
Rouquier dimension is Krull dimension for normal toric varieties
Rouquier 尺寸是普通复曲面品种的 Krull 尺寸
- DOI:
- 发表时间:
2023 - 期刊:
- 影响因子:0.6
- 作者:
David Favero;Jesse Huang - 通讯作者:
Jesse Huang
Reconstruction and finiteness results for Fourier–Mukai partners
Fourier-Mukai 合伙人的重构和有限性结果
- DOI:
10.1016/j.aim.2012.03.025 - 发表时间:
2012 - 期刊:
- 影响因子:1.7
- 作者:
David Favero - 通讯作者:
David Favero
David Favero的其他文献
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{{ truncateString('David Favero', 18)}}的其他基金
EAPSI: Investigating molecular connections between internal and external cues that affect seedling development
EAPSI:研究影响幼苗发育的内部和外部线索之间的分子联系
- 批准号:
1414471 - 财政年份:2014
- 资助金额:
$ 28万 - 项目类别:
Fellowship Award
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Research on Quantum Field Theory without a Lagrangian Description
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- 资助金额:0.0 万元
- 项目类别:省市级项目
Cell Research
- 批准号:31224802
- 批准年份:2012
- 资助金额:24.0 万元
- 项目类别:专项基金项目
Cell Research
- 批准号:31024804
- 批准年份:2010
- 资助金额:24.0 万元
- 项目类别:专项基金项目
Cell Research (细胞研究)
- 批准号:30824808
- 批准年份:2008
- 资助金额:24.0 万元
- 项目类别:专项基金项目
Research on the Rapid Growth Mechanism of KDP Crystal
- 批准号:10774081
- 批准年份:2007
- 资助金额:45.0 万元
- 项目类别:面上项目
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