Spectra, gaps, degenerations and cycles
光谱、间隙、简并和循环
基本信息
- 批准号:1201475
- 负责人:
- 金额:$ 24.3万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-07-01 至 2015-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The PIs will continue their foundational work on Homological Mirror Symmetry (HMS), to develop the structures and theories involved in HMS, and to build on applications of these theories. One notable direction is the theory of higher symplectic structures, which brings out the duality between the ``stacky'' directions and the ``derived'' directions of most moduli problems in algebraic geometry. Exploiting the full depth of these structures will require a careful study of the moduli of various kinds of categorical and higher categorical entities. This is one of the main areas of expertise of all the PIs. Kontsevich introduced one of the main tools, derived schemes. Katzarkov came up with the idea that moduli of LG models and its monodromy can be interpreted as stability conditions and spectra.These activities fit into a more general and global philosophy designed to accompany Geometry in the 21st Century. The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However,these objects, of various kinds, were uniformly viewed somehow as ``sets of points''. Along the way, the relationship with categorical structures grew steadily, leading to the many inputs into our program as discussed above. The PIs themselves played a pivotal role in much of the progress that was made at the turn of the century. With PI Kontsevich's introduction of HMS, a subtle change was introduced, in that ``Geometry'' began to be seen within a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as ``sets of points'' but rather as objects enclosing a higher structure. This project is highly connected with theoretical physics. As we head into the second decade of the 21st Century, elementary particle physics is on the crux of a profound revolution to be brought about by the new experimental results coming out of the LHC at CERN. These will serve to identify which of the multitude of theoretical possibilities which are currently open, best address quantum field theory at the high energy scale. And for those theories, to tell which are the right parameters. So there will soon be a lot of work to do on the theoretical side, and this will surely require new tools and a new approach. With the relationship between HMS and supersymmetric theories, with the relationship between higher categories and TQFT, with the relationship between partition functions and nonabelian cohomology, the kinds of geometrical objects which we are going to investigate in this project are becoming crucial for understanding these new panoramas in theoretical physics. The project has an educational component - conferences and educating postdocs, This component has been hugely successful in the past and with more funding we plan to bring it to the next level.
PI将继续他们在同调镜像对称(HMS)方面的基础工作,发展HMS中涉及的结构和理论,并建立在这些理论的应用基础上。一个值得注意的方向是理论的高级辛结构,它带来了“堆叠”的方向和“导出”的方向之间的对偶性的大多数模问题在代数几何。要充分利用这些结构的深度,就需要仔细研究各种范畴实体和更高范畴实体的模。这是所有首席执行官的主要专业领域之一。Kontsevich介绍了一个主要的工具,派生计划。 Katzarkov提出了LG模型的模及其单值性可以被解释为稳定性条件和谱的想法。这些活动适合于一个更普遍和全球性的哲学,旨在伴随几何学在21世纪。世纪的几何学研究在很大程度上致力于几何对象的分类和参数化,并取得了惊人的成功。然而,这些对象,各种各样的,被一致地视为某种“点集”。沿着这条路,与范畴结构的关系稳步增长,导致我们的程序中有许多输入,如上所述。在世纪之交取得的许多进步中,PI本身发挥了关键作用。随着PI Kontsevich对HMS的介绍,一个微妙的变化被引入,在“几何”中开始被视为一个范畴结构。与此同时,更高层次理论的发展意味着几何结构不再被视为“点的集合”,而是被视为包围更高层次结构的对象。该项目与理论物理学密切相关。当我们进入21世纪的第二个十年时,基本粒子物理学正处于一场深刻革命的关键,这场革命将由欧洲核子研究中心(CERN)大型强子对撞机(LHC)的新实验结果带来。这些将有助于确定目前开放的众多理论可能性中的哪一个,最好地解决高能尺度下的量子场论。对于这些理论,要知道哪些是正确的参数。因此,在理论方面很快就会有很多工作要做,这肯定需要新的工具和新的方法。 随着HMS和超对称理论之间的关系,随着更高范畴和TQFT之间的关系,随着配分函数和非交换上同调之间的关系,我们将在这个项目中研究的几何对象的种类对于理解理论物理中的这些新的几何对象变得至关重要。该项目有一个教育组成部分-会议和教育博士后,这个组成部分在过去取得了巨大的成功,随着更多的资金,我们计划把它带到一个新的水平。
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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Ludmil Katzarkov其他文献
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves THANKSREF="*" ID="*" DA was partially supported by NSF grant DMS-0244844. LK was partially supported by NSF grant DMS-0600800 and NSA grant H98230-04-1-0038. DO was partially supported by the Weyl Fund, the Civilian Research Development Foundation (CRDF grant No. RUM1-2661-MO-05), the Russian Foundation for Basic Research (No. 05-01-01034), and the Russian Science Support Foundation.
- DOI:
10.1007/s00222-006-0003-4 - 发表时间:
2006-07-11 - 期刊:
- 影响因子:3.600
- 作者:
Denis Auroux;Ludmil Katzarkov;Dmitri Orlov - 通讯作者:
Dmitri Orlov
Discriminants and toric emK/em-theory
判别式与环面 emK/em 理论
- DOI:
10.1016/j.aim.2024.109831 - 发表时间:
2024-09-01 - 期刊:
- 影响因子:1.500
- 作者:
R. Paul Horja;Ludmil Katzarkov - 通讯作者:
Ludmil Katzarkov
Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms
关于具有移位辛形式的导出概型上拉格朗日分布的严格化和胶合
- DOI:
10.1016/j.aim.2023.109477 - 发表时间:
2024-02-01 - 期刊:
- 影响因子:1.500
- 作者:
Dennis Borisov;Ludmil Katzarkov;Artan Sheshmani;Shing-Tung Yau - 通讯作者:
Shing-Tung Yau
Generalized toric varieties, LVMB manifolds and Lie groupoids
- DOI:
10.1007/s40879-024-00769-7 - 发表时间:
2024-09-27 - 期刊:
- 影响因子:0.500
- 作者:
Matheus Silva Costa;Lino Grama;Ludmil Katzarkov - 通讯作者:
Ludmil Katzarkov
Shifted symplectic structures on derived Quot-stacks II – derived emQuot/em-schemes as dg manifolds
导出商栈上的移位辛结构 II——作为 dg 流形的导出 emQuot/em 概型
- DOI:
10.1016/j.aim.2024.110092 - 发表时间:
2025-02-01 - 期刊:
- 影响因子:1.500
- 作者:
Dennis Borisov;Ludmil Katzarkov;Artan Sheshmani - 通讯作者:
Artan Sheshmani
Ludmil Katzarkov的其他文献
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{{ truncateString('Ludmil Katzarkov', 18)}}的其他基金
FRG: Collaborative Research: New Birational Invariants
FRG:合作研究:新的双理性不变量
- 批准号:
2245171 - 财政年份:2023
- 资助金额:
$ 24.3万 - 项目类别:
Continuing Grant
Conference on Homological Mirror Symmetry
同调镜像对称会议
- 批准号:
2001614 - 财政年份:2020
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
Categorical Kahler Geometry and Applications
分类卡勒几何及其应用
- 批准号:
2001319 - 财政年份:2020
- 资助金额:
$ 24.3万 - 项目类别:
Continuing Grant
Homological Mirror Symmetry Conference Miami 2015
2015 年迈阿密同调镜像对称会议
- 批准号:
1502578 - 财政年份:2015
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
Homological Mirror Symmetry and Categorical Linear Systems
同调镜像对称和分类线性系统
- 批准号:
1502162 - 财政年份:2015
- 资助金额:
$ 24.3万 - 项目类别:
Continuing Grant
Homological Mirror Symmetry MIAMI, Jan 27- Feb 1, 2014
同调镜像对称迈阿密,2014 年 1 月 27 日至 2 月 1 日
- 批准号:
1404779 - 财政年份:2014
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
Homological Mirror Symmetry Conference Miami
迈阿密同调镜像对称会议
- 批准号:
1303069 - 财政年份:2013
- 资助金额:
$ 24.3万 - 项目类别:
Standard 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:
1265230 - 财政年份:2013
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
Pan American Advanced Studies Institute on Wall Crossing, Stability Hodge Structures and TQFT- Natal, Brazil
泛美跨墙、稳定性 Hodge 结构和 TQFT 高级研究所 - 巴西纳塔尔
- 批准号:
1242272 - 财政年份:2012
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
Geometry and Physics Miami - Brazil - Mexico - Conference
几何与物理迈阿密 - 巴西 - 墨西哥 - 会议
- 批准号:
1201544 - 财政年份:2012
- 资助金额:
$ 24.3万 - 项目类别:
Standard Grant
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