Homological Mirror Symmetry and Functional Equations
同调镜像对称和函数方程
基本信息
- 批准号:0070967
- 负责人:
- 金额:$ 18.38万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2000
- 资助国家:美国
- 起止时间:2000-06-15 至 2003-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Abstract.Homological mirror symmetry is a conjecture, formulated by M.Kontsevich, which asserts the equivalence of certain categories associated to complex and symplectic structures on mirror dual Calabi-Yau manifolds. The investigator proposes to work on this conjecture in the case of elliptic curves. His previous results obtained in collaboration with E.Zaslow and D.Arinkin justify some part of this conjecture. He proposes to apply these results to the study of indefinite theta series. Another direction of research proposed here is related to a new class of functional equations associated to prehomogeneous vector spaces over local fields. Prehomogeneous vector spaces and their zeta-functions were studied extensively by M.Sato and his school. The investigator proposes to work on certain ``diagonalization'' of functional equations for Sato's zeta-functions. The next stage of this research would be to relate the constants in these functional equations to local L-factors. This would allow to find a new class of integrals for which the stationary phase approximation is exact.The first part of this project is aimed at proving a conjecture which originated from mathematical physics. This conjecture, which was proposed by M.Kontsevich in 1994, is expected to explain the phenomenon of mirror symmetry discovered by physicists about a decade ago. This discovery (along with other similar dualities in string theory) is an example of recent developments in theoretical physics which still lack solid mathematical foundation. The present work should be considered as a contribution to laying such a foundation. The second part of this project is devoted to some problems arising from number theory. It was known already in the 19-th century that some deep properties of numbers are encoded in certain functions of complex variable called zeta-functions. The proposed work is devoted to the study of a new class of functional equations satisfied by zeta-functions which arise in representation theory.
摘要:同调镜像对称是一个猜想,由M.Kontsevich提出,它断言与镜像对偶卡-丘流形上的复结构和辛结构相关的某些范畴是等价的。 研究者建议在椭圆曲线的情况下研究这个猜想。他以前与E.Zaslow和D.Arinkin合作获得的结果证明了这个猜想的一部分。 他建议将这些结果的研究不确定θ系列。 这里提出的另一个研究方向是与局部域上的预齐次向量空间相关的一类新的函数方程。 准齐次向量空间和它们的zeta-函数被M.Sato和他的学派广泛研究。 调查人员建议工作的某些"对角化“的功能方程佐藤的zeta功能。 这项研究的下一个阶段将是将这些函数方程中的常数与局部L因子联系起来。 这将允许找到一类新的积分,其中固定相近似是精确的。该项目的第一部分旨在证明一个猜想,起源于数学物理。 这一猜想由M.Kontsevich于1994年提出,有望解释物理学家在大约10年前发现的镜像对称现象。 这一发现(沿着弦论中其他类似的对偶性)是理论物理学最新发展的一个例子,但它仍然缺乏坚实的数学基础。 目前的工作应被视为对奠定这样一个基础的贡献。 本项目的第二部分致力于数论中的一些问题。 早在19世纪,人们就知道数的某些深层性质被编码在某些称为ζ函数的复变函数中。 所提出的工作是专门研究一类新的功能方程满足zeta函数出现在表示论。
项目成果
期刊论文数量(0)
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专利数量(0)
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Alexander Polishchuk其他文献
De Rham cohomology for supervarieties
超簇的 De Rham 上同调
- DOI:
10.1007/s40879-024-00736-2 - 发表时间:
2024 - 期刊:
- 影响因子:0.6
- 作者:
Alexander Polishchuk - 通讯作者:
Alexander Polishchuk
A tribute to Sasha Beilinson
- DOI:
10.1007/s00029-018-0399-x - 发表时间:
2018-02-16 - 期刊:
- 影响因子:1.200
- 作者:
Michael Finkelberg;Dennis Gaitsgory;Alexander Goncharov;Alexander Polishchuk - 通讯作者:
Alexander Polishchuk
$${\mathbb A}^{0|1}$$
A
0
|
1
$${mathbb A}^{0|1}$$ A 0 |
- DOI:
10.1007/s00220-023-04769-8 - 发表时间:
2023 - 期刊:
- 影响因子:2.4
- 作者:
Alexander Polishchuk - 通讯作者:
Alexander Polishchuk
Schwartz $\kappa$-densities for the moduli stack of rank $2$ bundles on a curve over a local field
局部场曲线上的阶 $2$ 束的模堆栈的 Schwartz $kappa$-密度
- DOI:
- 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
A. Braverman;D. Kazhdan;Alexander Polishchuk - 通讯作者:
Alexander Polishchuk
Moduli spaces of nonspecial pointed curves of arithmetic genus 1
- DOI:
10.1007/s00208-017-1562-y - 发表时间:
2017-06-30 - 期刊:
- 影响因子:1.400
- 作者:
Alexander Polishchuk - 通讯作者:
Alexander Polishchuk
Alexander Polishchuk的其他文献
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{{ truncateString('Alexander Polishchuk', 18)}}的其他基金
Derived Categories, Noncommutative Orders, and Other Topics
派生范畴、非交换顺序和其他主题
- 批准号:
2001224 - 财政年份:2020
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
Moduli of A-Infinity Structures and Related Topics
A-无穷大结构的模及相关主题
- 批准号:
1700642 - 财政年份:2017
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
A-infinity structures and derived categories in algebraic geometry
代数几何中的 A-无穷大结构和派生范畴
- 批准号:
1400390 - 财政年份:2014
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
Derived categories techniques in algebraic geometry
代数几何中的派生范畴技术
- 批准号:
1001364 - 财政年份:2010
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
Complex geometry of noncommutative tori and t-structures on derived categories
派生范畴上非交换环面和 t 结构的复杂几何
- 批准号:
0601034 - 财政年份:2006
- 资助金额:
$ 18.38万 - 项目类别:
Continuing Grant
Topics in Algebraic Geometry, Non-commutative Geometry and Representation Theory
代数几何、非交换几何和表示论专题
- 批准号:
0527042 - 财政年份:2004
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
Topics in Algebraic Geometry, Non-commutative Geometry and Representation Theory
代数几何、非交换几何和表示论专题
- 批准号:
0302215 - 财政年份:2003
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
Mathematical Sciences: Sheaves on Witt Schemes and Trace Formula with Application to Representation Theory
数学科学:维特方案和迹公式及其在表示论中的应用
- 批准号:
9700458 - 财政年份:1997
- 资助金额:
$ 18.38万 - 项目类别:
Standard Grant
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