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The Topology and Hodge Theory of Algebraic Maps

代数图的拓扑和霍奇理论

基本信息

批准号：
1901975
负责人：
Mark Andrea de Cataldo
金额：
$ 30万
依托单位：
SUNY at Stony Brook
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901975&HistoricalAwards=false
关键词：
Topology Hodge Theory Algebraic Maps

项目摘要

The award supports research in the field of algebraic geometry, the discipline devoted to the study of polynomial or algebraic equations. Algebraic equations are both beautiful and ubiquitous, as they describe many natural phenomena, from the motion of planets to the shape of leaves and flowers, to the behavior of microscopic particles. The goal of this research project is to study the deeper properties of the solutions to more complicated algebraic equations (called algebraic maps). The investigator plans to continue long-term investigation of the topology, Hodge theory, and cycle theory of algebraic maps. The close connection between the two main threads of the research, namely the discovery of new and deep aspects of the general theory and the study of fundamental examples, are the motivating principle behind the work. It is anticipated that the results will be of use to mathematicians (in algebraic geometry, combinatorics, and representation theory) and to mathematical physicists (in string theory). The investigator will explore the fundamental aspects of the general theory as well as important examples through five projects: to determine the supports of Hitchin fibrations in the case of the general linear group; to compactify the moduli space of Higgs bundles on a curve for every reductive group; to initiate a theory of perverse truncation and specialization with application to the algebraic topology of Hitchin systems; to determine, via the Ngo support theorem, the Betti and Hodge numbers of the only known remaining irreducible symplectic manifold for which these are not known; to establish and explain certain remarkable hidden symmetries on the cohomology of the moduli spaces of Higgs bundles.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.

该奖项支持代数几何领域的研究，该学科致力于研究多项式或代数方程。代数方程既美丽又无处不在，因为它们描述了许多自然现象，从行星的运动到树叶和花朵的形状，再到微观粒子的行为。这个研究项目的目的是研究更复杂的代数方程(称为代数映射)的解的更深层次的性质。研究人员计划继续对代数映射的拓扑学、霍奇理论和圈理论进行长期的研究。研究的两条主线之间的密切联系，即发现一般理论的新的和更深层次的方面，以及对基本实例的研究，是这项工作背后的激励原则。预计这些结果将对数学家(在代数几何、组合学和表示论中)和数学物理学家(在弦论中)有用。研究人员将通过五个项目来探索一般理论的基本方面和重要的例子：确定一般线性群情况下的Hitchin纤颤的支撑性；对每个约化群紧致曲线上的Higgs丛的模空间；应用于Hitchin系统的代数拓扑，建立倒截断和特殊化理论；通过Ngo支撑定理，确定已知的唯一未知的不可约辛流形的Betti数和Hodge数；建立和解释Higgs丛的模空间上同调的某些显著的隐藏对称性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。