The Topology and Hodge Theory of Algebraic Maps

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，从行星的运动或叶子和花朵的形状，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，即发现一般理论的新的和更深层次的方面和研究基本实例之间的密切联系，是这项工作背后的激励原则。研究人员将通过三个项目探索一般理论的基本方面和重要的例子：在莫尔斯理论和任意定义域上的代数几何的背景下，发展一种更灵活的可构造层理论；通过引入正特征域上的代数几何的新几何结果，为P=W猜想寻找新的证据；并研究Higgs丛和域上具有正特征的平坦联络的模空间的结构作为临界轨迹，并应用于这些模空间的结构和上同调。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。

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 or 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 the 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, is 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 the study of string theory.The investigator will explore the fundamental aspects of the general theory as well as important examples through three projects: to develop a more flexible theory of constructible sheaves in the contexts of Morse theory and of algebraic geometry over arbitrary fields of definition; to seek new evidence to the P=W Conjecture--one of the leading open questions in the geometry of Higgs bundles and flat connections on algebraic curves--by importing new geometric results from algebraic geometry over fields of positive characteristic; and to study the structure of moduli spaces of Higgs bundles and flat connections over fields of positive characteristic as critical loci, with applications to the structure and cohomology of these moduli spaces.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.