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。这个研究项目的目标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。研究者计划继续对代数映射的拓扑、Hodge理论和循环理论进行长期的研究。研究的两条主线之间的紧密联系，即发现一般理论的新的和深刻的方面和研究基本的例子，是工作背后的激励原则。预计这些结果将对数学家（代数几何、组合学和表示理论）和数学物理学家（弦理论）有帮助。研究者将通过五个项目探索一般理论的基本方面以及重要的例子：确定一般线性群情况下的希钦振动的支持；对于每一个约化群，曲线上希格斯束的模空间紧化；提出了一种反常截断和专门化理论，并应用于Hitchin系统的代数拓扑；通过Ngo支持定理，确定唯一已知的剩余不可约辛流形的Betti数和Hodge数；在希格斯束模空间的上同调上建立和解释某些显著的隐藏对称性。该奖项反映了美国国家科学基金会的法定使命，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。

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.