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。研究者将通过三个项目来探索一般理论的基本方面和重要的例子：确定希钦颤振的支持，建立和解释希格斯束模空间上同调的某些显著的隐藏对称性；识别和研究在有限域上分解定理成立的映射类；并通过将分解定理中的支撑点与弧的特殊子变体联系起来，建立了两族映射与弧空间的关系。

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 three projects: to determine the supports of Hitchin fibrations and to establish and explain certain remarkable hidden symmetries on the cohomology of the moduli spaces of Higgs bundles; to identify and study classes of maps for which the decomposition theorem holds over a finite field; and to establish a relation between birational maps and arc spaces by linking the supports in the decomposition theorem with special subvarieties of arcs.