Hodge Theory and Representation Theory

获奖：DMS 1309238，首席研究员：Colleen RoblesHodge理论提供了一些复杂代数变量的基本不变量，并在代数几何中产生了一些最深刻的结果。霍奇结构及其对称群MumfordñTate群的研究是复杂几何、表示理论和算术的交叉领域。提议的工作解决了霍奇理论和表征理论之间的关系，重点关注那些可以用复杂几何来描述的方面。（Hodge的代数变异理论将不被讨论：重点是作为独立兴趣对象的Hodge结构。）在经典情况下，Hodge域D是一个厄米对称空间，可以等价地嵌入到Siegel的上半部分空间中，Hodge理论与品种的几何和算术性质之间的关系是一个深入而广泛的研究课题。在非经典情况下，这一领域相当不发达，人们普遍认为，推广理论（算术性质、自同构形式、志村变分理论等）的主要障碍是我们对控制霍奇结构变化的微分方程系统的有限理解。这个项目的主要动机和目标是更好地理解这个系统，Hodge域D的复杂几何结构，以及非经典环境下的相关表示理论。霍奇理论处于几个数学学科的交叉点，包括代数几何、复几何、数论和表示论。这种丰富的融合使该学科成为一个丰富而有影响力的研究领域。研究Hodge结构的变化（本提案的主要主题）的潜在动机是了解代数变体的模。代数变量是多项式方程的解空间。理解和处理多项式方程组的解的能力在工程、科学和数学的许多领域都是必不可少的。

AbstractAward: DMS 1309238, Principal Investigator: Colleen RoblesHodge theory provides some of the basic invariants of a complex algebraic variety, and has yielded some of the deepest results in algebraic geometry. The study of Hodge structures and their symmetry groups, MumfordñTate groups, lies at the intersection of complex geometry, representation theory and arithmetic. The proposed work addresses the relationship between Hodge theory and representation theory, with a focus on those aspects that may be described using complex geometry. (The Hodge theory of algebraic varieties will not be addressed: the emphasis is on Hodge structures as objects of independent interest.) In the classical case that the Hodge domain D is a Hermitian symmetric space that may be equivariantly embedded in Siegel's upper-half space, the relation between Hodge theory and the geometric and arithmetic properties of a variety is a deep and extensively researched subject. The area is considerably less developed in the non-classical case, and it is generally felt that the principle obstacle to generalizing the theory (arithmetic properties, automorphic forms, theory of Shimura varieties, et cetera) is our limited understanding of the system of differential equations governing variations of Hodge structure. The principle motivation and objective of this project is to better understand this system, the complex geometry of the Hodge domain D, and the associated representation theory in the non-classical setting.Hodge theory lies at the crossroads of several mathematical subjects, including Algebraic Geometry, Complex Geometry, Number Theory and Representation Theory. This rich confluence makes the subject a fertile and influential area of research. The underlying motivation to study variations of Hodge structure (a dominant theme in this proposal) is to understand moduli of algebraic varieties. Algebraic varieties arise as the solution spaces to polynomial equations. The ability to understand and manipulate solutions of systems of polynomial equations is essential in many areas of engineering, science and mathematics.