Hodge Theory and Representation Theory

摘要奖：DMS 1309238，首席研究员：Colleen RoblesHodge理论提供了复杂代数簇的一些基本不变量，并在代数几何中产生了一些最深刻的结果。研究Hodge结构及其对称群，即MumfordñTate群，是复杂几何、表示论和算术的交叉学科。这项拟议的工作解决了霍奇理论和表示理论之间的关系，重点放在那些可以用复杂几何描述的方面。(霍奇的代数变种理论将不被讨论：重点是将霍奇结构作为独立感兴趣的对象。)在Hodge域D是可等变嵌入Siegel上半空间的Hermite对称空间的经典情况下，Hodge理论与簇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。这一领域在非经典情况下的发展相对较少，人们普遍认为推广该理论(算术性质、自同构形、Shimura变种理论等)的主要障碍是我们对控制Hodge结构变化的微分方程组的有限理解。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的复几何，以及非经典环境中的相关表示理论。Hodge理论位于几个数学学科的十字路口，包括代数几何，复几何，数论和表示论。这种丰富的融合使这一主题成为一个肥沃和有影响力的研究领域。研究霍奇结构变化的潜在动机(这一提议中的一个主要主题)是为了理解代数簇的模数。代数族作为多项式方程的解空间出现。在工程、科学和数学的许多领域，理解和处理多项式方程组的解的能力是必不可少的。

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.