Modular Forms and Topology

9870126 Hain称亏格g的黎曼曲面的模空间为亏格g的给定曲面上的所有共形结构的集合。Hain正在研究这种模空间的拓扑、算术和几何等各种问题。在亏格1(即，甜甜圈的边界)中，有一个在数论中非常重要的经典模形式理论。(事实上，这是安德鲁·怀尔斯在他的费马大定理证明中使用的主要工具之一。)目前，在高阶亏格上还没有很好的模形式理论。Hain的项目之一是寻找更高亏格中的模形式的自然例子，特别是那些推广了由判别式给出的权12的经典模形式的例子。他的基本工具是霍奇理论，特别是代数簇的基本群的理论。他进一步发展了基本变种群的Hodge理论，特别是那些与模形式和Galois理论有关的方面。亏格g的闭曲面是指有g个洞的圆环的边界的曲面。每个这样的表面都有额外的结构，通常称为保形结构或复杂结构。对表面和表面上这些丰富结构的研究可以追溯到19世纪中叶，当时伯恩哈德·里曼首次研究了它们。他们的研究仍然是数学中一个非常活跃和核心的领域。它在数论、拓扑学和几何学中都很重要。它也是弦理论的核心工具，弦理论是试图将广义相对论和量子力学结合起来的物理理论。***

9870126 Hain The set of all conformal structures on a given surface of genus g is called the moduli space of Riemann surfaces of genus g. Hain is studying various questions regarding the topology, arithmetic and geometry of such moduli spaces. In genus 1 (i.e., the boundary of a doughnut), there is a classical theory of modular forms that is of great importance in number theory. (In fact, it was one of the main tools used by Andrew Wiles in his proof of Fermat's Last Theorem.) At present, there is no good theory of modular forms in higher genus. One of Hain's projects is to find natural examples of modular forms in higher genus, especially ones that generalize the classical modular form of weight 12 given by the discriminant. His basic tool is Hodge theory, especially that of fundamental groups of algebraic varieties. He is further developing the Hodge theory of fundamental groups of varieties, especially those aspects related to modular forms and Galois theory. A closed surface of genus g is the surface bounding a doughnut with g holes. Each such surface has additional structure, often called a conformal structure or a complex structure. The study of surfaces and these enriched structures on them dates back to the middle of the 19th century, when they were first studied by Bernhard Riemann. Their study is still a very active and central area of mathematics. It is important in number theory, topology and geometry. It is also a central tool in String Theory, the physical theory that attempts to unite general relativity and quantum mechanics. ***