Modular Forms and Topology

9870126海恩 给定亏格g的曲面上的所有共形结构的集合称为亏格g的黎曼曲面的模空间。 海恩正在研究各种问题的拓扑结构，算术和几何等模空间。 在属1（即，一个甜甜圈的边界），有一个经典的理论，模形式是非常重要的数论。 (In事实上，它是安德鲁·怀尔斯证明费马大定理的主要工具之一。） 目前，高等属的模式还没有一个好的理论。 海恩的计划之一是寻找更高亏格的模形式的自然例子，特别是推广经典模形式的判别式给出的重量12。 他的基本工具是霍奇理论，特别是基本群体的代数品种。 他进一步发展霍奇理论的基本群体的品种，特别是那些方面有关的模块形式和伽罗瓦理论。 亏格g的闭曲面是包围一个有g个孔的圆环的曲面。每个这样的表面具有附加结构，通常称为共形结构或复杂结构。 对曲面和曲面上这些丰富结构的研究可以追溯到19世纪中叶，当时Bernhard Riemann首次研究了曲面。 他们的研究仍然是一个非常活跃和中心领域的数学。它在数论、拓扑学和几何学中很重要。 它也是弦论的核心工具，弦论是一种试图 统一广义相对论和量子力学 ***

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. ***