Discrete Groups in Algebraic Geometry

研究员将研究群论在代数几何中的应用。 更具体地说，他将研究代数几何对象的模空间，这些模空间可以用作用于厄米对称空间的离散群来理解。 他的工具将包括算术性质的团体和建设自守形式的对称spaces.Group理论是研究对称性，和例子离散群体的对称性的化学或矿物晶体结构。 模空间是在考虑一些几何对象（比如空间中的曲面）时产生的，如果其中两个对象只有一些微小的不同，比如视角的改变，那么它们就被认为是相同的。 令人惊讶和深刻的是，离散群可以帮助人们理解模空间，但许多不同的人已经发现了许多美丽的联系。 研究人员将寻找更多的这些联系，并试图更好地了解那些已经知道的。

The investigator will study applications of group theory to algebraic geometry. More specifically, he will study those moduli spaces of algebraic-geometric objects that can be understood in terms of discrete groups acting on Hermitian symmetric spaces. His tools will include the arithmetic properties of the groups and the construction of automorphic forms on the symmetric spaces.Group theory is the study of symmetry, and examples of discrete groups are the symmetries of a chemical or mineral crystal structure. Moduli spaces arise when one considers some geometric objects (say, surfaces in space), and considers two of them to be the same if they differ only in some minor way, such as a change in point of view. It is surprising and deep that discrete groups can help one understand moduli spaces, but many beautiful connections have been discovered by many different people. The investigator will look for more of these connections and try to better understand the ones already known.