Discrete Groups in Algebraic Geometry

研究者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。他的工具将包括群的算术性质和对称空间上自同构形式的构造。群论是对对称性的研究，离散群的例子是化学或矿物晶体结构的对称性。模空间的出现是当一个人考虑一些几何物体（比如空间中的曲面），并认为它们中的两个是相同的，如果它们只是在一些很小的方面不同，比如观点的改变。离散群可以帮助人们理解模空间，这是令人惊讶和深刻的，但许多不同的人已经发现了许多美丽的联系。研究者将寻找更多这样的联系，并试图更好地理解那些已知的联系。

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.