Mathematical Sciences: Deformations of Complex Structures

两名研究人员将研究克莱恩群和泰希穆勒空间。其中一个将分析变形空间和Klein群的上同调，黎曼曲面的模及其显式表示，自同构形和射影结构。第二部分将继续他对Klein群和双曲3-orbilold的结构的研究，重点是几何有限群的分类和组合定理。这两名调查人员将致力于解决与克莱恩群体研究相关的几个问题。这样一个组的一个例子是保持周期性平铺的平面运动的集合。作为群的元素，它们必须满足几个公理，其中一个公理要求存在一个与群的每个元素相关联的逆元素。如果一个元素将平面向“右”平移两个单位，则其反元素会将平面向“左”平移两个单位。对这种运动的研究引导数学家们解决了涉及晶体结构的问题。

Two investigators will study Kleinian groups and Teichmuller spaces. One will analyze deformation spaces and cohomology of Kleinian groups, moduli of Riemann surfaces and their explicit representations, automorphic forms, and projective structures. The second will continue his investigation into the structure of Kleinian groups and hyperbolic 3-orbifolds emphasizing the classification of geometrically finite groups and combination theorems. The two investigators will work toward solving several problems related to the study of Kleinian groups. An example of such a group would be a collection of motions of the plane which preserve a periodic tiling. As elements of a group these must satisfy several axioms including one which requires the existence of an inverse element associated with each element of the group. If one element translates the plane by two units to the "right," its inverse element would translate the plane two units to the "left." The study of such motions has led mathematicians to the solutions of problems involving crystal structures.