Mathematical Sciences: Deformations of Complex Structures

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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.