Mathematical Sciences: Deformations of Complex Structures

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9500557 Maskit This project is concerned with several different aspects of complex analysis and function theory: moduli of Riemann surfaces via Eichler cohomology of Kleinian groups, the Schottky problem, geometric formulation of 2-dimensional quantum gravity, and Hausdorff dimension and the boundary behavior of conformal mappings are among the topics proposed. Function theory is the study of functions of one independent complex variable, and has a classical origin. A notable aspect of function theory is the use of Riemann surfaces; given a locally defined complex function one associates an abstract (multi-sheeted) surface, called the Riemann surface of the function, via analytic continuation. This Riemann surface carries all the essential information about the original function and its possible regular extensions. The proposed project has to do with classifying and understanding the totality of such Riemann surfaces.