Mathematical Sciences: Discrete Groups, Quasiconformal Mappings, and Function Theoretic Inequalities

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Work to be done on this project focuses on questions arising at the interface of geometry and mathematical analysis in Euclidean spaces. The basis for the research lies within the geometric development of complex function theory, in that it emphasizes the group structure underlying analytic function theory and the infinitesimal behavior of such functions. It was their infinitesimal regularity that led to the notion of quasiconformal mappings in the 1930's. Decades passed before an intrinsic definition of these mappings was found which could be applied to mappings in higher dimensions. This work is about those mappings and their underlying domains. A quasiconformal mapping is a univalent mapping between two domains which maps infinitesimal circles into infinitesimal ellipses whose major and minor axes have bounded (the dilatation) ratios. An important question in the understanding of their structure concerns the decomposition of maps into iterates of maps with the same dilatation. Those with no such decomposition would form a basis of irreducible maps. From the geometric side, one would like to know how to determine when two domains are equivalent under quasiconformal mappings. Even when one of the two domains is a ball, this question is very difficult. The sufficient conditions given by Schoenflies type theorems and the necessary ones arising from local connectivity are very far apart. Some progress has been made in three dimensions when one of the domains is a slit domain and the other is a ball. Other work, concentrating on planar problems, will be concerned with Kleinian groups. This includes continuing work on measuring the distance of elements from the identity in several natural metrics. When applied to generators of groups, information about the singular set can be obtained. As this work develops further, higher dimensional analogues will be studied. Work will also continue on discrete convergence groups, a class of transformation groups introduced several years ago which reflect many properties of Mobius groups (they are conjugate to such groups by quasiconformal maps). Investigations into the fine structure of convergence groups will be conducted.