Each area of investigation relates to geometric questions in the theory of functions of complex and real variables. Functions of interest include both holomorphic and quasiconformal mappings. This work includes a continuing effort to find simple geometric conditions that can be used to determine if a domain in Euclidean space can support the Poincare inequality. This is actually a class of inequalities measuring the integral of (the absolute value of) a function against the integral of a power of the function's gradient. The inequality provides information as to whether or not certain boundary value problems in partial differential equations can be solved on the given domain. The best results to date have been obtained for two-dimensional domains. Evidence suggests that any general criterion will involve the behavior of a particular non-Euclidean metric attached to a domain. Work is also planned on the influence of differentiation on the distribution of roots of entire functions. More specifically, real entire functions having only real roots are under consideration. It is not true that roots of the derivative must also be real, and one main objective of this work is to measure the number of non-real roots that one can expect on successive differentiation. Ultimately, one would like to get asymptotic estimates on the number of roots, to attack a long-standing conjecture of Polya. A third area of investigation concerns spaces of holomorphic functions which are area integrable - Bergman spaces. Analysis of the mapping properties of such functions is pertinent to the theory of quasiconformal mappings. Many such mappings may share the same boundary values. Among them will be maps which have extreme dilatations. A characterization of the extremal mappings has never been obtained. This work seeks to use the Bergman spaces to show that for nonconstant extreme dilatations, the argument of such functions cannot be too limited.
