Deformations of Complex Structures

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Prof. Bishop will investigate several aspects of geometry and analysis related to non-smooth objects such as limit sets of Kleinian groups, Brownian motion and quasi-conformal mappings. Among the specific goals are to extend his earlier application of harmonic analysis and heat kernel techniques to compute the exact size (in terms of Hausdorff measures) of the limit sets of Kleinian groups to new examples (e.g., manifolds with thin parts) and new sets (the conical limit set, the set of escaping geodesics,...). He will also attempt to extend Bowen's famous dichotomy to all divergence type Fuchsian groups, extending work of Sullivan, Astala, Zinsmeister, Jones and himself. Part of this is to further study the ``beta''s, a technical measure of how close a set is to a line segment at different scales, which is closely related to the study of the traveling salesman problem. He will also continue his work investigating the removable and non-removable sets for quasi-conformal mappings, geometric properties of Brownian motion, and other geometric-analytic questions in classical complex analysis.The proposal deals with a variety of geometric questions which are united by a desire to understand traditional concepts and calculations (e.g. compute the area of a region) in non-traditional and highly non-smooth cases. This is motivated by (at least) two points of view. First, a better understanding of the classical case can be obtained by seeing how it succeeds or fails in a more general setting; even the failures lead to new, interesting phenomena to investigate. Second, many applications of mathematics (polymers, fractures, shock waves, wavelet analysis, crystal growth,...) involve highly irregular and fractal objects, and we need to understand and calculate with such objects the way we have always done with much smoother quantities. For example, Brownian motion is a mathematical object which models random movement and is of tremendous intrinsic interest, as well as being fundamental to understanding many other random processes (e.g., shapes of long polymers chains, growth of crystals by random accretion, ...). However, Brownian motion is an extremely non-smooth process and various simple geometric questions about its random paths are still unknown. Another example is with the Kleinian groups mentioned above. These are symmetries in non-Euclidean geometry; as such, they are fundamental objects in topology and have been an area of intense investigation for many years. They can be associated with certain fractal objects (the limit sets) involving non-linear rescalings, where the non-linearity is of a very special type, and present a theoretical stepping stone between self-similar sets with linear rescalings (which are fairly well understood) and much more complicated rescalings found in rational dynamics and chaos theory.