This collaborative proposal details a program that the researchers are pursuing to extract geometric, analytic and dynamic information concerning discrete groups of quasiconformal mappings acting on a compact metric space. The researchers' program consists of three ``themes'': The first theme concerns certain geometric properties of Kleinian groups that are induced from the existence of the ergodic Liouville measure on the unit tangent bundle of a geometrically finite hyperbolic manifold. A primary goal is to pair Liouville theory with quasiconformal deformation theory to the extent that it is revealing of the geometry of the deformation space of hyperbolic structures uniformizing a compact co-infinite hyperbolizable three-manifold. The second theme details an ongoing program to show that various analytical, dynamical and topological properties of Kleinian groups (e.g. the exponent of convergence, Hausdorff dimension, porosity, etc.) illuminate the action of the group when the group in question is generalized to a discrete convergence group. In this setting the researchers are primarily concerned with discrete quasiconformal groups. The third theme centers around certain non-equivariant interactions between complex analysis and hyperbolic geometry; it generalizes the first two themes. The motivating question is: How does the asymptotic geometry of the convex hull boundary of a Jordan curve reflect the conformal geometry of the curve? Under various stronger assumptions the researchers are investigating aspects of this question from both a geometric and analytic perspective.
The research described in this proposal resides at the intersection between hyperbolic geometry and conformal analysis. These topics form a vast and fundamental area of study in mathematics which dates back to the 18th century, when it was developed by such mathematicians as Gauss, Lobachevsky, Klein, and Poincare. Though this proposal does not directly address physics, we note that both hyperbolic geometry and conformal analysis (especially in the guise of Teichmuller theory) have found recent spectacular application in theoretical physics and cosmology. The proposers are dedicated researchers and educators, and Boston College and Wesleyan University both have strong dual identities as teaching and research institutions. An innovative component of the proposal is to use its multi disciplinary nature to introduce motivated undergraduates to areas of mathematics that are currently not well represented (e.g. geometry and analysis) in the undergraduate degree programs at the proposers' respective institutions. The researchers propose to initiate and maintain a ``Joint Boston College - Wesleyan University Working Groups for Undergraduates and Beginning Graduate Students,'' in part to entice talented undergraduates into consideration of a career in mathematical research.
