Quasiconformal Symmetries, Extremal Problems, and Patterson-Sullivan Theory

Wesleyan University, Middletown CT

Investigators

Abstract

The mathematics in this proposal lies at the intersection of geometric analysis, geometry, and low-dimensional topology. In particular, the researchers use geometric function theory in the form of the theory of quasiconformal mappings to probe analytic symmetries of hyperbolizable surfaces and $n$-manifolds. With equal emphasis, this proposal also details analytic questions in the theory of quasiconformal mappings. In particular the researchers tie together certain geometric invariants to the conjectured solution in dimensions three and above of the Teichmuller extremal problem. Another major theme in their work involves the interaction between dynamics and geometry as illuminated by Patterson-Sullivan theory. The researchers describe projects that study the generalization of the Patterson-Sullivan theory of Kleinian groups to both the setting of convex co-compact subgroups of the modular group, and to the purely analytic setting of discrete quasiconformal groups. The nexus of hyperbolic geometry, conformal analysis, and low-dimensional topology is a vast and fundamental area of study in mathematics. It dates back to the 19th century, when it was developed by such mathematicians as Gauss, Lobachevsky, Klein, and Poincare. These fields remain vital, as attested by the recent epochal results of G. Perelman on the Poincare and Geometrization Conjectures. As mathematics is inherently interconnected, surprising and beautiful applications are often found at the interfaces of mathematical fields. Recently, hyperbolic geometry has found application in the study of discrete geometry and machine vision. Further, in physics both hyperbolic geometry and conformal analysis (especially in the guise of Teichmuller theory) have become a standard tool in the exploration of theoretical physics and cosmology. Wesleyan University has a strong dual identity as a research and teaching institution, and the proposers are strongly dedicated to innovation in both research and education. A core objective of the researchers is to use the broad and integrative relationship between hyperbolic geometry and geometric analysis to increase both the interest and strength of those advanced undergraduate and beginning graduate students enrolled in graduate analysis and geometry courses at Wesleyan University.