Proposal: DMS-9970654 Principal Investigator: Nikola Lakic
Abstract: Lakic's project is concerned with four closely related topics in Teichmuller theory: asymptotic Teichmuller spaces, vector fields for holomorphic motions, substantial boundary points, and the variability set of a Riemann surface. The techniques he intends to use arise from two recently developed theories, asymptotic Teichmuller theory and the Teichmuller theory of closed planar sets. New insights into the hyperbolic geometry of plane domains are expected to emerge as a by-product of the research.
Teichmuller spaces of closed sets have many applications to dynamics and hyperbolic geometry. The recently developed asymptotic Teichmuller spaces, for instance, serve as parameter spaces in the study of dynamical systems. Modern dynamical system theory has had a number of important applications, ranging from the life sciences and physics to economics. Several complicated biological phenomena (the rate of change in the population of a single species, to cite a concrete example) are modelled by nonlinear dynamical systems, making it imperative to understand the long-term behavior of these mathematical systems if knowledge of the corresponding bio-system is to be advanced. The research conducted under this award should provide dynamicists with some valuable tools for the study of various infinite dimensional dynamical systems.
