Mathematical Sciences: "Dynamics, Hyperbolic Geometry and Quasiconformal Maps"

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The principal investigator will analyze how topology determines geometry in several settings. He will study deformation theory for open 3-manifolds, geometrization of automorphisms of free groups, and parallels between renormalization theory and hyperbolic 3-manifolds which asymptotically fiber over a circle. A central theme in modern dynamical systems is the restriction topology puts on allowable dynamics and geometry. For example, on the surface of a sphere, every vector field must have a singularity. On the earth, this means that the wind cannot blow everywhere at once; there must be at least one spot where the air does not move. In this vein, the principal investigator will develop relationships between complex dynamics and the geometry of three-dimensional hypersurfaces.