Date & Time: Thursday, January 15, 2009, 14:30-15:30.
Venue: Ramanujan Hall
Title: Holomorphic Motions and Holomorphic Families of Möbius Groups
Speaker: Sudeb Mitra, Queens College of CUNY
Abstract: In their study of the dynamics of rational maps, Mãné, Sad, and Sullivan introduced the idea of holomorphic motions. Since then, holomorphic motions have found several interesting applications in Teichmüller theory, complex dynamics, and Kleinian groups.
A central topic in the study of holomorphic motions is the question of extending a holomorphic motion. In this talk, we will show that a holomorphic motion of a closed set in the Riemann sphere, defined over infinite dimensional parameter spaces, can be extended to a quaiconformal motion of the sphere, in the sense of Sullivan and Thurston. Furthermore, if the holomorphic motion has a certain group equivariance property, the extended quasiconformal motion will also have the same property. We will then discuss a generalization of a theorem of Bers on holomorphic families of isomorphisms of Möbius groups. We will also outline some properties of quasiconformal motions of the Riemann sphere.
This talk will highlight the beautiful relationship between holomorphic motions, Teichmüller spaces, and Kleinian groups. All basic definitions will be given. The first part of this talk will be mainly expository, and the second part will be based on a recently completed joint work with Professor Hiroshige Shiga.