LOW-DIMENSIONAL MANIFOLDS AND HIGH-DIMENSIONAL CATEGORIES

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-1041217 Principal Investigator: Ian Agol, Robion C. Kirby This award will provide funding for a conference "Low-dimensional manifolds and high-dimensional categories". The title is somewhat ironic, in that the low and high dimensions are in fact the same: 3 and especially 4. The conference celebrates the occasion of Michael Freedman's 60th birthday and some of the current mathematical achievements and challenges influenced by Freedman's work. For manifolds of dimension 5 and higher, there is a well-established classification scheme, the so called surgery theory which includes the celebrated s-cobordism theorem that produces diffeomorphisms between manifolds from homotopy theoretic data. Dimensions below 5 are more difficult because there is less room to maneuver and, as a consequence, the s-cobordism theorem fails. For n-categories, dimensions n greater than 2 are difficult because of the complexity of the combinatorial relations which describe the various ways n-balls can be cut, glued and rotated. Thus dimensions 3 and 4 represent a shared frontier of the two subjects, though this frontier is approached from different directions. One of the main goals of this conference is to promote cross-fertilization between experts on 4-manifolds and experts on higher categories and quantum field theories. Michael Freedman made groundbreaking contributions to the study and classification of 4-dimensional spaces (manifolds), which are a central topic in the study of topology, and have connections with algebra, geometry and physics. Ideas from physics, in particular quantum field theory, imply that there ought to be certain constructions which describe the topology of 4-dimensional spaces, that is, intrinsic properties which do not depend on the measure of length or angles on the space. Mathematically, these theories are formulated in the algebraic language of category theory and require substantial new developments in that area. The object of the conference is to bring together experts on 3- and 4-dimensional manifolds together with experts in category theory and quantum field theory to explore the interactions between these topics. Additional geometric structures, such as broken Lefschetz fibrations, contact and symplectic structures, smooth structures, and gauge theories will also be explored at the conference.