LOW-DIMENSIONAL MANIFOLDS AND HIGH-DIMENSIONAL CATEGORIES

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AbstractAward: DMS-1041217Principal Investigator: Ian Agol, Robion C. KirbyThis award will provide funding for a conference "Low-dimensionalmanifolds and high-dimensional categories". The title is somewhatironic, in that the low and high dimensions are in fact the same:3 and especially 4. The conference celebrates the occasion ofMichael Freedman's 60th birthday and some of the currentmathematical achievements and challenges influenced by Freedman'swork. For manifolds of dimension 5 and higher, there is awell-established classification scheme, the so called surgerytheory which includes the celebrated s-cobordism theorem thatproduces diffeomorphisms between manifolds from homotopytheoretic data. Dimensions below 5 are more difficult becausethere is less room to maneuver and, as a consequence, thes-cobordism theorem fails. For n-categories, dimensions ngreater than 2 are difficult because of the complexity of thecombinatorial relations which describe the various ways n-ballscan be cut, glued and rotated. Thus dimensions 3 and 4 representa shared frontier of the two subjects, though this frontier isapproached from different directions. One of the main goals ofthis conference is to promote cross-fertilization between expertson 4-manifolds and experts on higher categories and quantum fieldtheories.Michael Freedman made groundbreaking contributions to the studyand classification of 4-dimensional spaces (manifolds), which area central topic in the study of topology, and have connectionswith algebra, geometry and physics. Ideas from physics, inparticular quantum field theory, imply that there ought to becertain constructions which describe the topology of4-dimensional spaces, that is, intrinsic properties which do notdepend on the measure of length or angles on the space.Mathematically, these theories are formulated in the algebraiclanguage of category theory and require substantial newdevelopments in that area. The object of the conference is tobring together experts on 3- and 4-dimensional manifolds togetherwith experts in category theory and quantum field theory toexplore the interactions between these topics. Additionalgeometric structures, such as broken Lefschetz fibrations,contact and symplectic structures, smooth structures, and gaugetheories will also be explored at the conference.