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.
This NSF grant supported a conference "Low-dimensional manifolds & high-dimensional categories" held at UC Berkeley, June 6-10, 2011. The bulk of the funds were spent supporting travel and lodging for the conference speakers and some graduate students and postdoctoral fellows. There were 19 talks (Dennis Sullivan had to cancel), and a panel discussion on the history of the 4-dimensional Poincare conjecture to replace the missing talk. Five of the talks were expository, aimed at the level of graduate students. David Gay spoke about Morse 2-functions, Cameron Gordon spoke about exceptional Dehn filling, Slava Krushkal discussed the A-B slice conjecture, Frank Quinn discussed the classification of topological 4-manifolds, Peter Teichner gave a primer on high-dimensional categories for low-dimensional topologists. The conference brought together experts on smooth 4-manifolds and TQFTs. For abstracts and videos of the talks, see the conference webpage: http://math.berkeley.edu/~ianagol/MF60/index.html There is a conference proceedings partially published by Geometry & Topology Monographs: http://msp.warwick.ac.uk/gtm/2012/18/ The videos of the talks (taken courtesy of David Auckley of MSRI) and the conference proceedings will contribute to the broader impact and dissemination of the conference talks. In the classification of smooth 4-manifolds, there has been a successful application of ideas from physics in the guise of Donaldson and Seiberg-Witten invariants to distinguish homemorphic smooth 4-manifolds. However, these invariants do not satisfy all of the axioms of a TQFT, which means that they cannot distinguish a homotopy 4-sphere from a standard 4-sphere. It is generally believed that there exist invariants called extended field theories, or just topological field theories, which are couched in the language of higher categories, which may be useful in distinguishing smooth homotopy 4-spheres or other 4-manifolds of small Euler characteristic. This conference brought together experts on these topics, and hopefully contributed to the education of students and postdocs who will lead the way in the future to unlocking the mysteries of these invariants in a way that will make them amenable to understanding 4-dimensional topology.