Synchronizing Smooth and Topological 4-Manifolds (16w5145)


(Indiana University)

(Michigan State University)

(Columbia University)

(Durham University)


The Banff International Research Station will host the "Synchronizing Smooth and Topological 4-Manifolds" workshop from February 21st to February 26th, 2016.

A significant rift in the field of 4-manifolds arises from the equally deep, yet starkly different approaches taken in the study of topological and smooth 4-manifolds. Both approaches originated in the early 1980's, with the work of Freedman and Donaldson, respectively. The former draws on delicate infinite constructions used to analyze the disc embedding problem for topological 4-manifolds, while the latter has roots in geometric analysis and, in particular, in the study of moduli spaces of solutions to non-linear PDEs. Due to the depth of both approaches, researchers in the respective areas quickly diverged, and very little communication took place -- or was even possible -- until recently. Recent and significant advances drive our understanding of both the topological and smooth concordance group of knots. The topological concordance group experienced advances through the calculus of gropes, a deeper understanding of duality, through the introduction of analytical invariants via Cheeger-Gromov theory, and an organized blending of these ideas through filtrations on the topological concordance group of knots. As our understanding of topological concordance of knots progressed, Ozsváth and Szabó developed their Heegaard Floer homology for 3-manifolds. Cobordisms between 3-manifolds induce maps on the Heegaard-Floer homology. Similarly a knot concordance induces maps on the related knot Floer homology. Obstructions extracted from this have proven extremely powerful. We now see these significant knot theoretic advances merging, providing a bridge to unite these previously disparate directions in 4-manifold theory. This workshop will bring together a diverse collection of researchers, with a particular emphasis on bridging a chasm between the knowledge bases of researchers studying 4-manifolds in the topological and smooth categories, respectively.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).