This conference will focus on connections between main approaches to low dimensional topology. Great successes have come from the classical topological 3-manifold program (surfaces, foliations and laminations), from the Ricci flow, Floer homology, and from algebraic invariants in knot theory and topological field theory. In dimension 4 the Donaldson, Seiberg-Witten, and Oszvath-Szabo theories have revealed structure of smooth manifolds. However these different threads remain largely unconnected and, with the notable exception of geometrization of 3-manifolds, few general structural features have emerged. This is in contrast with high dimensional topology where great advances between 1950-1980 revealed deep and relatively uniform structure dominated by homotopy theory, bundles, and stable algebra (K and L theory). We hope that a spirited discussion among a range of experts and new researchers will help reveal the deep structure of low dimensions. Can the Ricci flow be coupled to a 2-form or other gauge-theoretic object on a 4-manifold? Do nonsingular flows with special dynamics provide a ``singular homology'' analog of the Oszvath-Szabo ``cellular'' Floer homology? Do special metrics associated to foliations and laminations have implications for the Ricci flow? Taubes' relation between Seiberg-Witten and Gromov invariants of symplectic manifolds raised hopes that geometric representatives for Seiberg-Witten could be found more generally. So far this has been unsuccessful but perhaps it is time to revisit the question. We look forward to interactions among a wide range of experts and young researchers with fresh viewpoints.
Topology aims to understand the structure of objects that locally look like the ordinary Euclidean space but whose global shape may be rather complicated. Work in the early 20th century focused on dimensions 2 and 3, with the expectation that higher dimensions would be increasingly complicated and possibly beyond comprehension. Strangely this was not the case: dimensions above 4 are actually easier and a deep systematic theory was developed between 1950 and about 1980. For the last 30 years the focus has returned to dimensions 3 and 4. This has been one of the most active areas of mathematics, with deep relations to geometry, analysis, algebra and physics. However a unifying perspective analogous to the one achieved in higher dimensions is still lacking. The aim of the conference is to promote interactions among researchers working on different aspects of the subject, in hopes that unifying perspectives will begin to emerge.
