Novel Techniques in Low Dimension: Floer Homology, representation theory and algebraic topology

Recent advances have completely reshaped the landscape of geometric topology. On the one hand, Perelman's revolutionary work on Ricci flow confirms Thurston's geometrization conjecture, establishing that topological 3-manifolds can be decomposed into geometric pieces modelled on 8 possible geometries. On the other hand, long-standing conjectures due to Waldhausen and Thurston have now been resolved as the culmination of work of Kahn and Markovitch, Wise, and Agol. These results concern deep properties of 3-manifolds that become apparent on taking finite-sheeted covers and stem from a major step forward in our understanding of the structure of 3-manifold groups. At the same time, new machinery drawing on a vast array of mathematical subdisciplines has been developed and put to use, creating new vistas in low dimensions, posing important questions about the interaction between geometric structures on 3-manifolds and new invariants that draw from gauge theory, geometric representation theory, high-energy physics, symplectic geometry, and algebraic topology.

This CRG will take up problems spanning Thurston-style geometric topology and Floer-theoretic and representation-theoretic techniques in low dimensions, by working between subdisciplines in search of new structures. We are motivated by key open problems in low-dimensional topology, for instance, the characterization of those 3-manifolds with simplest-possible Floer homology. While at first glance this question appears internal to a particular subject, it is, in fact, far-reaching: as described further below, an alternate route to the Poincare conjecture - that is, one that does not appeal to Ricci flow - would be a consequence of a positive resolution to some of the conjectures described here.