Georgia Institute of Technology College of Sciences

There are many conjectured connections between Heegaard Floer homology and the various homologies appearing in low dimensional topology and symplectic geometry. One of these conjectures states, roughly, that if \phi is a diffeomorphism of a closed Riemann surface, a certain portion of the Heegaard Floer homology of the mapping torus of \phi should be equal to the Symplectic Floer homology of \phi. I will discuss how this can be confirmed when \phi is periodic (i.e., when some iterate of \phi is the identity map). I will recall how a mapping torus can be realized via Dehn surgery; then, I will sketch how the surgery long exact triangles of Heegaard Floer homology can be distilled into more direct surgery formulas involving knot Floer homology. Finally, I'll say a few words about what actually happens when you use these formulas for the aforementioned Dehn surgeries: a "really big game of tic-tac-toe".

We adapt techniques derived from the study of quasi-flats in Right Angled Artin Groups, and apply them to 2-dimensional Graph Braid Groups to show that the groups B_2(K_n) are quasi-isometrically distinct for all n.

The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.