Georgia Institute of Technology College of Sciences

We will describe some recent work of Lidman, Sivek, and Yasui as it pertains to the cabling conjecture. This is a question about which Dehn surgeries in S^3 yeild reducible 3-manifolds.

Existence of tight contact structures is a fundamental question of contact topology. Etnyre and Honda first gave the example which doesn't admit any tight structure. The existence of fillable tight structures is also a subtle question. Here we give some new examples of hyperbolic 3-manifolds which do not admit any fillable structures.

Given a plane field $dz-xdy$ in $\mathbb{R}^3$. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in $\mathbb{R}^3$, $J^1(S^1)$, and $\#^k(S^1\times S^2)$ as well as a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants.

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using an obstruction coming from Heegaard Floer homology, we will provide infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition. This is joint work with Lidman.

In honor of John Stallings' great paper, "How not to prove the Poincare conjecture", I will show how to reduce the smooth 4-dimensional Poincare conjecture to a (presumably incredibly difficult) statement in group theory. This is joint work with Aaron Abrams and Rob Kirby. We use trisections where Stallings used Heegaard splittings.

Given an action by a loop space on a structured ring spectrum we describe how to produce its associated quotient ring spectrum. We then describe how this structure may be leveraged to produce intermediate Hopf-Galois extensions of ring spectra, analogous to the way one produces intermediate Galois extensions from normal subgroups of a Galois group. We will give many examples of this structure in classical cobordism spectra and in particular describe an entirely new construction of the complex