Georgia Institute of Technology College of Sciences

Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. In this talk, we will consider the background necessary for an approach to this problem. Specifically, we will survey some essential notions and terminology related to low-dimensional contact and symplectic topology. These will involve Dehn surgery, tightness, overtwistedness, concave and convex symplectic fillings, and open book decompositions.

We show how to obtain a decomposition of an arbitrary closed, smooth, orientable 4-manifold from a loop of Morse functions on a surface or as a loop in the pants complex. A nice feature of all of these decompositions is that they can be encoded on a surface so that, in principle, 4-manifold topology can be reduced to surface topology. There is a good amount to be learned from translating between the world of Morse functions and the world of pants decompositions.

We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.

One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.  

One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.  

A geodesic metric space is said to be CAT(0) if triangles are at most as fat as triangles in the Euclidean plane. A CAT(0) cube complex is a CAT(0) space which is built by gluing Euclidean cubes isometrically along faces. Due to their fundamental role in the resolution of the virtual Haken's conjecture, CAT(0) cube complexes have since been a central object of study in geometric group theory and their study has led to ground-breaking advances in 3–manifold theory.

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent.

Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.

We study reducible surgeries on knots in S^3, developing thickness bounds for L-space knots that admit reducible surgeries and lower bounds on the slice genus of general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots. Our techniques involve the d-invariants and mapping cone formula from Heegaard Floer homology. This is joint work with Holt Bodish.