Georgia Institute of Technology College of Sciences

Heegaard Floer homology is a tool for studying three- and four-dimensional manifolds, using methods that are inspired by symplectic geometry. Bordered Floer homology is tool, currently under construction, for understanding how to reconstruct the Heegaard Floer homology in terms of invariants associated to its pieces. This approach has both conceptual and computational ramifications. In this talk, I will sketch the outlines of Heegaard Floer homology, with an emphasis on recent progress in bordered Floer homology.

We give a breezy overview of Teichmuller theory, the deformation theory of Riemann surfaces.

Algebraic torsion is a means of understanding the topological complexity of certain homomorphic curves counted in some Floer theories of contact manifolds.  This talk focuses on algebraic torsion and the contact invariant in embedded contact homology, useful for obstructing symplectic fillability and overtwistedness of the contact 3-manifold, but mostly left unexplored. We discuss results for concave linear plumbings of symplectic disk bundles over spheres admitting a concave contact boundary, whose boundaries are contact lens spaces.

The representations of quantum groups are important in topology, namely, they can be used to construct quantum invariants of links. This relationship goes both ways: for example, the equivariant tensor category of representations of $U_q(\mathfrak{sl}_2)$ can be understood as a category of tangles. We will discuss a landmark result by Kuperberg who constructed graphical calculuses which describe the representation theory of the rank-2 simple Lie algebras.

The classical Laudenbach-Poénaru theorem states that any diffeomorphism of $\#_n S^1 \times S^2$ extends over the boundary connect sum of $n$ $S^1 \times B^3$'s. This implies the familiar fact that in Kirby diagrams for closed 4 manifolds, you do not need to specify the attaching spheres for 3 handles; it is also the backbone result of trisection theory, which allows one to describe a closed 4 manifold by three cut systems of curves on a surface.

Dalton, Etnyre, and Traynor classified Legendrian cable links when the companion knot is both uniformly thick and Legendrian simple, and Etnyre, Min, and Chakraborty classified all cable knots of uniformly thick knots. Using convex surfaces, we build on these results to classify cable links of knots in $(S^3, \xi_\text{std})$ that are uniformly thick but not Legendrian simple, and address new questions that arise from their nonsimplicity. This is joint work with Rima Chatterjee, John Etnyre, and Hyunki Min.

Two prominent questions in low dimensional topology are: which knots are slice, and which $\mathbb{Q}$-homology $S^3$'s bound $\mathbb{Q}$-homology $B^4$'s? These questions are connected by a theorem that states if a knot $K$ in $S^3$ is slice, then the 2-fold branch cover of $S^3$ over $K$ bounds a $\mathbb{Q}$-homology $B^4$.

4-manifold topology is characterized by unexpected differences between the smooth and topological categories. For instance, it is the only dimension where there can exist infinitely many manifolds $Y_i$ which are homeomorphic to but not diffeomorphic to $X$. A natural question: how does one construct examples of this phenomenon? In this talk, we focus on the method of reverse engineering, which allows for the construction of “small” exotic 4-manifolds. Surprisingly, symplectic geometry is the main ingredient that makes this approach work!

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof.