Georgia Institute of Technology College of Sciences

The existence and nonexistence of tight contact structures on the 3-manifold are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structure, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not the smooth (2n − 1)-surgery along the T(2,2n+1) torus knot for any positive integer n.

Surprisingly, since then no other example of a 3-manifold without tight contact structure has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.  

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.  We explain our curve counting methods in terms of the Reeb dynamics and their parallel with the topological contact toric description of these lens spaces. This talk is based on joint work with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  Time permitting, we will discuss exploration of our methods to find nonfillable tight contact 3-manifolds obtained from more general plumbings.

In this talk, I will describe an excision construction for 3-manifolds and explain how (twisted) Heegaard Floer theory can be used to obstruct 3-manifolds from being related via such constructions. I will also discuss how the excision formula can be applied to compute twisted Heegaard Floer homology groups for specific 3-manifolds obtained by performing surgeries on certain links, including some 2-bridge links.

In their celebrated work, Gordon and Luecke proved that knots in the three-dimensional sphere are determined by their complements. Their result inspired several conjectures seeking to generalize the theorem, including "Cosmetic surgery conjecture" proposed by Gordon and the "Knot complement problem for null-homotopic knots" proposed by Boileau. In this talk, I will discuss applications of tools from Yang—Mills gauge theory to these Dehn surgery problems.