Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Unknotting number is one of the simplest, yet mysterious, knot invariants. For example, it is not known whether it is additive under connected sum or not. In this talk, we will construct lower bounds for the unknotting number using two homological knot invariants: knot Floer homology, and (variants of) Khovanov homology. Unlike most lower bounds for the unknotting number, these invariants are not lower bound for the slice genus and they only vanish for the unknot. Parallely, we will discuss connections between knot Floer homology and (variants of) Khovanov homology. One main conjecture relating knot Floer homology and Khovanov homology is that there is a spectral sequence from Khovanov homology to knot Floer homology. If time permits, we will sketch an algebraically defined knot invariant, for which there is a spectral sequence from Khovanov homology converging to it. The construction is inspired by counting holomorphic discs, so we expect it to recover the knot Floer homology. This talk is based on joint works with Eaman Eftekhary and Nathan Dowlin.

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1]. This is joint work with James Conway and John Etnyre.

Novikov homology was introduced by Novikov in the early 1980s motivated by problems in hydrodynamics. The Novikov inequalities in the Novikov homology theory give lower bounds for the number of critical points of a Morse closed 1-form on a compact differentiable manifold M. In the first part of my talk I shall survey the Novikov homology theory in finite dimensional setting and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my talk I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group associated to a cohomology class of a closed 1-form on M and its application to obtaining new lower bounds for the number of critical points of a Morse 1-form.