Georgia Institute of Technology College of Sciences

Real Heegaard Floer homology is a new invariant of branched double covers, introduced by Gary Guth and Ciprian Manolescu, and inspired by work of Jiakai Li and others in Seiberg-Witten theory. After sketching their construction, we will describe an extension of the "hat" variant to 3-manifolds with boundary, and the algorithm this gives to compute it when the fixed set is connected. We will end with some open questions.

Skein lasagna modules are smooth 4-manifold invariants constructed from functorial link homology theories. These invariants are capable of detecting exotic phenomena in dimension 4. Wall-type stabilization questions ask about the behavior of exotic smooth structures under various topological operations. In studying applications of these invariants to Wall-type stabilization problems, we construct an isomorphism between the skein lasagna invariant of a pair of the form (S^2xD^2, L), where L is a link in the boundary S^1xS^2, and the Rozansky-Willis homology of L in S^1xS^2 up to an extra tensor factor. In this talk, we will describe both invariants, describe their relationship, and discuss relevant properties. We will then briefly sketch how the properties of skein lasagna modules are used to establish functoriality for Rozansky-Willis homology and how to upgrade the theory to a new 4-manifold invariant.

One of the fundamental problems in contact topology is to classify contact structures on a given 3-manifold. In particular, classifying contact structures on surgeries along a given knot has been very poorly studied. The only fully understood case so far is that of the unknot  (lens spaces); for all other knots we have only partial results, or none at all. Several topological and algebraic tools have been developed to attack this problem. In this talk, we discuss recent developments and the strategy for classifying tight contact structures on surgeries along torus knots. This is joint work with John Etnyre, Bülent Tosun, and Konstantinos Varvarezos.

In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Σg. We will first apply various cyclic group actions on Σg for g>0, and then extend it diagonally to the product manifolds ΣgxΣg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2. Following the resolution process, we will determine the configurations of the singular fibers and the monodromy of the total space. In some cases, deformations of the Lefschetz fibrations give rise to nice applications using the rational blow-down operation, which provides exotic examples. This is a joint work with A. Akhmedov and M. Bhupal.