Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K) and showed that it is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain…

The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I…

We construct smooth compact 4-manifolds homotopy equivalent to S^2 which do not contain nicely embedded spheres realizing the homotopy equivalence.

Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational waves that were recently detected, and rotating black holes. The properties of a SFNGC induce a CR structure on the 3-dimensional leaf space of the…

Inspired by Bar-Natan's description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rank-statistic generating functions. Additionally, we present two main applications. The first, a categorification of certain generalized Vandermonde determinants gotten from the Bruhat order on the symmetric group by applying a special TQFT to…

In this overview talk, I will first explain several mathematical approaches of mirror symmetry. Then I will focus on the mathematical theory of Witten's gauged linear sigma model based on the recent joint work with Gang Tian, which settles the mathematical foundation of the approach of Hori and Vafa.

The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere.  One way to attack the S4PC is to examine a restricted class of 4-manifolds.  For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. …

The main goal of geometric topology is the classification of manifolds within a certain framework (topological, piecewise linear, smooth, simply-connected, symplectic, etc.). Dimension four is special, as it is the only dimension in which a manifold can admit infinitely many non-equivalent smooth structures, and the only dimension in which there exist manifolds homeomorphic but not…

Variational methods can be used to create numerical methods that respect conservation laws. I will discuss applications to electromagnetism, the Yang-Mills equations, and mean curvature flow. I will also discuss some new ideas about finite element spaces of differential forms.

I will describe a homological algebra construction which is fundamental in algebraic topology, algebraic geometry, and related areas: the spectral sequence. Originally developed by Jean Leray in the 1940s, a spectral sequence is a simultaneous higher-dimensional generalization of homology and long exact sequences. I will discuss a few examples of spectral sequences and their applications.

Exact couples and the Bockstein spectral sequence.

Computing Tor with spectral sequences.

Knot homology theories

A Floer homology is an invariant of a closed, oriented 3-manifold Y that arises as the homology of a chain complex whose generators are either the set of solutions to a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on Y…