In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loopspace of the configuration space of points in a Euclidean space.  We relate two different integration-based approaches to its cohomology, both encoded by complexes of graphs.  On the one hand, we can restrict configuration space integrals for…

The configuration space of some $n$ points in a space $X$ is a well-studied object in topology, geometry, and combinatorics. We present a generalization, the simplicial configuration space $M(S,X)$, which takes as its data a simplicial complex $S$ on $n$ points. In this talk, we will describe how $M(S,X)$ gives rise to polynomial and homological invariants of…

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…