We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we associate a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive…

I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. I will also talk about joint work with Prof. Luis Paris. We prove that on a compact, connected, nonorientable surface of genus at least 5, any superinjective simplicial map from the two-sided curve complex to itself is induced…

We will describe an adaptation of the theory of trisections to the setting of properly embedded, smooth, compact surfaces in smooth, compact, orientable four-manifolds with boundary.  The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection on the ambient four-manifold.  The trisection…

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…