Configuration spaces
Configuration spaces of points on manifolds are among the easiest to define and most versatile objects in geometric topology, and have been a central topic for several decades, together with their group-theoretic analogue, namely braid groups. Yet many fundamental questions about them and their homotopy type remain open. I have mostly worked on configuration spaces of surfaces, but I am in general interested in understanding these objects also in higher dimensions.
Moduli spaces
Moduli spaces are spaces of solutions to a given mathematical problem, or spaces of instances of a given mathematical definition. They are ubiquitous and their homotopy contains valuable information about the objects at hand, especially when these arise in families. For instance, the moduli space of finite sets of cardinality 10 is the classifying space of the 10th symmetric group: it is connected, accounting for the fact that 10 is a number, i.e. all finite sets of that cardinality are isomorphic with each other; but it is not contractible! And for another instance, the moduli space of real vector spaces of dimension 10 is the Grasmannian of 10-planes in R^infty: again a connected but quite non-trivial space!
I am mostly interested in moduli spaces of manifolds, and in my work so far I have mostly focused on moduli spaces of Riemann surfaces, whose group-theoretic analogue are mapping class groups. In particular I have developed a combinatorial model for moduli spaces of surfaces with parametrised boundary which is based on certain Hurwitz spaces with collisions. I have been able to employ this construction to recover known information, including the Mumford conjecture on the stable rational cohomology of moduli spaces of surfaces; I would like to extract more information from this construction.
String topology
String topology is the study of spaces of continuous maps from a given space X to a given manifold M, usually assumed closed and oriented. Functoriality in X and Poincaré duality of M give rise to a rich higher algebraic structure on the homology groups of such mapping spaces, in the form of string operations.
For instance, when X is a circle, we obtain the free loop space LM, and we may think of a circle in M as a "string", whence the name. The most studied string operations are the string product of Chas-Sullivan, making (a shift of) the homology of LM into an algebra, and the Goresky-Hingston coproduct, making it into a coalgebra.
In my work on the subject I have introduced a category of "graph cobordisms", which are roughly cospans of spaces Y-->W<--X together with the extra information about how W is obtained from X by attaching finitely many cells of dimension at most 1. Graph cobordisms can recover the Chas-Sullivan product and several other string operations known or expected to exist from the literature. Since they provably cannot recover the Goresky-Hingston coproduct, I would like to explore extensions of the construction of graph cobordisms that would cover that as well.
