Our domain of research is at the interaction between algebraic topology, quantum topology and algebraic geometry.
The principal objects of study in this project are moduli spaces of submanifolds. These play a central role in algebraic topology, having profound connections with knot theory (predicted by physics) and homotopy theory. In addition, the representation theory of their fundamental groups – motion groups – is very rich.
In dimension 0, these include configuration spaces and braid groups, whose study is a very well-known domain. In higher dimensions, they are more subtle, and their study has developed rapidly recently, leaving nevertheless many open problems.
The aim of our project is to create new interactions between the topology of moduli spaces, TQFTs and motivic and étale cohomology theories, having 4 main directions:
(1) The stable limit of the homology of moduli spaces of disconnected submanifolds. This is an open question for submanifolds of higher dimension.
(2) New kinds of moduli spaces of submanifolds, inspired by physics and number theory. More precisely, the complement of the submanifold is equipped with a 'field' taking values in a bundle.
(3) The definition of new topological representations of motion groups and extensions towards TQFTs. The aim is to create connections with quantum knot invariants.
(4) Algebraic-geometrical analogues of these moduli spaces, and the study of their étale and motivic cohomology.