Research Interests

Splicing knot complements

Heegaard Floer homology of the 3-manifold which is obtained by splicing a pair of knot complements may be described in terms of the knot Floer complexes associated with the two knots. This description may then be used to show that if a homology sphere contains an incompressible torus then its Heegaard Floer groups are non-trivial.

The sutured Floer complex

Associated with the boundary of a sutured 3-manifold, together with A. Alishahi, we define an algebra, which is generated over the integers by variables which are in correspondence with the sutures. The sutured Floer complex associated with the sutured manifold is then defined as a chain complex with coefficients in this algebra. We are currently working on defining a corresponding cobordism map.

Tautological ring

Together with I. Setayesh, we have been exploring the kappa ring of the Delign-Mumford compactifiaction of the moduli space of curves with markings. In particular, we have been able to compute the asymptotic growth of the rank of this ring.

Heegaard Floer homology and the fundamental group

In collaboration with A. Kamalinejad and N. Bagherifard we have started the study of the link between Heegaard Floer homology and the balanced presentations of the fundamental group of a given closed manifold.

Published Papers