Splicing knot complements
Heegaard Floer homology of the 3manifold 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 nontrivial.


The sutured Floer complex
Associated with the boundary of a sutured
3manifold, 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 DelignMumford 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.
