**
Sergei Gukov
**

**Topological strings and
categorification of knot invariants, part 1.
**

**Abstract for both talks
**

We start with a brief introduction into knot homology theories
and categorification of polynomial knot invariants. In particular,
we review the construction and the basic properties of a new homological
knot invariant, recently introduced by Khovanov and Rozansky.
To every knot diagram, it associates a bigraded chain complex
whose graded Euler characteristic is the quantum-group *sl(N)* invariant.
Motivated by the ideas from physics, we then present a reformulation of
the *sl(N)* knot homology in terms of new triply-graded knot invariants.
This leads to new conjectures on the structure of the homological
knot invariants and suggests a larger theory which unifies the *sl(N)*
Khovanov-Rozansky homology (for all *N*) with the knot Floer homology.
We argue that this unification should be accomplished by a triply
graded homology theory which categorifies the HOMFLY polynomial.
We also describe the geometric meaning of the new knot invariants
in terms of the enumerative geometry of Riemann surfaces with
boundaries in a certain Calabi-Yau three-fold.