the low-dimensional topology community has been energized in recent years by the introduction of a wealth of so-called ``homology-type" invariants. one associates to an object in low-dimensional topology (e.g., a link or a 3-manifold) an abstract chain complex whose homology is an invariant of the topological object. such invariants arise in two apparently different ways: ``algebraically," via the representation theory of quantum groups and ``geometrically," via constructions in symplectic geometry. i will discuss what is known about the relationship between two such invariants: khovanov homology, an ``algebraic" invariant of links and tangles defined by khovanov and heegaard-floer homology, a ``geometric" invariant of 3-manifolds defined by ozsvath-szabo. the portions of the talk describing my own work are joint with denis auroux and stephan wehrli.