New quantum obstructions to sliceness

A knot is called smoothly slice if it bounds a disc smoothly embedded into the four-ball. Requiring the disc's embedding to be locally flat instead of smooth leads to the definition of a topologically slice knot.
There are topologically slice knots that are not smoothly slice. Such knots could be used to construct a counterexample refuting the smooth four-dimensional Poincaré conjecture.
We will see a large family of obstructions to smooth sliceness, which are able to show the existence of such knots, arising from a theory which is combinatorial and has its roots in representation theory: the Khovanov-Rozansky knot homologies, categorifications of Reshetikhin and Turaev's sl(n)-polynomials.
The talk will begin with an introduction to the slice genus and Khovanov-Rozansky homologies, and finish with the spectral sequences that are our main tool in the analysis of those homologies.