we will read a paper by princeton graduate student jacob rasmussen, on khovanov homology and the slice genus. it is a breakthrough paper, proving the so called milnor conjectures by purely combinatorial means. these conjectures predict the 4-dimensional genus of torus knots, and their only known proof involves gauge theory, or more precisely, seiberg-witten theory. rasmussen gives a proof using only methods from combinatorial skein theory and a certain spectral sequence. he obtains a concordance invariant for knots, which is absolutely mind blowing. it can show that certain knots with trivial alexander polynomial are not smoothly slice, even though by freedmans theorem they are topologically slice. hence the invariant sees the difference between smooth and topological phenomena in dimension four. as usual, the talks in this seminar will be given by the participants, with two survey lectures at the beginning given by justin and peter. rasmussens paper, supplemented by some survey articles, will be used as reference for later talks.