**MIT Meeting**

Saturday, September 24, 9:30AM-12:15PM

Math Department,
4th Floor Lecture Room (Building
2, Room 449)

Title: How not to prove the smooth 4-dimensional Poincare conjecture

Abstract: This title is ripped off from Stallings' beautiful paper "How not to prove the Poincare conjecture", in which he reduces the 3-dimensional Poincare conjecture to certain group theoretic statements. I'll do the same in dimension 4, setting up a theory of "group trisections" whereby certain purely group theoretic objects are in one-to-one correspondence with diffeomorphism classes of smooth 4-manifolds.

Title: A generalization of the tau-invariant to rationally null-homologous knots

Abstract: Since it was introduced in 2004, the Ozsvath-Szabo tau-invariant has been a useful tool for studying genus and concordance of knots in the 3-sphere. Using Ni's construction of the Alexander grading for rationally null-homologous knots, I will show that one can define a collection of tau-invariants for any knot in a rational homology 3-sphere. In particular, these invariants are rational concordance invariants. Moreover, if K is a knot in the boundary of a negative definite four-manifold, the tau-invariants give a lower bound for the genus of any properly embedded surface with boundary K.

Title: Marked link invariants

Abstract: We study the instanton spectral sequence associated to a link with a singular bundle and, in particular, related it to a version of Khovanov homology with such data, in the case of alternating links. We will also explore the binary dihedral representations of alternating links with such bundle data.