Friday

The bordered Floer homology of a 3-manifold with torus boundary can be interpreted as a collection of immersed curves in the punctured torus. Similarly, the bordered Floer homology of a 3-dimensional cobordism M:T^2->T^2 give an operator F_M which eats an immersed curve on the punctured torus and spits out another such curve. (More formally, F_M is a functor from the Fukaya category of the punctured torus to itself.) I'll explain how to describe some interesting examples (the cabling operator, for example) using correspondences, without ever having to compute their bordered Floer homology. I will also discuss some relations with mirror symmetry. Based on joint work with Holt Bodish and James Pascaleff.

We will show methods for constructing exotic four-manifolds with fundamental group Z_2, and some extensions to further cases of non-trivial fundamental groups. Special attention will be paid to the case of definite manifolds, and in particular to ‘fake projective planes’. We also plan to point out several open questions along the way.

A pattern, or knot in a solid torus, induces a map on the set of knots modulo smooth concordance. In 2016, Hedden conjectured that essentially none of these maps are group homomorphisms--more precisely, that the only homomorphisms induced by satelliting are the identity map, the reversal map, and the zero map. In particular, this would imply that patterns with winding number not in the set {-1,0,1} cannot induce homomorphisms. I will discuss work with Randall Johanningsmeier and Hillary Kim in which we prove that if P is a pattern with winding number that is even but not divisible by eight, then P cannot induce a homomorphism on the smooth concordance group. This relies heavily on previous joint work with Tye Lidman and Juanita Pinzon-Caicedo, and is the first result that obstructs all patterns of a fixed winding number from inducing homomorphisms.

We use various aspects of Floer homology to study exotic smooth structures on four-manifolds.