Talk

In the early days of glory, geometric topology scored many astonishing achievements in the theory of manifolds and submanifolds, a few samples of which will be mentioned.

Let X be an elliptic surface with a section defined over a number field. Specialization theorems by N\'eron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran. 

A homotopy 4-sphere that is built without 1-handles can be encoded as a $n$-component link with an integral
Dehn surgery to $\#^n(S^1\times S^2)$.  I'll describe a program to prove that such spheres are smoothly standard

We will examine the interplay between Eliashberg's theory for embedding Stein domains and Freedman's theory of topological 4-manifolds. The speaker showed (J. Top. 2013) that an open subset $U$ of a complex surface $X$ is smoothly isotopic to a Stein open subset iff the induced almost-complex structure on $U$ is homotopic to a Stein structure. Using Casson handles, he showed (J. Symp. Geom. 2005) that $U$ is topologically isotopic to a Stein open subset iff it is homeomorphic to the interior of a 2-handlebody.

Given a cobordism between two knots in the 3-sphere, we present an inequality involving torsion orders of the knot Floer homology of the knots, and the number of critical points of index zero, one, and two of the cobordism. In particular, the torsion order gives a lower bound on the number of minima appearing in a slice disk of a knot. This has a number of topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, and on the fusion number of a ribbon knot.
 

David Gabai's smooth 4-dimensional "Light Bulb Theorem'' says that in the absence of involutions in the fundamental group of the ambient 4-manifold, homotopy implies isotopy for embedded 2-spheres which have a common geometric dual. In joint work with Peter Teichner we extend his result to orientable 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to the "homotopy implies isotopy'' question. The invariant takes values in an F2-vector space generated by the involutions in the fundamental group.

A knot is slice if it bounds an embedded disc in the 4-ball. There are (at least) two natural generalizations of sliceness: one might weaken either 'disc' to 'small genus surface' or 'the 4-ball' to 'any 4-manifold that is simple in some sense.'  In this talk, I'll discuss joint work with Jae Choon Cha and Mark Powell that gives new evidence that these two approaches measure very different things. Our tools include Casson-Gordon style representations of knot groups, L2 signatures of 3-manifolds, and the notion of a minimal generating set for a module.
 

Which 4d TQFTs and  4-manifold invariants detect the Gluck twist? Guided by questions like this, we will look for new invariants of smooth 4-manifolds and knotted surfaces in 4-manifolds.