The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. We show that the graded quotient of the bipolar filtration has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof uses higher order amenable Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term $d$-invariants simultaneously. This is joint work with Jae Choon Cha.

Developments in geometry and low dimensional topology have given renewed vigour to the following classical question: to what extent do the finite images of a finitely presented group determine the group? I'll survey what we know about this question in the context of 3-manifolds, and I shall present recent joint work with McReynolds, Reid and Spitler showing that the fundamental groups of certain hyperbolic orbifolds are distingusihed from all other finitely generated groups by their finite quotients.

The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^1) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.

I will talk about the diffeomorphism classification of 4-manifolds up to 
connected sums with the complex projective plane, and how the resulting 
equivalence class of a manifold can be detected by algebraic topological 
invariants of the manifold.  I may also discuss related results when one 
takes connected sums with another favourite 4-manifold, S^2 x S^2, instead.

The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of M. We associate the analog of a Lie group G(m) to the Monster Lie algebra m. We give generators for large free subgroups and we describe relations in G(m).

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.