I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.
 

The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension d can be organized into an E_d-monoidal n-category, where n is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free E_d-monoidal n-category with duals generated by a single object.

In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tangle hypothesis: Instead of showing that the E_d-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides supporting evidence for the tangle hypothesis, but can also be used to reduce the tangle hypothesis to a statement at the level of E_d-monoidal (n+1, n)-categories by means of obstruction theory.

I shall begin with a brief history of the problem of trying to understand infinite groups knowing only their finite quotients. I'll then focus on 3-manifold groups, describing the prominent role that they have played in advancing our understanding of this problem in recent years. The story for 3-manifold groups involves a rich interplay of algebra, geometry, and arithmetic. I shall describe arithmetic Kleinian groups that are profinitely rigid in the absolute sense -- ie they can be distinguished from all other finitely generated, residually finite groups by their set of finite quotients. I shall then explain more recent work involving products of Seifert fibered manifolds -- here we find groups that are profinitely rigid in the class of finitely presented groups but not in the class of finitely generated groups. This is joint work with McReynolds, Reid, and Spitler.

Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk I will discuss an upcoming work with Huneau and Stingo showing the nonlinear stability of such a Kaluza-Klein spacetime. The result is also connected to a claim of Witten.