Breakthrough Prizes 2021

Three days ago the Breakthrough Prize Foundation announced the recipients of the 2021 Breakthrough Prizes. I focus in this post only on the prizes in mathematics (there are also prizes in life sciences and physics). Martin Hairer is the recipient of the 2021 Breakthrough Prize in Mathematics for transformative contributions to the theory of stochastic … Continue reading "Breakthrough Prizes 2021"

Contractible 3-manifolds and positive scalar curvature, II

Let \((M,g)\) be a complete, contractible Riemannian \(3\)-manifold (without boundary). Chang-Weinberger-Yu (link) proved that if \((M,g)\) has uniformly positive scalar curvature, then \(M\) must be homeomorphic to \(\mathbb{R}^3\). Recently (arXiv:1906.04128), Wang proved that if \((M,g)\) has positive scalar curvature and \(M\) has trivial fundamental group at infinity, then \(M\) must be homeomorphic to \(\mathbb{R}^3\). Jiang … Continue reading "Contractible 3-manifolds and positive scalar curvature, II"

Spaces of positively curved Riemannian metrics

It is by now a classical topic in index theory to study on a (closed) Riemannian (spin) manifold the space of all Riemannian metrics of positive scalar curvature. We have several results showing that this space is usually highly complicated from a homotopy theoretic point of view (provided it is non-empty). Instead of studying positivity … Continue reading "Spaces of positively curved Riemannian metrics"