Wolf Prize in Mathematics 2019

The Wolf Prize in one of the most prestigious prizes in mathematics (Wikipedia entry). This year’s laureates are Prof. Gregory Lawler (Wikipedia, Homepage) and Prof. Jean Francois le Gall (Wikipedia, Homepage). The corresponding press release of the EMS may be found here.

The Whitehead manifold and positive scalar curvature

Recall that the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to Euclidean 3-space (Wikipedia entry). It was proven by Chang-Weinberger-Yu (link) that the Whitehead manifold can not admit a complete metric of uniformly positive scalar curvature. Last year their result was strengthened by Jian Wang in arXiv:1805.03544 to the statement that the Whitehead manifold can … Continue reading "The Whitehead manifold and positive scalar curvature"

Sir Michael Atiyah 1929 – 2019

Sir Michael Atiyah died three days ago on January 11th, 2019 (news of The Royal Society). A nice obituary (together with one for Jean Bourgain) may be found on this blog: link. Here is another obituary by the New York Times.

Coarse embeddings and non-positive curvature

Let \((X,d)\) be a complete, geodesic metric space. \((X,d)\) is called an Alexandrov space of global non-positive curvature if for every quadruple of points \(x,y,z,w\) such that \(w\) is a metric midpoint of \(x\) and \(y\), i.e., \(d(w,x) = d(w,y) = d(x,y)/2\), we have \[d(z,w)^2 + d(x,y)^2/4 \le d(z,x)^2/2 + d(z,y)^2/2.\] If the reverse inequality … Continue reading "Coarse embeddings and non-positive curvature"

Non-negative scalar curvature and mean convex boundaries

In a recent preprint ( arXiv:1811.08519 ) E. Barbosa and F. Conrado derive for manifolds with boundary topological obstructions to the existence of non-negative scalar curvature metrics with mean convex boundaries. The boundary of a Riemannian manifold is said to be mean convex, if the mean curvature of it with respect to the outward unit … Continue reading "Non-negative scalar curvature and mean convex boundaries"

MSJ Geometry Prize 2018

The Geometry Prize 2018 of the Mathematical Society of Japan was awarded to Shouhei Honda for his work on Geometric analysis on convergence of Riemannian manifolds and to Yuji Odaka for his work on Study on K-stability and moduli theory.

Selberg’s lemma and negatively curved Hadamard manifolds

Selberg’s lemma is a fundamental result about linear groups. It states that every finitely generated subgroup of \(\mathrm{GL}(n,K)\), where \(K\) is a field of characteristic zero, is virtually torsion-free (i.e., contains a torsion-free subgroup of finite index). Recently, Michael Kapovich proved that the conclusion of Selberg’s lemma can fail for finitely generated, discrete subgroups of isometry groups … Continue reading "Selberg’s lemma and negatively curved Hadamard manifolds"

Breakthrough Prize 2019

Ten days ago the winners of the Breakthrough Prize 2019 were announced (press release, AMS Blog). In mathematics the award goes to Vincent Lafforgue for “ground-breaking contributions to several areas of mathematics, in particular to the Langlands program in the function field case”. Further, the New Horizons Prize in mathematics goes to Chenyang Xu for “major … Continue reading "Breakthrough Prize 2019"

Classification of static vacuum black holes

In a series of two papers ( arXiv:1806.00818 and arXiv:1806.00819 ) Martin Reiris Ithurralde classifies all metrically complete solutions of the static vacuum Einstein equations with compact (but not necessarily connected) horizon. Main Theorem Any static vacuum black hole is either a Schwarzschild black hole, a Boost, or of Myers/Korotkin-Nicolai type. Basic definition A static vacuum … Continue reading "Classification of static vacuum black holes"