Non-linear wave equations (in 3+1 dimensions) are ubiquitous in physics, playing a vital role in many subfields of it. There are many methods available to show short-time existence of solutions for such equations, but global-in-time existence is a much more subtle problem. In the 1980s Klainermann introduced a so-called null condition on the nonlinearities of … Continue reading "Global existence for non-linear wave equations"
Ten years ago the first Polymath project was launched (link to the original proposal of Timothy Gowers). Polymath can be described as massively collaborative mathematics, i.e., a large groups of mathematicians works on a predefined problem and each member posts his or her (partial) ideas, so that the collective makes progress from each small step … Continue reading "Ten Years of Polymath"
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.
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 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.
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"
The DMV awarded the Cantor medal to Hélène Esnault from the FU Berlin (press release). The Cantor medal is the most important prize the DMV awards.
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"