Recently, there was a preprint posted on HAL (link) in which the authors provide an algorithm which runs in \(O(n\log(n))\)-time.

A nice article about this discovery may be found at the QuantaMagazine: link.

Further, it was also recently proven in another preprint (link) that \(O(n\log(n))\)-time is conditionally (i.e., if a certain conjecture in network coding is true) the best possible.

]]>In March 14th, 2019, the **L’Oréal-UNESCO For Women in Science International Awards** (homepage, UNESCO webpage, wikipedia) were presented to five women. One of the five laureates is

The other one is the **Ars legendi-Fakultätenpreis Mathematik** (webpage), which will be awarded to **Robert Rockenfeller** (homepage). Actually, since I’m quite interested in the art of teaching mathematics, I invited him next summer to Regensburg to our department colloquium to talk about what he did which earned him this prize; and I would actually recommend others to do the same – at least last year I was positively astonished by the talk (and what he did teaching-wise) by last-years laureate.

John Roe passed away last year. His personal webpage is still online for those who want to get a glimpse at all he was interested in.

Here are a few pictures from the meeting. In the first one you see Christopher Wulff giving his talk on solving some of John’s conjectures, the second one is Rufus Willett’s talk about *Being a student of John Roe*, and on the third one you see Qin Wang projecting a quote of John (you have to zoom in a bit to read it). (Unfortunately, I forgot to take a picture of Nigel Higson giving an overview of John’s mathematical work.)

But the most interesting talk for me was by Sara Zelenberg about *Mathematics for sustainability* – the book with which John was occupied at the end (Sara is one his co-authors of this book).

Let me finish this post by showing two pictures. The first one is the group picture of this special session, which was taken in front of a projection of an old conference picture (Paul Baum’s 60th). The second one is another picture from that conference (with naming provided by Nigel Higson).

]]>

… for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.

There is a nice article about it over at Quanta Magazine (link). A mathematical introduction to her field can be found at the following blog: link; and another one here: link.

]]>The Australian Academy of Sciences states:

Professor Williamson is a world leader in the field of geometric representation theory. Among his many breakthrough contributions are his proof, together with Ben Elias, of Soergel's conjecture—resulting in a proof of the Kazhdan-Lusztig positivity conjecture from 1979; his entirely unexpected discovery of counter-examples to the Lusztig and James conjectures; and his new algebraic proof of the Jantzen conjectures.]]>

In the 1980s Klainermann introduced a so-called *null condition* on the nonlinearities of a nonlinear wave equation which was shown to guarantee global-in-time existence of solutions for all sufficiently small initial data.

It is known that the null condition is only sufficient, but not necessary for the existence of global solutions. Fifteen years ago the *weak null condition* was introduced by Lindblad and Rodnianski, and all systems of non-linear wave equations, where we know that they have global existence of solutions for small initial data (i.e., also the ones which do not satisfy the null condition), satisfy the weak null condition.

A few months ago Joseph Keir posted a paper on the arXiv ( https://arxiv.org/abs/1808.09982 ) in which he shows that under an extra condition nonlinear wave equations in 3+1 dimensions satisfying the weak null condition do indeed have global solutions for small initial data. This extra condition that he imposes is satisfied by all currently known equations which are known to have global existence.

Since Joseph Keir’s paper is an astounding 372 pages long, it will probably take some time for the community to verify all his arguments. For the more interested reader, his paper actually also has a long and extensive introduction, so it might be worthwhile to read it.

]]>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 that everybody contributes (Wikipedia).

Up to now there were 16 projects (some of them are still running), and several of them were successfull (or made significant contribution to the solution of the problem). The results are published under the pseudonym D.H.J. Polymath. Here is the link to the general Polymath blog: link.

]]>The corresponding press release of the EMS may be found here.

]]>- 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 not admit a complete metric of positive scalar curvature that decays slowly enough (I have blogged about this result here: link).
- Now it seems that Jian Wang could improve the result even further to the desired one, namely that the Whitehead manifold does not admit any complete metric of positive scalar curvature (arXiv:1901.04605).

In the proof Wang uses minimal surfaces (as in the proof of the earlier result with the decay condition) and again the idea of filling curves by minimal discs. But this time there is a sequence of such curves and discs, and one has to discuss convergence issues. The connection to scalar curvature happens by the Cohn-Vossen inequality (Wikipedia entry) of which Wang proves a suitable version adapted to our needs here.

]]>