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.

]]>- \((X,d)\) 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 holds true for all such quadruples, then \((X,d)\) is called an
*Alexandrov space of global non-negative*curvature.

We will also need the notion of a coarse embedding. A (not necessarily continuous) map \(f\colon Y \to X\) between metric spaces \((Y,d_Y)\) and \((X,d_X)\) is called a *coarse embedding*, if there exist two non-decreasing functions \(\omega, \Omega\colon [0,\infty) \to [0,\infty)\) with \(\omega \le \Omega\) point-wise and \(\lim_{t\to\infty}\omega(t) = \infty\), such that \[\forall x,y\in Y \colon \omega(d_Y(x,y)) \le d_X(f(x),f(y)) \le \Omega(d_Y(x,y)).\]

A. Eskenazis, M. Mendel and A. Naor recently proved the following theorem (arXiv:1808.02179): *There exists a metric space \(Y\) that does not coarsely embed into any non-positively curved Alexandrov space \(X\).*

The interest in such a theorem stems from the following two observations:

- Admitting a coarse embedding into a Hilbert space (which are Alexandrov spaces of global non-positive curvature) has consequences for versions of the Novikov conjecture. And the known counter-examples to the Baum-Connes conjecture rely on spaces (so-called expanders) which were only considered in this situation,
*because*they do not embed into a Hilbert space. Generalizing then from Hilbert spaces to more general spaces allowed us to discover more phenomena of some problems related to the Novikov and Baum-Connes conjectures, hence studying exotic space which do not admit coarse embeddings into non-positively curved spaces might expose more interesting phenomena. - The theorem becomes wrong if we replace the words
*non-positively curved*by*non-negatively curved*: it follows from arXiv:1509.08677 that every metric space coarsely embeds into some non-negatively curved Alexandrov space (this is explained in Section 1.4.1 of the above cited paper arXiv:1808.02179).

Let me finish by mentioning that the cited paper of Eskenazis-Mendel-Naor has a nice introduction going into many details about the motivation and the history of the above discussed problems, hence is worth reading if you are interested in such things.

]]>The boundary of a Riemannian manifold is said to be mean convex, if the mean curvature of it with respect to the outward unit normal vector field is non-negative.

Instead of writing down here the results in their full glory, let us mention only the following example: the manifold \((S^1 \times T^\circ)\# N\), where \(T^\circ\) is the 2-dimensional torus with an open disc removed and \(N\) is a closed, connected and orientable 3-dimensional manifold, do not admit Riemannian metrics with non-negative scalar curvature and mean convex boundary.

]]>Recently, Michael Kapovich proved that the conclusion of Selberg’s lemma can fail for finitely generated, discrete subgroups of isometry groups of Hadamard manifolds (arXiv:1808.01602). Recall that a Hadamard manifold is a simply-connected, complete Riemannian manifold of non-positive sectional curvature.

Now what’s the relation of Kapovich’s result and Selberg’s lemma? Let us answer this in the case that the field \(K\) are the real numbers. Consider the maximal compact subgroup \(\mathrm{O}(n,\mathbb{R})\) of \(\mathrm{GL}(n,\mathbb{R})\) and form the homogeneous space \(\mathrm{GL}(n,\mathbb{R}) / \mathrm{O}(n,\mathbb{R})\). It comes with a natural Riemannian metric which turns it into a Hadamard manifold. Hence, if we have a finitely generated subgroup of \(\mathrm{GL}(n,\mathbb{R})\), then it acts on this Hadamard manifold by isometries. And now one can ask if it is actually important for Selberg’s lemma that the Hadamard manifold is exactly the homogeneous space \(\mathrm{GL}(n,\mathbb{R}) / \mathrm{O}(n,\mathbb{R})\). As Michael Kapovich shows, it is important.

]]>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 advances in the minimal model program and applications to the moduli of algebraic varieties”.**Karim Adiprasito**and**June Huh**for “the development, with Eric Katz, of combinatorial Hodge theory leading to the resolution of the log-concavity conjecture of Rota”.**Kaisa Matomäki**and**Maksym Radziwill**for “fundamental breakthroughs in the understanding of local correlations of values of multiplicative functions”.

The Breakthrough Prizes come with an award of 3.000.000 USD and the New Horizons Prizes with 100.000 USD. The awarding ceremony will be held on Sunday, November 4th.

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