- \((X,d)\) 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.

]]>Any static vacuum black hole is either

- a Schwarzschild black hole,
- a Boost, or
- of Myers/Korotkin-Nicolai type.

A static vacuum black hole is a pair \(\big((\Sigma,g),N\big)\) consisting of

- an orientable, complete Riemannian \(3\)-manifold \((\Sigma,g)\), possibly with boundary,
- a function \(N\) on \(\Sigma\) such that
- \(N\) is strictly positive on \(\Sigma \setminus \partial\Sigma\) and
- \(N\) satisfies the vacuum static Einstein equation \[N \cdot \mathrm{Ric} = \nabla \nabla N, \quad \text{and} \quad \Delta N = 0,\]

- and the boundary \(\partial\Sigma\) is compact and given by \(\partial\Sigma = \{N = 0\}\).

The following three facts are proved in the two articles (the first two points in the first article, and the third in the second one):

- \(\Sigma\) has only one end,
- every horizon is weakly outermost, and
- the end is either asymptotically flat or asymptotically Kasner.

A horizon \(H\) is a connected component of \(\partial \Sigma\). It is called weakly outermost, if there are no embedded surfaces \(S\) in \(\Sigma\) which are homologous to \(H\) and have negative outwards mean curvature.

Having proved the above three points, one can go on as follows: in the case the end is asymptotically flat one can invoke a well-known uniqueness theorem that it must be Schwarzschild. If the end is asymptotically Kasner, then it follows by previous results that it either is a Boost or every horizon is a totally geodesic sphere. In the latter case one can then go on and conclude that the static vacuum black hole is of Myers/Korotkin-Nicolai type.

]]>Die Förderung in den neuen Exzellenzclustern beginnt am 1. Januar 2019 und läuft sieben Jahre. Nach erfolgreicher Wiederbewerbung kann dies um weitere sieben Jahre verlängert werden.

]]>