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.

]]>The Shaw Prize comes with a 1.2 million USD monetary award and is hence one of the world’s biggest prizes for mathematics.

Luis Caffarelli has already won several other major prizes in mathematics and is a member of several learned societies (link, link).

]]>Thomas Friedrich contributed substantially to the development of Berlin mathematics, he was Editor-in-Chief of the journal *Annals of Global Analysis and Geometry* for more than three decades (and also one of the founding editors-in-chief), and in 2003 he received the Medal of Honor of the Charles University of Prague.

An obituary can be found on the webpage of the HU Berlin (link). There will be also a conference held in memory of Thomas Friedrich (link).

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