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).

]]>We consider a metric space \((X,d)\). On it we consider the space \(\ell^2(X)\) of all square-summable sequences indexed by points in \(X\). The metric \(d\) on \(X\) comes into play by the following definition:

A linear operator \(T\) acting on \(\ell^2(X)\) is said to have *finite propagation*, if there exists an \(R > 0\) such that for all \(v \in \ell^2(X)\) we have \[\mathrm{supp}(Tv) \subset \mathrm{Neigh}(\mathrm{supp}(v),R),\] where \(\mathrm{Neigh}(\mathrm{supp}(v),R)\) is the neighbourhood of distance \(R\) around \(\mathrm{supp}(v)\), i.e., all points in \(X\) of distance at most \(R\) to \(\mathrm{supp}(v)\).

If \((X,d)\) is the set \(\mathbb{Z}\) equipped with the usual distance, then operators of finite propagation are exactly the infinite band matrices. The smallest possible value of \(R\) in the above definition is called the propagation of \(T\).

In the following, given a linear operator \(T\) on \(\ell^2(X)\), its entry \(T_{x,y}\) is defined as \((T\delta_y)(x)\), where \(\delta_y \in \ell^2(X)\) denotes the sequence which has only one non-zero entry, namely \(1\) on the point \(y\).

In the starting post we noticed that if the entries of an infinite band matrix are uniformly bounded (in absolute value), then the matrix defines a bounded operator on \(\ell^2(\mathbb{Z})\) with norm bound \(\|T\| \le M\cdot K\), where \(M\) is the thickness of the band of non-zero entries and \(K\) is the uniform upper bound on the absolute values of the entries. But on a general metric space \((X,d)\), if we are given an operator of finite propagation with uniformly bounded entries it does not necessarily define a bounded operator on \(\ell^2(X)\). Morally, the reason for this is that the \(M\) in the estimate is actually not the thickness of the band, but it is the number of elements in a row (or column) of the band. And on a general metric space this might be infinite. Hence the following definition:

The metric space \((X,d)\) is said to have *bounded geometry*, if for every \(R>0\) we have \[\sup_{x\in X} \#\mathrm{Neigh}(x,R) < \infty.\]

Note that a metric space of bounded geometry is necessarily discrete, since every ball of finite radius must have only finitely many elements.

The following basic estimate is now immediate:

Let \((X,d)\) be a metric space of bounded geometry and let \(T\) be a linear operator of finite propagation on \(\ell^2(X)\). If the entries of \(T\) are uniformly bounded, then \(T\) defines a bounded operator on \(\ell^2(X)\) with \[\|T\| \le \sup_{x\in X} \#\mathrm{Neigh}(x,\mathrm{prop}(T)) \cdot \sup_{x,y \in X} |T_{x,y}|,\] where \(\mathrm{prop}(T)\) denotes the propagation of \(T\).

Note that the converse to this result is obvious: if \(T\) is a bounded operator on \(\ell^2(X),\) then its entries must be uniformly bounded: \(|T_{x,y}| \le \|T\|\).

In this generalized setup, the original question of approximating infinite matrices by band matrices becomes the question of investigating for a metric space \((X,d)\) of bounded geometry the operator norm closure of all bounded, linear operators of finite propagation on \(\ell^2(X)\). We will start this investigation in a future post.

]]>The preprint the QuantaMagazine refers to is arXiv:1710.01722 and was posted in October 2017. It is not yet published, hence one should probably treat it with some care (not claiming that being published makes articles automatically correct), but a reason for this might be that it is 217 pages long! That surely takes some time to review.

]]>The Fields medallists 2018 are

There are also many more prizes and medals that are awarded at the ICM:

- The Gauß Prize goes to David Donoho.
- The Chern Medal goes to Masaki Kashiwara.
- The Nevanlinna Prize goes to Constantinos Daskalakis.
- The Leelavati Prize goes to Ali Nesin.
- The Emmy Noether Lecture is given by Sun-Yung Alice Chang.

Preceding the ICM the K-theory foundation awards at a satellite conference of the ICM its prize.

- The prize winners of the K-theory foundation prize are Benjamin Antieau and Marc Hoyois.

Starting 2019, the IMU Executive Committee will consist of

- Carlos E. Kenig as president,
- Helge Holden as secretary general,
- Nalini Joshi and Loyiso G. Nongxa as vice-presidents, and
- the six members at large are