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

In fact, for any closed manifold \(X\) and any \(k \ge 3\) the manifold \(X \times \mathbb{R}^k\) admits a complete metric of uniformly positive scalar curvature by a result of Rosenberg and Stolz (link).

Now there exist contractible, open 3-manifolds which are not homeomorphic to \(\mathbb{R}^3\), e.g., the Whitehead manifold. It was proven by Chang-Weinberger-Yu (link) that no such manifold can admit a complete metric of uniformly positive scalar curvature.

In a recent preprint (arXiv:1805.03544) Jian Wang improved the above stated result of Chang-Weinberger-Yu. He proved that if a contractible 3-manifold admits a complete metric whose scalar curvature is positive and decays at most like \(d(x_0, -)^\alpha\), where \(\alpha \in [0,2)\) and \(x_0\) is a point in the manifold, then the manifold must be homeomorphic to \(\mathbb{R}^3\).

The basic ingredient in Wang’s proof is the theorem of Stallings that a contractible 3-manifold is homeomorphic to \(\mathbb{R}^3\) if and only if it is simply-connected at infinity. So assuming that the contractible 3-manifold is not simply-connected at infinity, Wang makes a clever choice of curve witnessing this and applies the solution of the Plateau problem to it, i.e., fills the curve with an area minizing disc. The scalar curvature condition comes in due to a result of Rosenberg (link) that this disc must be close to its boundary, i.e., to the chosen curve, contradicting the clever choice of that curve (since the filling constitutes a null-homotopy of the curve which is close to the curve itself, but the curve is chosen such that one has to go far away from it to contract it).

]]>The Kyoto Prize 2018 in the category Basic Sciences was awarded to Masaki Kashiwara from the RIMS at Kyoto University. (announcement)

The Kyoto Prize is awarded annually to “those who have contributed significantly to the scientific, cultural, and spiritual betterment of mankind” and is Japan’s highest private award for global achievement. (Wikipedia, official homepage)

Masaki Kashiwara (Wikipedia) was awarded the prize for his contributions to the theory of D-modules. The encomium reads “Dr. Kashiwara established the theory of D-modules, thereby playing a decisive role in the creation and development of algebraic analysis. His numerous achievements—including the establishment of the Riemann-Hilbert correspondence, its application to representation theory, and construction of crystal basis theory—have exerted great influence on various fields of mathematics and contributed strongly to their development.”

Mateusz Kwaśnicki from the Wrocław University of Science and Technology was awarded the 2018 EMS Gordin Prize for his outstanding contributions to the spectral analysis of Lévy processes. (announcement)

The EMS Gordin Prize honours the memory of Mikhail Gordin and is awarded to a junior mathematician from an Eastern Europe country working in probability or dynamical systems.

Herbert Edelsbrunner (Arts & Science Professor of Computer Science and Mathematics at Duke University, Professor at the Institute of Science and Technology Austria, and co-founder of Geomagic, Inc.) receives the Wittgenstein Award 2018. (announcement)

The Wittgenstein Award is named after the philosopher Ludwig Wittgenstein and is conferred once per year by the Austrian Science Fund. Awardees receive financial support up to 1.5 million euro to be spent over a period of five years. (Wikipedia)

Herbert Edelsbrunner is one of the world’s leading researchers in computational geometry and topology. (Wikipedia)

Volker Mehrmann, professor at TU Berlin and president-elect of the EMS, receives the W. T. and Idalia Reid Prize 2018. (announcement)

The W. T. and Idalia Reid Prize is an annual award presented by the Society for Industrial and Applied Mathematics (SIAM) for outstanding research in, or other contributions to, the broadly defined areas of differential equations and control theory. It was established in 1994 by Idalia Reid in honour of her husband W. T. Reid, who died in 1977.

]]>