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.

]]>The goal of this post is to provide an example of such a matrix.

We will consider equivariant matrices, i.e., matrices \(T\) such that for all \(m,n \in \mathbb{Z}\) we have \(T_{m,n} = T_{m+k,n+k}\) for all \(k \in \mathbb{Z}\). Such matrices are completely determined by their entries \(T_{0,n}\) for all \(n \in \mathbb{Z}\).

Since an equivariant band matrix \(T\) only has finitely many non-zero values \(T_{0,n}\), we get a map \[\{\text{equivariant band matrices}\} \to \mathbb{C}[\mathbb{Z}], \quad T \mapsto \sum_{n \in \mathbb{Z}} T_{0,n} \cdot n,\] where \(\mathbb{C}[\mathbb{Z}]\) denotes the complex group ring of \(\mathbb{Z}\). This is actually an isomorphism of \(\mathbb{C}\)-algebras.

We can use the above map to define a norm on \(\mathbb{C}[\mathbb{Z}]\) by using the operator norm of the corresponding equivariant band matrix. The completion of \(\mathbb{C}[\mathbb{Z}]\) under this norm is called the reduced group \(C^*\)-algebra of \(\mathbb{Z}\) and denoted \(C_r^*(\mathbb{Z})\). Taking the completion corresponds to taking the closure of the equivariant band matrices in the space of all infinite matrices with bounded operator norm, i.e., elements of \(C_r^*(\mathbb{Z})\) can be written as equivariant infinite matrices.

Forming Fourier series can be thought of as the map \[\ell^2(\mathbb{Z}) \to L^2(S^1), \quad (c_n)_{n \in \mathbb{Z}} \mapsto \sum_{n \in \mathbb{Z}} c_n \cdot e^{2\pi i t \cdot n},\] which is continuous.

On \(\ell^2(\mathbb{Z})\) we can act with (equivariant) band matrices and on \(L^2(S^1)\) we can act with \(C(S^1)\), i.e., with continuous functions on the unit circle, by point-wise multiplication. These actions are intertwined with each other by the operation of forming Fourier series: the assignment \[\sum_{n \in \mathbb{Z}} T_{0,n} \cdot n \mapsto \text{ point-wise multiplication by } \sum_{n \in \mathbb{Z}} T_{0,n} \cdot e^{2\pi i t \cdot n}\] defines a map \(\mathbb{C}[\mathbb{Z}] \to C(S^1)\) and the operator norm on \(\mathbb{C}[\mathbb{Z}]\) corresponds under it to the sup-norm on \(C(S^1)\). The image of this map is dense in \(C(S^1)\) and so we get an isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\) by the Stone-Weierstrass theorem. Note that \(C_r^*(\mathbb{Z})\) is viewed here as a subalgebra of \(B(\ell^2(\mathbb{Z}))\) and \(C(S^1)\) is viewed as a subalgebra of \(B(L^2(S^1))\), where \(B(-)\) denotes the bounded, linear operators.

What does all the above help us in our goal of providing an infinite matrix \(T\) which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have \(\|T^{(R)}\| \to \infty\)?

We assume that \(T\) is from \(C_r^*(\mathbb{Z})\). Then it is in the closure (in operator norm) of the band matrices with uniformly bounded entries and can be represented as \(\sum_{n \in \mathbb{Z}} T_{0,n} \cdot n\). The operators \(T^{(R)}\) correspond then to \(\sum_{-R \le n \le R} T_{0,n} \cdot n\).

Now we apply the isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\) that we discussed above: the operator \(T\) becomes the function \(\sum_{n \in \mathbb{Z}} T_{0,n} \cdot e^{2\pi i t \cdot n}\), the operators \(T^{(R)}\) the functions \(\sum_{-R \le n \le R} T_{0,n} \cdot e^{2\pi i t \cdot n}\), and the operator norm becomes the sup-norm.

This means that if we can find a continuous function \(f\) on \(S^1\) with Fourier series \(\sum_{n \in \mathbb{Z}} c_n \cdot e^{2\pi i t \cdot n}\) such that the continuous functions \(f^{(R)}\) defined by \(\sum_{-R \le n \le R} c_n \cdot e^{2\pi i t \cdot n}\) satisfy \(\|f^{(R)}\|_\infty \to \infty\), then we have our counter-example: we just have to transform \(f\) back to an operator from \(C_r^*(\mathbb{Z})\) via the isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\).

An example of such a continuous function can be found in Section 3.2.2 of the book “Fourier Analysis” by Stein and Shakarchi.

*I thank Rufus Willett for providing me the reference for the construction of such a continuous function with divergent Fourier series.*

Professor Daya Reddy was elected first president of the International Science Council (ISC). The ISC is newly founded and a merger of the International Social Science Council (ISSC) and the International Council for Science (ICSU). Professor Reddy holds the South African Research Chair in Computational Mechanics at the University of Cape Town. More information to be found in this press release: link.

]]>Recall that a surface is called minimal if it is a critical point of the area-functional, and that a minimal surface is called stable if the second-variation of the area-functional is non-negative for all smooth variations of the surface.

Last month there was a preprint posted on the arXiv (1806.03883) by Vanderson Lima containing the following two results:

- The above stated conjecture of Meeks-Pérez-Ros is wrong.
- The conjecture becomes correct if one replaces the word “embedded” by “immersed”.

The proof that the modified version of the conjecture is true heavily relies on the Geometrization Theorem for 3-manifolds. Assuming that N is not finitely covered by the 3-sphere, Lima considers the cases provided by geometrization and constructs in each one a corresponding surface. In most of the cases Lima constructs embedded surfaces, and in one case only immersed surfaces since he has ruled out the existence of embedded ones by the arguments he used to disprove the original conjecture of Meeks-Pérez-Ros.

But there is one case where Lima constructs in general only immersed surfaces, and the question whether one can construct embedded ones is still open: the case of orientable, irreducible, non-Haken, hyperbolic 3-manifolds.

]]>1978 he was an invited speaker at the ICM in Helsinki, and 1971-1973 he was president of the Brazilian Mathematical Society. Besides for his mathematical research he was also known for his textbooks.

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