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.

]]>Ein paar mehr Infos gibt es auch in diesem ScienceBlog: link.

]]>From the usual Euclidean plane we form the following graph: the points of the plane are the vertices of our graph, and two vertices are connected by an edge if they are exactly unit distance apart. The so-called Hadwiger-Nelson problem is to compute the chromatic number of this graph, i.e., the least amount of colors needed to color the graph such that any two vertices connected by an edge have different colors. This number is called the Chromatic Number of the Plane.

More information can be found on the Wikipedia page (link).

The problem was formulated over half a century ago, and it was quickly shown that the Chromatic Number of the Plane must be 4, 5, 6 or 7. But since then there was no progress on this problem at all.

Last month, a preprint appeared on the arXiv (link) claiming that the Chromatic Number of the Plane is not 4, i.e., it must be either 5, 6 or 7. It seems that the correctness of the arguments is accepted by now.

Being a problem which is easy to explain to the general audience and since this is the first progress on it since around 50 years, the result went quickly through the media. An example is the following article in the Quanta Magazine: link.

The proof that the Chromatic Number of the Plane is at least 5 is by providing a subgraph that is not 4-colorable. The subgraph proposed in the arXiv-article has 1581 vertices and checking its non-colorability requires computer assistance.

In order to simplify the arguments, there was a polymath proposal (link). The corresponding wiki-page is here and it seems that they managed already to find a non-4-colorable unit-distance-graph with 610 vertices. Another part of the project is about reducing the amount of computer assistance needed to check to non-4-colorability.

]]>But our first steps into coarse geometry will be very gently: we will be concerned with infinite matrices and their operator norms. We will see that if they are not band diagonal, then it becomes much more difficult to estimate their norms.

We consider an infinite band matrix \(T\) whose rows and columns are indexed by the integers \(\mathbb{Z}\). This means that \(T = (T_{m,n})_{m,n \in \mathbb{Z}}\) satisfies \[T_{m,n} = 0 \text{ for all } m,n \text{ with } |m-n| > N\] for some \(N \in \mathbb{N}\). If the entries of \(T\) are uniformly bounded, i.e., \[\sup_{m,n} |T_{m,n}| < K\] for some \(K \in \mathbb{R}\), then it is straight-forward to show that \(T\) acts continuously on bi-infinite vectors from \(\ell^2(\mathbb{Z})\) and the operator norm of \(T\) can be estimated from above as \[\|T\| \le 2N \cdot K.\] (The factor of \(2\) comes from the fact that the thickness of the band of non-zero entries of \(T\) is \(2N\).)

*But if \(T\) is not band diagonal, how to estimate its operator norm (in terms of its entries)?*

To show how complicated the above question is, let us consider the following related problem: we are given an infinite matrix \(T\) and we are told that it is in the closure (in operator norm) of the band matrices with uniformly bounded entries. How can we then actually construct a sequence of band matrices \(T^{(R)}\) with \(T^{(R)} \to T\)?

A natural first approach would be the following (at least, it was *my* first approach): we cut down \(T\) along thick diagonals and then try to show that the resulting sequence of band matrices actually approximates \(T\). Concretely, writing \(T = (T_{m,n})_{m,n \in \mathbb{Z}}\) and fixing \(R \in \mathbb{N}\), we define a band matrix \(T^{(R)}\) by \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & \text{ otherwise }\end{cases}\] The hope is that \(T^{(R)} \to T\) as \(R \to \infty\).

But the above approach is doomed to fail: it can even happen that \(\|T^{(R)}\| \to \infty\)!

So even if \(T\) is in the closure (in operator norm) of the band matrices, it does not mean that the terms off a thick-diagonal of \(T\) define an infinite matrix of small norm.

Now it is your turn – try to figure out how to solve the problem from the previous section, i.e., given an infinite matrix in the closure of the band matrices, how to construct a corresponding approximating sequence?

The above question about operator norms of infinite matrices is the starting point of our journey through coarse geometry. In a future post we will do our next step and address the following two questions:

- How to determine whether \(T\) is approximable (in operator norm) by band matrices?
- Provided \(T\) is approximable by band matrices, how to actually construct an approximating sequence?

Generalizing from bi-infinite vectors indexed by \(\mathbb{Z}\) to vectors indexed by a metric space \(X\) and from band matrices to so-called operators of finite propagation, we will see that (the answers to) the above questions lead us to a 30-year-old conjecture of John Roe and to coarse geometric notions like asymptotic dimension.

]]>His web page is still online ( http://sites.psu.edu/johnroe/ ) and can be visited to get a glimpse not only of his mathematical work, but also of his personal life and all the things in the world that mattered to him.

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