Horizons of Mathematics, with Máté Matolcsi (Alfréd Rényi Institute of Mathematics)
Máté Matolcsi is an expert in Fourier analysis, functional analysis and operator theory.
Fuglede’s conjecture is a fascinating proposal: it seeks to characterise so-called spectral sets –which appear in the area of analysis– by means of a geometrical property –tracing mathematical mosaics–. In 2003, thirty years after it was first proposed, and to the astonishment of the scientific community, Fields Medalist Terence Tao found an example showing that it was not true. Shortly afterwards, Máté Matolcsi (Alfréd Rényi Institute of Mathematics, Hungary) and Mihail Kolountzakis (University of Crete, Greece) finally disproved the conjecture in its general statement. However, in some interesting cases, it could still be true. Matolcsi discusses the latest developments on this question at the ICMAT-UAM-UCM-UC3M Joint Colloquium on 12 November at 12:00. His talk, ‘Spectral sets, weak tiling and Fuglede’s conjecture,’ takes place in the Aula Azul of the ICMAT and can be followed online at the following link. Beforehand, he presents some of the central ideas of the subject for non-specialists.
Ágata Timón García-Longoria (ICMAT)
Spectral sets are mathematical objects that allow us to analyse and understand the properties of functions, signals, operators, dynamical systems… ‘Although they were introduced in 1974 by Bent Fuglede in a paper studying partial differential operators, it is easier to understand the concept using the language of Fourier analysis, the main theoretical concept behind all signal transmission systems like phones, TVs, sound recordings…’ explains Máté Matolcsi (Alfréd Rényi Institute of Mathematics, Hungary).
Fourier analysis is used to decompose any signal into a combination of ‘pure’ waves (sines and cosines) with different frequencies and intensities. Using this tool, all signals defined over the interval [0,1] can be decomposed using only integers as frequencies. ‘In this sense, we say that the integers form a ‘spectrum’ of the interval [0,1]. In the same way, if the signal is defined over the unit cube, we can decompose it into pure waves using triplets of integers as frequencies. That is, the integer triplets form a spectrum of the cube,’ says Matolcsi. ‘Now, the concept of spectral sets is just one more step away: a set X is called spectral, if there exists a set of frequencies S such that all signals over X can be decomposed into a combination of pure waves with frequencies from S. The set S is called the spectrum of X’, he defines.
Not all sets have this property and many mathematicians have tried to find ways to identify the sets that do. Among them, Fuglede. ‘Fuglede conjectured that a set X is spectral if and only if it is possible to tile the space with copies of X,’ says Matolcsi. In mathematics, the idea of tiling a space is similar to that of making a mosaic. A covering is like a floor covered entirely with tiles: each tile (which would be a covering element) is placed in such a way that there are no gaps or overlaps. ‘For example, the unit cube tiles the space, while the ball does not. And indeed, the cube is spectral, while the ball is not. Like these, many examples were found that confirmed his conjecture. Fuglede himself showed that, if X tiles the space in a regular grid pattern, then X is spectral. This is the case, for example, for the regular hexagon in two dimensions,’ he adds.
Five different tiling patterns in dimension 3.
However, to the surprise of many, in 2003, Terence Tao (Fields Medal in 2006) found a counterexample to the conjecture: he found a spectral set that does not tile the space. ‘His tremendously ingenious example consists of a collection of cubes scattered in a particular arrangement,’ he says. Later, Matolcsi himself, together with Mihail Kolountzakis, found a set that tiles the space, but is not spectral. Getting to the counterexample was not easy: ‘Nothing seemed to work. And finally, after a long struggle, we felt we were close to the solution. However, one last piece was still missing. The weekend came, we put the problem aside and went to the beach with our families. The last piece of the puzzle then appeared. Sometimes it’s good to be distracted and do something else,’ says the mathematician.
So was Fuglede wrong? ‘Not completely. First of all, the counterexamples found only work in three or more dimensions. Therefore, the conjecture can still be true in dimension one and two. In fact, in one dimension, there are many indications that the conjecture is true, although a proof has not yet been found.
On the other hand, recently, Matolcsi, in collaboration with Nir Lev, has shown that the conjecture is true for all convex bodies –that is, those in which, if you take any two points inside it and join them with a straight line, that line also lies completely inside the set–. ‘Nir Lev and I started working on a particular case of the Fuglede conjecture at a conference in 2018. We couldn’t make much progress, but a new idea emerged: weak tiling. At that point we realised that we should use this concept for convex sets, and leave aside the original problem we were trying to solve. And indeed, we succeeded in proving the Fuglede conjecture for all convex bodies! But we still couldn’t solve the original problem we started with, a minor problem that would have generated only modest interest’.
What does this new type of tiling consist of? ‘Let’s say you have a large window and you want to block the light coming through it using a bunch of plastic pieces of a certain shape. It turns out that, if you glue just one layer, it’s too thin and some light can still get through. You need at least four layers to block the light. If the tile is square-shaped, then it’s easy: you simply do four layers of the covering, one on top of the other, in the same position. However, if the tile is shaped like a circular disc, you can’t just stack all four layers in the same position, because all those spots are covered four times, but there are gaps. It’s better to intelligently overlap the discs so that every point on the window is covered by exactly four layers,if you can,’ he explains.
Perhaps ‘the most interesting case of the conjecture that remains open’, according to Matolcsi, is that of finite cyclic groups. These are finite coverings of the integers. ‘These kinds of tilings always repeat the same pattern over and over again –they are called periodic–. Therefore, to understand them, it is enough to understand the repeating patterns’. To date, not all of these patterns are understood and there is a conjecture about them, which aims to characterise them. ‘If proven, it would imply that all mosaics are spectral in dimension one. If, in addition, we could prove that all weak tilings are proper tilings (in cyclic groups), the other direction of Fuglede’s conjecture in dimension one would also follow,’ he says.
Máté Matolcsi
Máté Matolcsi is a researcher and, since 2023, head of the Department of Analysis at the Alfréd Rényi Institute of Mathematics (Hungary). He is also a part-time lecturer at the Budapest University of Technology and Economics (BME).
He completed his PhD in Mathematics at Eötvös Loránd University (Hungary) in 2003, and since then has developed a career in the fields of functional analysis, Fourier analysis and additive combinatorics. His work has earned him numerous awards, including the ICBS Frontiers of Science Award in 2023 and the Academy Prize of the Hungarian Academy of Sciences in 2024.
ICMAT-UAM-UCM-UC3M Joint Mathematics Colloquium
‘Spectral sets, weak tiling and Fuglede’s conjecture’, by Máté Matolcsi (Alfréd Rényi Institute of Mathematics). 12 November 2024, at 12:00 in the Aula Azul (ICMAT) and online.
Abstract: A bounded measurable set X in a d-dimensional Euclidean space is called spectral if the function space L^2(X) admits an orthogonal basis of exponentials. The easiest example is the unit cube, where elementary Fourier analysis tells you that complex exponentials with integer frequencies form an orthogonal basis. Fuglede’s conjecture stated that a set X is spectral if and only if it tiles the space by translation. The conjecture was recently proved for all convex bodies in all dimensions in a joint work of Nir Lev and Máté Matolcsi. We will review the proof, which includes the notion of weak tiling as a key ingredient. Other results and open problems related to weak tiling will also be mentioned.