We highlight a few of the results obtained up to September 2025:
A convex body in n-dimensional space has constant width 2 if its projection onto any line is always a segment of length 2. The unit ball is the simplest example, but there are many others in every dimension. Motivated by Borsuk’s conjecture, Oded Schramm asked whether there are constant-width bodies in n-dimensional space whose volume is exponentially smaller than that of the unit ball, when the dimension increases.
We present an explicit construction of such a body with constant width 2 whose volume is less than an exponentially decreasing constant times the volume of the unit ball, thus answering Schramm’s question. Our construction also gives a new constant-width body in three dimensions with several interesting properties.
Our result about convex bodies of constant width has been presented by special articles in the leading popular science journals Quanta and New Scientist.
Over the past year, we have also investigated a class of problems involving common tiling domains: measurable sets that can tile space under translation by more than one lattice. Specifically, we asked: given two distinct lattices in the plane with the same density, is there a bounded, measurable set — a common tile — that tiles the plane perfectly when translated along both lattices? Our work shows that such a set indeed exists — not only in two dimensions but in any Euclidean space. That is, for any two lattices of equal density in n-dimensional space, there exists a common tile that works for both.
While this result is geometrically striking in its own right, it also has interesting implications in time-frequency analysis. In particular, it enables the construction of Gabor orthonormal bases with desirable localization properties. These bases are fundamental in signal processing, where they allow for efficient and precise representation of signals in both time and frequency domains.
Somewhat unexpectedly, the proof draws on recent results related to Meyer’s quasicrystals — mathematical structures that first emerged in the study of aperiodic order in materials, but which have since become central in harmonic and time-frequency analysis. Their role in the argument highlights a recurring theme in modern mathematics: that progress often arises not within isolated fields, but at the intersection of different mathematical areas, in this case geometry, harmonic analysis and signal analysis.
In classical function theory, Bohr’s theorem says something elegant: if a Dirichlet series converges close to infinity, but we know by other means that it extends to some larger half-plane as a bounded analytic function, then the series actually converges uniformly in any slightly smaller half-plane. In the usual setting, the Dirichlet series are formed from the ordinary integers.
We have studied a generalized world of “Beurling integer systems,” where one replaces the ordinary prime numbers by a more flexible sequence of Beurling primes, and builds “integers” by multiplying them in all possible ways. The question is: under what conditions does the conclusion of Bohr’s theorem still hold in these more exotic systems? We show that one can adjust any given Beurling prime system slightly to enforce Bohr’s theorem.
More strikingly, probabilistic methods can be used to produce a Beurling prime system that satisfies both Bohr’s theorem and the “Riemann hypothesis.” The Riemann hypothesis is the famous conjecture that the nontrivial zeros of the Riemann zeta function all lie on a single vertical line, a property deeply tied to the distribution of the prime numbers. For the usual integers, this remains the most important open problem of classical analysis.
The proposal consists of three topics: (A) Analysis on the Riemann zeta function, (B) multiplicative analysis, and (C) Fourier interpolation and quasicrystals. Parts (B) and (C) represent emerging fields of analysis and are interconnected via their link to part (A). The proposed research aims at shaping the new fields (B) and (C) and at the same time advancing related analytic methods pertaining to part (A).
Part (A) aims at developing analytic techniques and viewpoints that are of specific interest to the Riemann zeta function and other L-functions. The project will in particular pursue possible number theoretic applications of Fourier interpolation and quasicrystals in this context.
Part (B) will develop the theory of Hardy spaces of Dirichlet series and specifically study composition operators acting on such spaces and spectral resolution of Mellin type operators. The project aims at solving embedding problems, with special emphasis on clarifying the scope of contractive embeddings and describing suitable classes of Carleson measures. An additional goal of part (B) is to describe curves and measures for which Fourier restriction is admitted.
Part (C) will develop the general theory of Fourier interpolation and quasicrystals. The proposed research aims at clarifying the scope of Fourier interpolation and at unifying this emerging theory with classical time–frequency analysis. This involves obtaining a better understanding of the link between Fourier interpolation and crystalline measures, as well as determining whether the full structure inherent in a quasicrystal is indeed needed for universal sampling.
The project will support 9 man-years of research carried out by three PhD students.
