Combinatorial Methods in Analysis

Analysis is a branch of mathematics that studies various properties of functions, ranging from polynomials and elementary functions of one variable to solutions of partial differential equations or complicated functions in analytic number theory. The basic idea of harmonic analysis is to find useful ways of representing functions. Examples include basic harmonic expansions in signal analysis or wavelet decompositions in image processing. Recent significant progress in our work on geometric analysis of partial differential equations and analytic number theory has led us to formulate an ambitious program for COMAN, aiming to tackle several hard problems in classical analysis and related fields. The project is divided into four parts, each dealing with a more specific type of problems. We study quantitative properties of solutions of elliptic partial differential equations, in particular functions that describe membrane vibrations. The behavior of the Schrödinger evolution is another topic; we study evolution on discrete structures as web-like graphs. We also study Dirichlet series and the Riemann zeta function, which connect number theory and complex analysis. Finally, we investigate nonharmonic Fourier series and Meyer quasicrystals. The common approach to all problems is the search for distinct combinatorial structures and simple building blocks that are pivotal for solving the problem at hand. On the topic of elliptic partial differential equations, we study those which describe vibrations of thin metal plates. The main goal is to understand a variety of complicated geometric objects by looking at some of their distinct characteristics, or by approximating these objects by simpler ones. We work on the conjecture that patterns, which appear in the vibrations, are in many respects similar to relatively simple ones defined by algebraic polynomials. We have pursued the analysis of well-distributed point sets in the unit cube, and look at applications of these in signal analysis. Point sets with so-called low discrepancy have been widely used in quasi-Monte Carlo methods, and more recently it was recognized that the same sets are applicable in signal processing. It is well known that a shift of the integers by a Kronecker sequence gives a sequence with interesting sampling properties; this is the celebrated sampling theorem of Matei and Meyer for quasicrystals. We aim at determining whether a similar result is valid for a shift of the integers by a Halton sequence. As Halton sequences are much easier to handle numerically, this would be of significant interest. An additional central goal is to develop the theory of function spaces of Dirichlet series and the interaction with number theory and the theory of the Riemann zeta function. A recent breakthrough result allows us to view the classical Riemann-Weil formula for the Riemann zeta as a special case of a larger class of time-frequency representations of functions. We are currently investigating the analytic properties of such representations and possible number theoretic applications. We have also developed further our method for detection of extreme values of L-functions and found a dichotomy suggesting that the zeta function may attain significantly larger values than we have been able to verify in the past. A key result, closely related with the uncertainty principle, establishes a fundamental localization property of reproducing kernels in Hardy and Bergman spaces. Such kernels are central in quantum mechanics where they are known as coherent states. Our result verifies a conjecture about a certain entropy studied in mathematical physics. The Schrödinger evolution appears in quantum mechanics, and is closely related to Heisenberg's uncertainty principle. One of our aims has been to understand the concentration properties of solutions to the Schrödinger equation, known as wave functions, and we have managed to answer a long-standing question on the decay of stationary plane waves. Recently, we have suggested a new proof of the classical uncertainty principle of Hardy, using Schrödinger type evolutions with complexified time. The proof is more elementary than the classical one usually given in the textbooks. In another work related to uncertainty principles, we have obtained new results on concentration of signals on porous sets; this continues and extends a long line of research with various applications in time-frequency analysis and spectral geometry.

Prosjektet har utviklet kompetansen til flere unge forskere som nå hevder seg på høyt internasjonalt nivå. En viktig effekt av prosjektet er at forskningsmiljøet har styrket sin internasjonale synlighet og sitt internasjonale forskningssamarbeid.

COMAN is a 5-year FRIPRO Toppforsk project in mathematics at the Norwegian University of Science and Technology (NTNU). Its focus is basic research in analysis and its interaction with mathematical physics, PDEs, and number theory. The applicants are Eugenia Malinnikova (principal investigator), Andriy Bondarenko, Sigrid Grepstad, Yurii Lyubarskii, and Kristian Seip. Funding is sought for 3 PhD students, 3 postdoctoral fellows, and operating expenses. Collaboration with world-leading mathematicians is included. A perpetual trend in classical analysis is that major breakthroughs tend to rely on constructions or decompositions that are of a distinct combinatorial nature. The crux of the matter is often to identify and analyze certain discrete structures or simple building blocks that are instrumental in understanding and controlling the branching complexity of the problem at hand. In harmonic analysis, prominent examples stretch from the classical Calderón-Zygmund theory and Carleson's corona construction to wavelets, compressed sensing, and applications of Meyer quasicrystals. The recent significant progress in our work on geometric analysis of PDEs and analytic number theory has led us to formulate an ambitious program for COMAN, aiming to tackle several hard problems in classical analysis and related fields. The choice of topics rests on the scientific interests of our team members; the common thread in our scientific approach is to investigate in depth concrete and fundamental problems, in search for the basic underlying principles that are likely to be of a combinatorial nature and to arise from geometric, probabilistic, or number theoretic insight.