A mathematical law makes it impossible to perfectly filter a signal—isolating a specific moment in time and a specific frequency at the same time—and this project aims to measure exactly how close we can get. This matters because the trade-off between time and frequency precision, known as the uncertainty principle, is not just a quirk of quantum physics. It is a hard mathematical constraint that limits every device or algorithm that processes signals—from mobile phone antennas to medical imaging scanners. Engineers currently rely on approximate filters, called time-frequency localization operators, but lack a precise mathematical understanding of how well these filters can perform as the region of interest shrinks or changes shape. The project will develop new spectral theory to describe how the eigenvalues of these operators behave, particularly when both the size and the shape of the time-frequency region vary simultaneously—a problem that remains largely unsolved. This is fundamental mathematics, not applied engineering. If successful, the work will provide rigorous tools for two specific areas: signal processing (for example, quantifying how much noise a filter can suppress) and mathematical physics (modelling how charged particles distribute in a Coulomb gas). No immediate consumer product will emerge, but the mathematics of uncertainty has historically underpinned everything from radar to JPEG compression.
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Heisenberg's famous uncertainty principle states that we cannot know the position and momentum of a particle with perfect accuracy. This phenomenon goes beyond physics: it is hard-coded into mathematics itself. It is a fundamental fact relating functions and their frequency profile and is therefore ubiquitous. Aside from quantum mechanics, the uncertainty principle plays a key role in the analysis of signals in an area known as time-frequency analysis. In this context, the uncertainty principle presents itself in many forms, often as an obstruction ingrained in our mathematical models that goes against our intuitive understanding of reality. Many applications depend on quantitatively describing certain facets of the uncertainty principle. This project focuses on the study of time-frequency localization operators, that approximately filter a signal and mostly constrain it to a specific region in time and frequency. A perfect filter does not exist. It is impossible for a non-zero signal to be localized both in time and frequency simultaneously, due to the uncertainty principle. Thus it is important to be able to quantify how well localization operators concentrate a signal, that is, how far they are from matching their unattainable ideal counterparts. The techniques we will use to study localization operators belong to the area of spectral theory, which can be thought of as the study of vibrations. Mathematically, our objective is to study the behaviour of the eigenvalues of these operators and how they depend on the region of time-frequency space where we are localizing. Two classical and intensively studied phenomena include: the speed of decay of the eigenvalues, and the effects of resizing the region of localization. However, recent applications to signal processing and mathematical physics require a joint understanding of the interplay between them. This constitutes a highly delicate problem that remains underdeveloped and is the challenge we aim to address. The project will enable applications in mathematical physics and signal processing. In mathematical physics, it will advance a technique to study the distribution of particles in Coulomb gases, a model for charged particles interacting under the electrostatic force. In signal processing, localization operators are used in filtering methods, for example to suppress noise. The project has the potential to help refine the quantitative analysis of the performance of these methods.
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