Quantitative Estimates In Spectral Theory and Their Complexity

In a world where we increasingly rely on computers for anything from ordering groceries to designing space shuttles, it is important to know how fast they work, and whether it's guaranteed that their computations lead (or "converge") to the correct answer. This project aims to address both questions.

One of the most important fields that deals with rates of convergence is "ergodic theory". This field primarily deals with long-time averaged behaviour of physical systems. It is typically expected that this behaviour should converge to some averaged quantity (for example, the temperature of a jug of water slowly relaxes after it's placed in a refrigerator). The rate of this convergence is highly important in applications. For instance, it would be very useful to know how long it would take the jug of water to cool down to a certain predetermined threshold. In the first part of this project I propose a new method for obtaining such rates, using methods from a field in pure mathematics known as "spectral analysis". In a nutshell, spectral analysts study the spectrum associated to the particular problem at hand, which is akin to the DNA of the problem: it is an object that encodes all the significant properties of the physical system.

As an application, I intend to use this theory for studying physical phenomena such as plasmas and fluids. Many of the equations that govern their behaviour are amenable to the aforementioned analysis, and using these new tools I intend to understand some basic properties, such as long-time behaviour and stability. Plasma, for instance, is a form of charged matter which engineers hope to be able to harness to produce clean energy in fusion reactors. The main obstacle to this is the unstable nature of plasma.

However separately I have shown that it is not always guaranteed that approximations converge to the correct result. With my collaborators I provide some basic computational examples (for example, calculating spectra) where approximations (such as those a computer does) are doomed to fail and address this problem by introducing a new complexity theory that allows to compare the complexity of two problems that are "infinitely" complex. The second part of the proposed project is centered around understanding this new theory better and studying how "likely" it is for a given problem to be highly (or "infinitely") complex. The applications are crucial here too. I will apply the theory to some concrete physical problems that are solved using computers to see if these solutions might sometimes be wrong. I anticipate this to indeed be the case, and plan to develop warning mechanisms.

Complex computations are ubiquitous: from weather prediction, to modern material design, to simulations of the early universe. This project aims to introduce new techniques for obtaining rates of convergence for such calculations (i.e. determining how quickly they work), and for studying whether they are even guaranteed to converge correctly. This will have a direct impact on the applied sciences, engineering and technology in the UK and beyond. For instance, researchers at the Met Office (where there are extremely strong ties to academia) routinely use complex algorithms to predict the weather and weather patterns. Results that affect the efficiency at which they are able to do their calculations, and results warning them that in some instances their calculations may be wrong are extremely important. Likewise, companies such as Airbus (which has a strong presence in the UK) that deal with complex computations related to materials design and structural design, would benefit from both streams of proposed research, making their calculations faster, and guaranteeing their accuracy.