Quantum Chaos, Lattices, and Harmonic Analysis

The present research proposal concerns two types of questions (as well as arising applications): A) Distributional properties of sequences of numbers on fine scales B) the Geometry of Numbers in high-dimensional spaces. The goal of the former set of questions is to measure how randomly the distribution of certain (classical) sequences really is. The aforementioned sequences often arise in physics; to be more precise, they arise from a relatively recent research area which studies how chaos manifests in quantum mechanics. An answer to this question was proposed in the 1970s in a fundamental conjecture by the physicists Barry und Tabor. At present, this conjecture is only known in parts for truly special quantum systems Simply put, one is interested in understanding how the energy levels of a typical quantum system distribute. It is worth mentioning that here distribution is referring to distribution on fine-scales. This is in sharp contrast to, for instance, the classical theory of uniform distribution from number theory where the scale is a fixed quantity. Indeed, it is the explicit goal to sharpen results from that classical theory to statements involving smaller scales, whenever possible. The second type of questions concern lattices which one may think of as being higher dimensional versions of insect nets protecting ones window the key difference however is that a lattice can have an arbitrary high dimension instead of only three dimensions, like an ordinary insect net. What is this good for? On the one hand, there is a panoply of problems in mathematics which boil down to the existence or non-existence of an (interesting) object. On the other hand, the course of the about last 100 years have brought to light that the, so-called, theory of Geometry of Numbers provides a unifying (and at times simplifying) framework to translate existence problems to geometric problems. The latter are usually more amendable to a broader range of techniques. Consequently, the Geometry of Numbers plays a decisive role in, e.g., combinatorics, number theory, the theory of dynamical systems, and computer science. The present project focuses on a less understood aspect of the Geometry of Numbers, that is the dependence on the dimension in the following sense: Consider an infinite collection of compatible lattice point problems. Compatibe means here, roughly speaking, that the second lattice is contains a lower dimensional copy of the first and the third lattice a lower dimensional copy of the second and so froth. Is there always (and if so how many) solution to the lattice point problem in this given set of lattices as soon as the dimension is sufficiently large? The previously mentioned lattice point problems are directly motivated by applications in (algebraic) number theory and logic.

Our world is full of highly complex phenomena: one gram of water contains more than 10^22 molecules. Analysing the motion of water by attempting to compute the motion of each molecule is a fallacious endeavour. A powerful and versatile tool to describe complex systems is probability theory predicting that irregularities and difficulties on small scales cancel out beautifully on larger scales. Many laws of nature are quite successfully described in this way. My research follows this spirit. I study statistical properties of arithmetic data, showing that simple laws emerge from fine-scale randomness. Such statistics are central to various fields of mathematics and physics. My main focus is on statistics that are motivated by quantum chaos. Here, one investigates how chaotic behaviour and the quantum world fit together. Scientifically, chaos describes the future behaviour of a system is extremely sensitive to its initial state. This means predicting what happens in such a system needs in practise in-attainably precise information. Chaotic systems are all around us. A simple example is ball placed somewhat high over the peek a pyramid: deciding whether the ball falls to the right, or to the left of the peek needs extremely precise information on the initial position of the ball. In 1977, the renowned physicists Sir M. Berry and M. Tabor put forth a fundamental conjecture to clarify the mysterious relationship between chaos and quantum physics. A key point in their conjecture is the distribution of gaps between different energy levels of a quantum system. The random behaviour of the gaps should help us trace the fingerprints of chaos. Despite considerable efforts of the scientific community, the Berry-Tabor conjecture is still widely open - even in simple cases. The reason is clear: the available mathematical tools are insufficient. My research aims to create such tools. A distinguished starting point for understanding the gap distribution is to study the so-called pair correlation function. This function measures the statistical dependence of pairs of points. Mathematicians and physicists often expect the pair correlation function of data points, e.g. energy levels in a quantum system, to behave as if the data was (genuinely) randomly spaced. In that case, the pair correlation function is called Poissonian. A research highlight of this project is devising novel counting methods which deliver explicit examples (of sequences of mathematical interest), whose pair correlation function is Poissonian. This is the content of my collaboration with Lutsko and Sourmelidis. With Lutsko, I developed the techniques further to cover higher order correlation functions (involving more than two points at a time) provided sequences are sufficiently slowly growing. Such results are still rarities. These works were featured in a recent article in Scientific American that Lutsko wrote.