FWF - QCLaHA - 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 questions is to measure how randomly the distribution of certain (classical) sequences really is. The aforementioned sequences arise often mostly from 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 one’s 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.