Randomness in Equidistribution Theory

Natural and artificial systems often exhibit random behavior, and pro- bability theory offers the tools for analyzing and making predictions about them. Surprisingly, randomness also appears in the seemingly most organi- zed and deterministic mathematical structures. Prime numbers, for instance, appear in a completely chaotic fashion among the positive integers: we can only make probabilistic predictions about how many of them we will find in a given interval unless we actually count them one by one. The aim of this project is to identify mathematical structures which, even though deterministic by construction, exhibit random behavior. Understan- ding the delicate statistical principles governing such systems allows us to understand their most important and robust properties; the point is that we can avoid a painstaking full analysis which is typically algorithmically infeasible. Random behavior has been observed even in extremely simple systems such as circle rotations with an irrational angle. The statistics of this system is connected to surprisingly deep arithmetic properties of the given angle, which is still not fully understood. The ultimate goal is to construct more complicated systems in which an arithmetic input (such as an irrational angle) governs the long-term behavior. At the intersection of number theory, analysis and probability theory, this emerging field has recently attracted considerable attention. Another particularly simple model is trigonometric products. They play an important role in several very different areas including numerical analysis, number theory, knot theory and quantum physics. These different viewpoints make it an ideal model to work with, and offer a combination of tools to achieve the strongest results in the field. The key to unlock several deep conjectures relating quantum knot invariants to number theory is through a better understanding of trigonometric products, including their behavior under certain transformations ubiquitous in contemporary number theory, such as Mobius transformations. The final part of the project is to study such systems on abstract spaces, such as high-dimensional surfaces called Riemannian manifolds. Objects such as functions or point configurations on these spaces can be decomposed into oscillating wave-like components; a far-reaching abstract generalizati- on of the decomposition of the sound made by a vibrating chord such as a violin string into musical harmonics. The harmonic components contain valuable information on the distribution properties of the object. The aim of the project is to develop a unified approach to random and deterministic systems on abstract spaces based on harmonic analysis.

The main goal of the project was to gain a deeper understanding of the random behavior of mathematical structures, which can shed light on why randomness inevitably appears in nature and in certain artificial systems. The tools of probability theory then make detailed predictions about the robust behavior of the system without the need for understanding the fine details. Working with a system that is defined in terms of a single irrational number, we established results about the likelihood of certain extreme events taking place, when the system exhibits atypical behavior. These results connect the fields of dynamical systems and number theory to extreme value theory and probability. A key ingredient was finding the precise distribution of number theoretical properties of best rational approximations to a random real number. We found further instances when the statistics of the system is sensitive to the arithmetic properties of the irrational parameter. In an influential paper from 2010, Zagier found that the topological properties of knots are described by certain trigonometric products, and made several deep conjectures about them. We proved one of these deep conjectures, which in particular led to the understanding of the random behavior of trigonometric products. We further connected the framework of Zagier, called quantum modular forms, to the mathematical field of ergodic theory. We also studied point sets on compact spaces in which the points repel each other, inspired by fermionic particles in physics and by the eigenvalues of large matrices. We established the random behavior of such point sets by following a harmonic analysis approach, in which functions are decomposed into oscillating wave-like components the same way as musical tones are decomposed into harmonics. The harmonic analysis approach will likely lead to further advances in the area, and serve as a unifying framework for equidistribution theory.