# Density and approximation in automorphic families

#### What

Number theory is one of the oldest branches of mathematics, rooted in our quest to understand the integers and related objects. The area has developed immensely over time, influenced and influencing other areas of not only mathematics, but also physics, computer science, and engineering. The focus of my project lies in the area of automorphic forms and their relation to spectral theory which has been a particularly fruitful and exciting area of number theory, and lies right at the intersection of not only various mathematical fields but also mathematical physics. In this project we want to study properties of automorphic forms - certain functions carrying a lot of information about the objects they live on - from several related perspectives.

#### Why

The goal of this project is to study families of automorphic forms and representations in a relatively wide sense. Automorphic forms and representations lie at the heart of modern number theory, in particular the Langlands program. They often appear naturally in families, for example by varying their spectral parameters, or the level of ramification. In the arithmetic setting 'harmonic' families come with 'symmetry types' which are supposed to reflect deep structural properties connected to the Langlands conjectures and predict how the spectral data distribute in the family. The Weyl law, limit multiplicity theorems, Sato-Tate equidistributions, and low-lying zeros in families of L-functions are all examples of conjectures and theorems in this context.

#### How

In this project we want to combine methods from various areas of mathematics to push beyond the current state of knowledge. Some of the central tools will be trace formulae in various incarnations, in particular the so-called Arthur-Selberg trace formula and the Kuznetsov trace formula. Trace formulae allow us to relate two different kinds of properties of our objects (like manifolds or automorphic forms) with each other, namely spectral property, that is, energy eigenstates physically speaking, with geometric properties. This also allows us to bring in methods from other areas such as analytic numbers theory or statistics to obtain a deeper understanding of our objects.

#### SSR

Our goal with this project is to uncover fundamental results at the intersection of various mathematical areas, all centered around number theory. With research in pure mathematics it is often hard to tell whether practical applications lie ahead in the future. Historically, research in number theory has been quite fruitful in this regard, for example, without research in number theory, we would not be able to communicate safely online, and currently there are indications that our research might have some applications in the area of quantum computing, for example. It is thus of huge importance to continue research in such abstract areas even without concrete applications in mind as those might emerge at a later and often unexpected point.