In this talk I will give a gentle introduction to some aspects of the theory of hypergeometric functions as a natural language for addressing various integrals appearing in quantum field theory (QFT). In particular I will focus on the so-called intersection pairings as well as the differential equations satisfied by the integrals, and I will show how these aspects of the mathematical theory can find a natural interpretation in concrete QFT applications. I will mostly focus on Feynman integrals as paradigmatic example, where the language will shed new light on our most powerful method for computing Feynman integrals as well as their non-local symmetries. I will then give an outlook how these methods could allow us to also learn about integrals appearing in other places in field and string theory, such as Coulomb branch amplitudes, celestial holography and AdS (supergravity and string) amplitudes.

Even in a totally discrete space $X$ you need to know how to move between distinct points. A path $P_{x,y}$ between two points $x,y \in X$ is a sequence of points in $X$ that starts with $x$ and ends with $y$. A path system is a collection of paths $P_{x,y}$, one per each pair of distinct points $x, y$ in $X$. We restrict ourselves to the undirected case where $P_{y,x}$ is $P_{x,y}$ in reverse.

Strictly metrical path systems are ubiquitous. They are defined as follows: There is some spanning, connected graph $(X, E)$ with positive edge weights $w(e)$ for all $e\in E$ and $P_{x,y}$ is the unique $w$-shortest $xy$ path. A metrical path system is defined likewise, but $w$-shortest paths need not be unique. Even more generally, a path system is called consistent  (no $w$ is involved here) if it satisfies the condition that when point $z$ is in $P_{x,y}$, then $P_{x,y}$ is $P_{x,z}$ concatenated with $P_{z,y}$. These three categories of path systems are quite different from each other and in our work we find quantitative ways to capture these differences.

Joint work with Daniel Cizma.

A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction. No prior knowledge of stacks is needed for this talk!