School "Modular forms, periods and scattering amplitudes"

Ruth Britto, Introduction to Feynman integrals

An overview of Feynman integrals of interest to physicists and some approaches to computing them, including differential equations, iterated integrals, and on-shell factorization. I will discuss their evaluation to generalized hypergeometric functions and a conjectured coaction on these functions that is related to operations on Feynman graphs.

Claude Duhr, Elliptic polylogarithms for practitioners

The focus of the lecture will be to introduce iterate integrals on elliptic curves in a way that makes them useful for explicit computations, in particular for multi-loop Feynman integrals.

Javier Fresán, Introduction to Feynman motives

I will explain, assuming as less background as possible, how Feynman amplitudes arise as periods of motives and how the study of these motives can give precious information about the kind of numbers one gets. I will mainly focus on the case of primitive log-divergent graphs, one of the goals being to explain the coaction conjecture of Panzer and Schnetz.

Nils Matthes, Around integrals of modular forms for SL₂(Z)

Given a modular form a natural question is whether it has a primitive which also transforms "like a modular form''. It turns out that the answer is "no'' if one looks among holomorphic functions, the defect being measured by the periods of the modular form. These are certain complex numbers carrying important arithmetic information and the first goal of this course will be to give a concise introduction to these objects in the special case of modular forms for SL₂(Z). Time permitting, we will then show how to construct the "correct'' primitives of modular forms among a more general class of real-analytic functions. These turn out to have several nice properties, namely, they are eigenfunctions of the hyperbolic Laplacian and have interesting, presumably transcendental Fourier coefficients.