From Logarithms to Calabi-Yau Periods and Beyond --- Exploring the Landscape of Special Functions in Perturbative QFT

Perturbative quantum field theory (QFT) provides an extremely powerful framework for making predictions in particle physics. However, working out these predictions beyond leading order requires evaluating increasingly complicated integrals over the (unobserved) momentum flowing through virtual loops. At one loop, these integrals are under good control and can be evaluated in terms of special functions defined just in terms of logarithmic integrations. However, at two loops more complicated types of special functions begin to appear, which involve things like Calabi-Yau manifolds and higher-genus curves. In this talk, I will describe what we've begun to learn about the broader landscape of special functions that appear at high loop orders in perturbative QFT, and the interesting ties they exhibit to various areas of contemporary mathematics.