Abstract：

To prove that zeta(3) is irrational, Apéry realized this number as the limit of the ratio of two solutions of a linear recurrence with polynomial coefficients. Similar recurrences and their "Apéry limits" have been studied in mirror symmetry. In this talk, I will sketch a method to realize Feynman integrals as Apéry limits, using a combinatorial graph invariant from arXiv:2304.05299. Examples include fourth order recurrences for zeta(3,5) and a third order recurrence for zeta(5) and zeta(9). The mechanism behind this method is a general theory (applicable beyond Feynman integrals) based on a Mellin transform, and furthermore connecting "diagonal" coefficients of powers of a polynomial (combinatorics) with point-counts over finite fields (arithmetic). This is work in progress together with Francis Brown.

About the Speaker:

Erik Panzer

University of Oxford

Research interests

Feynman graphs and Feynman integrals, hyperlogarithms, (elliptic) polylogarithms, (elliptic) multiple zeta values, motivic periods, combinatorial Hopf algebras, renormalization, resummation

Personal website

https://people.maths.ox.ac.uk/panzer/