Fri, 27.01.2017, 14:30
Fri, 27.01.2017, 15:30
What is the Galois theory of algebraic integrals?
Oliver Schnetz (FAU)
Algebraic integrals are (multiple) integrals of rational differential forms with integer coefficients. The cycle of integration is given by polynomial inequalities with integer coefficients. Kontsevich and Zagier called numbers 'periods' when they are algebraic integrals. Here we also consider the case of functions (like log(z) which is the integral of 1/x dx over 1<x<z).
A deep mathematical theory going back to Grothendieck claims that there should exist a Galois theory for algebraic integrals. The Galois group acts on the pair (differential form, integration cycle). By duality the Galois action turns into a coaction on algebraic integrals. Knowlege of the coaction mostly determines the integral. This makes the Galois theory a powerful tool for algebraic integration. In general, the Galois theory of integrals is considered very complicated and basically only one explicit result (due to Goncharov and Brown) is known.
In this talk I will discuss the possibility that (inspite of the deep background) at least in the linear case there might exist simple, elementary, and explicit formulas for the Galois coaction of integrals.