Project funded by the ANR


Periods are a class of complex numbers obtained by integrating algebraic differential forms over algebraically defined domains which only involve rational coefficients. Examples include logarithms of integers, multiple zeta values, and certain amplitudes in string and quantum field theory.

From the modern point of view, periods appear as entries of a matrix of the comparison isomorphism between algebraic de Rham and Betti cohomology of varieties over number fields. Thanks to this interpretation, the theory of motives becomes a powerful tool to predict all algebraic relations among these numbers and, in some favourable cases, to prove them. It should be thought of as a higher analogue of the Galois theory of algebraic numbers. Indeed, all recent breakthroughs in the study of periods, namely Ayoub's theorem (a relative version of the Kontsevich-Zagier conjecture) and Brown's theorem (every multiple zeta value can be written as a linear combination of those having only 2 and 3 as exponents), were the reflection of the emergence of new ideas and techniques in motivic Galois theory.

This JCJC project aims at gathering together young researchers working on the theory of periods and motives form different points of view, the interaction of which seems particularly promising.