Motivic multiple zeta values

by Prof. Jose Burgos I Gil (ICMAT-CSIC, Spain)

Monday, November 3, 2014 from 16:00 to 17:30 (Asia/Kolkata)
at Mumbai ( AG-77 )

Description

The famous Riemann zeta function is one of the most studied objects in mathematics. But it still hides many mysteries and enigmas such as the Riemann hypothesis. The values of this function at positive integers is another example of these mysteries. Euler already knew that the values of the zeta function at even integers can be expressed in terms of powers of $\pi$ and rational numbers.

However, nobody has been able to express the values of the zeta function at odd integers in terms of known numbers like $\pi$. In fact, it is expected, that this should not be possible. More precisely we \emph{expect that the numbers $\pi$, $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, etc., are algebraically independent over $\mathbb{Q}$}.

As it is often done in mathematics, in order to simplify a problem, we turn to a generalization of it. In this case we introduce multiple zeta values (MZVs for short) that have the advantage that the product of two MZVs is a linear combination of MZVs. Therefore we have reduced the problem of algebraic dependence between zeta values to a problem of $\mathbb{Q}$-linear dependence between MZVs. MZVs have deep properties, and have appeared in recent years in connection with many topics of surprising diversity, including knot invariants, Galois representations, periods of mixed Tate motives, and calculations of integrals associated to Feynman diagrams in perturbative quantum field theory (pQFT).

The MZV are classified by its weight. We denote by $V_{n}$ the
$\mathbb{Q}$-vector space generated by the MZVs of weight $n$, with the convention that $1$ is a multiple zeta value of weight zero, and we write $V$ for the subspace of $\mathbb{R}$ generated by all MZVs. Zagier has conjectured that there are no relations among MZVs of different weights, and that the dimensions $\dim V_{n}$ are given by the numbers $d_{n}$ satisfying the recurrence relation $d_{0}=1$, $d_{1}=0$, $d_{2}=1$ and $d_{n}=d_{n-2}+d_{n-3}$.

Thanks to work of Goncharov Deligne and Terasoma, we know half of this. conjencture. Namely that $\dim V_n\le d_n$. Surprisingly, the only known method of proof of this result uses the theory of motives. The aim of this short course is to give an outline of the connection between MZV, periods and motives, explaining the basic ideas of such proof.