Chandrasekhar Hall

Multiple zeta values

J. Oesterle

University of Paris VI

Multiple zeta values are the real numbers of the form
$$
\zeta(k_1, \cdots, k_r) = \sum_{n_1 > \cdots > n_r} n_1^{-k_1} \cdots n_r^{-k_r}
$$
where $(k_1, \cdots, k_r)$ is a finite sequence of non negative integers with $k_1 > 1$.
They satisfy various polynomial relations over $\mathbb{Q}$, some of them already
known to Euler and some others only discovered quite recently. Francis Brown proved
last year that they are all $\mathbb{Q}$ -linear combinations of those for which
the exponents $k_i$ belong to $\{2,3\}$; these latter
ones are believed to be $\mathbb{Q}$-linearly independent.



During the first two weeks, we shall describe the history of
the subject and state the main results. Brown's method consists in
constructing motivic analogues of the multiple zeta values for
which he is able to prove the analogous results. This construction
involves Hopf algebras, iterated integrals, unipotent group schemes,
cohomology and periods, motivic versions of the fundamental group,
mixed Tate motives. Describing these tools and giving the proofs
will be the main task of the course.