CMI-IMSc Lecture series in Number Theory

Prof. Joseph Oesterlé University of Paris VI, France will be giving a series of 24 lectures on

Multiple Zeta Values, the recent advancements

at the Institute of Mathematical Sciences (IMSc) during 5th October to 30th December, 2012.

Multiple zeta values are the real numbers of the form $$\zeta(k_1,..., k_r) = \sum_{n_1 > ... > n_r} n_1^{-k_1} ... n_r^{-k_r}$$, where $(k_1, ..., k_r)$ is a finite sequence of non negative integers with $k_1 > 1$. They satisfy various polynomial relations over the field of rational numbers, some of them already known to Euler and some others only discovered quite recently. Last year Francis Brown proved that they are all rational linear combinations of those for which the exponents $k_i$ belong to $\{2,3\}$; these latter ones are believed to be linearly independent over rational numbers. The lectures in the first two weeks are part of the Panorama lecture series meant for a general audience. In these two weeks, Professor Oesterlé will describe the history of the subject and state the main results.

The lectures in the later weeks will be an attempt to provide the relevant details of Brown's work. As we will see, 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.

Support towards travel and local hospitality for researchers is available for a limited number of people from all over India. People from outside India can only be provided local hospitality.