Multiple Zeta Values

Time and location: Trinity term 2019, Tuesdays 10-12, C4 (Math. Inst.)

No lecture on 11 June! Replacement: Monday, 10 June, 10-12, C4.

Course description: Multiple zeta values are natural generalizations of special values of the Riemann zeta function. They have been studied at least since Euler who found many of their algebraic properties. Having been seemingly forgotten for more than 200 years, multiple zeta values were rediscovered by many mathematicians and theoretical physicists since the 1980s in several different contexts (mixed Tate motives, quantum groups, moduli spaces of vector bundles, scattering amplitudes, resurgence theory, etc.). An important open problem is to determine the dimensions of the space of multiple zeta values of a given weight and depth and to find explicit bases for these spaces.

A crucial feature of multiple zeta values is that they can be written as iterated period integrals on the projective line minus three points. With considerable work one can use this integral representation to construct motivic multiple zeta values. These form a graded Q-algebra which maps surjectively onto the Q-algebra of multiple zeta values, and which is endowed with a coaction of a certain motivic Hopf algebra. This coaction, which is not known to exist for multiple zeta values, is a very important piece of structure: its systematic usage led to a partial resolution of Hoffman's conjecture which gives a basis for multiple zeta values (Brown). Another important use of motivic multiple zeta values is to give an upper bound on the space of multiple zeta values of a given weight. This was predicted by Zagier and was originally proved independently by Goncharov and Terasoma.

The goal of this course is to give an introduction to (motivic) multiple zeta values and to develop the necessary tools to prove the above-mentioned results of Goncharov, Terasoma and Brown. Since this requires a formidable amount of technical machinery (Tannakian categories, unipotent fundamental groups, etc.) and very deep results (mostly related to the existence and basic properties of the category of mixed Tate motives over Z), some parts of the construction of motivic multiple zeta values will only be sketched.

Lecture notes

Prerequisites: None, but some previous exposure to algebraic geometry (in particular Tannakian categories and Hodge theory) would be useful.

Literature:

• Y. André: Chapter 25 of Une introduction aux motifs. Panoramas et Synthèses, 17. Société Mathématique de France, Paris, 2004. xii+261 pp.

• F. Brown: Mixed Tate motives over Z. Ann. of Math. (2) 175 (2012), no. 2, 949–976. pdf

• F. Brown: On the decomposition of motivic multiple zeta values. Galois-Teichmüller theory and arithmetic geometry, 31–58, Adv. Stud. Pure Math., 63, Math. Soc. Japan, Tokyo, 2012. pdf

• J. Burgos, J. Frésan: Multiple zeta values: From numbers to motives. pdf

• P. Deligne, A. Goncharov: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 1–56. pdf

• P. Deligne: Multizêtas, d'après Francis Brown. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058. Astérisque No. 352 (2013), Exp. No. 1048, viii, 161–185. pdf