There is a fascinating interplay between the mathematical fields of polylogarithms, associators

and mixed Tate motives. At the intersection of these subjects lie the multiple zeta values, a certain

class of real numbers first introduced by Euler as generalizations of special values of the classical

Riemann function. They possess a very rich algebraic structure, which is an artifact of the

above-mentioned three fields, and is also the subject of many beautiful conjectures, some of them

due to Broadhurst--Kreimer, Goncharov, Hoffman and Zagier.

In this talk, we will try to explain an analogous picture in genus one. We will focus on the elliptic

multiple zeta values, introduced by Enriquez. Those are defined from the monodromy of the (universal)

elliptic Knizhnik--Zamolodchikov--Bernard connection on a once-punctured elliptic curve. An important

difference is that elliptic multiple zeta values are functions (of the modulus), rather than numbers, which

in particular that their transcendence properties are easier to understand. We will see how elliptic multiple

zeta values relate to modular forms for SL_2(Z) and to multiple zeta values, and if time permits, also

indicate a relation with Brown's multiple modular values.

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