On the algebra of multiple zeta values

Classical multiple zeta values (MZV’s) first studied by Euler in the eighteenth century have been investigated intensively over the past four decades due to their deep connections to various branches of mathematics and physics, including arithmetic geometry, knot theory and high-energy physics. We review central conjectures on the algebra of these values and report on recent progress. We then introduce the analogues of MZV’s in positive characteristic, following the pioneering work of Carlitz and Thakur. After reviewing their basic properties and outlining the main goals of the theory, we report on several new results on their algebraic structure. This is based on joint work with Nguyen Chu Gia Vuong and Pham Lan Huong.