The arithmetic of multiple zeta values

“Multiple zeta values” are the numbers defined by $\zeta(k_1,\dots,k_n)=\sum m_1^{k_1}\cdots m_n^{k_n}$, where the summation goes over all $0<m_1<\cdots<m_n$, all the $k_i$'s are positive integers, and $k_n>2$. They were first studied (in the case $n=2$) by Euler and have become an object of great interest in recent years because of connections with many branches of mathematics, ranging from arithmetic algebraic geometry to quantum field theory. The talk will give a survey of some of these connections and of some recent results.