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Mathematics Colloquium


Date:  Tuesday, September 23, 2008

Title:  Elliptic curves and Multiple zeta numbers

Abstract:  Multiple zeta numbers generalize the values of the Riemann zeta function at integers larger than 1. They were first considered by Euler and have recently resurfaced in the works of Zagier, Goncharov and others. The multiple zeta value $\zeta(n_1,\dots,n_r)$ is defined by $$ \zeta(n_1,\dots,n_r) = \sum_{01$. Multiple zeta numbers occur as periods of the {\em mixed Tate motives}, constructed by Deligne and Goncharov. They satisfy many interesting combinatorial identities; and some of their transcendence properties are controlled by the algebraic $K$-theory of the integers. After surveying these results, I will discuss some mysterious identities between multiple zeta values that arise from cusp forms for $\SL_2(\Z)$ that go back to Goncharov and are due to Gangl, Kaneko and Zagier. The final goal is to explain why the theory of {\em elliptic motives} should explain these mysterious relations.

Speaker:  Richard Hain
Institution:  Duke University


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