XXXV Workshop on Geometric Methods in Physics | 26.06-2.07.2016 |

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## Ivan Horozov## Iterated Integrals and Periods over Totally Real Number RingsRiemann zeta values and multiple zeta values were defined by Euler. Kontsevich expressed each of them via iterated integrals, which led to many algebra-geometric properties of Riemann zeta values and of multiple zeta values. Deligne showed that multiple zeta values are (mixed Tate) period over the integers. More recently F. Brown showed that essentially they are all mixed Tate periods over the integers. Dedekind zeta values (Dedekind)are a number theoretic analogue of Riemann zeta values. They are (mixed Tate) periods over a number ring (Borel, Beilinson). In 2014, I defined a number theoretic analogue of multiple zeta values, which I called multiple Dedekind zeta values. These objects are again defined via iterated integrals. They will be the central object of the talk together with a recent result (2016): multiple Dedekind zeta values give examples of mixed Tate periods over totally real number rings. |

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