*Periods in Geometry, Arithmetic and Physics*

Durham University (1st November 2007 - 31st January 2008)

**Introduction**

Periods are, roughly speaking, a distinguished class of numbers obtained by integrating an algebraic deRham form along a semi-algebraic set or a chain in the singular homology. Arithmetically interesting examples of periods include zeta and multiple zeta values and special values of *L*-series. Recently, there has been a resurgence of activity about periods which involve several disparate areas of mathematics and mathematical physics: the moduli space of stable genus 0 curves, combinatorics of algebraic cycles as well as evaluation of Feynman graphs in quantum field theory to "arithmetically interesting" numbers. The unifying theme here is the still-mysterious theory of *motives* originally conceived by Grothendieck in the 1960s.

During 1st November 2007 till 31st January 2008, the Mathematical Sciences Department of Durham University will host several different activities related to the study of periods and special values of multiple polylogarithms. Topics of emphasis includes: moduli spaces, algebraic cycles, motives and computational quantum field theory. Time permitting we would also like to cover: periods and modular forms, motivic aspects of mirror symmetry, vertex operator algebras and cluster algebras.

**Organizers and participants**

The program is being organized by Herbert Gangl and the three EU Marie Curie Early Stage Researchers in-residence at Durham for the period: Christian Bogner (Mainz), Abhijnan Rej (Bonn) and Ismael Souderes (Paris). Local participants include members of the number theory group as well as several interested mathematical and theoretical physicists.

Other than the local participants, we also expect several visitors during this period. Notably, we expect Dirk Kreimer during the second week of January who will give a serie of lectures on *Hopf algebras and numbers: the recursive structure of Green functions*. We also expect Francis Brown (Chevaleret), Hidekazu Furusho (Paris) and Qingxue Wang (Cambridge, UK) during the first week of December and Isabella Bierenbaum (Zeuthen) and Kai Keller (Hamburg) during the second week.

**Planned activities (with links to individual webpages)**

- A special (partly pedagogical) seminar
*Feynman graphs, periods and polylogarithms.* - The Durham number theory seminar-the Arithmetic Study Group- will mostly focus on research talks around periods, polylogarithms and motives. An archive for this term is available here.
- A special
*Polylogarithms and Moduli Spaces Day*on 5th December. - A special
*Feynman Graphs and Periods Day*on 12th December. *Algebraic cycles and algebraic K-theory*: A course for advanced undergraduates and beginning graduate students.- A special week (8th - 10th of January) with a series of lectures by D. Kreimer and D. Broadhurst.