Arithmetic Study Group

Schedule and abstracts of talks 

Unless announced otherwise, the seminar meets every Thursday 2:15 pm at E102

November 1st

Herbert Gangl (Durham) Overview

Physicists and mathematicians alike have encountered "periods" (arising from integrating an algebraic integrand against an algebraically defined domain), on the one hand from considering Feynman graphs, and on the other hand from associating interesting invariants to "motives". This term in the seminar series, we would like to understand work that has been done relating the two--rather different--points of view. As a main example, Bloch-Esnault-Kreimer have linked the period zeta(3) arising from the "wheel of spokes graph" in physics to algebraic-geometric constructions. 

November 8th

Ismael Souderes (Paris and Durham) Multiple zeta values and the geometry of moduli spaces of curves

A.B. Goncharvov and Manin have shown that the moduli spaces of curves of genus 0 with n marked point are natural spaces to see the multiple zeta values as periods and to build motivic avatar (over $\Z$) of the multiple zeta values. In this talk, we will describe the geometry of thoses spaces needed in order to sketch a proof of their main result on the algebraic aspect of their work. The motivic part of the article will be discussed in a later talk. (slides)

November 15th 

 

November 22th

Abhijnan Rej (Bonn and Durham) Motives: A colloquial introduction 

In this talk, we present a bird's-eye view of the theory of motives. We begin with an overview of the theory of pure motives based on correspondences on algebraic cycles (as envisioned by Grothendieck in the 1960s in-order to prove the so-called "standard conjectures".). We then introduce mixed Hodge structures and using the definition of pure (Tate) motives and mixed Hodge structures over the rationals, we explain what mixed Tate motives are, and list the desirable properties of the conjectural abelian category of mixed motives of which mixed Tate motives are a subcategory. (All through this we treat Voevodsky's construction of a derived triangulated category of mixed motives as a "black-box"- in a later talk, we will return to Voevodsky's theory.) We finish by mentioning a few applications of mixed Tate motives to questions about special values of zeta and multizeta functions, especially with a teaser on the recent work of Bloch-Esnault-Kreimer. 

November 29th

Abhijnan Rej (Bonn and Durham) Motives: A colloquial introduction 
Part 2
 

December 6th

 Francis Brown (CNRS-Paris VI) Dedekind zeta functions and products of hyperbolic manifolds
 

December 13th

 Ismael Souderes (Paris VII and Durham University) Double shuffle and moduli spaces of curves
(slides)

 

January 17th

Matthew Kerr (Durham University) The Abel-Jacobi and regulator map for higher Chow groups

 

We discuss an explicit formula for a map from the motivic cohomology of an algebraic variety to its rational Deligne homology. This generalizes the Abel-Jacobi map of Griffiths to (essentially) certain "relative algebraic cycles" living over X. We will work out some simple (but interesting) examples related to polylogarithms and hypergeometric integrals. While arithmetic issues are necessarily involved, the flavor of the talk will be primarily algebro-geometric (in characteristic 0).

January 24th

Yasuo Ohno (Kinki University/Bonn) Relations among non-strict multiple zeta values

Euler, the father of multiple zeta values, mainltreated non-strict multiple zeta values (MZSVs) in his article. The Q-algebras spanned by strict (ordinary) multiple zeta values (MZVs) and MZSVs are the same to each other. I am planning to review and compare various relations among MZVs and MZSVs and explain an advantage of MZSVs in studying the explicit structure of the algebra. I will also introduce new identities and prospects which are proper to MZSVs.

January 31th

Abhijnan Rej (Bonn and Durham) Some recent results on Feynman graph hypersurfaces