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Main lectures
Francis Brown (CNRS Paris)
Multiple zeta values from algebraic geometry to physics
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Multiple zeta values from algebraic geometry to physics
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These lectures will be divided into two parts: the first half will be a purely mathematical account of multiple zeta values
from the point of view of the general philosophy of periods and motivic Galois theory. In particular, I will explain how multiple zeta values
are periods of certain hyperplane configurations related to the moduli space of curves of genus zero, following Goncharov and Manin.
In the second half, I will try to explain the role of multiple zeta values in Quantum Field Theory. After reviewing properties of graph polynomials,
I will give an account of Bloch-Esnault-Kreimer's construction which exhibits convergent scalar Feynman integrals as periods. This has very many similarities with the corresponding construction for multiple zeta values. Finally, I will explain how, by counting points over finite fields, we can show that multiple zeta values do not in fact suffice to describe Feynman amplitudes in Quantum Field Theory.
José Burgos (ICMAT Madrid)
Multiple zeta values, polylogarithms and elliptic curves
(show abstract [here])
Multiple zeta values, polylogarithms and elliptic curves
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Multiple zeta values appear as periods of certain algebraic varieties.
These varieties are of a very special kind, namely, their associated
motives are mixed Tate motives. Thus, in a first sight, they seem
unrelated to more complicated motives such as elliptic motives.
Nevertheless there are interesting connections between MZV and the
theory of elliptic curves. In this course we will explore two of these
connections.
The first connection arises when considering the relations among MZVs.
It is believed that all such relations are given by the structure of
double shuffle algebra of the space generated by MZVs. Nevertheless
this structure is very difficult to use because relations among low
weight MZV's may appear as factors of relations of higher weight. The
first of these exotic relations was discovered by Gangl, Kaneko and
Zagier and is related to a cusp modular form. Several authors have
further studied this relation between cusp modular forms and MZVs.
The second connection is through the elliptic analogues of
polylogarithms, introduced by Brown and Levin and of multiple zeta
values, introduced by Enriquez. We will give an account of the
definition of these analogues and their basic properties.
Dirk Kreimer (HU Berlin)
From the structure of renormalizable quantum fields to periods of Feynman graphs
(show abstract [here])
From the structure of renormalizable quantum fields to periods of Feynman graphs
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In the first of the three lectures we will outline the basic algebraic structures underlying
a renormalizable quantum field theory, with a focus on the Hopf algebra of graphs,
the accompanying Hochschild cohomology and the rise of the renormalization group equations.
The second lecture will introduce the two Kirchhoff polynomials, and discuss the computation
of Feynman graphs in scalar field theory in examples. Emphasis will be given to understand how periods emerge from
the computation of graphs which are primitive elements in the above Hopf algebra, in comparison to the general situation.
The third lecture will focus on gauge theories. It will introduce a third graph polynomial, which enables to transfer results from
scalar field theory to gauge theories. Such theories are renormalizable quantum field theories believed to underly particle physics as we know it.
The arithmetics of their Feynman graphs raises vexing questions on cross cancellations between periods of different graphs which are poorly understood at present.
Oliver Schnetz (U Erlangen-Nürnberg)
Multiple zeta values and single-valued multiple polylogarithms
(show abstract [here])
Multiple zeta values and single-valued multiple polylogarithms
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In the first introductory lecture I will give a brief overview over the
structure of multiple zeta values.
I will cover the integral representation, double shuffle relations, the
Brown-Goncharov coaction, and the f-alphabet.
In the second lecture I will give a short account of Chen's iterated
integrals and their properties using generating
functions. Thereafter I define (multi-valued) multiple polylogarithms
and determine their monodromies.
The third lecture will treat Brown's construction of single-valued
multiple polylogarithms and their evaluations to
'single-valued' multiple zeta values.
We give two residue formulas and mention possible generalizations
of single-valued multiple polylogarithms.
The fourth lecture will apply the previous results to Feynman periods. We define graphical functions and
explain why some periods are single-valued multiple zeta values. Finally, the K5 conjecture suggests a complete
description of all primitive divergent phi^4 graphs with single-valued multiple zeta period of maximum weight.
Talks by docs/postdocs
Dzmitry Doryn (HU Berlin)
Multiple zeta values and graph hypersurfaces
(show abstract [here])
Multiple zeta values and graph hypersurfaces
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First Talk: Counting rational points on graph hypersurfaces
There are interesting numbers appearing in the computation of Feynman periods in
Φ4 theory. These numbers are related to multiple zeta values in many cases
and should have a motivic meaning, so we need to understand the geometry of the
hypersurface associated to a Feynman graph, that is quite hard. The
simplest idea is to count points on these hypersurfaces. We can define the c2
invariant as the coefficient of q2 in the q-expansion of the number of rational
points over Fq of a graph hypersurface. This discrete characteristic
surprisingly contains some information about the period of the graph.
Second talk: Cohomology of graph hypersurfaces
The next natural step in understanding the geometry and in the motivic direction is
to compute the cohomology of graph hypersurfaces. This can be done in some cases.
For denominator reducible graphs we can construct the framing of the relative
cohomology and this allows us to interpret our interesting
MZV-combinations in Φ4 as periods of mixed Tate motives.
Clément Dupont (Paris 6)
Periods of hyperplane arrangements and their motivic Galois theory
(show abstract [here])
Periods of hyperplane arrangements and their motivic Galois theory
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This talk is an introduction to the philosophy of motivic periods and their Galois theory. Following Beilinson et al., we introduce Aomoto polylogarithms, which are periods indexed by hyperplane arrangements in projective space. As special cases, one recovers multiple zeta values and values of multiple polylogarithms. We discuss the computation of their motivic coproduct, focussing on concrete examples.
Erik Panzer (HU Berlin) [PDF]
Relations among multiple zeta values
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Relations among multiple zeta values
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In this introductory lecture we give examples of the many known relations between MZVs, as well as the few known non-relations (irrationality results). We mention the conjectures of Zagier and Broadhurst-Kreimer regarding the weight- and depth-filtrations and some partial results on these.
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