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Computer algebra and combinatorial problems at the interplay
of number theory and quantum algebra: the example of Multiple
Zeta Values
Georges Racinet
Fachbereich Mathematik und Informatik, Universität Münster
Motivation
During the last fifteen years, some links have been established
between at first sight quite far fields of mathematics, namely number
theory, topology and quantum algebra. One of the common features of
these directions lies in the combinatorial flavor of the objects they
deal with. It is therefore relevant to ask what computer algebra can
bring to the subject. The experience that the presenter would like to
share is that it provides not only stronger evidence for conjectures
but can also be used in the quest for proofs. A better understanding
of the picture gives then in return more efficient computational
methods.
This calculus is often highly non-commutative and comes down in the
cases that we will consider to non-commutative polynomials
manipulations, hence words combinatorics. We will mostly stick to the
objects connected to the multiple zeta values (MZV's), a multi-variable
generalization of the values of the Riemann zeta function at positive
integers. This has to be considered both as an example and a starting
point. We will try to give some insight about the applications of the
encountered bestiary.
Outline
Each topic will be covered with a view towards computations and
examples will be provided.
- Basic tools and concepts.
Computations in free Lie algebras, Lyndon words, shuffle product,
and Hopf algebras.
- First objects of calculations.
The Lie algebras of infinitesimal pure braids, Ihara's
bracket and Drinfeld's Lie algebra. Statement of the Deligne-Drinfeld
conjecture. This is were most correlations occur.
- Combinatorics of MZV's.
Definition of MZV's, non-commutative symmetric functions,
double shuffles, proofs having been guided by computer, review of
results by numerical, polynomial and linear methods.
- Prospective issues (whether possible).
Quivers, Trees (operads) and graph (Feynman diagrams)-styled material.
Prerequisites
Linear algebra including familiarity with tensor products.
Elementary calculus.
Target audience
Computer algebra researchers interested by this possible field of
developments, mathematicians interested by computational
aspects towards the above outlined applications.
Last modified on Fri Jul 5 11:46:51 MET DST 2002