Résumé : This small account aims at presenting a project which deals
with combinatorial aspects of Quantum Mechanics (or
Physics).
This project brings together specialists in Mathematics, Informatics
and Physics.
The general subjects are: Hopf algebras, representation theory,
deformations, discrete operators
and evolution equations.
Since the accomplishment of Connes and Kreimer, the Hopf algebras entered
Physics as an efficient tool to solve or to render compact computation
problems of composition and decomposition. As many of these Hopf
algebras are defined by diagrams (the language of R. Feynman),
the problem of implementing them in order to compute at a high order
induces many fundamental questions (polynomial realizations, data
structures, finite orbits, etc.). These are indeed the problems of
(multiple) indexation which appear in ``special sums'' like MZV, EZS,
Jack polynomials, MacDonald polynomials or in geometric problems like
the combinatorial presentation of Schubert cycles, combinatorics of
flag manifolds, or the computation and measure of entangled states.
Quantum Mechanics rests almost entirely on commutation relations
(of Heisenberg-Weyl type or deformed) and on evolution equations which
give new insights respectively for orthogonal polynomials and for exact
solutions of Fokker-Planck or non-commutative evolution equations ($KZ_n$).
The strength of the project relies not only on its scientific unity but
also on a well coordinated team scattered among several communities.
The qualities and complementarity of the different centers will be
commented throughout this small talk.
Dernière modification : Monday 28 January 2013 | Contact : Cyril.Banderier at lipn.univ-paris13.fr |