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 : Thursday 21 November 2024

Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr