Algebraic Combinatorics - Jean-Philippe Labbé

Title: Lineup polytopes and applications in quantum physics

Abstract:  To put it simply, Pauli's exclusion principle is the reason why we can't walk through walls without getting hurt. Pauli won the Nobel Prize in Physics in 1945 for the formulation of this principle. A few years later, this principle received a geometrical formulation that is still overlooked today. This formulation uses the eigenvalues of certain matrices (which represent a system of elementary particles, for example electrons). These eigenvalues form a symmetric geometric object obtained by cutting a hypercube: it is a hypersimplex.

To represent systems of particles with a non-zero temperature, it is necessary to generalize the hypersimplex to obtain what is called "lineup polytopes". These polytopes are defined using classical notions of combinatorics and discrete geometry. Moreover, they produce new exclusion principles which refine Pauli's principle that shall be put to the test by experimentalists. During this talk, we will see the history behind the introduction of these polytopes and give a presentation of some properties.

This is joint work with physicists Julia Liebert, Christian Schilling and mathematicians

Eva Philippe, Federico Castillo and Arnau Padrol.