back to N-Cube Days XXII

Cordian Riener (UiT)

Title: The wonderful geometry of the Vandermonde map: Symmetry at infinity and applications

Abstract: The Vandermonde map is given by $d$ power sum polynomials in $n$ variables. We study in detail the image of the probability simplex $\Delta_n$ under this map. We will see that this image possesses a combinatorial structure resembling a cyclic polytope. After analyzing the image in finitely many variables, we concentrate on the limit as the number of variables approaches infinity. We explain how the geometry of the limit plays a crucial role in undecidability results in nonnegativity of symmetric polynomials, deciding validity of trace inequalities in linear algebra, and extremal combinatorics.