Title: On Differentiating Symmetric Functions

Abstract: The ring of symmetric functions has several bases, each of them natural for a different reason. In this talk we will present the basis that is most natural from the analytic point of view, of differentiation. The resulting symmetric coordinates, and the associated differential operators on symmetric functions, have many interesting properties: Their formulae can be obtained via divided difference operators like the ones from the combinatorial theory of Schubert polynomials; They have good vanishing properties along the total diagonal; They can be used for completing the image of the non-surjective map from the tangent space of R^N to that of Sym^N R; And they produce higher-order differential operators that satisfy the Leibniz rule along the total diagonal.