Interpolation in several variables
Compared to the case of functions of one variable, or to the case of symmetric functions, there are not many algebraic tools
to manipulate polynomials in several variables. The symmetric group can greatly help in that matter. Newton found how to transform a discrete set of data into an algebraic function, introducing divided differences. This leads to different algebras of operators on polynomials, and different linear bases of polynomials. I shall illustrate Schubert polynomials, and Macdonald polynomials, both families being defined by vanishing conditions, recursions being provided by divided differences.
Alain Lascoux (Univ. Paris-Est, France)
February 10, 2010