We will discuss how the theory of symmetric functions can be extended to the supersymmetric case, involving anti-commuting variables in addition to those
that commute. We shall see how beautiful combinatorics arises from the super-extension of classical symmetric functions.

Recently, we developped an approach to study the concentration of the
first eigenfunction of a positive second order operator on Riemannian
Compact manifolds. The set of limit measures will be described and can be
characterized explicitly. In particular in some cases, the first
eigenfunction sequence concentrates along manifolds of dimension
k(=0,1,2). Explicit formula can be given for the restriction of the
invariant measure on the invariant manifolds.