I work at the crossroads of geometry and probability theory. I am interested the typical behaviour of random submanifolds in closed smooth manifolds.
Let us choose a normally distributed random function in a finite-dimensional space of smooth functions on a closed manifold. Under some technical assumptions, the zero set of this function is, almost surely, a smooth hypersurface (this generalises to higher codimension submanifolds obtained as the zero sets of random sections of a vector bundle).
I study the distribution of random variables related to these submanifolds, such as their volume, their Euler characteristic or their Betti numbers.
This framework covers two interesting special cases. First, the algebraic submanifolds of the real projective space defined as the zero sets of random homogeneous polynomials of fixed degree. In this case, we choose independent centered Gaussian coefficients. The variances are chosen so that the joint distribution of the coefficients is invariant under the action of the orthogonal group. The second special case is concerned with zeros of random spherical harmonics, or more generally Riemannian random waves.