The geometry of Hilbert scheme of points on manifolds

While the Hilbert scheme of points on surfaces is extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be notoriously complex, exhibiting wild and pathological properties. Our understanding of their topology and geometry is very limited.

In this survey I will present recent developments in various aspects of their geometry. I will discuss the distribution of torus fixed points within the components of the Hilbert scheme (work with J. Svendsen), a new formula for tautological integrals over geometric components (work with A. Szenes),  
and p-adic integration on Hilbert schemes leading to new invariants of hypersurface singularities (work with I. Rossinelli).