Topology of random real hypersurfaces

TL;DR

This paper studies the topology of random real hypersurfaces in projective spaces, providing estimates for the expected Betti numbers and the number of components diffeomorphic to a given hypersurface.

Contribution

It offers new lower and upper bounds on the expected Betti numbers and component counts of random real projective hypersurfaces, focusing on the case of projective spaces.

Findings

Estimated from above and below the expected Betti numbers for degree d hypersurfaces.

Provided lower bounds for the expected number of components diffeomorphic to a given hypersurface.

Focused on the case of projective spaces and lower estimates.

Abstract

These are notes of the mini-course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July $2014$. The subject was my joint work with Damien Gayet on the topology of random real hypersurfaces, restricting myself to the case of projective spaces and focusing on our lower estimates. Namely, we estimate from (above and) below the mathematical expectation of all Betti numbers of degree $d$ random real projective hypersurfaces. For any closed connected hypersurface $\Sigma$ of $\mathbb{R}^n$, we actually estimate from below the mathematical expectation of the number of connected components of these degree $d$ random real projective hypersurfaces which are diffeomorphic to $\Sigma$.