4. The zonoid ring and the nonarchimedean world. In the last lecture I will explain a ring-theoretical interpretation of the computations in random algebraic geometry, using a recently discovered ring structure on special convex bodies. This leads to the construction of a probabilistic version of Schubert calculus. In the final part of the lecture I will export some of the ideas from the previous lectures to the case $\mathbb{K}=\mathbb{Q}_{p}$, leaving with some open questions.[-]

In the last years there has been an increasing interest into the statistical behaviour of algebraic sets over non-algebraically closed fields: when the notion of 'generic' is no longer available, one seeks for a 'random' study of the objects of interest. In this course, divided into four lectures, I will present the major ideas in the subject (lecture notes will be made available):

1. Generic and random. In the first lecture I will discuss how to switch from the notion of generic, from classical algebraic geometry, to the notion of random. Of course, this depends on the choice of the probability distribution on the 'moduli space' of the objects of interest. I will discuss what are the reasonable choices in the case $\mathbb{K}=\mathbb{C}$ (where still these questions make sense, and 'random' and 'generic' are synonymous) and in the case $\mathbb{K}=\mathbb{R}$ (where spherical harmonics play a crucial role).[-]

Odd symplectic Grassmannians are a family of quasi-homogeneous varieties with properties nevertheless similar to those of homogeneous spaces, such as the existence of a Schubert-type cohomology basis. In this talk based on joint work with Ryan Shifler, I will explain how to construct their curve neighbourhoods. Curve neighbourhoods were first introduced by Buch, Chaput, Mihalcea and Perrin in the homogeneous setting: it is the union of all rational curves of fixed degree passing through a given Schubert variety. Potential applications include the computation of minimal degrees in quantum cohomology.[-]

3. The square-root law and the topology of random hypersurfaces. In the third lecture I will focus on the case $\mathbb{K}=\mathbb{R}$ and explain in which sense random real algebraic geometry behaves as the 'square root' of complex algebraic geometry. I will discuss a probabilistic version of Hilbert's Sixteenth Problem, following the work of Gayet & Welschinger (introducing a local random version of Nash and Tognoli's Theorem and of Morse theory for the study of Betti numbers of random hypersurfaces) and of Diatta $\&$ Lerario (showing that 'most' hypersurfaces of degree $d$ are isotopic to hypersurfaces of degree $\sqrt{d \log d}$ ).[-]