Random algebraic geometry - lecture 1
Appears in collection : Real Algebraic Geometry / Géometrie algébrique réelle
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):
- 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).
