Introduction to Random Algebraic Geometry

This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with random coefficients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial $$p(x) = a_{0} +· · ·+a_{d}x^{d}\: , where\: a_{0}, . . . , a_{d}$$ are random Gaussian variables – this problem is “classical” and was pioneered by Kac in the 40s. More generally algebraic geometers might be interested, for example, in the number of components of a random real plane curve of degree d, or in the expected number of real solutions of more advanced counting problems (e.g. enumerative problems). This is a very fresh and modern approach to real algebraic geometry: when the outcome is highly sensitive to the choice of the parameters (in general, there is no notion of “generic” in the real world), the attention is shifted to the “typical” situation. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with convex geometry, random matrix theory and random fields.

Date and time info
Tuesday, 09:00 - 10:00, from February 6 to March 27

Keywords
Real Algebraic Geometry, Random Matrix Theory, Integral Geometry

Prerequisites
Basic knowledge from Differential Geometry, Probability and Algebraic Geometry

Audience
MSc students, PhD students, Postdocs