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 whose coefficients are Gaussian random variabl2s – this problem was pioneered by Kac in the 1940s. More generally algebraic geometers might be interested, for example, in the number of components of a random real plane curve, or in the expected number of real solutions of more advanced counting problems. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with random matrix theory and random fields.

Date and time info
Tuesdays, 11:15-12:45

Keywords
Algebraic Geometry, Probability, Expected Counting Problems, Average Topology

Prerequisites
Basic understanding of algebraic and differential geometry and probability

Language
English