Introduction to Random Algebraic Geometry

  • Lecturer: Antonio Lerario
  • Date: Tuesday, 09:00 - 10:00, from February 6 to March 27
  • Room: G3 10
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: Real Algebraic Geometry, Random Matrix Theory, Integral Geometry
  • Prerequisites: Basic knowledge from Differential Geometry, Probability and Algebraic Geometry

Abstract:

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.

Regular Lectures (Winter 2018/2019)

06.11.2018, 11:06