Stochastic Geometry

Venue and schedule

Tuesdays 11am-1pm, room 134

Thursdays, 2.30pm-4.30pm, room 134

duration: 40h (two cycles)

Course description

This course is an introduction to the classical theory of random polynomials, with applications to the study of statistical properties of their zero sets.

We will start by addressing the basic question "How many zeroes of a random polynomial are real?”.

Specifically, we take a polynomial whose coefficients are real random variables and consider the random variable number of real zeroes. We will discuss two general methods for dealing with this type of questions: (1) the Kac-Rice formula and (2) the Integral Geometry Formula.
The Kac-Rice formula is more analytic in its flavour and allows to compute expectation of the number of zeroes of general systems of random equations; the Integral Geometry Formula computes the average of functions defined over Orthogonal groups (w.r.t. Haar measure).
These techniques can also be applied to the complex case, where typical means
generic, i.e. nondegenerate (using this idea we will give a probabilisitc proof of the Fundamental Theorem of Algebra).

In the second part of the course we will take a broader point of view and study the geometry of the space of polynomials (of several variables). We will concentrate on the concepts of volumes and discriminants, generalizing to higher dimensions the results of the first part. We will conclude with the Weyl tube formula, for describing the volume of tubes around hypersurfaces in spaces with constant curvature.

(An alternative end of the course, depending on the interest of the audience, might be the study of topological properties of random algebraic varieties.)

Tentative list of topics

Part 1. "How many zeroes of a random polynomial are real?"

1. Gaussian random polynomials, Kac's counting formula and Kac-Rice formula

2. Proof of Kac-Rice formula (gaussian case)

3. Kostlan statistic using the Kac-Rice formula, Random trigonometric polynomials

4. Kac-Rice formula for random fields on manifolds I

5. Kac-Rice formula for random fields on manifolds II

6. The moment curve ("1-dimensional Veronese embedding") and the integral geometry formula

7. Kac's, Kostlan's and other statistics revised.

8. Preliminaries on the Proof of the Integral Geometry formula 

9. Proof of the Integral Geometry formula I

10. Proof of the Integral Geometry formula II and applications

11. A probabilistic approach to the Fundamental Theorem of Algebra

Part 2. Geometry in the space of polynomials. 

12. Real versus Complex case, discriminants, genericity, the Bombieri Weyl scalar product, projectivization

13. Spherical harmonics and invariant scalar products (dual description of Gaussian distributions).

14. The topological interpretation of the FTA and its "real" counterpart using spherical harmonics (Maxwell's poles.)

15. complexity of Bezout Theorem I

16. complexity of Bezout Theorem II

17. Eckart-Young and the loss of numerical precision (linear case)

18. Discriminant and condition number: the non-linear case

19. Weyl's tube formula I

20. Weyl's tube formula II