Gaussian distributions and invariant metrics, random complex polynomials, random real polynomials, integral geometry and tube formulas, asymptotic methods, distribution of zeroes (univariate case), large deviation principles, random matrices, systems of random equations, Kac-Rice formulas, statistics for geometrical quantities (volume, curvature..), statistics for topological quantities, incidence and enumerative problems, random convex bodies.

Outline.

In the course I will discuss the main ideas and techniques coming from geometry and asymptotic analysis for the study of problems in stochastic geometry.

The course might be of interest to analysts (1, 2, 3, 5, 7.2), numerical analysts (1, 2, 4), geometers (1, 2, 3, 7, 8) and mathematical physicists (1, 2, 4, 5, 8).

1. Basic questions in stochastic geometry

1.1 The integral geometry formula in spheres and projective spaces

1.2 "How many roots of a random polynomial are real?"

1.3 Volumes and degree in real and complex projective spaces

2. Geometry in the space of polynomials

2.1 Complex polynomials: invariant measures and statistics

2.2 Real polynomials: invariant measures and spherical harmonics

2.3 Complexity of Bezout's Theorem

3. Tubes

3.1 Weyl's tube formula

3.2 The curvature polynomial and the generalized Gauss-Bonnet theorem

3.3 Focal points, characteristic classes and Schubert varieties

4. Condition numbers and the loss of numerical precision (linear case)

4.1 Eckart-Young theorem

4.3 Random matrices

4.4 Volume of the set of singular matrices

4.3. Discussion on the nonlinear case

5. Asymptotic methods

5.1 Multidimensional Laplace method

5.2 Logarithmic potential theory (discussion)

5.3 Large deviations techniques (basics)

6. Kac-Rice formulas

6.1 "How many roots of a random polynomial are real?", revised

6.2 System of random equations

6.3. Applications

7. Topology (the case of plane curves)

7.1. Upper bound: Kac-Rice approach

7.2. lower bound: the barrier method

8. Enumerative geometry

8.1 Geometry of real grassmannians

8.2 Integral geometry on real grassmannians

8.3. "How many lines intersect four random curves in space"? decoupling

8.4 "How many lines intersect four random curves in space"? asymptotics

9. Open questions

References.

R. Adler, J. E. Taylor, "Random fields and geometry", Springer

L. Blum, F. Cucker, M. Shub, S. Smale, "Complexity and Real Computation", Springer

A. Edelman, E. Kostlan, "How many zeros of a random polynomial are real?", Bull. Amer. Math. Soc. 32 (1995), 1-37

A. Gray, "Tubes", Birkhauser

P. Burgisser, F. Cucker, "Condition: the geometry of numerical algorithms", Springer

M. Shub, S. Smale: "Complexity of Bezout's theorem, II" Computational Algebraic Geometry, Volume 109 of the series Progress in Mathematics pp 267-285