Special topics course on "High-dimensional geometry and probability" offered in Spring 2018 by Galyna Livshyts.

No prerequisites aside from Calculus I-III and an introduction to undergraduate Probability Theory

R. Vershynin, High-dimensional Probability, (2017) (available online for free)

S. Artstein-Avidan, A. Giannopoulos, V. D. Milman, Asymptotic Geometric Analysis, Part I, Mathematical Surveys and Monographs 202, Amer. Math. Society (2015).

S. Brazitikos, A. Giannopoulos, P. Valettas, B-H. Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs 196, Amer. Math. Society (2014).

V. D. Milman, G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Springer Verlag, New York (1986).

R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993, 490 pp. ISBN: 0-521-35220-7.

- Convex bodies and Helly’s theorem
- Duality
- Steiner symmetrization and Blascke-Santalo inequality
- Brunn-Minkowski inequality and isoperimetric type inequalities
- Banach-Mazur distance and John’s theorem
- Classical positions and reverse isoperimetric inequality

- Concentrations of sums of independent random variables
- Random vectors in high dimensions
- Subgaussian distribution
- Random matrixes
- Concentration of measure on the sphere
- Gaussian isoperimetric inequality
- Concentration of measure for Lipschitz functions
- Johnson-Lindenstrauss lemma

- Isotropic position
- The geometry of isotropic convex bodies
- Slicing problem and Klartag’s upper bound on the isotropic constant

- Klartag’s Central Limit Theorem for convex sets

- Gaussian processes
- Chaining and comparison inequalities
- Gordon’s minimax theorem
- Generic chaining
- Escape through a mesh theorem

- Euclidean subspaces of L_p spaces

- The L-position
- Pisier’s inequality
- MM^* estimates
- Mahler’s conjecture and the Bourgain-Milman inequality
- Milman’s reverse Brunn-Minkowski inequality