High-dimensional geometry and probability

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Not regularly scheduled

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

Course Text: 

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. 

Topic Outline: 
In this course we shall discuss geometric properties of high dimensional objects, and the asymptotic of their quantitative parameters as the dimension tends to infinity. It is natural to think that geometry in high dimensions is much more complicated than, say, the geometry of two or three-dimensional objects. However, many nice, and sometimes surprising, properties arise in high dimensions. Such properties are informally called ``high-dimensional phenomenon’’. The most classical example is the Central Limit Theorem, which suggests that a normalized sum of independent random variables has distribution very close to Gaussian; this makes the sum of many random variables a nicer object than the sum of, say, five random variables, whose distribution may be tedious to compute. In this course we shall understand why the central limit theorem, and other key facts and objects from Probability theory can be understood via geometry.    
The course will cover some classical results in finite-dimensional functional analysis, such as:
  • 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
After that, the course will focus on the geometric aspects of Probability theory, and the following topics shall be discussed:
  • 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
Next, a very modern topic of volume distribution in convex bodies shall be discussed:
  • Isotropic position
  • The geometry of isotropic convex bodies
  • Slicing problem and Klartag’s upper bound on the isotropic constant
As an important application of the material learned, we shall discuss an important modern result
  • Klartag’s Central Limit Theorem for convex sets 
Further, the course shall explore the geometric aspects of Gaussian and Subgaussian random processes, and cover the following topics:
  • Gaussian processes
  • Chaining and comparison inequalities
  • Gordon’s minimax theorem
  • Generic chaining
  • Escape through a mesh theorem
As a very important application, a modern proof of Milman-Dvoretzky theorem shall be discussed:
- Milman-Dvoretzky theorem
  • Euclidean subspaces of L_p spaces
Further, if time permits, several important facts in high-dimensional geometry shall be discussed:
  • The L-position
  • Pisier’s inequality
  • MM^* estimates
  • Mahler’s conjecture and the Bourgain-Milman inequality
  • Milman’s reverse Brunn-Minkowski inequality