Juhan Aru: Catalogue data in Spring Semester 2017

Name Dr. Juhan Aru
DepartmentMathematics
RelationshipLecturer

NumberTitleECTSHoursLecturers
401-3597-64LConcentration of Measure4 credits2VJ. Aru, T. Lupu
Abstract
Objective
ContentThe basic examples of the concentration of measure phenomena are the following:

1) The visual distance of a N-dimensional unit sphere is only of order N^{-0.5}. In other words, more than 99% of the measure on the sphere lies at distance of at most O(N^{-0.5}) of a fixed hyperplane through the origin.

2) The suprema of a centred Gaussian process G(t) even with a possibility infinite index set T is always concentrated around its expected value with a Gaussian tail that only depends on the highest variance among the Gaussians G(t).

In this course we will try to understand these two slightly puzzling examples and related phenomena. We try to approach and understand the concentration of measure phenomena from different directions: through elementary martingale inequalities like Azuma-Hoeffding or McDiarmid inequality; through exact isoperimetry; through Poincaré and log-Sobolev inequalities.

On the way we aim to discuss several applications and connections to different topics, including Dvoretzky's theorem, convergence of Markov chains to their stationary measure, entropy, threshold phenomena, empirical processes etc...