Concentration of Measure in High Dimensions

by Mr Rakesh Venkat (School of Technology and Computer Science)

Tuesday 5 Apr 2011, 17:00 → 18:00 Asia/Kolkata

A-212 (Colaba Campus)

We know that the Gaussian distribution concentrates sharply around its mean, ie. the probability mass outside a few standard deviations decreases exponentially in the number of steps taken. Such a concentration result can actually be derived for areas and volumes in higher dimensions, as purely geometrical facts. For instance, for the unit sphere in n-dimensions, most of the volume is concentrated around every slice through the equator. Such results have various interesting implications. For example, any Lipschitz function defined on the n-dimensional sphere is more or less constant! I will talk about some basic concentration of measure results, and related questions like isoperimetric inequalities, the Brun-Minkowski inequality and possibly something on metric embeddings, depending on what time permits.