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.
