Abstract. Geometry in high-dimensional spaces comes with a different set of rules, governed by probabilistic principles. The need to better understand such phenomena led to the development of one of the most fundamental tools in modern probability theory — the concentration of measure phenomenon. Its impact now extends to many areas of mathematics and applied fields. This is especially true in data sciences, where many problems can be approached using probabilistic and geometric techniques.
The purpose of this talk is to discuss some of these techniques, e.g. in classification problems in machine learning, signal recovery in compressed sensing, dimension reduction in computer science. A common feature of all these problems is that they depend essentially on concentration properties in high-dimensions. Recently, a demand for stronger accuracy in concentration principles has led to the notion of super-concentration. I will discuss new types of concentration of measure, the super-concentration phenomenon, and their interplay with high-dimensional data analysis and geometry.