In metric spaces a distance is defined between each pair of points. In topological spaces, distances are replaced by only a certain notion of nearness. This abstract setting sheds new light upon basic questions: What is continuity? When should we consider two spaces to be equal?
This course provides an introduction to probability theory - the mathematical study of randomness. The aim is to get to know the basic mathematical framework of probability theory, and to learn to think and argue in this framework.
Introduction to the theory of martingales in discrete time, in particular convergence and optional sampling theorems. Application to branching processes. Introduction to Brownian motion and its main properties.
This year we will be looking at topics in high-dimensional probability, i.e. properties of large random systems.