Loïc Chaumont: Self-similar processes via Lévy processes
An Rd-valued stochastic process is said to be self-similar if its image by any linear time change has the same law as its image by a linear dilation of space. They often appear in practice as scaling limits of time changed stochastic processes. The self-similarity of Brownian motion was highlighted in the 1940s by Paul Lévy who also, later on, considered the example of stable processes. This notion was then defined in a very general context by Lamperti in the early 1960s.
Self-similar processes enjoy particular distribution or trajectory properties depending on whether they are Markovian, have independent increments or have stationary increments. In each case, Lévy processes are involved in their construction through path representations. These lectures aim to introduce the main classes of self-similar processes of which a common element is a Brownian motion. We will pay particular attention to the case of Markovian processes that we will approach through the famous Lamperti-Kiu representation. The example of stable Lévy processes and some of their conditioned versions will be studied in detail.
Nicolas Dutertre: Topology and geometry of semi-algebraic sets
The aim of this masterclass is to extend to the class of (possibly singular and/or non-compact) semi-algebraic sets some classical results of differential topology and differential geometry, such as the Poincaré-Hopf theorem or the Gauss-Bonnet theorem. We will provide the students with tools and techniques of semi-algebraic geometry and apply them to get interesting results on the topology and geometry of (possibly singular and/or non-compact) semi-algebraic sets. The content of the course will be the following:
- Background in differential topology and differential geometry
- Basic properties of semi-algebraic sets and maps
- Results on the topology and geometry of (singular and/or non-compact) semi-algebraic sets
The organization board will cover accommodation, breakfast, and lunch, but not dinner. As for the travel expenses, we will do our best in the limit of our budget. The registration deadline is November 2, 2021.