There will be two main courses lasting thefor the entire school, and three mini-courses.
Elchanan Mossel: Influences and noise stability in product space.
Functions of independent random variables are of major interest in probability, statistics, functional analysis and theoretical computer science. In this course we will explore probabilistic and analytic tools for studying such functions with focus on notions of influences (the effect of re-sampling individual variables on the function) and stability (with respect to correlated sampling of all the variables). We will explore connections to isoperimetric problems as well as examples concerning random walks, voting, testing and percolation.
Asaf Nachmias: Random walks on random fractals.
We will explore the geometry and random walk behavior on two popular random fractals: critical percolation clusters and random planar maps. In particular, our goal will be to prove that the spectral dimension of critical percolation in high dimensions is 4/3 and that the random walk on the uniform infinite planar triangulation (UIPT) is recurrent. We will study from scratch most of the different probabilistic and geometric tools required to prove these results such as electric networks, critical exponents, extremal length and Koebe's circle packing theorem.