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. Supplementary sessions will be run by Yuval Filmus on Mondays at 4pm.
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.
- Alison Etheridge: Stochastic models of evolution (June 6-10)
There will be three lectures, each essentially self-contained.
- 1. Genealogies and pedigrees
- We review the classical Kingman coalescent, which provides a simple and elegant description of the way in which, under suitable conditions, genes sampled from individuals in a population are related to one another. However, in reality, in a population such as our own, in which individuals each have two parents, genetic ancestry is constrained by the `pedigree' which relates individuals to one another. We investigate some relationships betwen genealogies and pedigrees.
- 2. Spatially distributed populations
- We consider modelling frequencies of different genetic types in populations which are spatially structured. Classical approaches are beset by Felsenstein's `pain in the torus', but we describe a framework which overcomes that, while also allowing us to incorporate large-scale demographic events, and explain some of the patterns that emerge from the resulting models.
- 3. Selection
- We discuss models of genetic types in populations which are subject to natural selection and the ways in which selection distorts genealogies. We are interested in populations both with and without spatial structure. In particular we shall describe some recent and ongoing work which is concerned with the interaction between natural selection and spatial structure.
- Allan Sly: Phase transitions for random constraint satisfaction (June 10-13)
- Ofer Zeitouni: Log correlated Gaussian fields and branching random
walks (June 23-27)
This mini course will be devoted to recent advances in the understanding of logarithmically correlated fields, the prime example being the two dimensional Gaussian Free Field. An important component in this progress is the similarity, both in proof methods and in phenomenology, between logarithmically correlated fields and branching random walks. I will discuss these proof methods, point out some of these recent results concerning both the maximum and the extremal process, and indicate some remaining exciting research directions. No previous knowledge of branching random walks or of the GFF will be assumed.