# 49th Probability Summer School Saint-Flour (France), 07 - 19 July 2019

## Lectures 2019

Nicolas Curien
Université Paris-Sud Orsay (France)
Random discrete surfaces.

These lectures will be an introduction to the study of random discrete surfaces obtained by gluing of polygons along their sides to create 2D surfaces. They include random trees and random planar maps as special cases. Our focus will be on the geometry of those objects (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walk, self-avoiding random walk...). In order to get understanding of these random discrete surfaces, we will explore them in a "Markovian way" and relate these explorations to one-dimensional random walk processes. This goes under the name of "peeling exploration" in the random planar map theory and can be seen as generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). We will see that different type of Markovian explorations can yield to different type of information on the surface.

The lectures will contain exercises as well as open problems (some of them accessible) for those eager to learn more.

Elchanan Mossel
Massachusetts Institute of Technology (USA)
Probabilistic aspects of voting, intransitivity and manipulation.

Marquis de Condorcet, a French philosopher, mathematician, and political scientist, studied mathematical aspects of voting in the eighteen century. Condorcet was interested in studying voting rules as procedures for aggregating noisy signals and in the paradoxical nature of ranking $3$ or more alternative. The course will survey some of the main mathematical models, tools, and results in a theory that studies probabilistic aspects of voting. Our journey will take us through major results in mathematical economics from the second half of the 20th century, through the theory of boolean functions and their influences and through recent results in Gaussian geometry, and functional inequalities.

Philippe Rigollet
Massachusetts Institute of Technology (USA)
Statistical optimal transport.

Optimal transport is a fundamental concept in probability theory which has recently seen an explosion of interest in machine learning and statistics as a tool for analyzing high-dimensional data. However, the key obstacle in using optimal transport in practice has been its high statistical and computational cost. In these lectures, we review different notions of regularization that can lead to better statistical rates and faster algorithms.