École d'été de probabilités de Saint-Flour

Concentration inequalities and model selection

Exponential and Information Inequalities.- Gaussian Processes.- Gaussian Model Selection.- Concentration Inequalities.- Maximal Inequalities.- Density Estimation via Model Selection.- Statistical Learning.

Oracle inequalities in empirical risk minimization and sparse recovery problems

The purpose of these lecture notes is to provide an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. In recent years, there have been new developments in this area motivated by the study of new classes of methods in machine learning such as large margin classification methods (boosting, kernel machines). The main probabilistic tools involved in the analysis of these problems are concentration and deviation inequalities by Talagrand along with other methods of empirical processes theory (symmetrization inequalities, contraction inequality for Rademacher sums, entropy and generic chaining bounds). Sparse recovery based on l_1-type penalization and low rank matrix recovery based on the nuclear norm penalization are other active areas of research, where the main problems can be stated in the framework of penalized empirical risk minimization, and concentration inequalities and empirical processes tools have proved to be very useful.

Some mathematical models from population genetics

This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

Disorder and critical phenomena through basic probability models

1 Introduction.- 2 Homogeneous pinning systems: a class of exactly solved models.- 3 Introduction to disordered pinning models.- 4 Irrelevant disorder estimates.- 5 Relevant disorder estimates: the smoothing phenomenon.- 6 Critical point shift: the fractional moment method.- 7 The coarse graining procedure.- 8 Path properties.

Topological complexity of smooth random functions

1 Introduction.- 2 Gaussian Processes.- 3 Some Geometry and Some Topology.- 4 The Gaussian Kinematic Formula.- 5 On Applications: Topological Inference.- 6 Algebraic Topology of Excursion Sets: A New Challenge

Coarse Geometry and Randomness

Isoperimetry and expansions in graphs.- Several metric notions.- The hyperbolic plane and hyperbolic graphs.- More on the structure of vertex transitive graphs.- Percolation on graphs.- Local limits of graphs.- Random planar geometry.- Growth and isoperimetric profile of planar graphs.- Critical percolation on non-amenable groups.- Uniqueness of the infinite percolation cluster.- Percolation perturbations.- Percolation on expanders.- Harmonic functions on graphs.- Nonamenable Liouville graphs.

Ecole d' eté de probabilités de Saint-Flour XXI, 1991

Measure-valued Markov processes.- Processus de Markov: Naissance, retournement, regeneration.- Nine lectures on random graphs.

Ecole d'été de probabilités de Saint-Flour XVIII, 1988

Theorie du Potentiel sur les Graphes et les Varietes.- Random fields and inverse problems in imaging.- Probabilistic methods in the study of asymptotics.

Ecole d'été de probabilités de Saint-Flour XX, 1990

Semi-linear pdes and limit theorems for large deviations.- Some properties of planar brownian motion.

École d'été de probabilités de Saint-Flour XV-XVII, 1985-87

Large deviations and applications.- Applications of non-commutative fourier analysis to probability problems.- Random fields and diffusion processes.- Waves in one-dimensional random media.- Remarks on the point interaction approximation.- Geometric aspects of diffusions on manifolds.- Stochastic mechanics and random fields.