Yves Benoist, Random walks on Lie groups, Mondays March 3, 10 at 10:00-12:00 and Fridays March 7, 14 at 14:00-15:30

Abstract: This mini-course will be an introduction to the book with Jean-François Quint Random walk on reductive groups. The aim will be to describe limit laws for the product of identically distributed independent matrices:
  • . Stationary measure on projective space,
  • . Law of large norms,
  • . Simplicity of the spectrum,
  • . Central limit theorem,
  • . Local limit theorem.
Notes lecture 1, lecture 2, lecture 3, lecture 4 (notes taken by Frédéric Mathéus, after notes taken by Julien Bled and Jean Lécureux).

Emmanuel Breuillard,  Random walks, expander graphs and approximate groups, Tuesdays February 4, 11, 18; March 4, 11, 18; April 1 at 14:00-17:30 in Amphithéâtre Darboux

Notes lecture 1 (lecture notes taken by Alba Marina Málaga Sabogal)

Notes lecture 2 (lecture notes taken by Alba Marina Málaga Sabogal, first 15 minutes missing)

Notes lecture 2 (lecture note taken by Nicolas Matte Bon)

Notes lecture 3 (lecture note taken by Nicolas Matte Bon)

Notes lecture 4  (lecture note takien by Lucile Devin)

Maxim Kontsevich, Lyapunov exponents for Kähler random walks, Mondays February 10, 17 at 10:15-12:30 in Amphithéâtre Darboux

Abstract: Lyapunov exponents are important characteristics of equivariant vector 
bundles (a.k.a. matrix-valued 1-cocycles) on ergodic dynamical 
systems. For example, the entropy of a diffeomorphism preserving 
a smooth Borel probability measure is equal to the sum of positive 
Lyapunov exponents for the tangent bundle (Pesin's formula). Unfortunately, 
the exact value of the leading Lyapunov exponent or of the sum 
of positive exponents is known only in a handful of situations. Besides 
homogeneous examples like geodesic fl
ows on locally symmetric spaces, 
the main source of explicit formulas is a random walk on a Riemannian 
surface, with 1-cocycle given by a representation of the fundamental 
group. The sum of positive exponents is known for so-called variations 
of Hodge structures of weight 1. Such matrix-valued 1-cocycles appear 
in algebraic geometry, e.g. for any algebraic 1-parameter family 
of algebraic curves. Values of Lyapunov exponents in this case have 
applications to the zero-entropy dynamics of interval exchange maps, 
via the Teichm
üller geodesic fl

In my lectures I will discuss general properties of the Lyapunov spectrum, then I will explain the mechanism underlying the explicit formula based on Kähler geometry and pluriharmonicity. At the end I will describe the hypothetical extension to some variations of Hodge structure of higher weights, coming from recent computer experiments with hypergeometric functions. Conjecturally, the sum of positive exponents is given by a topological expression if and only if a certain multi-valued holomorphic function does not vanish.

Prerequisites: besides basic measure theory, I will assume some knowledge of topology (manifolds, fundamental group, homology). Relevant notions from complex algebraic geometry and Hodge theory will be explained during the course.

The lectures will be followed by informal discussions moderated by Alex Eskin on the same days at 14:00 in Amphithéâtre Darboux

Steve LalleyLocal limit theorems for random walks on hyperbolic groups, January 20,21,23,24 at 14:00-15:30 in Amphithéâtre Darboux, except for Tuesday, in room 001 at ENSCP (the same campus as IHP)

Abstract: This course will discuss in some detail recent results concerning the asymptotic behavior of return probabilities for random walks on discrete hyperbolic groups and their relation to Martin boundary theory. The course will include an introduction to symbolic dynamics for hyperbolic groups, to Gibbs states and thermodynamic formalism, to Ancona inequalities, and to the use of Karamata's Tauberian theorem.

Volodymyr Nekrashevych, Hyperbolic dynamics and groups, Fridays February 7, 14, 21 at 14:00-15:30

Abstract: We will talk about groups and pseudogroups appearing naturally in the study of expanding and hyperbolic dynamical systems: iterated monodromy groups, Ruelle groupoids, topological full groups, etc. We will show how dynamical systems can be used to study geometric and asymptotic properties of groups; and how group theory can be used to answer questions in dynamical systems. As one of examples, we will discuss applications of random walks to proving amenability of non-elementary amenable groups.

Notes lectures 1 and 2 (lecture notes taken by Elisabeth Fink)

Notes lecture 3 (lecture notes taken by Elisabeth Fink)

François Quint, Mesures stationnaires et fermés invariants des espaces hom­ogènes (a course at IHÉS), Thursdays February 6, 13, 20; March 6, 13, 20 at 14:00-16:00

Résumé: Dans ce cours, je présenterai des résultats que j'ai obtenus récemment en collaboration avec Yves Benoist. Nous avons démontré que, pour certaines actions de groupes sur des espaces homogènes, les adhérences d'orbites sont toutes des sous-variétés. Cet énoncé fait suite à de célèbres travaux de Furstenberg, Ratner, Margulis, Dani, Lindenstrauss, Katok, Einsiedler, etc. qui obtiennent des résultats proches, pour des actions de groupes différents. L'idée nouvelle que nous avons introduite, consiste à montrer que, sous nos hypothèses, l'adhérence de l'orbite d'un point peut s'obtenir en tirant au hasard les éléments du groupe qu'on lui applique successivement. Notre résultat découle alors des propriétés de la chaine de Markov ainsi construite, pour la description de laquelle nous utilisons la théorie des produits de matrices aléatoires, due à Furstenberg, Kesten, Kifer, Guivarc'h, Raugi, Gol'dsheid, Margulis, etc.

Paul Schupp, TBA, Friday March 21 at 10:00-12:00 and 14:00-16:00 

Nikolai Vavilov, The yoga of commutators: commutator formulas and commutator width in algebraic groups, Monday March 17 at 14:00-15:30 and Thursday March 20 at 14:00-15:30

Abstract: As we teach our students, in an abstract group an element of the commutator subgroup is not necessarily a commutator. However, the famous Ore conjecture, recently completely settled by Ellers—Gordeev and Liebeck—O’Brien—Shalev–Tiep, asserts that any element of a finite simple group, or, more generally, of an adjoint elementary Chevalley group over a field, is a single commutator.

On the other hand, from the work of van der Kallen, Dennis and Vaserstein it was known that nothing like that can possibly hold in general, for commutators in classical groups over rings. Actually, these groups do not even have bounded width with respect to commutators.

In this series of lectures, we address this topic in a broader context, and outline the proofs of the recent results asserting that exactly the opposite holds: over any commutative rings commutators have bounded width with respect to elementary generators.

One of the most powerful ideas in the study of classical groups and [the groups of points of] algebraic groups over rings is localisation. Localisation allows to reduce many important problems over commutative rings (or, more generally, rings subject to commutativity conditions) to similar problems for semi-local rings. Localisation comes in a variety of versions. The two most familiar ones are localisation and patching, proposed by Quillen and Suslin, and localisation–completion, proposed by Bak.

As a matter of fact, both methods rely on a large body of common calculations, and technical facts, known as conjugation calculus and commutator calculus. Their objective is to obtain explicit estimates of the modulus of continuity in s-adic topology for conjugation by a specific matrix, in terms of the powers of s occuring in the denominators of its entries, and similar estimates for commutators of two matrices.

I plan to outline recent powerful versions of localisation methods developed by Roozbeh Hazrat, Alexei Stepanov, Zhang Zuhong, and myself, and their applications to the study of commutators in [the groups of points] of algebraic and algebraic-like groups over rings.

One of the most striking corollaries of these results is that in the algebraic groups over rings there are very few commutators. The only reason, why it appears that there are many commutators in the groups of points over zero-dimensional rings (such as fields or local rings) is that in these cases there exist very short expressions of arbitrary elements in terms of elementary generators.

Also, I plan to discuss some further applications of our methods, such as multiple commutator formulae, nilpotency of K1, etc., as well as some further related asymptotic problems, and connections with geometry, arithmetics and complex analysis.