Khovanov and Seidel introduced in the early 2000's an action of the braid group by autoequi-valences on the homotopy category of projective modules over the zig-zag algebra. This categorical action descends to the Burau representation, one of the most famous braid representations, but unlike the classical story, the lifting is faithful. It is interesting to notice that simultaneously, the Burau representation was also extended into a faithful finite-dimensional linear representation by Lawrence, Krammer and Bigelow, proving the linearity of the braid group.
I will review the basic constructions, both at the level of vector representations and at the ca-tegorical level. We will discuss possible extensions of these from classical braids (type A) to larger Artin-Tits groups, spherical or not, and try to relate Khovanov-Seidel's construction to Soergel bimodules and categorified quantum groups. I will also try to emphasize several metric aspects that appear in an elegant way from the categorical setting, with an emphasis on Bridgeland's stability conditions. Along the way, I would like to list several open questions and problems that I care about, hoping that someone in the audience will come up with a good idea.[-]

Random walks is a topic at the crossroads of probability, ergodic theory, potential theory, harmonic analysis, geometry, and graph theory. Its roots can be traced back to the famous article by Pólya in 1921, which characterizes the recurrence of random walks on $\mathbb{Z}^{d}$ in terms of the dimension $d$. When random walks take place on a group, or more generally on a homogeneous space, it provides an even richer framework for study. From a probabilistic point of view, this additional structure serves as an extra tool, facilitating the study of the behaviour of the random walk on the underlying space. Regarding groups and their actions, random walks offer a means to explore generic or non-generic parts of groups and, at times, even to demonstrate intrinsic geometric properties, as is clearly shown by Kesten's amenability criterion (1959). This is an introductory course on the topic. Emphasis will be given on the interplay between probability and the structure of the group. The course will also provide insights into current research questions. Here is an outline of each session :
(1) Equivalent of Pólya's criterion for random walks on groups and rigidity theorems : does walking randomly on a given group in two different ways affect the recurrence of the walks ?
(2) Kesten's probabilistic criterion of the amenability of a finitely generated group ; defined in this course in terms of isoperimetric profile. The tools in 1) and 2) are essentially coming from analysis on groups.
(3) Tools coming from subadditivity to study the behaviour of a random walk on a group (drift, entropy and expansion of the random walk, etc.)[-]

Random walks is a topic at the crossroads of probability, ergodic theory, potential theory, harmonic analysis, geometry, and graph theory. Its roots can be traced back to the famous article by Pólya in 1921, which characterizes the recurrence of random walks on $\mathbb{Z}^{d}$ in terms of the dimension $d$. When random walks take place on a group, or more generally on a homogeneous space, it provides an even richer framework for study. From a probabilistic point of view, this additional structure serves as an extra tool, facilitating the study of the behaviour of the random walk on the underlying space. Regarding groups and their actions, random walks offer a means to explore generic or non-generic parts of groups and, at times, even to demonstrate intrinsic geometric properties, as is clearly shown by Kesten's amenability criterion (1959). This is an introductory course on the topic. Emphasis will be given on the interplay between probability and the structure of the group. The course will also provide insights into current research questions. Here is an outline of each session :
(1) Equivalent of Pólya's criterion for random walks on groups and rigidity theorems : does walking randomly on a given group in two different ways affect the recurrence of the walks ?
(2) Kesten's probabilistic criterion of the amenability of a finitely generated group ; defined in this course in terms of isoperimetric profile. The tools in 1) and 2) are essentially coming from analysis on groups.
(3) Tools coming from subadditivity to study the behaviour of a random walk on a group (drift, entropy and expansion of the random walk, etc.)[-]

Random walks is a topic at the crossroads of probability, ergodic theory, potential theory, harmonic analysis, geometry, and graph theory. Its roots can be traced back to the famous article by Pólya in 1921, which characterizes the recurrence of random walks on $\mathbb{Z}^{d}$ in terms of the dimension $d$. When random walks take place on a group, or more generally on a homogeneous space, it provides an even richer framework for study. From a probabilistic point of view, this additional structure serves as an extra tool, facilitating the study of the behaviour of the random walk on the underlying space. Regarding groups and their actions, random walks offer a means to explore generic or non-generic parts of groups and, at times, even to demonstrate intrinsic geometric properties, as is clearly shown by Kesten's amenability criterion (1959). This is an introductory course on the topic. Emphasis will be given on the interplay between probability and the structure of the group. The course will also provide insights into current research questions. Here is an outline of each session :
(1) Equivalent of Pólya's criterion for random walks on groups and rigidity theorems : does walking randomly on a given group in two different ways affect the recurrence of the walks ?
(2) Kesten's probabilistic criterion of the amenability of a finitely generated group ; defined in this course in terms of isoperimetric profile. The tools in 1) and 2) are essentially coming from analysis on groups.
(3) Tools coming from subadditivity to study the behaviour of a random walk on a group (drift, entropy and expansion of the random walk, etc.)[-]

Kazhdan projections are usually considred objects relevant in operator algebras. In particular, they played a central part in the construction of counter-examples to the Baum-Connes conjecture.
In this talk I shall explain how, in the general setting of a family of representations on Banach spaces, one can reformulate the Kazhdan property "almost invariant implies invariant vectors" in terms of Kazhdan projections, providing also an explicit formula of the latter, using Markov operators associated to a random walk on the group. I will then explain some applications of this new approach.
This is joint work with Piotr Nowak.[-]

Given a nontrivial conjugacy class $g$ in a free group $F_{N}$, what can we say about the typical growth of g under application of a random product of auto-morphisms of $F_{N}$? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the surface. This is partly joint work with François Dahmani.[-]

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right ) $ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.[-]

Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the general case of a torsion-free hyperbolic group $G$; we show that every element in $G$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. In addition, we show the following dichotomy: every element of $G$ grows either exponentially fast or polynomially fast under iteration of $f$.
This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.[-]