John's theorem states that if the Euclidean unit ball is the largest volume ellipsoid in a convex body K in R^d, then there is a set of unit vectors u₁,...,u_m on the boundary of K such that the identity operator I on R^d is a positive linear combination of the diads u_i⊗u_i. Put in another way, I is the expectation of a probability distribution on the set of n×n real matrices supported on a certain set of rank one matrices.
Motivated by geometric applications, it is natural to ask if the average of few of these random matrices is close to I. Our main interest is whether the known positive answer to this question extends from diads to larger classes of matrices.
Joint work with Grigoriy Ivanov and Alexander Polyanskii.

In this joint project with Jiří Černý we study level-set percolation of the zero-average Gaussian free field on a class of large d-regular graphs with d larger equal 3, containing d-regular expanders of large girth and typical realisations of random d-regular graphs. Through suitable local approximations of the zero-average Gaussian free field by the Gaussian free field on the infinite d-regular tree we are able to establish a phase transition for level-set percolation of the zero-average Gaussian free field which occurs at the critical value for level-set percolation in the infinite model.

The Kuramoto model is the archetype of nonlinear heterogeneous systems of coupled oscillators. Its phenomenology (in the continuum limit) strongly relies on the nonlinear stability of its stationary states. To understand and to rigorously assert stability in this infinite-dimensional setting have been long-standing challenges, and show similar features of the Landau damping in the Vlasov equation. In this talk, I will review results on stability conditions and asymptotic stability of various stationary states, that mathematically confirm the intuited phenomenology and its dependence on parameters.

We define an inhomogeneous percolation model on `ladder graphs'  obtained as direct products of an arbitrary graph G=(V,E) and the set of integers (vertices are thought of as having a `vertical' component indexed by an integer). We make two natural choices for the set of edges, producing an non-oriented and an oriented graph. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite `column' are open with probability q, and all other edges are open with probability p. For all fixed q one can define the critical percolation threshold p_c(q). We show that this function is continuous in (0,1). Joint work with D. Valesin.

The question about existence of groups of intermediate growth (super-polynomial but sub-exponential) was raised by Milnor in the 60s. First examples of such groups were constructed by Grigorchuk in the early 1980s. We will discuss some probabilistic ideas in studying such groups, and explain near optimal volume growth lower estimates of Grigorchuk groups coming from random walks with nontrivial Poisson-Furstenberg boundary on these groups. Joint with Anna Erschler.

The random-cluster model (or Fortuin-Kasteleyn percolation) plays a key role in studies of models on lattices, as it is connected to many of them, and the results obtained for RCM can be though applied for other models.
In this talk I will present another proof of the well-known fact that for the square lattice the critical probability of the random-cluster model p_cr is equal to √q/(1+√q) for q in [1,4]. Unlike other proofs, this one involves the method of parafermionic observables applied to exploration paths in boxes and strips of growing size.
This result was presented in a joint work with E. Mukoseeva during my PhD under the supervision of H. Duminil-Copin.

What does a random independent set look like? This is an important problem at the intersection of probability theory, statistical mechanics, and theoretical computer science. I will introduce this problem, also known as the hard-core model, and explain various ways in which the question can be answered. In particular, I will describe a recent algorithm for producing approximate samples of high-density independent sets on lattices. This is the first known algorithm in the high-density regime, and standard algorithms, like MCMC using Glauber dynamics, are known to fail.
Based on joint work with Will Perkins and Guus Regts.

We study mean field coupled map systems of uniformly expanding circle maps. We first consider N globally coupled doubling maps of the circle with diffusive coupling. Reconsidering and extending the results of Fernandez we prove ergodicity breaking for N=3 and N=4 and showcase some synchronization phenomena for various values of N in case of strong coupling. We then introduce the continuum limit of the system, where we generalize the doubling map to a smooth uniformly expanding circle map T. Now the state of the system is described by a density function and the evolution of an initial density with respect to the transfer operator of the coupled dynamics is studied. We show that for weak enough coupling, a unique, asymptotically stable invariant density exists in a suitable function space. Furthermore, we show that this invariant density depends Lipschitz continuously on the coupling parameter. For sufficiently strong coupling, we prove convergence to a point mass which can be interpreted as chaotic synchronization. To conclude, we provide some outlook on the case of discontinuous T.

In this talk, we consider a random smooth Gaussian function from the plane to ℝ and, given a level u, we colour the points where the function is larger than u in black and the points where the function is less than u in white. By relying on recent works by V. Beffara and D. Gayet, we study the percolation properties of this random colouring and try to compare it with Bernoulli percolation. Joint works with S. Muirhead and A. Rivera.

Consider a Poisson point process in the plane, construct its Voronoi tiling, and colour each tile in black with probability 1/2 and in white with probability 1/2. This defines a critical Voronoi percolation configuration. We prove that, if we let each point move according to a long range stable Lévy process, then there exist atypical (random) times with an unbounded monochromatic component. To this purpose, we study a continuous spectral object - the annealed spectral sample - which is a continuous analogue of the spectral sample studied by Garban, Pete and Schramm.

The main aim of the Thesis is to demonstrate the diverse applicability of fractals in different areas of mathematics. Namely,

• widen the class of planar self-affine carpets for which we can calculate the different dimensions especially in the presence of overlapping cylinders,
• perform multifractal analysis for the pointwise Hölder exponent of a family of continuous parameterized fractal curves in R^d including deRham's curve,
• show how hierarchical structure can be used to determine the asymptotic growth of the distance between two vertices and the diameter of a random graph model, which can be derived from the Apollonian circle packing problem.

I will present the results in an informal way, illustrated with plenty of examples, and some hints about the heuristics of the proofs.
Thesis advisor: Károly Simon

The longest increasing subsequence in a random permutation, the second class particle in TASEP, and semi-discrete polymers at zero temperature have the same scaling limit: a random function with Holder exponent 2/3. This limit can be described in terms of the directed landscape, a random metric at the heart of the Kardar-Parisi-Zhang universality class.
In this talk I will give some insight into the construction of the directed landscape, which is joint work with Duncan Dauvergne and Janosch Ortmann.