Noticias en español
On fraudulent stochastic algorithms.
Resumen: We introduce and analyse the almost sure convergence of a stochastic algorithm for the global minimisation of smooth functions. This diffusion process is called fraudulent because it requires the knowledge of minimal value of the function. Nevertheless, its investigation is not without interest, since in particular it appears as the limit behaviour of non-fraudulent and time-inhomogeneous swarm mean-field algorithms for global optimisation or in stochastic gradient descent algorithms in over-parametrised deep learning applications. The talk is based on collaborations with Benaïm,...
Read MoreAsymptotic behavior and rigidity in one-phase free boundary problems.
Abstract: In this talk, we will present some results concerning the behavior of solutions to the one-phase Bernoulli problem that are modeled –either at infinitesimal or at large scales–by one-homogeneous solutions with isolated singularity. We address the uniqueness of blow-up and blow-down limits provided that the one homogeneous solution has additional symmetries (integrability through rotations), and establish a rigidity type theorem, in the spirit of Simon-Solomon, given suitable conditions on the linearized operator around the one-homogeneous solution.
Read MoreQuenched limits for sub-ballistic random walks in random conductances: high and low dimensions.
Resumen: Consider a random walk amongst elliptic conductances with a deterministic directional bias. For all dimensions larger or equal than 2, Fribergh proved that the walk is ballistic if and only if the mean of a conductance is finite. In the infinite mean case, under proper regularity conditions, Fribergh and Kious showed the convergence of the rescaled process towards Fractional Kinetics, in the annealed setting. I will explain how to obtain a quenched limit by exploiting a celebrated idea of Bolthausen and Sznitman. I will highlight the difference between the high dimensional case (d...
Read MoreRandom burning of the Euclidean lattice.
Resumen: The burning number of a graph is the minimal number of steps that are needed to burn all of its vertices, with the following burning procedure: at each step, one can choose a point to set on fire, and the fire propagates constantly at unit speed along the edges of the graph. In joint work with Alice Contat, following Mitsche, Prałat and Roshanbin, we consider two natural random burning procedures in the discrete Euclidean torus $\mathbb{T}_n^d$, in which the points that we set on fire at each step are random variables. Our main result deals with the case where at each step, the...
Read MoreOptimal d-Clique Decompositions for Hypergraphs.
Abstract: We determine the optimal constant for the Erdős-Pyber theorem on hypergraphs. Namely, we prove that every n-vertex d-uniform hypergraph H can be written as the union of a family F of complete d-partite hypergraphs such that every vertex of H belongs to at most (n choose d)/(n lg n) graphs in F. This improves on results of Csirmaz, Ligeti, and Tardos (2014), and answers an old question of Chung, Erdős, and Spencer (1983). Our proofs yield several algorithmic consequences, such as an O(n lg n) algorithm to find large balanced bicliques near the Kővári-Sós-Turán threshold. Moreover,...
Read MoreThe Ramsey Number of Multiple Copies of a Graph
Abstract: Let H be a graph without isolated vertices. The Ramsey Number r(nH) is the minimum N such that every coloring of the edges of the complete graph on N vertices with red and blue contains n pairwise vertex-disjoint monochromatic copies of H of the same color. In 1975, Burr, Erdős and Spencer established that r(nH) is a linear function of n for large enough n. In 1987, Burr proved a superexponential upper bound for when the long-term linear behavior starts. In 2023, Bucic and Sudakov showed that the long-term linear behavior starts already when n is an exponential function of |H|...
Read More