Thomas Kruse, University of Giessen
Multilevel Picard approximations for high-dimensional semilinear parabolic partial differential equations

We present new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of one of the proposed methods grows polynomially both in the dimension and in the reciprocal of the required accuracy. We illustrate the efficiency of the approximation methods by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe Von Wurstemberger.

Cécile Mailler, University of Bath
Degree distribution in random simplicial complexes

In this joint work with Nikolaos Fountoulakis, Tejas Iyer and Henning Sulzbach (Birmingham), we study a random graph model introduced by Bianconi and Rahmede in 2015. This model is a simplicial complex model that generalises Apollonian networks and the random recursive trees, by, in particular, adding random weights to the nodes. For this general model, we prove limiting theorems for the degree distribution, and confirm the conjecture of Bianconi and Rahmede on the scale-free propoerties of this random graph.

Andreas Basse-O’Conner, Aarhus University
On infinite divisible laws on high dimensional spaces

Infinite divisible laws play a crucial role in many areas of probability theory. This class of laws corresponds exactly to all the possible weak limits of triangular arrays of random variables, due to the generalized central limit theorem, and hence infinite divisible laws are natural generalizations of the Gaussian laws. In this talk we will discuss properties and representations of infinite divisible probability measures on Banach spaces, with particular focus on their Lévy-Khintchine representations and their shot noise representations. We will see that these representations depend heavily on the Banach space under consideration. In particular, both the geometry of the Banach space and its “size” will play a big role. Many examples of high dimensional probability measures come from stochastic process theory, and we will illustrate the results within this framework.

Manuel Cabezas, Pontificia Universidad Católica de Chile
The totally asymmetric activated random walk model at criticality

Activated random walks is a system of particles which perform random walks and can spontaneously fall asleep, staying put. When an active particle falls on a sleeping one, the sleeping particle becomes active and continues moving. The system displays a phase transition in terms of the density of particles. If the density is small, all particles will eventually sleep forever, while, if the density is high, the system can sustain a positive proportion of active particles. In this talk we describe the critical behavior of the model in the totally asymmetric case. Joint work with Leo Rolla.

Coalescent models for trees within trees

Phylogenetic gene trees are contained within the branches of the species trees. In order to model genealogy backwards in time, of both, gene trees and species trees, simple exchangeable coalescent (snec) process are defined and characterized in talk. In particular, we study the coming down form infinity property for the so called nested Kingman. Finally, we present a model to include population structure in gene lineages.

On the profile of trees with a given degree sequence

For a given (plane) tree $\tau$, let  $N_i$ be the quantity of individuals with $i$ descendants and define its degree sequence  as $s=(N_i)_{i\geq 1}$. We will be interested in the uniform distribution on trees whose degree sequence is $s$. We give conditions for the convergence of the profile (aka the sequence of generation sizes) as the size of the tree goes to infinity. This gives a more general formulation and a probabilistic proof of a conjecture due to Aldous for conditioned Galton-Watson trees.  Our formulation contains results in this direction obtained previously by Drmota-Gittenberger and Kersting. The technique, based on path transformations for exchangeable increment processes, also gives us a (partial) compactness criterion for the inhomogeneous continuum random tree.
Joint work with Osvaldo Angtuncio. 

Olivier Hénard, Université Paris-Sud
Parking  on critical GW trees

Lackner and Panholzer (2015) introduced the parking process on trees as a generalization of the classical parking process on the line. 
In this model, a random number of cars with mean m and variance σ² arrive independently on the vertices of a critical Galton–Watson tree with finite variance Σ² conditioned to be large. The cars go down the tree and try to park on empty vertices as soon as possible.  We show a phase transition depending on Θ:=(1−m)² −Σ²(σ² +m² −m), confirming a conjecture by Goldschmidt and Pryzkucki (2016). Specifically, if Θ > 0, then most cars will manage to park, whereas if Θ < 0 then a positive fraction of the cars will not find a spot and exit the tree through the root. 
The proof relies on tools in percolation theory such as differential (in)equalities obtained through increasing couplings combined with the use of many-to-one lemmas and spinal decompositions of random trees.
This is joint work with Nicolas Curien (Paris Saclay).