Abstract: In this talk we look at a higher-dimensional generalization of the Erdős--Rényi random graph model to random simplicial complexes, namely the Linial--Meshulam model. In this high-dimensional setting there will often be many nonequivalent ways to topologically generalize common graph properties. Several of these will be surveyed and then new work will be discussed establishing an anticollapsibility threshold in the Linial--Meshulam model as a generalization of path connectivity of random graphs. This work is motivated by questions about high-dimensional trees. This talk will assume familiarity with random graphs but all relevant topological notions will be defined.
This is joint work with Davide Lofano.

Abstract: In this talk we discuss the asymptotic shape and the spread rate for growing stochastic particle systems. We start with the classical models and results and then proceed to discuss more recent developments. We will highlight the differences between discrete-space and continuous-space models. We will also briefly discuss computer simulations.

Abstract: Bensmail, Harutyunyan, Le and Thomassé conjectured that for a fixed tree T, the edge set of any graph G of size divisible by size of T with sufficiently high degree can be decomposed into disjoint copies of T, provided that G is sufficiently highly connected in terms of maximal degree of T. They proved the conjecture for trees of maximal degree 2 (i.e., paths). In particular, they showed that in the case of paths, the conjecture holds for 24-edge-connected graphs.
We improve this result showing that 3-edge-connectivity suffices, which is best possible. We disprove the conjecture for trees of maximum degree greater than two, prove the conjecture for certain forests and prove a relaxed version of the conjecture that concerns decomposing the edge set of a graph into disjoint copies of two fixed trees of coprime sizes.
Joint work with S. Thomassé.

Abstract: We review the concept of factor of iid processes that allows to capture the behavior of local algorithms on large networks. We investigate how much a global decision can or cannot be made using these processes. In particular, how much independent the output values need to be at far away vertices when applied on the limit case of the d-regular infinite tree.
Joint work with Viktor Harangi.