Alessandra Cipriani (University College London)

Title: Moving sand: the Abelian sandpile

Abstract:

The Abelian sandpile model is a simple dynamical system that displays surprisingly rich collective behavior. In the first introductory talk, we will explore its basic rules, the emergence of self-organized criticality, and its deep connection with other probabilistic models such as the uniform spanning tree and the (fermionic) Gaussian free field, among others.

In the second half we will explore limit results for some observables of the system, such as height fields. Based on joint (sometimes ongoing) works with S. Baldassarri, L. Cattani, L. Chiarini, R. S. Hazra, A. Rapoport, W. Ruszel.

Alexandre Stauffer (King's College London)

Title: Spread of infection among moving particles

Abstract:

We consider a collection of particles moving on an infinite graph, such as $\mathbb{Z}^d$. Each particle is either infected or healthy, and infected particles transmit the infection to healthy particles when they meet at the same vertex. This simple model gives rise to a number of natural mathematical questions concerning the spread of infection.

In the first lecture, we will review the state of the art on this problem, focusing on the main known results and some open problems. In the second lecture, we will present more recent results, in particular for a non-monotonic variant of the model. We will explain the main challenges in this setting and discuss the proof techniques.