Miércoles Jueves Viernes
10:00-11:00 Rivero Cabezas Kyprianou
11:00-11:30 Café Café Café
11:45-12:30 Kyprianou Rivero Rivero
12:30-15:00 Almuerzo Almuerzo Almuerzo
15:00-16:00 Cabezas Kyprianou Cabezas
16:00-16:30 Café Café Café
16:30-17:30 Mena Tapia Lopes


Andreas Kyprianou, University of Bath

Exploration of R^d by the isotropic α-stable process
In this mini-course we will review some very recent work on isotropic stable processes in high dimension. The recent theory of self-similar Markov and Markov additive processes gives us new insights into their trajectories. Combining this with classical methods, we revisit some old results, as well as offering new ones.

Víctor Rivero, Centro de Investigaciones Matemáticas, Guanajuato

Excursion theory and fluctuation theory for Lévy and Markov additive processes
Fluctuation theory of Lévy and Markov additive processes aims at obtaining path properties by decomposing the paths at the instants were it reaches a new suprema, respectively a new minimum. Most of the applications of these processes are based in results from fluctuation theory. The purpose of this course is to give an introduction of fluctuation theory for these processes based on the techniques from excursion theory from the supremum, respectively from the infimum. In order to explain the key ideas, we will start by the simplest case of a real valued random walk or a discrete time Markov additive process, where many technicalities arising in the continuous time case are avoided. Then we will tackle the continuous time case and we will make some connections with the course of Pr. Kyprianou.

Manuel Cabezas, Pontificia Universidad Católica de Chile

The ant in the labyrinth: Random walks on critical high-dimensional graphs
The ant in the labyrinth is a term coined in 1976 by Pierre-Gilles de Gennes to refer to the simple random walk on a critical percolation cluster of $\mathbb{Z}^d$.
He proposed to study this model since it is the canonical example of diffusion in critical environments.
The goal of this mini-course is to present the history of this model and to communicate some of the recent progress towards understanding this model in the high-dimensional case. In particular, we will present a very detailed result obtained for the simple random walk on critical branching random walks in $\mathbb{Z}^d$. This simplified model is strongly believed to share a common scaling limit with the critical percolation case, a behaviour that is expected to be universal in high dimension.
Presenting this topic will lead us to discuss a wide variety of subjects: random walks in random environments, critical trees, critical graphs, the super-brownian motion …

Charlas invitadas:

Fabio Lopes, Pontificia Universidad Católica de Valparaíso

Cows on the move
Consider an epidemic that spread in a population of moving particles on Z^d. The particles may infect each other upon direct contact, but may also be infected via contaminated locations. We formulate a simple model incorporating this phenomenon and demonstrate that the presence of site contamination may have an impact on the epidemic spread. Specifically, site contamination can cause a subcritical model to become supercritical, and measures aimed at controlling site contamination thereby has the potential to suppress large outbreaks. We also discuss some open problems regarding this model. (joint work with M. Deijfen and T. Britton.)

Gonzalo E. Mena, Columbia University

Optimal transport and applications to Data Science
Optimal transport (OT) provides rich representations of the discrepancy between probability measures supported on geometric spaces. Recently, thanks to the development of computational techniques, OT has been used to address problems involving massive datasets, as an alternative to usual KL-divergence based approaches. In this talk I will introduce the OT problem and comment on its elementary duality properties. Then, I will present the entropy regularized problem and its (fast) solution via Sinkhorn iterations. Finally, I will overview two applications to Data Science: first, dimensionality reduction via Wasserstein Barycenters and Wasserstein PCA. Second, parameter inference in generative models defined through complex nonlinear transformation of a noise distribution.

Nikolas Tapia, Universidad de Chile

Construction of geometric rough paths
This talk is based on a joint work in progress with L. Zambotti (UPMC). First, I will give a brief introduction to the theory of rough paths focusing on the case of Hölder regularity between 1/3 and 1/2. After this, I will address the basic problem of construction of a geometric rough path over a given ɑ-Hölder path in a finite-dimensional vector space. Although this problem was already solved by Lyons and Victoir in 2007, their method relies on the axiom of choice and thus is not explicit; in exchange the result is more general. In an upcoming paper, we provide an explicit construction clarifying the connection between rough paths theory and free (nilpotent) Lie algebras. In particular, we use an explicit form of the Baker–Campbell–Hausdorff formula due to Loday in order to provide explicit expressions and bounds to achieve such a construction.