| Monday | Tuesday | Wednesday | Thursday | Friday | |
| 10:00-11:15 | Inés | Amitai | Inés | Amitai | Luis |
| 11:15-11:45 | Coffee | Coffee | Coffee | Coffee | Coffee |
| 11:45-13:00 | Luis | Inés | Amitai | Luis | Amitai |
| 13:00-15:00 | Lunch | Lunch | Lunch | Lunch | Lunch |
| 15:00-16:00 | Roberto | Luis | Inés | Pablo | |
| 16:00-16:30 | Coffee | Coffee | Coffee | ||
| 16:30-17:30 | Cecilia* | Gonzalo | Julian | Sani | |
| Reception | Dinner |
*This talk starts 15 minutes earlier
Mini-Courses
Inés Armendáriz – Universidad de Buenos Aires
Random permutations and the boson point process.
Luis Fredes – Universidad de Bordeaux
Heaps of Pieces and Applications to Probability Theory
In this course, we will begin by introducing the theory of heaps of pieces, which we will later apply to demonstrate various theorems of probabilistic significance. First, we will present and prove the matrix tree theorem, which, through the graph interpretation of Markov chains, states that the invariant measure at a node u of an irreducible Markov chain can be computed as the sum over weighted spanning trees rooted at u. Next, we will examine a particular class of random walks on trees, where we will apply the theory of heaps of pieces and the matrix tree theorem to study its invariant measure(s), recurrence/transience criteria, and other properties.
Amitai Linker – Universidad Andrés Bello
The contact process on dynamic graphs
In this mini-course, we will provide a brief introduction to the contact process, presenting some of its standard results and the main techniques used for its analysis. We will then move on to studying the contact process on networks undergoing dynamical percolation. Specifically, we will analyze how the speed of the background dynamics influences the infection process, focusing on results for cases where the underlying network is either the d-dimensional lattice or a long-range percolation network. If time allows, we will conclude the mini-course by discussing results for the contact process on certain scale-free dynamic networks.
Talks
Julian Amorim– IMPA
Quantitative Hydrodynamics for a Generalized Contact Model
We use the framework of Quantitative Hydrodynamics to derive CLT around its hydrodynamic limit for an interacting particle system inspired by integrated-and-fire neuron models. The hydrodynamic limit of this model was originally derived by Chariker, De Masi, Lebowitz and Presuitti, and as an important intermediate step we show that this convergence holds at optimal L^2-speed. Joint with Milton Jara and Yangrui Xiang.
Pablo Araya – Universidad de Chile
Characterisation of Markovian Properties on Maps
In this talk, we revisit the Markov property of decorated planar maps, providing a comprehensive characterization of random submaps that satisfy the Markov property and demonstrating that they are not restricted to those obtained through a peeling procedure. Additionally, we identify the class of laws for decorated planar maps that exhibit both the Markov property and reroot invariance, establishing that they correspond to Boltzmann-type maps.
Sani Biswas – Universidad de Chile
Milstein-type schemes for McKean–Vlasov SDEs driven by Brownian motion and Poisson random measure (with super-linear coefficients)
We present a general Milstein-type scheme for McKean-Vlasov stochastic differential equations (SDEs) driven by Brownian motion and Poisson random measure and the associated system of interacting particles where drift, diffusion and jump coefficients may grow super-linearly in the state variable and linearly in the measure component. The strong rate of L^2-convergence of the proposed scheme is shown to be arbitrarily close to one under appropriate regularity assumptions on the coefficients. For the derivation of the Milstein scheme and to show its strong rate of convergence, we provide an It\^o formula for the interacting particle system connected with the McKean-Vlasov SDE driven by Brownian motion and Poisson random measure. Moreover, we use the notion of Lions derivative to examine our results. The two-fold challenges arising due to the presence of the empirical measure and super-linearity of the jump coefficient are resolved by identifying and exploiting an appropriate coercivity-type condition.
Cecilia De Vita – Universidad de Buenos Aires
Synchronization in Random Geometric Graphs
The Kuramoto model is a system of ordinary differential equations that describes the behavior of coupled oscillators. In this talk, we consider the case where the coupling is given by a random geometric graph on the unit circle. We ask whether it is possible to guarantee global synchronization (for almost any initial condition, the system converges to a state where all phases coincide) or if there exist other stable equilibria. To address this question, we work with the energy function of the Kuramoto model on these graphs and prove the existence of at least one local minimum for each winding number (with high probability). There is a correspondence between these states and the explicit ‘twisted states’ found in cycle graphs, although in this case without an explicit formula.
Roberto Cortez – Universidad Andrés Bello
Fractal opinions among interacting agents
Abstract TBA
Gonzalo Mena– Carnegie Mellon University.
Statistical properties of the rectified transport.
The problem of finding a transformation mapping one distribution into another is a relevant mathematical problem with several applications in physics, genomics, etc. When this transformation is assumed to be monotonic, the above problem corresponds to finding the so-called optimal transport map, for which a rich mathematical regularity theory is available and for which a recent non-parametric estimation theory has been established. These statistical results indicate that plug-in estimators of such maps converge faster than expected for Kernel density estimators, a consequence of the extra degree of smoothness of the optimal map compared to the original densities. Moreover, a central limit theorem has been established for such estimators under suitable bandwidth selection, enabling uncertainty quantification. The main drawback is that their computation is typically intractable as it relies on solving an optimal transport problem in the continuum, for which we can only obtain approximated solutions.
To deal with these issues, we propose rectified transport as an alternative to optimal transport. The rectified map (Liu et al., 2022) is a relaxation of optimal transport that recovers optimal transport if sufficient constraints are added. Unlike optimal transport, the rectified map is computed pointwise by solving an ordinary differential equation with a velocity field given by a conditional expectation. Moreover, the computation of a plug-in estimator for the rectified flow amounts to solving a sequence of non–parametric regression problems. Rectified maps are typically used in diffusion models, although little is still known about their regularity and sample properties.
We establish an elementary regularity theory, showing that the population rectified map also has more regularity than the underlying densities. Based on this theory, we derive a bias and variance analysis for this estimator, as well as a central limit theorem. Our results indicate that this estimator benefits from the enhanced regularity of the transport map. However, the benefits are more modest compared to optimal transport, presumably because the rectified map is typically less structured than the optimal transport.
This is joint (ongoing) work with Arun Kumar Kuchibhotla and Larry Wasserman.