Plenary session 3

Wednesday, 7/16, 9:00-12:00, Congress Centre

Transport in Disordered Media

Alessandra Faggionato (Department of Mathematics. Sapienza University of Rome)

Random resistor networks are widely used in Physics to analyze transport in disordered systems, such as mixtures of conducting and non-conducting materials, doped semiconductors, and disordered superconductors. For a broad class of random geometries large-scale conduction can be described using deterministic coefficients derived from stochastic homogenization. Within this universal description, a material behaves as a conductor or an insulator depending on the statistical properties of macroscopic crossings in the resistor network. We will describe sufficient conditions for the emergence of macroscopic conduction. As an application, we will discuss Mott variable-range hopping in doped semiconductors in the low temperature regime which amplifies the effects of disorder, giving rise to the physics Mott’s law.

Time permitting, we will also provide insights into universal laws for AC conduction via random electrical circuits and into the analysis of transport through stochastic interacting particle systems in random environments - moving beyond the mean-field approximation at the basis of random resistor networks.

Keywords: percolation theory, random graphs, stochastic homogenization, interacting particle systems.

On estimating Fréchet means

Roberto Imbuzeiro Oliveira (IMPA (Rio de Janeiro, Brazil))

A Fréchet mean (or barycenter) of a distribution \(P\) over a metric space \((\mathcal{X},d)\) is any point \[\mu\in {\rm arg}\min_{m\in\mathcal{X}}\mathbb{E}_{X\sim P}\,d^2(X,m).\] This definition makes sense for any \((\mathcal{X},d)\), and coincides with the expectation of \(P\) if \(\mathcal{X}\) is Euclidean space. This talk considers the problem of estimating Fréchet means from an i.i.d. sample from \(P\), possibly contaminated by adversarial noise. The goal is to obtain optimal or-nearly optimal high probability bounds under weak moment assumptions on \(P\).

Two main settings are considered. The first one consists of Alexandrov spaces where geodesics are “bi-extendible” in some precise quantitative sense, which is rich enough to include the space of probability measures over \(\mathbb{R}^d\) with the Wasserstein-\(2\) distance. Our main theorem for this case refines an analysis of \([1]\), and obtains bounds with the “right” variance parameters The second setting we consider is that of uniformly convex Banach spaces of power type \(2\), where our results seem to be completely new. Both results are obtained via an analysis of certain “trimmed empirical processes”  that is based on previous work with Lucas Resende \([2]\).

Bibliography

\([1]\) T. Le Goı̈c, Q. Paris, P. Rigollet and A. Stromme. “Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space.” Journal of the European Mathematical Society (2022), Vol 25(6), 2229–2250.
\([2]\) R. I. Oliveira and L. Resende. “Trimmed sample means for robust uniform mean estimation and regression.” arXiv:2302.06710.

What AI will not tell you about white noise

Krzysztof Burdzy (University of Washington)

This lecture is a brave attempt to show that human intelligence might still be useful in the era of artificial intelligence. White noise and Brownian motion are closely related fundamental concepts in science and mathematics. Their attraction stems in part from their paradoxical mixture of simplicity and complexity.

The lecture will be illustrated and accessible to a wide audience.