Privacy concerns are becoming a major obstacle to using data in the way that we want. It's often unclear how current regulations should translate into technology, and the changing legal landscape surrounding privacy can cause valuable data to go unused. How can data scientists make use of potentially sensitive data, while providing rigorous privacy guarantees to the individuals who provided data? A growing literature on differential privacy has emerged in the last decade to address some of these concerns. Differential privacy is a parameterized notion of database privacy that gives a mathematically rigorous worst-case bound on the maximum amount of information that can be learned about any one individual's data from the output of a computation. Differential privacy ensures that if a single entry in the database were to be changed, then the algorithm would still have approximately the same distribution over outputs. In this talk, we will see the definition and properties of differential privacy; survey a theoretical toolbox of differentially private algorithms that come with a strong accuracy guarantee; and discuss recent applications of differential privacy in major technology companies and government organizations.[-]

This presentation will be kept at a basic level, both continuous and algebraic versions of the methods will be given in their most common variants and the main ingredients of domain decomposition methods will be presented. The content will follow the lines of the chapters 1 and 3 from the domain decomposition book. A short introduction to Freefem software will be given which will allow the students to use quickly the codes illustrating the methods.
Outcomes: At the end of this first lecture, students will have a basic understanding of the methods but also of their implementation.[-]

An early motivation of smooth ergodic theory was to provide a mathematical account for the unpredictable, chaotic behavior of real-world fluids. While many interesting questions remain, in the last 25 years significant progress has been achieved in understanding models of fluid mechanics, e.g., the Navier-Stokes equations, in the presence of stochastic driving. Noise is natural for modeling purposes, and certain kinds of noise have a regularizing effect on asymptotic statistics. These kinds of noise provide an effective technical tool for rendering tractable otherwise inaccessible results on chaotic regimes, e.g., positivity of Lyapunov exponents and the presence of a strange attractor supporting a physical (SRB) measure. In this talk I will describe some of my work in this vein, including a recent result with Jacob Bedrossian and Sam Punshon-Smith providing positive Lyapunov exponents for f inite-dimensional (a.k.a. Galerkin) truncations of the Navier-Stokes equations.[-]

We consider stochastic models of scalable biological reaction networks in the form of continuous time pure jump Markov processes. The study of the mean field behavior of such Markov processes is a classical topic, with fundamental results going back to Kurtz, Athreya, Ney, Pemantle, etc. However, there are still questions that are not completely settled even in the case of linear reaction rates. We study two such questions. First is to characterize all possible rescaled limits for linear reaction networks. We show that there are three possibilities: a deterministic limit point, a random limit point and a random limit torus. Second is to study the mean field behavior upon the depletion of one of the materials. This is a joint work with Lai-Sang Young.[-]

3. The square-root law and the topology of random hypersurfaces. In the third lecture I will focus on the case $\mathbb{K}=\mathbb{R}$ and explain in which sense random real algebraic geometry behaves as the 'square root' of complex algebraic geometry. I will discuss a probabilistic version of Hilbert's Sixteenth Problem, following the work of Gayet & Welschinger (introducing a local random version of Nash and Tognoli's Theorem and of Morse theory for the study of Betti numbers of random hypersurfaces) and of Diatta $\&$ Lerario (showing that 'most' hypersurfaces of degree $d$ are isotopic to hypersurfaces of degree $\sqrt{d \log d}$ ).[-]

We study the statistical behavior of empirical estimators of entropy-regularized optimal transport couplings between compact probability measures. These couplings were first proposed in a thought experiment of Schrödinger as a model for diffusing particles observed at different times, and have become popular in the last 10 years as computationally efficient proxies for optimal transport couplings. We review progress in characterizing rates of estimation for these estimators as well as their asymptotic limits. In particular, we describe a recent proof of a functional CLT conjectured by Harchaoui, Liu, and Pal (2020). Our proof is based on a stronger CLT for the dual solutions to the entropy-regularized problem in a suitable Hölder space. These CLTs also allow us to propose asymptotically valid goodness-of-fit tests based on the Sinkhorn divergence, a popular measure in machine learning. Based on joint work with E. del Barrio, A. González Sanz and J.-M. Loubes, and with G. Mena.[-]