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.[-]

We study an optimal reinsurance problem under the criterion of maximizing the expected utility of terminal wealth when the loss process exhibits jump clustering features and the insurance company has restricted information about the claims arrival intensity. By solving the associated filtering problem we reduce the original problem to a stochastic control problem under full information. Since the classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite dimensionality of the filter, we choose an alternative approach based on Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize the value process and the optimal reinsurance strategy in terms of a BSDE driven by a marked point process. The talk is based on a joint work with M. Brachetta, G. Callegaro and C. Sgarra (arXiv:2207.05489, 2022).[-]

In many physical fluid systems, the constituent particles present a parity-breaking intrinsic angular momentum: this is the case, for instance, of quantum fluids and super-fluids, polyatomic gases, chiral active matter and vortex dynamics. In such situations, only the skew-symmetric component of the total viscous stress tensor, often dubbed odd viscosity, is non-zero, implying that the viscosity becomes non-dissipative.
At the level of the mathematical model, the odd viscosity term is responsible for a loss of regularity, as it involves higher order space derivatives of the velocity field and, in the case of non-homogeneous fluids, of the density.
In this talk we consider the dynamics of non-homogeneous incompressible fluids having odd viscosity and we set up a well-posedness theory in Sobolev spaces for the related system of equations. The proof is based on the introduction of a set of suitable 'good unknowns' for the system, which allow to put in evidence an underlying hyperbolic structure and to circumvent, in this way, the loss of derivatives created by the odd viscosity term.
The talk is based on a joint work with Rafael Granero-Belinchón (Universidad de Cantabria) and Stefano Scrobogna (Università degli Studi di Trieste).[-]