Filippo Quatrocchi et Xavier Lamy

 

 

10-11h Filippo Quatrocchi

 

Title: Asymptotics for quantization of probability measures

 

Abstract: Quantization is the problem of approximating (or compressing) a given probability measure using discrete measures, supported on a prescribed number n of points. One of the fundamental questions is: At what rate does the minimal quantization error (measured in Wasserstein distance) tend to zero as n grows? A classical answer is given by Zador's theorem. I will discuss this result and present new findings for a variant of the problem in which the discrete approximating measures are additionally constrained to be uniform. This talk is based on the work arXiv:2408.12924.

 

11h-12h Xavier Lamy

Title : Hyperbolic regularization effects for degenerate elliptic equations

Abstract : I will report on joint work with Riccardo Tione, where we establish partial regularity results for Lipschitz solutions of general 2D equations $\mathrm{div}\: G(Du)=0$ with highly degenerate ellipticity. Under the assumption that the strictly monotone field $G$ is fully degenerate only along curves, we can use tools from the theories of Hamilton-Jacobi equations and hyperbolic conservation laws and recover some regularity. This extends all previously known results, where the degeneracy set was required to be zero-dimensional.