Coloquios Departamento

Deep learning has become a prominent approach for many applications, such as computer vision or neural language processing. However, the mathematical understanding of these methods is still incomplete. A recent approach is to consider neural networks as discretized versions of differential equations. I will first give an overview of this emerging field and then discuss new results on residual neural networks, which are state-of-the-art deep learning models.

Hilbert’s 19th problem asked whether minimizers of elliptic functionals are always analytic. In this lecture I will describe progress made on the problem since the late fifties. After explaining the celebrated result of De Giorgi and Nash, we will focus on minimal surfaces, from the developments in the late sixties to the recent important work of Chodosh and Li. In the last part of the lecture, I will concentrate on a recent result (joint with Figalli, Ros-Oton, and Serra) for reaction-diffusion equations. As in minimal surfaces theory, smoothness of stable solutions (and of minimizers) only holds up to a certain critical space dimension.

Hilbert’s 19th problem asked whether minimizers of elliptic functionals are always analytic. In this lecture I will describe progress made on the problem since the late fifties. After explaining the celebrated result of De Giorgi and Nash, we will focus on minimal surfaces, from the developments in the late sixties to the recent important work of Chodosh and Li. In the last part of the lecture, I will concentrate on a recent result (joint with Figalli, Ros-Oton, and Serra) for reaction-diffusion equations. As in minimal surfaces theory, smoothness of stable solutions (and of minimizers) only holds up to a certain critical space dimension.