Alberto Farina, Université Picardie Jules Verne, A Bernstein-type result for the minimal surface equation, Wednesday, January 25, 2017, time 17:15 o'clock, Aula seminari 6° piano
Abstract:Abstract:
We prove the following Bernstein-type theorem: if $u$ is a solution to the minimal surface equation over $R^N$, such that $N-1$ partial derivatives are bounded on one side (not necessarily the same), then $u$ is an affine function. Besides its novelty, our theorem also provides a new, simple and self-contained proof of celebrated results of J. Moser and of E. Bombieri e E. Giusti.

Gianluca Vinti, Università di Perugia, Risultati di approssimazione e loro applicazioni al Digital Image Processing per la diagnosi di patologie vascolari , Wednesday, January 18, 2017, time 17:15 o'clock, Aula seminari 6° piano
Abstract:Abstract:
Viene mostrato come alcuni risultati di approssimazione tramite operatori di tipo campionamento svolgono un ruolo importante nell’ambito della ricostruzione di segnali/immagini e vengono discusse alcune loro applicazioni ad immagini CT per la diagnosi di patologie vascolari dell’aorta addominale.

Nicola Soave, Politecnico di Milano, On entire solutions of an elliptic system modelling phase separation, Wednesday, December 14, 2016, time 13:15 o'clock, Aula seminari 3° piano
Abstract:Abstract:
We consider a system of elliptic equations defined in the whole space, arising in the analysis of phase separation phenomena in singularly perturbed problems driven by strong competition. We present several results concerning existence and qualitative properties of the solutions, with particular emphasis to their symmetry and rigidity.

Alessandro Audrito, Università degli Studi di Torino, Universidad Autonoma de Madrid, The Fisher-KPP problem with doubly nonlinear diffusion, Wednesday, November 30, 2016, time 13:15 o'clock, Aula seminari 3° piano
Abstract:Abstract:
The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 <= u(x, t) <= 1 of the problem posed in the real line. The existence of propagation waves with finite speed has been confirmed in some related models and disproved in others. We investigate here the corresponding theory when the linear diffusion is replaced by the “slow” and “fast” doubly nonlinear diffusion. In the first case, we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions (we show a TW asymptotic behaviour also in the critical case that we call “pseudo-linear”, i.e., when the operator is still nonlinear but has homogeneity one. In the second one, we prove the non-existence of travelling waves while the propagation is exponentially fast in space for large times. Moreover, in the “fast” diffusion case, we show precise bounds for the level sets of general solutions.

Silvia Villa, Politecnico di Milano, Convergence of proximal gradient methods, Wednesday, November 16, 2016, time 13:15 o'clock, Sala del consiglio 7° piano
Abstract:Abstract:
First order methods have recently been widely applied to solve convex optimization problems in a variety of areas including machine learning and signal processing. In particular, proximal gradient algorithms (a.k.a. forward-backward splitting algorithms) and their accelerated variants have received considerable attention. These algorithms are easy to implement and suitable for solving high dimensional problems thanks to the low memory requirement of each iteration. In this talk I will present some recent convergence results for this class of methods.

Giovanni Alberti, ETH, Zurigo, Absence of critical points of solutions to the Helmholtz equation in 3D, Wednesday, October 12, 2016, time 16:15 o'clock, Aula seminari 3° piano
Abstract:Abstract:
In this talk I will discuss a method to prove the absence of critical points for the Helmholtz equation in 3D. The key element of the approach is the use of multiple frequencies in a fixed range, and the proof is based on the spectral analysis of the associated problem. This question is strictly connected with the Rado-Kneser-Choquet theorem, whose direct extension to the Helmholtz equation or to three dimensions is not possible.