AM^2 Seminars

We present new results about the statistical properties of non-autonomous (sequential) dynamical systems, in particular: the Central Limit Theorem, the Almost Sure Invariance Principle, and the Extreme Value Theory. These results are proved even in the non-uniformly expanding setting.

Evolutionary Games (EG) represent the attempt to describe the evolution of populations by Game Theory. A long list of applications of EG spans from biology to economy, where several complex systems and phenomena may be identified. For instance, the immune system, financial markets, and even crowds, or ants swarms, constitute emblematic examples of complex systems composed by a huge amount of elements, whose interactions drive the whole system towards particular states or equilibria. On the other hand, statistical physics constitutes the natural framework to study dynamics of complex systems in and out of equilibrium. Therefore, the application of statistical physics to EG seems a natural step which allows, in principle, to study several complex systems by analytical and computational approaches. During this talk two main examples will be presented: the spatial prisoner's dilemma and poker. The former has been widely investigated during last years, in particular to identify mechanisms that may lead a population to cooperate. Instead the latter, although represents an open problem in different communities (artificial intelligence, game theory, economy), has been investigated with less emphasis by statistical physicists.

The last decade has seen a huge progress in a wide variety of applications related to the area of artificial intelligence, from speech recognition to computer vision, from natural language understanding to machine translation. Most of these improvements have been achieved by Deep Learning, a big trend in recent AI research, which goes back to multi-layered neural network architectures inspired by the human brain. Big companies such as Facebook, Google, Microsoft and Baidu are investing more and more resources in order to push forward with this novel technology. In this seminar, I will present some of the models and algorithms developed in the context of Deep Learning, together with some applications and a few criticisms which have recently been raised.

The last decade has seen a huge progress in a wide variety of applications related to the area of artificial intelligence, from speech recognition to computer vision, from natural language understanding to machine translation. Most of these improvements have been achieved by Deep Learning, a big trend in recent AI research, which goes back to multi-layered neural network architectures inspired by the human brain. Big companies such as Facebook, Google, Microsoft and Baidu are investing more and more resources in order to push forward with this novel technology. In this seminar, I will present some of the models and algorithms developed in the context of Deep Learning, together with some applications and a few criticisms which have recently been raised.

The Mumford-Shah model is one of the most important image segmentation models, and has been studied extensively in the last twenty years. In this talk, we propose a two-stage segmentation method based on the Mumford-Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford-Shah model. Once g is obtained, then in the second stage, the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle-Pock method. We prove that our method is convergent and the solution g is always unique. In our method, there is no need to specify the number of segments K (K>= 2) before finding g. We can obtain any K-phase segmentations by choosing (K -1) thresholds after g is found in the first stage; and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multi-phase segmentation methods for very general images, including anti-mass, tubular, MRI, noisy, and blurry images; and for very general noise models such as Gaussian, Poisson and multiplicative Gamma noise. We will also mention the generalization to color images.

In 1984 A.P. Veselov and S.P. Novikov obtained explicit formulas for real finite zone, two--dimensional potential Schroedinger operators. In particular5 they introduced a new integrablòe equation, known thereafter as the Veselov--Novikov equation whose real finite--gap solutions are associated to algebro-geometric data on Prym varieties. In this talk we shall review some of their results and discuss generalizations.

In questa relazione si presenta dapprima una breve rassegna storica della problematica che nasce con l’avvento della termodinamica e della delicata questione, ancora non risolta, che riguarda la transizione tra la dinamica delle particelle che obbediscono alle leggi di Newton reversibili e la dinamica dei gas che obbediscono al principio di entropia che condanna i processi alla irreversibilità (VI Problema di Hilbert). Si passa quindi a descrivere la termodinamica del non-equilibrio in cui, nonostante l’irreversibilità, possono nascere sistemi ordinati e complessi. Infine, si discutono recentirisultati della cosiddetta Termodinamica Estesa in cui si cerca di creare un legame tra l’approccio della Meccanica dei Continui e la Teoria Cinetica per descrivere la fisica dei gas rarefatti. Questa teoria Fisico-Matematica apre nuove frontiere per la comprensione del comportamento dei gas nei processi di non-equilibrio con molteplici applicazioni sia in ambito classico che relativistico.