Conferenze

Alan Cigoli

From groups to semi-abelian categories

The notion of abelian category was one of the earliest and most striking successes of category theory, providing an axiomatic context for homological algebra of abelian structures. On the other hand, an equally effective categorical understanding of non-abelian algebra, and in particular of group theory, was missing for a long time. The notions of Maltsev, protomodular, and semi-abelian category, that appeared in the nineties, finally provided such insight and categorical non-abelian algebra largely developed since then. In this talk, I will outline this development and my contribution to the field.

Giuseppe D'Onofrio

The first-passage-time statistics of the Jacobi process and applications to neuronal modeling

Nonlinear dynamical systems are often affected by different sources of noise and the usual belief is that the presence of noise can hinder or deteriorate the signal transmission in the system. However it has been observed, both in theoretical models and experiments, that random fluctuations can sometimes improve information processing. Mathematical models in neuroscience are one of the most prominent examples of phenomena for which the noise is of primary importance or even a part of the signal itself rather than a source of inefficiency and unpredictability.

The aim of this talk is to contribute to the discussion on the role of noise, focusing on diffusion processes with multiplicative noise. First, I will give an overview on the probabilistic problem of first-passage-time of a Jacobi diffusion process through a constant threshold. Then I will discuss the effects of a jump component on the dynamics of the Jacobi process using the so-called intertwining relations. Finally, I will propose applications to single neuron models, together with the description of the occurrence of a phenomenon of coherence resonance and other counterintuitive effects due to the presence of the multiplicative noise and its dependence on the input parameters.

Based on joint works with Petr Lansky (Czech Academy of Sciences), Pierre Patie (Cornell University), Laura Sacerdote (Università di Torino) and Massimiliano Tamborrino (University of Warwick).

Alessandro Iacopetti

On some results on PDEs arising in Geometry and Physics

In this talk we show some recent results concerning the asymptotic analysis, the existence and regularity of solutions to some classes of elliptic partial differential equations. In particular we focus on the prescribed mean curvature equation for spacelike hypersurfaces in the Lorentz-Minkowski space, which plays a role in Relativity and in the Born-Infeld model of Electromagnetism.

Then we present some results concerning the symmetry breaking for an overdetermined problem in cones, which are obtained by studying the minimizers of a related shape optimization problem.

Elena Issoglio

Stochastic Equations with singular coefficients and their links to Partial Differential Equations

In this talk I will introduce a class of Stochastic Equations with singular coefficients (SDEs and BSDEs), i.e. with coefficients that are generalised functions, in particular elements of Besov/Sobolev spaces with negative order. I will give a meaning to these equations via their links to Partial Differential Equations. The latter will also have singular coefficients, and will be solved using semigroup theory and fixed point theorems in function spaces. Afterwards, I will investigate theoretical questions of existence and uniqueness of solutions to the singular Stochastic Equations, which turn out to be (relatively) smooth stochastic processes. If time permits, I will present some related topics (numerical scheme for singular SDEs and/or blow up questions for singular nonlinear PDEs).

Riccardo Moschetti

Hurwitz space, monodromy groups, and the degree of the forgetful map

Hurwitz spaces parametrise coverings of the projective line with some data fixed, among them the so-called monodromy group. I will talk about some results concerning the number of connected components of some particular Hurwitz spaces, and about the problem of counting how many covering structures a fixed curve can be endowed with.