CVGMT Seminarshttp://cvgmt.sns.it/seminars/en-usFri, 23 Mar 2018 18:29:14 +0000Limiting behaviour of rescaled nonlocal perimeters and of their first variationshttp://cvgmt.sns.it/seminar/631/2018-03-28: <a href="/person/2437/">V. Pagliari</a>.<p>We introduce a class of integral functionals known as nonlocal perimeters, which can be thought as interactions between a set and its complement that are weighted by a positive kernel. In the first part of the talk, we summarise the main features of these functionals and then we study the asymptotic behaviour of the family associated with mass-preserving rescalings of a given kernel. Namely, we prove that when the scaling parameter approaches $0$, the rescaled non local perimeters $Gamma$-converge to De Giorgi's perimeter, up to a multiplicative constant. In the second part of the talk, we show that a similar result holds for nonlocal curvatures, i.e. for the first variations of the nonlocal perimeters; time permitting, we shall hint at possible applications of this to dislocation dynamics.</p>http://cvgmt.sns.it/seminar/631/Notions of complexity for minimal subvarietieshttp://cvgmt.sns.it/seminar/630/2018-04-17: A. Carlotto.<p>The recent, remarkable advances in the construction of minimal hypersurfaces in general ambient manifolds, either via min-max methods in the spirit of Almgren-Pitts as developed by Marques and Neves, or studying the interfaces arising as suitable singular limits of solutions to the Allen-Cahn equations as proposed by Guaraco, suggest an incredibly rich scenario and have inspired an impressive body of research directions, related to the problems of classifying minimal hypersurfaces under various sorts of ambient curvature conditions, studying their moduli spaces and the associated Weyl's laws, proving generic finiteness results and topological uniqueness theorems among others.</p><p>After a broad-spectrum introduction, I will outline a general project aimed at comparing different ways of quantifying the "complexity" of a minimal subvariety: I will present universal comparison results relating analytic data (like the Morse index, the value of the p-th eigenvalue of the Jacobi operator etc...), geometric data (the area, the Yamabe invariant etc...) and topological invariants (like e. g. the Betti numbers). We shall be concerned about the interactions between these different pieces of information and investigate whether these measures of complexity are equivalent in some suitable sense.For instance, I will discuss the state of the art about Schoen's conjecture asserting that the Morse index of any (closed, embedded) minimal surface inside a 3-manifold of positive Ricci curvature should be bounded from below by an affine function of the genus (with universal coefficients) and, on the other hand, mention partial results for the conjecture by A. Ros classifying the possible topological types of index one minimal surfaces inside 3-manifolds of positive scalar curvature.</p><p>Analogous results have also been obtained for free boundary minimal hypersurfaces inside Riemannian domains.</p><p>This lecture is based on various results that have been obtained in collaboration with Lucas Ambrozio, Reto Buzano and Benjamin Sharp.</p>http://cvgmt.sns.it/seminar/630/Modelli matematici della visione e applicazioni al trattamento di immaginihttp://cvgmt.sns.it/seminar/629/2018-05-09: E. Provenzi.<p>In questo seminario presenterĂ² alcuni modelli della visione a colori e la loro implementazione nell'ambito del trattamento di immagini digitali. ComincerĂ² con una formulazione variazionale in ambito discreto, seguita da un modello di statistica delle immagini naturali, per terminare con l'analisi di una descrizione dello spazio dei colori percepiti che utilizza strumenti di geometria differenziale e di analisi armonica.</p>http://cvgmt.sns.it/seminar/629/