Il link al quale collegarsi per seguire da remoto il Congresso è il seguente:
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Informazioni sull'evento:
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Questo convegno ha lo scopo di riunire giovani scienziati ed esperti internazionali nell'ambito delle equazioni differenziali alle derivate parziali ed in particolare degli operatori pseudo-differenziali e dell'analisi di Fourier.
Il Comitato scientifico e organizzativo: A. Ascanelli, C. Boiti
The purpose of this workshop is to bring together young researchers and Italian experts on the subject of vector fields and related geometric structures in singular settings, such as Carnot-Carathéodory or Gaussian spaces, where usual Euclidean analysis techniques meet several difficulties.
Meetings will take place at Aula 6 of Dipartimento di Matematica e Informatica, via Machiavelli 30, Ferrara.
11:00 -- 11:40 Addona
11:50 -- 12:30 Vittone: On the rank-one theorem for BV functions
Lunch break
14:30 -- 15:10 Comi: The Gauss-Green theorem in stratified groups
15:20 -- 16:00 Menegatti: Sobolev classes and bounded variation functions on domains of Wiener spaces
Coffee break
16:40 -- 17:20 Stefani: Heat and entropy flows in Carnot groups
9:00 -- 9:40 Bruè: Approximation in Lusin’s sense of Sobolev functions by Lipschitz functions and applications.
9:50 -- 10:30 Buffa: BV Functions in Metric Measure Spaces: new insights into integration by parts formulæ, and traces
Coffee break
11:10 -- 11:50 Lunardi: Funzioni BV in spazi di Hilbert
Si introducono e si studiano funzioni a variazione limitata definite su uno spazio di Hilbert dotato di una misura di probabilità "buona", ovvero che permetta di fare integrazioni per parti lungo direzioni opportune. Particolare attenzione è dedicata alle funzioni caratteristiche di insiemi misurabili, e quindi agli insiemi di perimetro finito. Si stabiliscono proprietà e caratterizzazioni di base, e si danno esempi in alcune situazioni significative.
In 1993 G. Alberti proved a celebrated result, conjectured by L. Ambrosio and E. De Giorgi, concerning a rank-one property for the singular part of the derivative of a vector-valued map with bounded variation. We will discuss a recent elementary proof of this result together with some applications to BV functions in sub-Riemannian Carnot groups. These are joint works with S. Don and A. Massaccesi.
The Gauss-Green formula is of significant relevance in many areas of mathematical analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields. In the Euclidean setting it has been shown by Silhavy (2005) and Chen, Torres and Ziemer (2009) that Gauss-Green formulas hold for sets of finite perimeter and L^{\infty}-divergence measure fields, i. e. essentially bounded vector fields whose distributional divergence is a Radon measure. We extend these results to the context of stratified groups. In particular, we prove the existence of generalized normal traces on the reduced boundary of sets of locally finite h-perimeter without requiring De Giorgi's rectifiability theorem to hold. This is a joint work with V. Magnani.
We consider problems connected to W^{1,p}(O) and BV(O) for O convex set in a Wiener space (Banach separable space with Gaussian measure); we focus our analysis on the approximation of functions with regularizing sequences, in particular by considering an extension of a result obtained by Barbu and Röckner in the Euclidean case.
After the work of Ambrosio, Gigli and Savaré, it is well-known that in any CD(K,+\infty) space, i.e. a space with Ricci curvature bounded from below in the sense of Sturm-Lott-Villani, the gradient flow of the Boltzmann entropy and the heat flow coincide. In 2014 Juillet proved that this correspondence holds also in the Heisenberg groups of any dimension, although these groups are not CD(K,+\infty) spaces. It was an open problem to establish the same correspondence in any Carnot group. In this talk, we give a positive answer to this question. This is a joint work with L. Ambrosio.
We say that a real valued function f, defined in a metric measure space, is approximable in a Lusin sense by Lipschitz functions if, for every \epsilon>0, there exists a Lipschitz function that coincides with f outside a set of measure less than \epsilon. In Euclidean spaces, more generally in metric measure spaces satisfying the doubling and Poincarè inequality, Sobolev functions fulfill this approximation property in a quantitative form.
In a joint work with L. Ambrosio and D. Trevisan we extend these results to a class of non-doubling metric measure structures. Our strategy relies upon pointwise estimates for heat semigroups and applies to Gaussian and RCD(K,\infty) spaces. As a consequence, we prove a first quantitative stability estimate for regular Lagrangian flows associated to Sobolev vector fields in an infinite dimensional setting.
TBA
We adapt the tools from the differential structure developed by N. Gigli in order to give a definition of BV functions on RCD(K,\infty) spaces via suitable vector fields and then establish an extended Gauss-Green formula on a class of "regular" domains, which features the "normal trace" of vector fields with finite divergence measure. Then, we pass to the more classical context of a doubling metric measure space supporting a Poincaré inequality, where we reformulate the theory of "rough traces" of BV functions (after V. Maz'ya)in comparison with the Lebesgue-points characterization, and discuss the conditions under which the respective notions of trace coincide. Based on a joint work with M. Miranda Jr.
Relatore
Professore di Intelligenza Artificiale all’Università di Bristol, UK
A seguire Panel di discussione con la Professoressa Silvia Borelli e il Professor Enrico Maestri del Dipartimento di Giurisprudenza
Modera la Professoressa Evelina Lamma del Dipartimento di Ingegneria.
Non possiamo più vivere senza l’infrastruttura digitale su cui incontriamo vari tipi di agenti intelligenti che prendono decisioni sottili e importanti che ci influenzano direttamente, e quindi dobbiamo imparare a conviverci. Ma senza una comprensione dei principi alla base di questa tecnologia non è possibile regolamentarla, o nemmeno prevederne i rischi. Lo spazio tra la cultura scientifica, quella umanistica e quella tecnica, è il luogo in cui si sta formando questa nuova comprensione delle macchine intelligenti.
L'evento è aperto al pubblico con ingresso gratuito fino ad esaurimento posti.
- Lunedì 29 Maggio, ore 11
- Martedì 30 Maggio, ore 11
- Giovedì 1 Giugno, ore 11
- Lunedì 5 Giugno, ore 11
(computer vision, image processing, medical imaging). Tale problema può essere formulato
attraverso la minimizzazione del funzionale di Mumford-Shah, uno degli approcci più utilizzati
in questo ambito. Negli anni recenti nuove tecniche sono state studiate, basate su approcci
di tipo Deep Learning. In questo seminario proponiano un approccio non supervisionato per la
segmentazione semantica basato sul framework Deep Image Prior: il funzionale di Mumford Shah
viene riparametrizzato attraverso i pesi di una rete convolutoria. I primi esperimenti, eseguiti su
immagini "naturali" e mediche mostrano come questo approccio fornisca risultati molto affidabili,
anche in presenza di rumore. Porremmo anche le basi per l'estensione al modello di Ambrosio-Tortorelli.
Tutti gli interessati sono invitati a partecipare.
Abstract:
A classical tool to get numerical invariants of a curve singularity is the study of its value
semigroup. In case of a one branch singularity this semigroup is a numerical semigroup
(i.e. a submonoid of N with finite complement in it); in case the singularity has h branches,
this semigroup is a subsemigroup of Nh, belonging to the class of the so-called ”good semi-
groups”. Despite their name, the combinatoric of good semigroups is quite problematic;
moreover, for h ≥ 2, it is an open problem to understand which good semigroups can be
realized as value semigroups.
In case of a plane singularity with one branch, an old result of Ap ́ery shows that there
is a particularly strict connection between the value semigroups of the singularity and
of its blowup; this connection is obtained using a particular set of generators of the
semigroup, named ”Ap ́ery set”. In fact, using that result, it is possible to show very
easily, that the equisingularity classes given by the multiplicity sequence and by the
value semigroup coincide. In particular, this method allows to reconstruct the numerical
semigroup associated to a plane branch singularity starting from the multiplicity sequence.
When the singularity has more than one branch, in order to generalize the Ap ́ery result,
two main problems arise: firstly, the Ap ́ery set becomes an infinite set; secondly, in the
process of blowing-up it is necessary to deal with semilocal rings, that cannot be presented
as quotients of a power series ring in two variables, as it happens in the local case. These
problems where partially solved twenty years ago in the two branch case, but the general
case is still open.
In my talk, after describing some key definitions and results on value semigroups and
good semigroups of a curve singularity with h branches, I will explain explaining the
Ap ́ery process for a plane branch and then, I will present some recent results obtained in
a joint project with F. Delgado de la Mata, L. Guerrieri, N. Maugeri and V. Micale, that
are a significant progress toward a complete solution of this problem.
Dipartimento di Matematica e Informatica - University of Catania
V.le A. Doria 6, 95125
Catania
Italy
E-mail address: marco.danna@unict.it
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A wide range of phenomena in science and technology may be described by nonlinear partial differential equations, characterizing systems of conservation laws with source terms.
Well known examples are hyperbolic systems with source terms, kinetic equations and convection-reaction-diffusion equations. This class of equations fits several fundamental physical laws and plays a crucial role in applications ranging from plasma physics and geophysics to semiconductor design and granular gases.Recent studies employ the aforementioned theoretical background in order to describe the collective motion of a large number of particles such as: pedestrian and traffic flows, swarming dynamics, opinion control, diffusion of tumor cells and the cardiovascular system.
Goal of the present Workshop is to present some recent numerical results for these problems with a particular focus on multiple scales.
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