Archivio 2017

In this talk we will present the basics of harmonic measure (with focus on the probabilistic interpretation). We will give a (very brief) overview of its history in analysis, properties and discuss some related results (mostly about the geometrical properties of its support).

General topology plays an important role in the study of Banach spaces. In this talk, after giving a glance to the main relations between general topology and Banach spaces, the classes of Corson, Valdivia and non-commutative Valdivia compact spaces, will be presented. Special care will be given to the class of set-theoretical compact trees. This class will be used to address open questions in this field.

Le leggi naturali e i dispositivi artificiali "lineari" il cui funzionamento non varia col tempo sono esprimibili in termini di convoluzione, quindi le trasformate di Fourier, che traducono le convoluzioni in prodotti, hanno una particolare utilità pratica e concettuale. Una particolare incarnazione della trasformata di Fourier è quella che associa a una successione a=(a_n: n=0,...) (un segnale discreto) la funzione olomorfa f=Za avente quella come successione dei coefficienti (in combinatoria Za è la "funzione generatrice dalla successione"). La legge naturale o il dispositivo artificiale possono essere quindi compresi in termini di operazioni algebriche tra funzioni olomorfe, se del caso introducendo operatori di proiezione. Molte proprietà delle funzioni olomorfe diventano proprietà del sistema sotto esame. A partire dalla fine degli anni '70, queste idee sono state applicate al "controllo" di un sistema lineare. Lo scopo di questi seminari è di percorrere questo circolo di idee, tenendo a mente anche le connessioni con diverse aree della matematica pura (analisi in spazi di Hilbert di funzioni olomorfe, combinatoria, teoria dei numeri), privilegiando gli esempi (da ingegneria e matematica) e le interazioni alla teoria generale.

The simplicial volume (or Gromov norm) is a homotopy invariant of compact manifolds introduced in 1982 by Gromov in his pioneering paper "Volume and Bounded Cohomology". Roughly speaking, the simplicial volume measures how it is difficult to describe the manifold in question in terms of real singular chains. Despite its topological meaning, the simplicial volume turns out to be a fundamental tool for understanding rigidity phenomena, i.e. to establish obstructions to the existence of some geometric structures in terms of topological invariants. More precisely, working with negatively curved manifolds, simplicial volume provides useful information about their Riemannian volume. The aim of this talk is to give an accessible overview about the notion of simplicial volume and to discuss the main techniques involved in this context (some key words are: degree of maps, amenability and negative curvature). If there will be enough time, I will discuss some classical applications like Mostow rigidity and the study of the variation of the volume during a hyperbolic Dehn filling.

Le leggi naturali e i dispositivi artificiali "lineari" il cui funzionamento non varia col tempo sono esprimibili in termini di convoluzione, quindi le trasformate di Fourier, che traducono le convoluzioni in prodotti, hanno una particolare utilità pratica e concettuale. Una particolare incarnazione della trasformata di Fourier è quella che associa a una successione a=(a_n: n=0,...) (un segnale discreto) la funzione olomorfa f=Za avente quella come successione dei coefficienti (in combinatoria Za è la "funzione generatrice dalla successione"). La legge naturale o il dispositivo artificiale possono essere quindi compresi in termini di operazioni algebriche tra funzioni olomorfe, se del caso introducendo operatori di proiezione. Molte proprietà delle funzioni olomorfe diventano proprietà del sistema sotto esame. A partire dalla fine degli anni '70, queste idee sono state applicate al "controllo" di un sistema lineare. Lo scopo di questi seminari è di percorrere questo circolo di idee, tenendo a mente anche le connessioni con diverse aree della matematica pura (analisi in spazi di Hilbert di funzioni olomorfe, combinatoria, teoria dei numeri), privilegiando gli esempi (da ingegneria e matematica) e le interazioni alla teoria generale.

The notions of extremal metric and extremal length can be develop on the discrete context of finite graphs. In particular, following an article by Oded Schramm (1993), they are the main tool to build a correspondence between the 1-skeleton of triangulations of a quadrilateral and square tilings: the squares are associated to the vertices in a combinatorial fashion to fill a rectangle with no overlaps. The extremal metric expresses the length of the edge of the squares. Furthermore, extremal length can be considered on trees. It is in some sense the reciprocal of the notion of capacity from potential theory.

Interpolating sequences for the holomorphic Dirichlet space and its multiplier space were characterized in two preprints of 1994: Marshall and Sundberg, and Chris Bishop. Both articles never were published, but they were very influential; especially the second one, which is linked below. The plan is going through the details of the proof, following a third route opened a few years later by B. Boe. The exposition will be rather technical (the generalities about interpolation by sequences were already covered in previous seminars), but some interesting open problems will be mentioned.

I will present a class of boundary conditions for Kramers-Fokker-Planck operators which guarantees subelliptic estimates similar to the whole space problem.

The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. However, it also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with corners, edges, and conical points. This surge in attention is owed to the connection with resonances of transmission/scattering problems used to model surface plasmons in nanoparticles. I aim to give an overview of recent developments, with particular focus on the NP operator’s action on the energy space of the domain. I will also present recent work for domains in 3D with conical points featuring rotational symmetry. In this situation, we have been able to describe the spectrum both for boundary data in L^2 and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval. Based on joint work with Johan Helsing and Mihai Putinar.

The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. However, it also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with corners, edges, and conical points. This surge in attention is owed to the connection with resonances of transmission/scattering problems used to model surface plasmons in nanoparticles. I aim to give an overview of recent developments, with particular focus on the NP operator’s action on the energy space of the domain. I will also present recent work for domains in 3D with conical points featuring rotational symmetry. In this situation, we have been able to describe the spectrum both for boundary data in L^2 and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval. Based on joint work with Johan Helsing and Mihai Putinar

In this talk we will discuss questions about the boundedness and continuity of classical and fractional maximal operators acting on Sobolev spaces and spaces of functions of bounded variation.

I'll start with presenting some known results about the boundedness and compactness properties of the generalized Cesaro operator, Tg, on Hardy spaces in the unit disc, as well as some of its applications. In the second part we introduce a variant of this operator which depends on an analytic symbol g, and we prove the analogous results for this operator. As an application we generalize a theorem of J. Rattya about complex linear differential equations, and we prove a result about factorization of derivatives of Hardy functions.

The study of invariant subspaces of Hilbert space operators is a classical problem in analysis. It is an open question whether every operator on an Hilbert space has an invariant subspace other than the trivial ones, the zero subspace and the whole space. A. Beurling completely characterized the invariant subspaces of the unilateral shift $(a_0,a_1,\ldots)\mapsto(0,a_0,a_1,\ldots)$ on $\ell^2(\mathbb{N})$ modeling the shift operator on $\ell^2(\mathbb{N})$ with the multiplication by $z$ on the Hardy space of the unit disc. Few years later P. Lax proved an analogous result, that is, he characterized the translation invariant subspace of $L^2(0,\infty)$. In this seminar I will illustrate Beurling and Lax's result. Time permitting, I will also present an analogous of Beurling's result in the setting of the quaternionic Hardy space of the unit ball. This result was recently obtained in a joint work with G. Sarfatti.

Why doing analysis on trees, besides the intrinsic interest? We show as application the characterization of the Carleson measures (or "trace measures") for the Dirichlet space. The seminar requires virtually no prerequisite on holomorphic functions.

We discuss interpolating sequences for RKHS, with special emphasis on the Dirichlet space. The general plan is to cover in a few hours: - interpolation in RKHS in general; - the case of RKHS with the complete Nevanlinna-Pick property; - the Dirichlet space; -the problem of "onto interpolation" on the Dirichlet space; - the weighted Dirichlet spaces (with the construction of Peter Jones). It would be nice to finish with the recent result of Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter (https://arxiv.org/abs/1701.04885) characterizing the interpolating sequences for the spaces having the complete Nevanlinna-Pick property and their multiplier spaces (volounteers are welcome!).

On a metric space (X,d) one can define a set function called capacity which has motivations coming from Physics and which plays a deep role in potential theory and geometric measure theory. It is well known that to any compact subset E of X can be associated a probability measure m on X, called equilibrium measure for E, such that m(E)=cap(E). These measures at present are not well understood. We will present a characterization of equilibrium measures when X is a locally finite tree of infinite depth.

Interpolating sequences for the Nevanlinna class will be used to discuss a natural problem on finitely generated ideals in the class

Moving a continuous problem to a discrete setting is a popular and everpresent approach, that (from time to time) allows to single out the geometric and combinatorial issues of the question ad hand. A particular case we are aiming to investigate is the representation of the unit disc (polydisc) by a dyadic tree (cartesian product of dyadic trees), and its connection to Dirichlet type spaces. Following Arcozzi, Rochberg, Sawyer and Wick we give a (very brief) introduction to the potential theory on a polytree, and then present a very incomplete list of related problems.

In 1983 Tom Wolff proved a surprising inequality which proved to be a pivotal tool in Nonlinear Potential Theory. I will go through Wolff's proof, but in the dyadic setting. Hedberg, L. I.; Wolff, Th. H. Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187.

In questo seminario introduciamo le derivate frazionarie di Riemann-Liouville e di Caputo, con alcune delle loro principali proprietà. Concludiamo illustrando alcuni risultati di regolarità massimale per problemi misti al contorno, in cui compaiono tali derivate.

Enabling visually-guided behaviors in artificial agents implies picking-up and organizing appropriate information from the visual signal at multiple levels. The question arises about how to carefully define which feature to extract, or, from a different perspective, which kind of representation to adopt for the visual signal itself. It is well known that receptive fields (RFs) in the early stages of the primary visual cortex behave as band-pass linear filters performing a multichannel representation of the visual signal (cf. the Gabor jets). Typically, visual features are direcly derived, as symbols, from the outputs of such front-end RFs. Here, I want to emphasize the advantages of thinking early visual processes in terms of signal processing, pointing out the key role played by a full harmonic representation of the visual signal and how highly informative properties of the visual signal are efficiently and effectively embedded in the local image phases and their relationships. Accordingly, instead of directly extracting "classic" spatial features (such as edges, corners, etc.) and then looking for correspondences, we can follow a complementary approach: the visual signal is described in frequency bandwidths in terms of local amplitude, phase and orientation, and more complex visual features are derived as "qualities" based on local phase properties e.g., such as phase conguency, phase difference, and phase constancy, for contrast transitions, disparity and motion, respectively. Notably, phase-based interpretation of the visual signal allows direct links between consolidated machine vision computational techniques and the ascertained properties of visual cortical cells. The issue of direct phase-based measurements vs. distributed population coding of visual features will be discussed in relations to motion and stereo perceptual tasks.

We establish new fractional Poincaré inequalities encoding geometry of conformally flat manifolds with finite total Q-curvature. The method of proof is based on some improvement of the standard Poincare inequality and harmonic analysis techniques. We will give a description of the underlying geometry and in particular the role of the Q-curvature.

The aim of Geometric Group Theory is to get insights on the structure of groups looking at their actions on suitable topological spaces. A significant type of these spaces are classifying spaces for families of subgroups. Classifying spaces for families have been widely studied in the case of the families of finite subgroups and virtually cyclic subgroups, due to their connections with the celebrated Baum-Connes Conjecture and Farrell-Jones Conjecture, respectively. But the definitions are stated for all families of subgroups. The purpose of this seminar is to convince that even off the main road of the two standard families of finite and virtually cyclic subgroups there is fascinating mathematics, that still waits to be explored. In particular, this talk is intended to be an accessible invitation to the investigation of "exotic" classifying spaces in the realm of knot theory. We show how to build a classifying space for a significant family of subgroups of a prime knot group and we investigate its cohomological properties. While playing a little with it, interesting analogies with algebraic number theory will pop out.

Some years ago M.G.Kuhn and T.Steger introduced a class of unitary representations of free groups which they called multiplicative. This class is large enough to include many examples of unitary representations constructed henceforth. Moreover, multiplicative representations have the very interesting property of being tempered, i.e. weakly contained in the regular representation. After recalling the basic notions needed to enter into the field, we outline some algebraic and analytic properties of these representations and we advance some ideas for extending such a study to surface groups and hyperbolic groups.

abstract: We consider the subelliptic eikonal equation, i.e. the eikonal equation associated with a family of (real) smooth vector fields satisfying the Hoermander bracket generating condition on a neighborhood of an open bounded set with smooth boundary. We study the regularity and the singularities of the viscosity solution of the homogeneous Dirichlet problem for such an equation.

COLLOQUIO DI DIPARTIMENTO C’è una rivoluzione in corso, la rivoluzione digitale: la quantità di dati che produciamo raddoppia ogni anno; nel 2016 abbiamo generato tanti dati quanti ne erano stati prodotti nell’intera storia dell’umanità fino al 2015. Con IoT (Internet of Things) entro 10 anni avremo 150 miliardi di sensori connessi in rete, 20 volte più che il numero di persone sulla Terra. Allora la quantità di dati raddoppierà ogni 12 ore. È la quinta rivoluzione dell’IT: dopo i grandi computer, i pc, internet e il web 1.0, i cellulari e il web 2.0, i Big Data – una rivoluzione dovuta allo tsunami di dati, dove tutto quello che facciamo lascia una traccia digitale. Una rivoluzione paragonabile a quella avvenuta con l’invenzione della stampa. I bits faranno molto più di quanto i caratteri mobili di Gutenberg abbiano fatto in termini di spostamento degli equilibri del potere e di trasferimento della conoscenza dalle mani di pochi a comunità sempre più allargate. L’intelligenza artificiale sta facendo progressi impensati, soprattutto attraverso l’analisi dei dati. L’AI non si programma più riga per riga, ma è ora capace di imparare e di automigliorarsi continuamente: sono ormai standard algoritmi in grado di completare compiti che richiedono ‘intelligenza’ meglio degli uomini. Fra il 2020 e il 2060 i super-computer sorpasseranno le capacità umane in moltissime aree. In questo quadro, da un lato i Big Data dall’altro l’AI impongono compiti di manipolazione dei dati che sono strenui sia per la computer science (nuovi paradigmi computazionali; computazione interattiva; la sfida del 'beyond Turing') che per la 'data analytics' (nuove metodologie di approccio al 'data mining'; analisi dei dati topologica; inferenza causale non lineare) per affrontare problemi complessi, nelle scienze di base (scienze della vita, clima, scienze della terra, …) come in quelle sociali, con la data science (A.I., data mining, machine learning, deep learning, teoria topologica del campo dei dati) e la scienza della complessità (teoria delle reti). Ne segue la necessità di una nuova, forte alleanza che combinando metodi e conoscenze della fisica statistica, della matematica, della computer science permetta alla scienza di affrontare in modo vincente questa sfida epocale. Mario Rasetti è Professore Emerito di Fisica Teorica al Politecnico di Torino ed è Presidente di ISI Foundation, Torino e ISI Global Science Foundation, New York.

Il Soap Bubble Theorem (SBT) stabilisce che una superficie compatta con curvatura media costante è una sfera. Per dimostrare questo risultato, A. D. Alexandrov ha inventato il suo principio di riflessione, che è stato in seguito perfezionato da J. Serrin nel metodo dei piani mobili, per ottenere la simmetria radiale per una classe di problemi sovra-determinati. H. F. Weinberger ha fornito una dimostrazione del risultato di Serrin basata su alcune identità e disuguaglianze integrali. R. C. Reilly ha infine fatto vedere come il metodo di Weinberger può essere usato per ottenere un'altra dimostrazione del SBT. Nel mio seminario, seguendo le orme di Weinberger e Reilly, farò vedere come i due risultati di simmetria discendano da due identità integrali per la rigidità torsionale di una sbarra. Le due identità saranno poi usate per ottenere risultati di stabilità della configurazione sferica nei due problemi ed in altri problemi analoghi.

In this seminar some recent results concerning Harnack inequalities will be presented for several classes of sub-elliptic operators. We will start by considering a class of sub-elliptic operators, in divergence form, with low-regular coefficients under global doubling and Poincaré assumptions; for these operators a non-homogeneous invariant Harnack inequality will be shown. As a consequence, we will prove the solvability of the Dirichlet problem (in a suitable weak sense). In the second part, we will consider a class of hypoelliptic non-Hormander operators for which we have been able to construct a Green function; with a completely different approach with respect to the case of doubling metric spaces, we will conclude by showing (by means of techniques of Potential Theory) how the solvability of the Dirichlet problem has been a fundamental tool in order to prove a homogeneous Harnack inequality in the framework of harmonic spaces.

In this presentation, we will analyze a p-Laplacian problem set in a ball of R^N, with homogeneous Neumann boundary conditions. The equation involves a nonlinearity g which is (p-1)-superlinear at infinity, possibly supercritical in the sense of Sobolev embeddings. The nonlinearity allows the problem to have a constant non-zero solution. In this setting, we prove via shooting method the existence, multiplicity, and oscillatory behavior (around the constant solution) of non-constant, positive, radial solutions. We show that the situation changes drastically depending on p>1. For example, in the prototype case g(s)=s^{q-1}, if p>2, the problem has infinitely many solutions for q>p. While, if p=2, the problem admits at least k non-constant solutions provided that q-2 is bigger than the (k+1)-th radial eigenvalue of the Laplacian with Neumann boundary conditions. Finally, for 1<p<2 a surprising result is found, as non-constant solutions with the same oscillatory behavior appear in couples when the radius of the domain is big enough. We will try to give a unified description and motivation for these three different situations. This is a joint work with Alberto Boscaggin (Università di Torino) and Benedetta Noris (Universitè de Picardie Jules Verne). [A. Boscaggin, F. Colasuonno, B. Noris, Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions, preprint] [F. Colasuonno, B. Noris, A p-Laplacian supercritical Neumann problem, Discrete Contin. Dyn. Syst., Vol. 37 n. 6 (2017) 3025-3057]

We present some recent results in the study of the fractional Allen-Cahn equation. In particular, we are interested in the analogue, for the fractional case, of a well known De Giorgi conjecture about one-dimensional symmetry of bounded monotone solutions. In dimension n=2 and for any fractional power 0<s<1 of the Laplacian, the conjecture is known to be true. In this seminar, we will address the 3-dimensional case. Depending wheter s is below or above 1/2, we need to exploit different techniques and ingredients in the proof of the one-dimensional symmetry. In particular, when s<1/2, some properties of the so-called nonlocal minimal surfaces, will play a crucial role. This talk is based on several papers in collaboration with X. Cabré, J. Serra, and E. Valdinoci.

The aim of this seminar is to present some results about the isoperimetric problem in Carnot-Carathéodory spaces connected with the Heisenberg geometry. The Heisenberg group is the framework of an open problem about the shape of isoperimetric sets, known as Pansu’s conjecture. We start by studying the isoperimetric problem in Grushin spaces and Heisenberg type groups, under a symmetry assumption that depends on the dimension. We emphasize a relation between the perimeter in these two types of structure. We conclude by presenting some recent results about constant mean curvature surfaces (hence about isoperimetric sets) in the Riemannian Heisenberg group, focusing our attention on the subriemannian limit.

In this talk we analyze the local solvability property of a class of degenerate second order partial differential operators with smooth and non-smooth coefficients. The class under consideration exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. In particular we shall show the local solvability property of the class in the neighborhood of a set where the principal symbol of the operator can possibly change sign (which is a property that can negatively affect the local solvability of the operator).

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

It is well known that real regular bounded KP (n-k,k)-line solitons are associated to soliton data in the totally non-negative part of the Grassmannian Gr(k,n) and that, in principle, they may be obtained in a certain limit from regular real quasi--periodic KP solutions. The latter class of KP solutions correspond to algebraic geometric data a la Krichever on regular M-curves according to a theorem by Dubrovin-Natanzon. In this talk I shall present some new results recently obtained in collaboration with P.G. Grinevich (LITP-RAS and Moscow State University). The purpose of our research is the connection of such two areas of mathematics using the real finite gap theory of the KP equation. I shall explain how we associate to any KP soliton data in the real totally nonnegative part of Gr(k,n) the rational degeneration of an M-curve of genus g=k(n-k) and the effective KP divisor.

Necessaria iscrizione a finanza@dm.unibo.it

One of the most classical results in Hodge theory is Griffiths' description of the Hodge filtration of a smooth projective hypersurface in terms of a very explicit polynomial algebra, the so-called Jacobian ring. This turns to be extremely useful in solving Torelli-type problems, amongst others. Griffiths' result has been generalised to the smooth projective complete intersection case by Dimca et al., but not much other progress has been made so far. In this talk we present two different generalisations of Griffiths' theory. First we show how to attach to a smooth projective variety (with no hypotheses on the codimension) a graded module that controls (part of) its Hodge theory and deformation theory (joint work with Carmelo Di Natale/Domenico Fiorenza). Then we analyze the case of smooth hypersurfaces in Grassmannians, and show how to construct an explicit analogue of the Jacobian ring in this case.

Dopo aver introdotto e descritto le principali caratteristiche del mapping class group di una superficie, si indagherà il problema dell'esistenza di rappresentazioni lineari fedeli per tali gruppi. Si presenteranno i risultati noti in quest'ambito, con particolare riguardo alla dimostrazione di linearità per i mapping class group del disco puntato (i.e. i gruppi treccia) e si discuteranno le possibili tecniche per affrontare i casi aperti.

Let $\Omega$ be a domain in ${\mathbb{R}^N$. A density with the mean value property for non-negative harmonic functions in $\Omega$ is a positive l.s.c. function $w$ such that, for a suitable $ x_0 \in \Omega $, $$ u(x0) = \frac{1}{w(Ω)} \nt_{\Omega} u(y)w(y)dy $$ for every non-negative harmonic function $u$ in $\Omega$. In this case we say that $(\Omega,w,x_0)$ is a $\Delta$-triple. Existence of $\Delta$-triples on every sufficently smooth domain has been proved in 1994-1995, by Hansen and Netuka, and by Aikawa. Very recently, we have given positive answers to the following inverse problem: “Let $ (\Omega,w,x_0)$ and $(D,w',x_0)$ be $\Delta$-triples such that $\frac{w }{w(\Omega)= \frac {w'}{w'(D)} in $D ∩Ω$. Then is it true that $ \Omega = D$?” Our result contains, as particular cases, several classical potential theoretical characterizations of the Euclidean balls. Densities with the mean value property for solutions to wide classes of Picone’s elliptic-parabolic PDEs have appeared in literature since the 1954 pioneering work by B.Pini on the mean value property for caloric functions. In this talk we present an abstract inverse problem Theorem allowing to extend the previously recalled result on the $ \Delta$-triples to elliptic, parabolic and sub-elliptic PDEs. The results have been obtained in collaboration with Giovanni Cupini (Universita' di Bologna).

We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on (−∞;∞). Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too. Keywords: Inverse problem; First-Order problem, Second-Order problem, c0−semigroup, Periodic Solution. Joint work with: Mohammed AL Horani; Mauro Fabrizio; Hiroki Tanabe

Along the last years the technological advancements have been fundamental to improve the recording capability from brain areas and neural populations. For example multi-site recordings can be achieved from thousands of channels (sites) with a good spatial and temporal resolution yielding a good description of the underlying network dynamics. Given that, the brain operates on a single trial basis such recordings are becoming important to understand the neural code. As a first step, multi-site recordings allow to quantify the information flow in the network. The anatomical wiring (i.e. Structural Connectivity, SC) clearly plays a fundamental role to understand how cells communicate among them but it is often not well known neither it can by itself explain the overall network activity. Multi-site recordings can be used to infer statistical dependencies (i.e. Functional Connections, FC) among the recorded units and to track the information flow in the network. On the other hand the Effective Connectivity (EC) denotes the directed causal relationship between the recorded sites. Experimentally, the EC is typically estimated by stimulating one cell and studying the effects on the connected elements. Alternatively the EC can also be studied by using a causal mathematical model between the recorded units data. Importantly, multi-site recordings raise some limitations that need to be evaluated carefully before any further analysis. First, the experimental sessions are often limited in time. Second, the high dimensional data sets involve a set of numerical and mathematical problems that would be hard to face even with long enough recording sessions. These issues are common to different fields and have been coined as “curse of dimensionality”. In order to capture nonlinear interactions between even short and noisy time series, we consider an event- based model. Then, we involve the physiological basis of the signal, which is likely to be mainly nonlinear. Specifically, we suppose that we are able to observe the dynamical behaviours of individual components of a neuronal networks and that few of the components may be causally influencing each other. The variables could be time series from different parts of the brain. In order to introduce our method we have considered a simulated cerebellar granule cell network capturing nonlinear interactions between even short and noisy time series. Although the proposed EC algorithm cannot be applied straightforwardly to the experimental data, our preliminary results are quite promising. This is a joint work with G. Aletti, T. Nieus, and M. Moroni.

In this talk we discuss a rigidity result for a class of real hypersurfaces in C^2 with constant Levi curvature. Following old techniques due to Jellett, we consider the boundaries of starshaped domains which satisfy a suitable condition. We provide as application an Aleksandrov-type result for domains with circular symmetries. This is a joint work with V. Martino.

Nel seminario presenteremo un risultato di minimalità locale per un’energia ottenuta come limite del modello di Ohta-Kawasaki. Utilizzando tale risultato mostreremo che le configurazioni tridimensionali periodiche, strettamente stabili per il funzionale dell’area, sono esponenzialmente stabili sia per il flusso non locale di Mullins-Sekerka che per quello di Hele-Shaw.

Enunciata negli anni '50 del secolo scorso per una varietà compatta di Kaehler, la congettura di Calabi è stata dimostrata circa venti anni dopo da Shing-Tung Yau, il quale ha costruito metriche di Kaehler Ricci piatte su varietà compatte con fibrato canonico banale. Tali varietà prendono il nome di varietà di Calabi-Yau e in dimensione complessa uno o due sono tutte diffeomorfe. Al contrario, in dimensione tre non è nemmeno noto se il valore assoluto della loro caratteristica di Eulero è limitato. Se una soluzione a questo problema sembra ancora molto difficile, ha senso porsi la stessa domanda per le varietà log Calabi-Yau. Una volta ricordata la loro definizione nel corso del seminario, mostreremo una costruzione, realizzata in collaborazione con il dott. Filippo F. Favale, di una famiglia numerabile di varietà log Calabi-Yau, per cui l'insieme delle rispettive caratteristiche di Eulero è illimitato inferiormente.

Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means. Joint work with Amalia Culiuc and Yumeng Ou and partly with Jose Manuel Conde-Alonso, Yen Do and Gennady Uraltsev.