Archivio 2018

TBA

Migratory fluxes of humans and of insects of various species have favoured the spreading of diseases world-wise. It is important to stress that those epidemics can have strong social and economical impacts if not seriously controlled. Only in 2010 in Brazil, one million infected individual of which 80,000 where hospitalised. I shall present the SIR-Network model, introduced in Stolerman et al (2015), and revisite the SIR model with birth and death terms and time-varying infectivity parameter β(t). In the particular case of a sinusoidal parameter, we show that the average Basic Reproduction Number Ro, already introduced in Bacaer et al. (2006) is not the only relevant parameter and we emphasise the role played by the initial phase, the amplitude and the period. For a quite general slowly varying β(t) (not necessarily periodic) infectivity parameter all the trajectories of the system are proven to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given and comparison with real data from Dengue epidemics in Rio de Janeiro allow us to estimate the infectivity rate and make predictions about what are the periods more at risk of infection.

Emma Castelnuovo ha lasciato un metodo molto efficace di insegnamento della Matematica, mostrando in modo dinamico proprietà e armonie nascoste, in modo diretto e semplice. E' prezioso lo studio dei tabelloni preparati dalle classi di Scuola Media di Emma Castelnuovo per due celebri Esposizioni di Matematica, a Roma negli anni '70. E' istruttivo ricostruire il percorso per lo studio della cicloide, particolarmente esemplare.

Random walks in random sceneries (RWRS) have been first studied by Borodin, Kesten and Spitzer, and Bolthausen. We will present the historical results, as well as the most recent ones completing and detailing the former. We will also be interested in a model introduced by Matheron and de Marsily to describe the trajectory of a particle in an inhomogeneous stratified medium, which consists in a type of random walk in Z^2 with random orientation of the horizontal lines. We will see how this model is related to RWRS. We will also present a natural extension of RWRS: the U-statistics indexed by random walks. Finally, we will present a model of a one dimensional random walk on the line with randomly located roundabouts. These results have been obtained in collaboration with Fabienne Castell, Nadine Guillotin-Plantard and Bruno Schapira (RWRS and Matheron-de Marsily's model); Brice Franke and Martin Wendler (U-statistics indexed by random walks); Alessandra Bianchi and Marco Lenci (random roundabouts).

In 1991 Bendersky and Gitler give a spectral sequence converging to the cohomology of the configuration space MΓ depending on a graph Γ. Later, Baranovsky and Sazdanovi´c show that the E1 page of this spectral sequence is a graph cohomology complex. We first compare this complex with the Kontsevich type of graph complex and then with the differential graded algebra given by Kriz and Totaro as model for the rational homotopy type of the configuration space. In particular we show that there is a quasi equivalence between the dual of the Baranovsky and Sazdanovi´c’s graph complex and the dg algebra (as complex) given by Kriz and Totaro.

In 1991 Bendersky and Gitler give a spectral sequence converging to the cohomology of the configuration space MΓ depending on a graph Γ. Later, Baranovsky and Sazdanovi´c show that the E1 page of this spectral sequence is a graph cohomology complex. We first compare this complex with the Kontsevich type of graph complex and then with the differential graded algebra given by Kriz and Totaro as model for the rational homotopy type of the configuration space. In particular we show that there is a quasi equivalence between the dual of the Baranovsky and Sazdanovi´c’s graph complex and the dg algebra (as complex) given by Kriz and Totaro.

In this talk I will explain a connection between Commutative Algebra and Linear and Integer Programming. In the first part, it is explained how one can translate the problem of bounding the index of stability of the Castelnuovo-Mumford regularities of the integral closures of powers of a monomial ideal into an Integer Linear Programming. The second part is devoted to the asymptotic behavior of Linear and Integer Programming with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integral coefficients depending linearly on $n$. It is shown that the optima of such Linear Programming problems are a linear function of $n$, while the optima of the corresponding Integer Programming problems are a quasi-linear function of $n$, provided $n\gg 0$. In the last part I give bounds on the indices of stability of the Castelnuovo-Mumford regularities of the integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.

We briefly discuss the common physical features of peculiar phenomena of quantum ultra-cold matter, as, e.g., Bose-Einstein condensation, superfluidity and superconductivity, as well as the mathematics of the corresponding models used in physics. We stress the role of effective theories and comment on their mathematical derivation. We then focus on the specific example of the Ginzburg-Landau theory of superconductivity and the phenomenon of the response of a superconducting material to an external magnetic field. We review some recent results on the surface superconductivity regime and in particular describe how the boundary curvature affects the energy and distribution of Cooper pairs in the material.

Le geometrie di Cartan sono una generalizzazione della geometria secondo il programma di Erlangen di Klein. Come `e noto, nell’ambito della geometria differenziale, le strutture geometriche che rientrano nello schema di Klein sono le variet`a omogenee G/H. La general- izzazione proposta da Cartan nella prima meta` del secolo scorso consiste euristicamente nel considerare strutture geometriche su variet`a che sono deformazioni di modelli omogenei as- segnati a priori. In questo approccio emerge un nuovo concetto di curvatura, piu` generale di quello classico di Gauss e Riemann, in base al quale lo spazio di Klein G/H su cui una geome- tria di Cartan `e modellata `e piatto per definizione. La curvatura misura quanto la struttura geometrica in considerazione devia dal suo modello. In questo corso ci poniamo l’obbiettivo di introdurre i fondamenti delle geometrie di Cartan quali il concetto di assoluto parallelismo e la curvatura normale

Le geometrie di Cartan sono una generalizzazione della geometria secondo il programma di Erlangen di Klein. Come `e noto, nell’ambito della geometria differenziale, le strutture geometriche che rientrano nello schema di Klein sono le variet`a omogenee G/H. La general- izzazione proposta da Cartan nella prima meta` del secolo scorso consiste euristicamente nel considerare strutture geometriche su variet`a che sono deformazioni di modelli omogenei as- segnati a priori. In questo approccio emerge un nuovo concetto di curvatura, piu` generale di quello classico di Gauss e Riemann, in base al quale lo spazio di Klein G/H su cui una geome- tria di Cartan `e modellata `e piatto per definizione. La curvatura misura quanto la struttura geometrica in considerazione devia dal suo modello. In questo corso ci poniamo l’obbiettivo di introdurre i fondamenti delle geometrie di Cartan quali il concetto di assoluto parallelismo e la curvatura normale

In questo seminario presenterò brevemente i principali modelli utilizzati nell'indagine teorica delle reti neurali, ovvero il modello di Hopfield e la macchina di Boltzmann. In seguito, ne darò una descrizione matematica riassumendo i risultati fondamentali e privilegiando una prospettiva meccanico-statistica. In questo contesto darò particolare risalto alle tecniche di interpolazione introdotte da Guerra e al loro successo nell’esplorazione dei sistemi disordinati (tra cui vetri di spin e reti neurali). Infine, presenterò alcuni risultati recenti (in collaborazione con A. Barra, F. Guerra, D. Tantari) in cui consideriamo una generalizzazione a molti partiti del modello di Hopfield. Va sottolineato che questo sistema contiene due degli ingredienti principali delle moderne architetture di reti neurali profonde: le interazioni hebbiane per memorizzare informazioni e più strati per codificarne i diversi livelli di correlazione. Il modello è completamente risolvibile nel regime di basso carico attraverso un’opportuna estensione della tecnica Hamilton-Jacobi, nonostante l'Hamiltoniana possa essere una forma quadratica nei parametri d’ordine con segno non-definito.

We will discuss some recent developments in the mathematical characterization of glassy transitions. In particular we will focuse on the role of Almeida Thouless line in the replica symmetry breaking solution of the Sherringhton-Kirkpatrick model.

Le geometrie di Cartan sono una generalizzazione della geometria secondo il programma di Erlangen di Klein. Come `e noto, nell’ambito della geometria differenziale, le strutture geometriche che rientrano nello schema di Klein sono le variet`a omogenee G/H. La general- izzazione proposta da Cartan nella prima meta` del secolo scorso consiste euristicamente nel considerare strutture geometriche su variet`a che sono deformazioni di modelli omogenei as- segnati a priori. In questo approccio emerge un nuovo concetto di curvatura, piu` generale di quello classico di Gauss e Riemann, in base al quale lo spazio di Klein G/H su cui una geome- tria di Cartan `e modellata `e piatto per definizione. La curvatura misura quanto la struttura geometrica in considerazione devia dal suo modello. In questo corso ci poniamo l’obbiettivo di introdurre i fondamenti delle geometrie di Cartan quali il concetto di assoluto parallelismo e la curvatura normale.

The talk focuses on a stationary problem in a rarefied gas confined between two coaxial cylinders. The boundary values introduced generate a helicoidal motion of the fluid together with an heat transfer. The physical problem is described using the equations of Rational Extended Thermodynamics. The results are compared with the prediction of the corresponding Classical Thermodynamics model and significative differences are observed. The role of the helicoidal motion is also analyzed through a comparison with the cases of only rotation or only axial flows.

Optimization Programming Language (OPL) is a powerful, yet intuitive programming language that allows to solve Mixed Integer Linear Problems (MILP) via ILOG CPLEX - one of the most performing general-purpose MILP solvers available on the market. OPL allows to efficiently solve a wide variety of MILP involving thousands of variables and constraints within reasonable computing times. In this series of two seminars, we will show how some classical combinatorial optimization problems (e.g., Traveling Salesman Problem, Generalized Assignment Problem, etc) can be formulated and solved in OPL.

Optimization Programming Language (OPL) is a powerful, yet intuitive programming language that allows to solve Mixed Integer Linear Problems (MILP) via ILOG CPLEX - one of the most performing general-purpose MILP solvers available on the market. OPL allows to efficiently solve a wide variety of MILP involving thousands of variables and constraints within reasonable computing times. In this series of two seminars, we will show how some classical combinatorial optimization problems (e.g., Traveling Salesman Problem, Generalized Assignment Problem, etc) can be formulated and solved in OPL.

In 1882 Lord Rayleigh developed a model related to the study of the (lack of) equilibria of liquid droplets, provided of an electric charge. In the seminar we provide a precise variational description of such a model. We will then discuss about the existence of minimizers and their regularity. Eventually we will provide a full solution of the problem in the bidimensional setting. The talk is based on collaborative works with M. Goldman (Paris Diderot university), M. Novaga (Università di Pisa) and C. Muratov (New Jersey Institute of Technology)

Raffaella Simili : Donne e scienza: una storia da fare? Clara Silvia Roero: Il successo europeo delle "Instituzioni analitiche" di M.G. Agnesi Alberto Parmeggiani: La “versiera” di Maria Gaetana Agnesi Miriam Focaccia: Bologna nel Settecento: un paradiso per le donne

Seminario riservato al gruppo di ricerca su varietà di Fano e IHS.

Nell'ambito della rassegna Scienza al Cinema la dott. Tiziana Primori, laureata in matematica e amministratore delegato di Eataly World, commentera' il film "Il diritto di contare", parlando delle possibilità professionali offerte dalla laurea in matematica, in particolare alle donne.

Si tratta di un seminario nell'ambito del Workshop PLS "Incertezza, rischio e previsioni: il ruolo della scienza nel prendere decisioni " Un giudice che pronuncia una sentenza può commettere due tipi diversi di errore: 1) condannare un innocente; 2) assolvere un colpevole. Situazioni analoghe si presentano in molti campi: per esempio, un talent scout può scartare un futuro campione oppure può reclutare una giovane promessa che si rivelerà un brocco. Davanti a una proposta di matrimonio, si rischia di accettare un partner sbagliato oppure di declinare la persona giusta. Quando la soluzione di un problema si riduce a rispondere si/no, è cruciale saper distinguere fra i due tipi di errore e valutarne le loro conseguenze. Dopo una leggera scossa sismica, il Sindaco chiede alla Protezione Civile se bisogna ordinare l'evacuazione del paese: una risposta positiva rischia un falso allarme e una risposta negativa rischia una catatrofe. Il principale ruolo della scienza è aiutarci a non prendere decisioni sbagliate. Per questo tipo di problemi, è possibile soppesare in modo rigoroso il rischio di commettere questi due tipi di errore.

TBA

The notion of noncommutative principal bundle is not as well understood as that of noncommutative vector bundle. Main examples, like the noncommutative instanton bundle, are algebraically understood as Hopf-Galois extensions. This definition is particularly useful when the noncommutative base space is affine (the function algebra being given in terms of generators and relations). We relax it by presenting a sheaf theoretic approach that allows to consider noncommutative principal bundles over non affine base spaces; examples include projective spaces.

In noncommutative geometry Hopf-Galois extensions of algebras are regarded as principal bundles, with Hopf algebras playing the role of structure groups. This seminar will present results, obtained in collaboration with P. Aschieri and G. Landi, on the study of the group of gauge transformations in this algebraic framework.

Con metodo delle orbite quantico si intende una corrispondenza tra foglie simplettiche di una varietà di Poisson e *-rappresentazioni irriducibili della sua algebra quantizzante. Mostreremo come l'approccio alla quantizzazione di spazi omogenei di Poisson tramite la integrazione simplettica permetta di costruire esplicitamente tale corrispondenza e quali informazioni aggiuntive (di natura topologica) possa dare.

TBA

will discuss a double geometry formulation of the isotropic rigid rotator, as a dynamical system whose configuration space is usually chosen to be the group manifold of SU(2). I will generalize the carrier space of the dynamics in terms of the double group of SU(2) and discuss the emerging geometry. The construction is thus extended to principal chiral models as an instance of $1+1$ field theory with non-Abelian duality.

Starting with the classical duality between spaces and algebras of functions on these spaces, the idea in noncommutative geometry is to forget the commutativity of the algebras of functions and replace them by appropriate classes of noncommutative associative algebras. In this talk we present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang--Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical ``quaternionic torus” SU(2) x SU(2) in parallel with the action of the torus U(1) x U(1) on a ``complex” noncommutative torus.

Si considera un sistema di spin con interazione aleatoria di campo medio in condizioni di equilibrio termico multiscala, cioè quando l'equilibrio termodinamico viene raggiunto attraverso una successione di termalizzazioni. Il modello è stato introdotto in fisica teorica negli anni '90 quando ne sono state studiate le proprietà dinamiche di metastabilità e piccola produzione di entropia. Il seminario presenterà la soluzione esatta del modello mostrando come il limite termodinamico della pressione (funzionale generatore) per particella si riduca a un problema variazionale sullo spazio delle distribuzioni di probabilità sull’intervallo unitario. Si presentano le idee principali alla base della dimostrazione quali le stime dal basso (con metodi di positività per interpolazione) e dall’alto (con metodi di sincronizzazione).

Seminario riservato ai membri del progetto GNAMPA 2018 (Cupini)

A Lorentz gas consists of a set of non-interacting point particles that move along a certain type of trajectory (for example, straight or circular trajectories) and a set of fixed obstacles in space with which particles collide specularly. The obstacles can be distributed in different ways. Probably the simplest nontrivial case is where the trajectories are straight and the obstacles are 2D disks distributed periodically. This system is equivalent to the Sinai billiard, which was proved to be chaotic. If we consider circular trajectories instead of straight ones on this periodic Lorentz gas, some trajectories become non-chaotic. A simple way to find if there are non-chaotic trajectories is with a Poincaré map. A chaotic trajectory will fill most of the phase space, except maybe for some "islands" where drifting trajectories appear, which move effectively in only one direction with a constant velocity. Because the islands have a positive measure, the diffusion behavior is ballistic, i.e. <x^2(t)> ~ t^2, where <x^2(t)> is the mean square displacement, and t the time. If the islands disappear, then the diffusion becomes normal, i.e. <x^2(t)> ~ t. When the obstacles have a Poisson distribution, the diffusion is always normal if there is no localization of particles. If the density of obstacles is high enough, then the maximum diffusion coefficient is for a radius of trajectories different from infinity. Quasiperiodic arrays of obstacles have an angular symmetry as the periodic arrays, but there is no longer a translational symmetry as with random distributions. In this talk, we will first summarize some of results for the periodic and random arrays of obstacles with circular trajectories, and then we will show numerical computations of the diffusion coefficient for a Lorentz gas with quasiperiodic array of obstacles and circular trajectories. The obtained results are unexpected for high densities of obstacles, where we find more than one local maximum in the diffusion coefficient. We also studied a Poincaré map, finding in some cases islands similar to the periodic case. However, those islands do not correspond to drifting trajectories. Contrary to the periodic case, all trajectories produce normal diffusion.

The marginal stability of the oscillations exhibited by the Lotka-Volterra rate equations has been blamed for the apparent failure of stochastic predator-prey dynamics to replicate a stationary regime of stable oscillations; either of the prey or predator populations typically go extinct after a finite time. I will show that a simple modification of the master equation which adequately accounts for the conservation law underlying the rate equations is sufficient to prevent extinction. In the large system-size limit, the observed regime of oscillations is distributed according to the canonical equilibrium distribution of the thermostatted rate equations. Applications to standard time series arising in population ecology will be discussed.

The notion of noncommutative principal bundle is not as well understood as that of noncommutative vector bundle. Main examples, like the noncommutative instanton bundle, are algebraically understood as Hopf-Galois extensions. This definition is particularly useful when the noncommutative base space is affine (the function algebra being given in terms of generators and relations). We relax it by presenting a sheaf theoretic approach that allows to consider noncommutative principal bundles over non affine base spaces; examples include projective spaces.

We show how to recover quantum mechanics by using parallel section of the contact-Fedosov connection

We introduce the notion of Fedosov connection on Contact manifolds

The rogue (anomalous) waves in nature are actively studied now. The generation of such giant waves is an essentially non-linear phenomena, and the focusing Nonlinear Schrodinger equation is treated as one of the basic mathematical models. In the spatially periodic setting anomalous waves correspond to special solutions associated with almost degenerate spectral curves. In this seminar results obtained in collaboration with P.M. Santini will be presented. We show that these special theta-functional solutions admits a simple explicit approximation by elementary functions. P.G. Grinevich was supported by RSF grant No 18-11-00316.

Il ragionamento matematico deduttivo, com’è percepito attraverso le pubblicazioni ufficiali e, più in generale, la “comunicazione” del messaggio logico-matematico, non sempre riflette tutte le caratteristiche del pensiero del matematico in corso d’opera (non comunicato) e gli approcci metodologici che portano alla formulazione finale dei risultati. “Affinare la mira”, valutare il rischio dell’imbarcarsi in un tentativo, la convenienza nell’ottimizzare, nel generalizzare o meno l’ipotesi di un teorema al massimo, o viceversa specificarne una tesi, sono fasi inevitabili della storia della lotta ai problemi che non appaiono di norma nel prodotto finito. Riteniamo che l’abitudine alla categorizzazione, all’inquadramento formale di problemi complessi e la lungimiranza di pensiero che la disciplina scientifica fornisce, possano rendere menti esercitate a tale scopo anche idonee a sistematizzare altre teorie, prestando un contributo unico, per esempio, alle scienze umane e sociali, non solo nell’analisi quantitativa di dati ma nella propria concezione e ideazione della modellazione, nell’elaborazione e raffinamento della metodologia, nell’attribuzione di significato e interpretazione qualitativa dei risultati, oltre che nascondere un grande potenziale di valutazione decisionale. La ricerca interdisciplinare si costruisce con lungo sforzo dal contatto di mondi destinati, a volte per molto tempo, a rimanere fisicamente incomunicanti. Ma quando la distanza è rilevante anche nei contenuti, essere pionieri equivale a fare un salto nel vuoto e spostarsi interamente e radicalmente. Seguendo le orme della propria esperienza personale, l’autore, un matematico di prima formazione, si propone di presentare una serie di studi di linguistica, inerenti al suo attuale ambito di ricerca, l’intonazione, che hanno visto la concretizzazione di alcune sue contribuzioni speciali, sebbene non strettamente “matematici” nei contenuti. Presenteremo un problema di analisi del cambiamento linguistico nel tempo, un problema di rianalisi e rigrammaticalizzazione intonativa, un progetto di recupero di informazione linguistica data per morta attraverso l’intonazione e infine uno studio statistico che mostra la stabilizzazione di una “logica ritmica” specifica in alcune varietà linguistiche. La speranza è che il contatto con il pubblico, oltre a fruire di un diversivo ed aprire gli orizzonti conoscitivi, possa contribuire a un dibattito proficuo su possibili applicazioni di modelli matematici preesistenti a supporto di problemi nuovi.

We propose new modules in programming environment MATHEMATICA for the Black-Scholes model taking into account the sensitivity coefficients and considering discrete dividends taxed at rate tax. Then Black Scholes model with Leland corrections is presented in the case when we have discrete dividends and discrete taxes. The modules presented in this talk are component of web-based application, realized in the program environmental with central mathematical kernel and in some sense realizes the problem proposed above, and the build software instruments can be used for research investigations, as well as for training.

This talk deals with the regularity properties (including propagation and interaction of nonlinear waves) of the solutions of the Cauchy problem to 2D semilinear wave equation with the removable singularities of the solutions of fully nonlinear hyperbolic systems arising in the mechanics of compressible fluids with constant entropy, and with the regularizing properties of the multidimensional wave equation with dissipative term. We shall first discuss the machinery of the pseudodifferential, respectively paradifferential operators which is applied. More precisely, "radially smooth" initial data having singularities on a "massive" set of angles in the plane, including the Cantor continuum set, yield singularities propagating as in the linear case. There is a big difference between the 2D case and the multidimensional case (3D) when the interaction of several (for example four) characteristic hyper-planes could produce singularities on a dense subset of the compliment of the light cone of the future located over the origin. A result of Bony for the triple interaction of progressing linear waves in the 2D case is commented too as then new effect appears: new born wave propagating along the cone of the future with vertex at the origin. We assume that the first variation of the nonlinear system under consideration is linear, symmetric and positive one in the sense of Friedrichs. A microlocal version of the Moser's condition on the existence of global solutions on the torus of the same system enables us to prove the nonexistence of isolated singularities at each characteristic point of the main symbol of the first variation. For symmetric quasilinear hyperbolic systems we study the propagation of regularity. As usual, the strength of the singularities is measured both in Sobolev space and microlocalized Sobolev spaces. An example from fluid mechanics will be presented in order to illustrate our results.

The Cauchy-Riemann geometry (CR in short) is modelled on the three sphere and the group of its biholomorphic transformations. In 2008 Falbel makes use of ideal triangulations to shows that the figure eight knot complement admits a (branched) CR structure. This three manifold belongs to a larger class of important three manifolds that are fiber bundles over the circle, with fiber space the once-punctured torus. In this talk we introduce the audience to these manifolds and show that almost every once-punctured torus bundle admits a (branched) CR structure.

I will review some old and new results concerning existence, multiplicity and asymptotic behaviour of solutions to the classical Lane-Emden equation on a planar domain when the exponent of the non-linearity is large.

Il seminario è riservato ai partecipanti al progetto

We describe recent results obtained in collaboration with Daniela De Silva and Fausto Ferrari on two-phase free bounday problems in presence of distributed sources. The focus will be mainly on higher regularity and related open problems.

Existence and uniqueness of a mild solution to the dynamic programming equation corresponding to optimal control associated with the Heston stochastic volatility control is studied. The approach is based on nonlinear semigroup theory in the space $L^{1}$.

One of B. Pini's most quoted papers deals with his (and Hadamard's) contemporary derivation of a theorem analogue to a theorem by Harnack on harmonic functions. In the talk I will focus on Harnack's contributions to potential theory and his solution to Dirichlet problem as presented in his 1887 book as well as to Kellog's 1929 book on potential theory to which Pini himself referred in his paper

Miguel A. Herrero Real Academia de Ciencias and Universidad Complutense, Madrid, Spain. Consider a society consisting of a large number of individuals (about 10 13, several orders of magnitude larger than today´s world population) organized in hundreds of different social groups (current count of UN countries being about 200) and employed in many different jobs. Assume further a huge immigration rate (about 1011 new arrivals per day) coupled to a high mortality rate, so as to balance the previous figure. Such society enjoys full employment, and wealth is shared by all individuals. Order is maintained by an extremely efficient police force that keeps threatening aliens at bay. What kind of government could possibly be up to the task of ruling such society? The answer is simple: none. Anarchy prevails in the society we have summarily described, which is not located at the remote island of Utopia: it is just you (or me) and no central organ of control is in charge of its operation. Its efficient performance is an emergent property, resulting from individual decisions of its cells, and is not centrally regulated from any commanding headquarters. We shall describe in this lecture some features of one of the cornerstones of this complex structure, the immune system, and will shortly remark on other cell regulation properties which show a distinct emergent character as well.

In Hilbert or Banach spaces $X$ endowed with a good probability measure $\mu$ there are a few "natural" definitions of Sobolev spaces and of spaces of bounded variation functions. The available theory deals mainly with Gaussian measures and Sobolev and BV functions defined in the whole $X$, while the study and Sobolev and BV spaces in domains, and/or with respect to non Gaussian measures, is largely to be developed. As in finite dimension, Sobolev and BV functions are tools for the study of different problems, in particular for PDEs with infinitely many variables, arising in mathematical physics in the modeling of systems with an infinite number of degrees of freedom, and in stochastic PDEs through Kolmogorov equations. In this talk I will describe some of the main features and open problems concerning such function spaces.

Motivated by an application to stratified turbulent flows in oceanography, I shall discuss some singular solutions of nonlinear PDE's of evolutionary type, in particular discontinuous and Radon measure valued solutions. We focus on different regularizations of backward parabolic equations, first order scalar conservation laws, and, if time permits, a toy problem for a Hamilton-Jacobi equation. Most of this lecture is based on collaborations with L. Giacomelli, F. Smarrazzo, A. Terracina and A. Tesei.

I will give an overview of some results concerning the validity of sign propagation property u ≤ 0 on ∂Ω, F(x,u,Du,D2u) ≥ 0 in Ω implies u ≤ 0 in Ω in an unbounded domain Ω satisfying either measure-type or geometric conditions related to the directions of ellipticity of the (possibly) degenerate fully nonlinear mapping F.

Il seminario è riservato ai partecipanti al progetto

We describe the pioneering work of Bruno Pini towards the modern Potential Analysis of linear second order parabolic Partial Differential Equations. We mainly focus on the caloric Harnack Inequality discovered by Bruno Pini in 1954, jointly, and independently, with Jacques Hadamard. Pini made of this inequality the crucial tools in his construction of a Wiener-type solution to the ''Dirichlet problem'' for the Heat equation. To this end he also introduced an average operator on the level set of the Heat kernel, characterizing caloric and sub-caloric functions, in analogy with the classical Gauss-Koebe, Blaschke-Privaloff and Saks Theorems for harmonic and sub-harmonic functions. Pini also established, and used, the notion of caloric capacity to study the boundary behavior of his Wiener-type solution to the first boundary value problem for the Heat equation.

The mathematical analysis and numerical simulation of acoustic and electromagnetic wave scattering by planar screens is a classical topic. The standard technique involves reformulating the problem as a boundary integral equation on the screen, which can be solved numerically using a boundary element method. Theory and computation are both well-developed for the case where the screen is an open subset of the plane with smooth (e.g. Lipschitz or smoother) boundary. In this talk I will explore the case where the screen is an arbitrary subset of the plane; in particular, the screen could have fractal boundary, or itself be a fractal. Such problems are of interest in the study of fractal antennas in electrical engineering, light scattering by snowflakes/ice crystals in atmospheric physics, and in certain diffraction problems in laser optics. The roughness of the screen presents challenging questions concerning how boundary conditions should be enforced, and the appropriate function space setting. But progress is possible and there is interesting behaviour to be discovered: for example, a sound-soft screen with zero area (planar measure zero) can scatter waves provided the fractal dimension of the set is large enough. This research has also motivated investigations into the properties of fractional Sobolev spaces (the classical Bessel potential spaces) on non-Lipschitz domains. Accurate computations are also challenging because of the need to adapt the mesh to the fine structure of the fractal. As well as presenting numerical results, I will outline some outstanding open questions. This is joint work with Simon Chandler-Wilde (Reading) and David Hewett (UCL).

** IL SEMINARIO SI TERRA' ALLE ORE 16 ** I will discuss a certain infinite product of random, positive 2×2 matrices appearing in the exact solution of some 1 and 1+1 dimensional disordered models in statistical mechanics, which depends on a deterministic real parameter ε and a random real parameter with distribution μ. For a large class of μ, we prove a prediction by B. Derrida and H. J. Hillhorst (1983) that the leading Lyapunov exponent behaves like C ε^2α in the limit ε→0, where α ∈ (0,1) is determined by μ. The proof is made possible by a contractivity argument which makes it possible to control the error involved in using an approximate stationary distribution similar to the original proposal, along with some refinements in the estimates obtained using that distribution. A limiting procedure gives a continuum process whose leading Lyapunov estimate admits an exact formula, which also allows us to reformulate part of the argument by McCoy and Wu for the presence of an essential singularity in the free energy of the two-dimensional Ising model with columnar disorder in a form which is closely related to the results obtained for the random matrix product, but which does not yet provide a proof.

I will present several examples of derived and L-equivalent pairs of non-birational Calabi-Yau type manifolds aiming at understanding the relation between these two notions of equivalence. We will try to find a common geometric framework governing these examples. The talk will include joint work with Marco Rampazzo and with Grzegorz Kapustka and Riccardo Moschetti.

We will study the Morin problem and present a method of classification of finite complete families of incident planes in ℙ5 as a result we prove that there is exactly one, up to Aut(P^5), configuration of maximal cardinality 20 and a unique one parameter family containing all the configurations of 19 planes. The method is to study projective models of appropriated moduli spaces of twisted sheaves on K3 surfaces. This is a joint work with A. Verra.

We consider the Cauchy problem for hyperbolic operators with characteristics of variable multiplicities r ≤ 3 assuming that the fundamental matrix of the principal sym- bol has two non-vanishing real eigenvalues. The last condition is necessary for the Cauchy problem to be well posed for every choice of lower order terms. The operators with this pro- perty are called strongly hyperbolic and it was conjectured that every effectively hyperbolic operator is strongly hyperbolic. In this talk we present a survey of the results in the case r = 3. The proofs are based on the energy estimates with a big loss of derivatives depending of lower order terms. This is a joint work with T. Nishitani.

In una vulgata abbastanza comune, si sostiene una derivazione diretta dell'informatica dai lavori teorici di Turing ed altri sulla calcolabilità; e si ritiene che vi sia stata un'influenza significativa della logica matematica sul progetto dei linguaggi di programmazione ad alto livello negli anni '60. Argomenterò come si tratti di una ricostruzione per molti versi fallace: pur con le dovute eccezioni degli "early giants", la "teoria matematica della computazione" per i linguaggi di programmazione è una creazione degli anni '60, che contribuisce alla costituzione dell'"informatica" come disciplina scientifica accademica. Esemplificherò alcuni dei rapporti tra logica matematica e progetto dei linguaggi di programmazione attraverso la nozione di "tipo di dato", che sembra avere un ovvio corrispettivo nel concetto di "tipo" della logica matematica.

Questa presentazione ha l’obiettivo di presentare un lavoro di ricerca a metà strada tra la logica e l’algebra, due aree della matematica dove trovare spiragli per sfornare nuovi risultati è una sifda ardua, lontano dall’essere una pratica di tutti i giorni. Più precisamente esploreremo il concetto di “definibilità del primo ordine”, cioè dell’esistenza di una formula del primo ordine capace di descrivere un sottinsieme o una sottostruttura di una struttura data in un linguaggio fissato (nel nostro caso la teoria degli anelli) come l’insieme degli elementi per cui la formula è vera. È risaputo, in questo senso, che l’anello degli interi è definibile in Q (Robinson, 1949), non è definibile in R (Poonen, 2008) ma è definibile in R[x] (Robinson, 1951, Shlapentokh, 1990) quando R è un dominio di integrità. Il nuovo risultato che presenteremo permette di definire gli interi razionali (immagine del morfismo unitale) dentro R[x], per tutti gli anelli R in una classe più ampia di quella dei domini, quella degli anelli ridotti e indecomponibili, estendendo così il risultato già noto. Una tecnica per definire gli interi, che sarà presentata, consiste nell’utilizzo strategico dell’elemento x per definire l’insieme delle sue potenze e da esse “estrarre gli esponenti” mediante un artificio logico. Ma il lavoro più grande e innovativo consisterà nell’eliminare la necessità di valersi di un simbolo specifico (costante del linguaggio) per nominare tale elemento, mediante una quantificazione su un insieme definibile di elementi con le sue stesse proprietà. La dimostrazione, che sarà un’opportunità per mostrare alcune tecniche di costruzione di formule nell’interfaccia tra logica ed algebra, sarà basata su tre risultati fondamentali, di natura algebrica e logica. Dal punto di vista algebrico, proveremo che nella classe di anelli considerata, x é un elemento irriducibile e i polinomi che sono costanti come funzioni sono anche costanti come polinomi. In logica, troveremo un modo di scrivere, nel primo ordine, che “due potenze di basi distinte hanno lo stesso esponente” e, data l’impossibilità di definire con una formula finita il concetto di potenza, definiremo un concetto molto simile (“potenza logica” o “multiplo puro”) che per una certa classe di elementi, sufficiente al nostro scopo, coinciderà col concetto di potenza.

Kinematical Lie groups are generalisations of the relativity groups and include the group of galilean transformations, as well as the isometries of Minkowski and (anti)de Sitter spacetimes. We will review the recent classification of kinematical Lie algebras in arbitrary dimension and the classification of the corresponding homogeneous spacetimes. Most of these spacetimes do not have invariant metrics, but rather Newton-Cartan and/or carrollian structures, which we will review.

In this seminar we discuss how by considering eye movements, and in particular the resulting sequence of gaze shifts, a stochastic process, a wide variety of tools become available for analyses and modelling beyond conventional statistical methods. We first give a brief, though critical, probabilistic tour of current computational models of eye movements and visual attention, and we lay down the basis for gaze shift pattern analysis. Then we discuss their links to the concepts of Markov Processes, the Wiener process and related random walks within the Gaussian framework of the Central Limit Theorem Eventually we will deliberately violate the fundamental assumptions of the Central Limit Theorem to elicit a larger perspective, rooted in statistical physics, for analysing and modelling eye movements in terms of anomalous, non-Gaussian, random walks and modern foraging theory.

We will discuss a couple of characterizations of spherical sectors inside cones. We will consider partially overdetermined problems in conical domains and constant mean curvature hypersurfaces with boundary attached to a smooth cone. For the case of convex cones, we will present respectively a Serrin-type and an Aleksandrov-type result. We will focus on two aspects: the role of the convexity of the cone, and some gluing assumptions for the intersection between the relative boundary of the domain and the cone. We will also show a rigidity result for constant mean curvature surfaces in starshaped sectors related to non-convex cones. This is a joint work with F. Pacella.

We present some recent results on the double bubble problem for the anisotropic perimeter Pα, α ≥ 0 associated with the Grushin plane. The problem consists in finding the best configurations of two regions in the plane enclosing given volumes, in order to minimize their total anisotropic perimeter. When $\alpha=0$, the Grushin plane is just the Euclidean one. If $\alpha\neq 0$, this is a Riemannian structure that degenerate to a sub-Riemannian one on an axis. We prove existence of minimizers and characterize them, in the case of two equal given volumes, and under the assumption that the interface between the bubbles lays on one axis. In particular, we characterize the angles between the bubbles, providing a nice relation with the regularity theory for (Riemannian) perimeter minimizers. In conclusion, in the case $\alpha=1$, minimal double bubbles with interface on the vertical axis have perimeter strictly greater then the ones having interface on the horizontal one: we interpret this fact in terms of isoperimetric sets. Joint work with G. Stefani (SNS, Pisa).

The concept of number has traditionally been considered as a prototypical instance of abstract(ed) knowledge. It denotes the size of any arbitrary set of objects, thus seemingly preventing systematic correlations with sensory or motor features. Yet, numerosity does co-vary with physical parameters in perception and action. Importantly, number symbols preserve this association. In this presentation, I describe how number processing obligatorily activates sensory and motor features: both sensory and motor processing are improved in left vs. right space following the presentation of small vs. large numbers. These links are bi-directional and suffice to identify numbers as embodied concepts. Moreover, these space-magnitude associations influence mental arithmetic and everyday quantitative reasoning (cf. Fischer & Shaki, 2014). Implications for research and theorizing will be discussed. Reference: Fischer, M. H. & Shaki, S. (2014). Spatial Associations in Numerical Cognition: From single digits to arithmetic. Quarterly Journal of Experimental Psychology, 67(8), 1461-1483.

In this talk, I will discuss several generalized versions, dependent on different boundary conditions, of the classical Gaffney-Friedrichs inequality for differential forms in Heisenberg groups. In the first part I will consider horizontal differential forms and the horizontal differential. In the second part, I will illustrate the counterpart of these results in the context of the Rumin’s complex. The results presented in this talk are obtained jointly with B. Franchi and E. Serra of the University of Bologna.

We recall the notion of nonlocal minimal surfaces and we discuss their qualitative and quantitative interior and boundary behavior. In particular, we present some optimal examples in which the surfaces stick at the boundary. This phenomenon is purely nonlocal, since classical minimal surfaces do not stick at the boundary of convex domains.

In this talk, we show some estimates of the Poincaré constant on a convex set in terms of geometric quantities: inradius, diameter, perimeter, Cheeger constant and so on. We deal with the case of Poincaré inequalities for functions vanishing at the boundary, as well as for zero-mean functions.

We study a geometric flow driven by the fractional mean curvature (FMC). The notion of fractional mean curvature arises naturally when performing the first variation of the fractional perimeter functional (introduced by Caffarelli, Roquejoffre, and Savin). More precisely, we show the existence of surfaces which develope neckpinch singularities, in any dimension n ≥ 2. Interestingly, in dimension n=2 our result gives a counterexample to Greyson Theorem for the classical mean curvature flow. The result has been obtained in collaboration with C. Sinestrari and E. Valdinoci.

In this talk I will present some results concerning a localized Steiner-type formula for tubular neighborhoods in the Carnot-Carathéodory metric and a possible definition of horizontal Gauss curvature for smooth surfaces in the first Heisenberg group. The talk is based on a joint works with Z.M. Balogh, F. Ferrari, B. Franchi, J.T. Tyson and K. Wildrick.

The aim of this talk is to provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived as well as new characterizations of rotationally symmetric solutions and domains. The talk is based on a joint work with L. Mazzieri and M. Fogagnolo.

I will consider the problem of prescribing the Gaussian curvature (under pointwise conformal change of the metric) on surfaces with conical singularities. This question has been first raised by Troyanov [TAMS,1991] and it is a generalization of the Kazdan-Warner problem for regular surfaces, known as the Nirenberg problem on the sphere. From the analytical point of view, this amounts to solve a singular Liouville-type equation on the surface. Initially, in the supercritical regime, only the case of positive prescribed Gaussian curvature has been attacked. In this talk I will present some new results (obtained in collaboration with T. D`Aprile, I. Ianni, S. Kallel, R. López-Soriano and D. Ruiz) concerning the sign-changing case.

We will present here some different characterizations of Besov and Sobolev spaces on Carnot groups. This characterizations involve the use of the fractional heat kernel and the fractional Poisson kernel. As an application of the previous results, we prove diverse commutator results involving the fractional sub-Laplacian.

We propose the use of image symmetries, in the sense of equivalences under image transformations, as priors for learning symmetry adapted representations, i.e., representations that are invariant to these transformations. We show how Deep Convolutional Neural Networks implement such representations for the translation group and propose a new regularization term to extend the learning to other groups. Further, from a computational neuroscience point of view, we show in which sense the ventral stream architecture can be mapped to a class of Deep Convolutional Neural Networks.

Sia X una varietà proiettiva, complessa, liscia e Fano di dimensione arbitraria n. Una fibrazione conica f : X -> Y è una contrazione di tipo fibrato con fibre di dimensione uno. Denotiamo con N_1(X) lo spazio vettoriale reale degli 1-cicli a coefficienti in R, modulo equivalenza numerica, la cui dimensione è il numero di Picard \rho(X). Dato un divisore primo D in X, l'inclusione i : D -> X induce il pushforward di 1-cicli i_* : N_1(D) -> N_1(X). Consideriamo N_1(D;X) := i_*(N_1(D)) in N_1(X), quindi il sottospazio lineare di N_1(X) generato dalle classi di equivalenza numerica di curve contenute in D. Casagrande ha introdotto il seguente invariante, chiamato Lefschetz defect: Delta_X := max {codimN_1(D;X), con D divisore primo}. In questo seminario, osserveremo dapprima che data una fibrazione conica f : X -> Y con r := \rho(X) - \rho(Y) > 1, sussiste un legame tra r e Delta_X. Ad esempio, è possibile trovare dei lower-bounds per Delta_X in termini di r. Successivamente ci focalizzeremo sul caso in cui n = 4 e Delta_X = 3, presentando un risultato di caratterizzazione in termini di Delta_X di Fano 4-folds che ammettono una fibrazione conica f : X -> Y con \rho(X)-\rho(Y) = 3. Come conseguenza, osserveremo che tali varietà sono razionali.

Lo spazio di Minkowski è l’analogo Lorentziano dello spazio Euclideo, ed è noto che esiste un’immersione isometrica del piano iperbolico nello spazio di Minkowski di dimensione 2+1, la quale è analoga all’immersione isometrica della sfera nello spazio Euclideo. A differenza del caso Euclideo, questa immersione isometrica non è unica a meno di isometrie globali. Presenterò alcuni risultati (i più recenti in collaborazione con Francesco Bonsante e Peter Smillie) sul problema della classificazione di tali immersioni isometriche, sottolineando le connessioni con altri argomenti, come le mappe armoniche tra varietà Riemanniane, le equazioni di Monge-Ampère e la teoria di Teichmüller.

Quando si costruisce un modello matematico per descrivere il comportamento di un sistema fisico, si deve spesso affrontare il problema che alcuni parametri del modello (coefficienti, forzanti, condizioni al bordo, forma del dominio etc) non sono noti esattamente, ma al contrario sono affetti da un certo grado di incertezza, e quindi descritti in maniera naturale in termini di variabili aleatorie/campi aleatori. La necessita` di stimare l'affidabilita` delle simulazioni numeriche tenendo conto di tale aleatorieta` ha portato all'introduzione di tecniche di Quantificazione dell'Incertezza (Uncertainty Quantification) nel calcolo scientifico. Obiettivi classici di questo tipo di analisi sono a) il calcolo di indici statistici (ad es media e varianza) per quantita` di interesse legate alla soluzione dell'equazione considerata (ad esempio, il valore della soluzione in un punto, il suo integrale sul dominio di calcolo, o il flusso in uscita) b) il miglioramento della descrizione statistica dei parametri del modello basandosi su osservazioni sperimentali di tali quantita` di interesse. Il primo tipo di analisi e` tipicamente conosciuto come "Forward uncertainty Quantification", mentre il secondo "Inverse Uncertainty QUantification". Uno degli ostacoli principali in UQ e` rappresentato dal fatto che in molte applicazioni sono necessarie numerose variabili aleatorie (a volte dell'ordine di decine o centinaia) per ottenere rappresentazioni accurate dell'incertezza del modello. Gli schemi numerici adottati per eseguire l'analisi di UQ devono quindi essere tali da limitare il piu` possibile il peggioramento della performance quando il numero di parametri aumenta - un fenomeno noto come "curse of dimensionality". In questo seminario introdurro` le basi della metodologie di UQ per PDE con parametri aleatori e discutero` la loro applicazione a qualche problema (semplificato) di interesse ingegneristico (stampa 3d, flussi in mezzi porosi, bacini sedimentari)

Durante il seminario cinque ex studenti di matematica, laureati negli ultimi anni, verranno a raccontare la loro esperienza nel mondo del lavoro. Sarà un’occasione per scoprire le reali possibilità occupazionali per un laureato in matematica, le difficoltà che al termine del percorso gli studenti si troveranno ad affrontare e le potenzialità che sono più apprezzate. I cinque giovani matematici faranno una breve presentazione sulla loro esperienza per poi lasciare spazio alle domande.

This study is intended to provide a continuous-time equilibrium model in which overconfidence generates disagreements among two groups regarding asset fundamentals. Every agent in trading wants to sell more than the average stock price in the market. However, the overconfident agent drives a speculative bubble with a false belief that the stock price will tend to move to the average price over time. I represent the difference between a false belief and a stochastic stationary process which does not change when shifted in time. The gap of beliefs shows how to accommodate dynamic fluctuations as parameters change such as the degree of overconfidence or the information of signals. By showing how changes in an expectation operator affect the stochastic variance of economic fundamentals, speculative bubbles are revealed at the burst independently from the market.

The last few years have witnessed the rise of the Sharing Economy, a collection of decentralized online platforms whose users exchange knowledge, goods, and resources on a peer-to-peer basis. Sharing Economy platforms are often praised for their meritocratic approach, where all participants, regardless of their gender or ethnicity, receive the same opportunities to emerge through digital peer review mechanisms. Yet, they have recently come under fire due to reports of discriminatory behaviours and manipulations of their reputation systems. This raises an important question: are Sharing Economy platforms fair marketplaces, where all participants operate on a level playing field, or are they large-scale online aggregators of offline human biases? In this talk I will address this question on a number of examples, showing how online platforms can be represented in terms of networks, and how this allows to detect and measure some of the biases that might affect their users' behaviour. In particular, I will present clear evidence of avoidance between users from different ethnic backgrounds on Airbnb, and I will show how user reputation scores are distorted by the widespread practice of reciprocating highly positive ratings in a variety of platforms. I will conclude by discussing how these findings can be used to provide platform design recommendations, aimed at exposing and possibly reducing the biases we detect, in support of a fairer and more inclusive growth of Sharing Economy platforms.

We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the Central Limit Theorem and Large Deviation Principles for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps, and strongly ergodic Markov chains. We primarily use spectral techniques to obtain these results.The work on CLT is joint with Carlangelo Liverani (Rome) and the work on LDPs is joint with Pratima Hebbar (Maryland).

In several real-word imaging applications such as microscopy, astronomy and medical imaging, transmission and/or acquisition faults result in a combination of multiple noise statistics in the observed image. Classical data discrepancies models dealing with this scenario linearly combine standard data fidelities used for single-noise removal or consider exact log-likelihood MAP estimators which are difficult to deal with in practice. In this talk, we derive a statistically consistent variational model for combining mixed data fidelities associated to single noise distributions in a handy infimal convoution fashion by which individual noise components in the data are modelled appropriately and separated from each other after a Total Variation smoothing. Our analysis is carried out in function spaces. For the numerical solution of the resulting denoising model, we propose a semismooth Newton-type scheme and show preliminary results in the context of bilevel learning for blind mixed denoising.

We will discuss some recent results and perpsectives on mean field disordered models.In particular we will focus on convexity properties of bipartite spin glasses

Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. Our results are obtained through the interplay of Hodge theory, deformation theory, and a further example of Verbitsky's technique which might go under the name "how to deduce beautiful consequences from ugly behavior of moduli spaces".

The mean integral of harmonic functions on balls centered at x equals the value of these functions at x. This is the well known Gauss mean value theorem. In 1972 Kuran proved the reverse: if D is a bounded open set containing x, such that the mean integral of harmonic functions on D equals the value of these functions at x, then D is a ball centered at x. Two questions may be raised: (1) similar rigidity results can be proved for weighted mean integrals? (2) is the Gauss mean value formula stable? That is: if the mean integral of harmonic functions on D centered at x is almost equal to the value of these functions at x, then D is almost a ball with center x? In this talk I will discuss recent results on these issues obtained in collaboration with E. Lanconelli (1) and with N. Fusco, E. Lanconelli and X. Zhong (2)

We describe the problem of the regularity of sub-Riemannian minimizers. We give some regularity results, in the real-analytic framework, obtained in collaboration with A.Bove.

In this seminar I will present some results on the electrostatic Born-Infeld equation set in the whole R^n. This equation is governed by the Lorentz-Minkowski mean curvature operator and was introduced, in the theory of nonlinear electromagnetism, as a generalization of the Poisson equation for the electrostatic potential. I will consider the case of a superposition of (possibly non-symmetrically distributed) point charges and discuss sufficient conditions to guarantee that the minimizer of the action functional is a solution of the problem. I will also present an approximation of the considered problem, governed by a sum of 2m-Laplacians, and show some qualitative properties of the approximating solutions, such as their behavior near the charges. This is a joint work with Denis Bonheure (Université Libre de Bruxelles) and Juraj Foldes (University of Virginia) available at arXiv:1707.07517.

Abstract: We shall see how symmetry plays a role on the estimates of the principal eigenvalue. We shall briefly recall some results for the Pucci operators and then show some new surprising results for the Harvey Lawson truncated laplacian. We shall also see some applications concerning the regularity of solutions in domains which are convex but not strictly convex.

In this talk we present some recent results concerning the classification of stable solutions to some semilinear nonlocal problems, such as the fractional Allen-Cahn equation. Crucial ingredients in the proof of the main results will be given by density and energies estimates for stable solutions.

We discuss some recent results on nonlocal phase transitions modelled by the fractional Allen-Cahn equation, also in connection with the surfaces minimising a nonlocal perimeter functional. In particular, we consider the "genuinely nonlocal regime" in which the diffusion operator is of order less than 1 and present some rigidity and symmetry results.

In this talk we will discuss the validity of Harnack inequalities for two classes of linear second order equations in nondivergence form. The first class is formed by degenerate-elliptic operators which are horizontally elliptic with respect to Heisenberg-type vector fields. The second one constitutes a class of evolution operators of Kolmogorov-Fokker-Planck type. The analogous of the Krylov-Safonov Harnack inequality for these classes of Hörmander operators with bounded measurable coefficients is still unknown, due to the absence of proper Aleksandrov-Bakelman-Pucci type estimates. We will show a perturbative approach to prove invariant Harnack inequalities for operators with coefficients satisfying either a Cordes-Landis assumption or a continuity hypothesis. This talk is based on joint works with F. Abedin and C.E. Gutiérrez.

In this talk, I will present a recent joint work with C. Mouhot. It concerns a toy non-linear model in kinetic theory. We will see that the problem is globally well-posed in Sobolev spaces. The construction of solutions rely on some recent Hoelder estimates obtained with F. Golse, C. Mouhot and A. Vasseur combined with appropriate Schauder estimates.

We will explains some details of the proof of the previous seminar, concerning the quantization rule of resonances when the trapped set of the corresponding Hamiltonian system consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. (joint work with Jean-Fran¥c cois Bony (Bordeaux), Thierry Ramond (Paris XI) and Maher Zerzeri (Paris XIII)).

It is well known as Hörmander's theorem that the (semiclassical) wave front set of the solution to a pseudo-differential equation propagates along Hamiltonian flows of the real principal type symbol. We extend this theorem to fixed points of the Hamiltonian vector field. To a hyperbolic fixed point, associated an outgoing and an incoming stable manifolds, and we show that if the semiclassical wave front set is empty on the incoming stable manifold (except at the fixed point), then it is also empty on the outgoing one. We also show how such theorems are applied to a scattering problem of the Schr¥"odinger operator. We give the quantization rule of resonances when the trapped set of the corresponding Hamiltonian system consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. This is a joint work with Jean-Fran¥c cois Bony (Bordeaux), Thierry Ramond (Paris XI) and Maher Zerzeri (Paris XIII)

L'intenzione e' di discutere le note di Fei Fei Li: http://cs231n.stanford.edu/ [cs231n.stanford.edu] Non sono necessari requisiti particolari eccetto qualche conoscenza di programmazione, preferibilmente python.

We will discuss some examples of random walks and their relation to asymptotic mean value properties, and viscosity solutions to p-Laplace equations in the Heisenberg group.

I will review the role of sub-Riemannian geometry in neurogeometry of the visual cortex (SE(2) group) and show how it is engendering visual perception. Particularly horizontal curves and geodesics will account for a number of geometric phenomena on the visual field. A mean field equation defined on the SE(2) geometry will model neural activity and its symmetry breaking will provide visual percepts. An open problem about heterogeneous neurogeometry will be preliminarily discussed. (Joint work with Giovanna Citti)

The problem of restriction of the Fourier transform to hypersurfaces was posed by Stein in the seventies. This operator, in its adjoint form, gives the solution of dispersive equations (Schrödinger, wave, etc) in terms of the Fourier transform of the initial data. There are many open problems about dispersive equations for which it can be a powerful tool. Also, the restriction operator can be thought as a model case for more complicated oscillatory integral operators, such as, for instance, the operators of spherical summation of the Fourier transform. We will make a review of the problem, which is still open. We will present some new results for the case of surfaces with negative curvature. This part is joint work with Detlef Müller and Stefan Buschenhenke.

For an algebraic or geometric object L, its automorphisms form a group Aut(L). If in addition Aut(L) has a topology, we denote its identity component by Int(L), and define the quotient group Out(L) = Aut(L)/Int(L). The members of Int(L) are called inner, and the remaining are called outer. Let L be a complex semisimple Lie algebra with Dynkin diagram D. It is well known that Out(L) = Aut(D). We extend this result to the real forms of L, and discuss the classification of real forms up to Int(L). We also consider possible extensions of these results to contragredient Lie superalgebras.

Geometric Measure Theory in the sub-Riemannian Heisenberg group leave unanswered several fundamental questions. The main issue is the regularity of area-minimizing surfaces, because sets of finite sub-Riemannian perimeter may have fractal behaviours. I will present some recent results obtained in collaboration with Francesco Serra Cassano and Manuel Ritoré.

As a young Ph.D. student, the first nonlinear operator I met was the p-Laplacean; later on, the first visitor I invited to Parma was Juan Manfredi (in 1997, if I remember correctly). And it is definitely not by chance the two things are related. This talk is the occasion to celebrate both of them, with some recent regularity results coming form nonlinear potential theory.

Every closed differential form \omega on a Euclidean ball has a primitive whose L^q norm is bounded by the L^p norm of \omega (for suitable exponents p and q). We prove an analogous result for Rumin's exterior differential on Heisenberg balls. This is used to prove vanishing of \ell^{q,p}-cohomology of Heisenberg groups. Extension to other Carnot groups will be discussed.

Seminario riservato ai membri del progetto MANET

Vengono introdotti i tensori di Grassmann per proiezioni P^k - - -> P^h_j, j=1,...r, utilizzati in Computer Vision per la ricostruzione proiettiva di scene statiche e dinamiche. Dopo aver ricordato i risultati classici (ossia nel caso di proiezioni da P^3), si considera il caso di due proiezioni tra spazi di dimensione qualsiasi e si accenna a qualche risultato per piu' proiezioni tra spazi di dimensione "piccola". I tensori di Grassmann vengono poi utilizzati per lo studio dei luoghi critici per la ricostruzione.

There are two alternative definitions of discrete connections on triangulated manifolds. The most known one associates a group element to each edge. An alternative approach uses first-order operators on simplexes of higher dimension. We show that in dimension two such connections are associated with self-adjoint second order operators, and the self-adjointness is equivalent to existence of two factorizations. We also show that Laplace transformations can be interpreted as the star-triangle transformation used in electrical circuits.

In contrast with 1+1 dimensional systems, integrable 2+1 systems are usually non-local, and integration based on the scattering transform corresponds to a special choice of integration constants. For the KP equation this study turned out to be rather non-trivial. We show that in case of the so-called Pavlov equation, with is a dispersionless 2+1 dimensional integrable model, the answer can be explained using some lemma from integral geometry.

Se X e' una varieta' iperkaehler di tipo Kummer, il gruppo di coomologia H^3(X) ha dimensione 8, e quindi la Jacobiana intermedia J^3(X) e' un toro complesso compatto di dimensione 4, proiettivo se X e' proiettiva. Faro' vedere come ricostruire esplicitamente J^3(X) a partire dalla struttura di Hodge su H^2(X). In particolare seguira' che, se X e' proiettiva, allora J^3(X) e' una varieta' abeliana di tipo Weil. Lo studio di J^3(X) suggerisce come (tentare di) costruire famiglie esplicite localmente complete di varieta' iperkaehler di tipo Kummer proiettive.

A simplicial complex of dimension d - 1 is said to be Cohen-Macaulay in codimension t, 0 <= t <=d -1, if it is pure and the link of any face with cardinality at least t is Cohen-Macaulay. This generalizes several concepts on simplicial complexes such as Cohen-Macaualyness, Buchsbaum property, S_r condition of Serre, and locally Cohen-Macaulayness. Most results on the simplicial complexes with aforementioned properties naturally extend to the case of Cohen-Macaulayness in codimension t. In particular, the Eagon-Reiner theorem, the local behavior, and the homological vanishing properties are suitably retained. Furthermore, characterizations of certain families of Cohen-Macaulay simplicial complexes carry over characterizations of these families of simplicial complexes which are Cohen-Macaulay in codimension t. This talk is based on recent joint works with H. Haghighi, S. A. S. Fakhari and S. Yassemi. 1

A simplicial complex of dimension d - 1 is said to be Cohen-Macaulay in codimension t, 0 <= t <=d -1, if it is pure and the link of any face with cardinality at least t is Cohen-Macaulay. This generalizes several concepts on simplicial complexes such as Cohen-Macaualyness, Buchsbaum property, S_r condition of Serre, and locally Cohen-Macaulayness. Most results on the simplicial complexes with aforementioned properties naturally extend to the case of Cohen-Macaulayness in codimension t. In particular, the Eagon-Reiner theorem, the local behavior, and the homological vanishing properties are suitably retained. Furthermore, characterizations of certain families of Cohen-Macaulay simplicial complexes carry over characterizations of these families of simplicial complexes which are Cohen-Macaulay in codimension t. This talk is based on recent joint works with H. Haghighi, S. A. S. Fakhari and S. Yassemi. 1