Archivio 2015

Dimostriamo come, anche su campi di caratteristica positiva, il prodotto di due curve ellittiche non ammetta  FM partner non triviali. Questo e' un lavoro in collaborazione con K. Honigs (University of Utah) and L. Lombardi (Stonybrook).

Geometria Algebrica Reale e Tensori Giovedì 10 dicembre 2015 alle ore 14:30, si terrà presso il Dipartimento di Matematica dell'Università di Bologna, aula Seminario VIII piano, il primo seminario di studio di un ciclo su Geometria Algebrica Reale e Tensori organizzato congiuntamente da E. Angelini, C. Bocci, A. Calabri, L. Chiantini, M. Mella, G. Ottaviani, E. Rubei. Referenze bibliografiche: - J. M. Landsberg, "Tensors: Geometry and Applications", AMS, Graduate Studies in Mathematics, vol 128, 2012; - J. Bochnak, M. Coste, M. F. Roy, "Real Algebraic Geometry", Springer, 1998. Giovedì 10 dicembre 2015 il prof. Luca Chiantini (Università di Siena) terrà il seguente seminario: Titolo: Cosa cambia sui reali? Tutti gli interessati sono invitati a partecipare. Per ulteriori informazioni sul ciclo di seminari si visiti il link: <http://web.math.unifi.it/gruppi/algebraic-geometry/GeometriaAlgebricaRealeeTensori20152016.html> che sarà disponibile a breve. Gli organizzatori

Nell’accezione che si è affermata tra gli storici moderni, il Risorgimento italiano è un lungo processo che attraversa tutto l’Ottocento, ha inizio con le campagne napoleoniche e si conclude con la Prima Guerra mondiale. Di quel processo i matematici furono protagonisti attivi, alla guida delle istituzioni nelle “Repubbliche sorelle” della Francia, combattenti nelle battaglie del Quarantotto, e infine protagonisti della ‘rivoluzione’ – come la chiamò Manzoni – che ha portato all’Unità. Nella conferenza sarà messo in luce soprattutto l’impegno dei matematici della ‘generazione risorgimentale’ – in particolare di Brioschi, Menabrea, Cremona, e Betti – nella vita politica e nelle istituzioni del nuovo Stato. Figure di primo piano all’epoca, ma dimenticate e scomparse dai manuali di storia. Tuttavia le loro vicende, solo a prima vista lontane nel tempo, invitano a guardare alla recente storia del nostro paese da una prospettiva inedita, che offre ancora oggi molti elementi di riflessione.

How many repelling periodic points of any period does a transcendental function have? For a generic rational function, the number of periodic points can be easily counted using the degree, and by the Fatou Shishikura inequality all but finitely many are repelling. Entire functions in general do not even need to have fixed points in general,see for example tan(z)+z. However we will be able to show--with a rather elementary proof--that for several important classes of transcendental functions there are--as expected--infinitely many repelling periodic points of any given period,and give some more information on the way they are distributed in the dynamical plane.

I problemi di ottimizzazione possono essere suddivisi in due categorie: quelli con variabili /continue/ e quelli con variabili /discrete/, che vengono piu` comunemente denominati come problemi di ottimizzazione combinatoria. Nei problemi continui la soluzione corrisponde a un insieme di numeri reali, mentre nei problemi combinatori la soluzione e` un insieme di numeri interi. Molti problemi reali come, ad esempio, nei campi della logistica e della supply chain sono formulabili come problemi di ottimizzazione combinatoria la cui soluzione richiede lo sviluppo di algoritmi specifici. Nella prima parte del seminario verranno illustrati alcuni dei problemi applicativi tipici dell'ottimizzazione combinatoria quali il Traveling Salesman Problem, alcune varianti del Vehicle Routing Problem ed il Fixed Charge Transportation Problem. Nella seconda parte del seminario verranno illustrati alcuni dei metodi generali di soluzione dell'ottimizzazione combinatoria che vengono impiegati per la risoluzione dei problemi descritti nella prima parte, quali: gli algoritmi branch and bound, i metodi cutting plane, i metodi column generation e gli algoritmi branch-and-cut-and price.

We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in \R^2 is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. Moreover we establish BV estimates and energy estimates in every dimension for the more general notion of stable critical sets. This is a joint work with Joaquim Serra and Enrico Valdinoci.

Stellar dynamics is a very old and much studied field of theoretical astrophysics, yet several core-problems are still unsolved or even not fully understood. In this talk I will illustrate a selection of such open problems, ranging from the construction of self gravitating figures of equilibrium strongly departing from spherical symmetry (in configuration space, through the Poisson equation, and in phase-space, by using the Collisionless Boltzmann Equation), to the long-term evolution of N-body systems.

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review results on the derivation of effective models to describe statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular, we will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

We study the location of the eigenvalues of the generator $G$ of a contraction semigroup $V(t) = e^{t G}, \: t \geq 0,$ related to the wave equation with dissipative boundary conditions. The spectrum of $G$ in $\Re z < 0$ is formed by isolated eigenvalues with finite multiplicity and the solutions $u(t, x) = e^{\lambda t} f(x)$ with $Gf = \lambda f, \Re \lambda < 0,$ are called asymptotically disappearing. The location of the eigenvalues of $G$ is important for the inverse scattering problems. We show that the eigenvalues are located in parabolic neighborhoods of the negative real axis or the imaginary one. For strictly convex obstacles we obtain a sharper result.

The minimisation of a smooth objective function subject to a matrix rank constraint can sometimes be very effectively solved by methods from Riemannian optimisation. This is for instance the case with the low-rank matrix completion problem or the solution of PDEs on square domains like the Lyapunov equation. However, the theory of Riemannian optimisation leaves some questions unanswered regarding the practical application of such algorithms. I will focus on two such questions. The first is how the metric has a significant impact on the convergence of the numerical methods. This is related to how Newton’s method can be seen as a variable metric in numerical optimisation and to general preconditioning techniques. The second topic is rank adaptivity. In rank-constrained optimisation, one does typically not know the rank a priori but may be searching for the smallest rank satisfying a certain criterion, like a small residual. I will explain how the geometry of the tangent cone of the variety of matrices of bounded rank can be incorporated so as to obtain rank adaptive algorithms that stay true to the manifold philosophy.

For a smooth algebraic variety X over C, the periods of X are certain complex numbers obtained by integration of differential forms on X along topological cycles drawn on X_an. As shown by Deligne, algebraic regular connections on X give rise to periods, and when varying in families, periods satisfy a differential equation with regular singularities. This talk is an introduction to the ideas and notions that revolve around these results. We will also explain a recent progress on what can be said for periods of general algebraic connections. No prerequisite on D-modules is necessary to follow this talk.

General domain Hankel and Toeplitz operators is a class of operators that significantly extends the classical counterparts to several variables. I will discuss various results concerning their structure, positive semidefinitness and finite rank. It turns out that their symbols are then sums of exponential functions, and these operators therefore have a potential for playing a key role in multidimensional frequency estimation / approximation by sparse exponential sums. I will elaborate on this connection and show some numerical results. Potential applications range from seismic imaging to chemistry (NMR) and medicine (e.g. MRI).

Two simple involutions on the set of Dyck words will be considered. The first one corresponds to the mirror image of the path coded by the word, and the second one is related to the rank parameter on graphs introduced by Baker and Norine. The composition of these two involutions is a permutation which has an interesting behavior. Some conjectures and results on this behavior - obtained in collaboration with M. Barnabei and F. Bonetti - will be presented.

In the second part, we consider in particular how to apply tug-of-war games to obtain regularity results. As an application, we establish regularity results for p-harmonic functions. The arguments are based on choice of strategies in the game, and are thus quite different from the proofs using PDE tools.

The fundamental works of Doob, Hunt, Kakutani, Kolmogorov, Lévy and many others have shown a powerful connection between the classical linear potential theory and the corresponding probability theory. The idea behind the classical interplay is that harmonic functions and martingales share a common origin in mean value properties. This approach turns out to be useful in the nonlinear theory as well. This talk introduces basic theory of tug-of-war games and their connection to p-harmonic functions.

I will present an overview of Sarnak’s conjecture on the disjointness of the Möbius function from any deterministic sequence and the related Chowla’s conjecture on the self-correlations of the Möbius function. Some progress towards weaker versions of these conjectures have been made recently, and I plan to illustrate them.

We prove a quantitative version for a weighted isoperimetric inequality in cones (recently proven by Cabré, Ros, Serra), for weights which satisfies some concavity conditions in the cone. The proof uses the ABP method applied to solutions of a certain Neumann problem, combined with a weighted trace inequality, which let us the get the optimal exponent on the isoperimetric deficit. This is a joint work with X. Cabre’, A. Pratelli, X. Ros and J. Serra.

We describe all the finite dimensional complex irreucible  representations of  SU(1|1), we give a reducibility result for representations not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end, we prove a version of the Peter-Weyl theorem for SU(1|1).

We  discuss the highest weight kr-finite representations of the pair (gr, kr) consisting of gr, a real form of a complex basic Lie superalgebra of classical type g, and the maximal compact subalgebra kr of gr0.

A seguito del lavoro di Khovanov, molta ricerca è stata fatta sulla categorificazione di vari tipi di algebre di Heisenberg. In questo seminario verra' illustrato un approccio generale di categorificazione tramite i prodotti intrecciati di un'algebra di Frobenius fissata. Come casi speciali di questa costruzione si possono riottenere risultati di Khovanov, di Cautis e Licata, e di Hill e Sussan. I risultati sono frutto di una collaborazione con Alistair Savage (Università di Ottawa).

The algebra of invariant differential operators on a multiplicity-free representation of a reductive group has a concrete basis, usually referred to as the Capelli basis. The spectrum of the Capelli basis on spherical representations results in a family of symmetric polynomials (after \rho-shift) which has been studied extensively by Knop and Sahi in the early 1990's. In this talk, we generalize some of the Knop-Sahi results to the symmetric superpair GL(m,2n)/OSp(m,2n). We prove that in the Frobenius coordinates of Sergeev-Veselov, our polynomials turn into the shifted super Jack polynomials. This talk is based on joint work with Siddhartha Sahi.

We will consider polygonal billiards with a non-standard reflection law such that the angle of reflection is smaller than the angle of incidence. We will present some recent results concerning the existence of hyperbolic attractors and SRB measures for such billiards. The talk is based on joint work with J. Lopes Dias, P. Duarte and J.P. Gaivão.

A bird's-eye view of the development of integer and combinatorial optimization through the lens of the author work on disjunctive programming. The focus will be on explaining the significance of the revolution in the state of the art of integer optimization that took place roughly between 1990 and 2005.

In the study of multiplicity free actions of reductive Lie algebras, one is naturally lead to invariant polynomial coefficients differential operators. The abstract and concrete Capelli problems of Howe and Umeda ask for the realization of invariant differential operators by means of the centre of the universal enveloping algebra. We will review this classical theory and describe our new results for Lie superalgebras. This talk is based on joint work with Siddhartha Sahi.

We are capable to create chess programs that easily defats the best humans player in the world. However, we still don’t know how to create machines that naturally discuss with common human people. The talk will discuss all the sophisticated natural language processing techniques and the awful dirty hacks used at Aldebaran not only to solve this difficult and fascinating problem, but also to build a real producer.

In the late 70s, V. Rittenberg and D. Wyler introduced multi-graded (also called color) algebras. These are generalizations of the notion of superalgebra, in which the $\mathbb{Z}_2$-grading is replaced by $\mathbb{Z}_2^n$-gradings, $n>1$, and the considered sign rule is modified accordingly (i.e. is determined by the scalar product of the involved $\mathbb{Z}_2^n$-degrees). The development of $\mathbb{Z}_2^n$-superalgebra theory, and the corresponding geometry, recently acquired new relevance: V. Ovsienko and S. Morier-Genoud observed that a remarkable class of classical non-commutative algebras, namely quaternions and more generally Clifford algebras, are in fact $\mathbb{Z}_2^n$-commutative for some $n$. In this talk, I will report on joint works with N. Poncin and J. Grabowski, concerning a first study of the notion of manifold in this multigraded setting. Contrary to a widespread belief, this $\mathbb{Z}_2^n$-supergeometry present important differences with the classical supergeometry,  and provides a sharpened viewpoint.

Le interazioni onda particella nei plasmi magnetizzati di interesse fusionistico giocano un ruolo fondamentale in quanto possono essere causa di instabilità e, al tempo stesso, governare la dinamica nonlineare delle fluttuazioni. In particolar modo, sono interessanti le condizioni in cui l’interazione è non-perturbativa, ossia in grado di modificare da un lato la distribuzione delle particelle e dall’altro la struttura delle fluttuazioni. Ne conseguono fenomeni di autorisonanza e phase-locking, che danno luogo a fenomeni nonlineari complessi. In generale, lo studio di questi fenomeni porta a descriverli in termini di equazioni modello tipo Schroedinger nonlineare integro-differenziale. Si discuteranno esempi di soluzione di queste equazioni modello in casi semplici, in grado di spiegare i risultati ottenuti da simulazioni numeriche. Tuttavia, la struttura matematica delle equazioni e le proprietà delle soluzioni corrispondenti rimane da esplorare.

Presentiamo una generalizzazione del teorema di Meyers-Serrin, in cui sostituiamo le derivate deboli in sottoinsiemi aperti di $R^m$ con famiglie finite di operatori differenziali lineari, definiti su sezioni regolari di fibrati vettoriali su una varietà (non necessariamente compatta) X.

Un compito di base in robotica e' di trovare una sequenza di mosse che parte da una posizione iniziale e fa navigare il robot verso una posizione finale desiderata. Questa sequenza puo' essere interpretata come un cammino nello spazio di tutte le posizioni possibili (detto spazio delle configurazioni) del robot. Il piano generale di navigazione di un robot e' quindi una mappa che ad ogni coppia di configurazioni (iniziale, finale) assegna un cammino nello spazio di configurazioni che collega i due estremi. Usando considerazioni elementari e' possibile dimostrare che un piano di navigazione puo' essere dato da una mappa continua se e solo se lo spazio di configurazioni e' contraibile. Per trattare il caso generale M. Farber (2003) ha definito la complessita' topologica di uno spazio di configurazioni (o per estensione, di uno spazio topologico X ) che misura il grado minimo di discontinuita' (o instabilita') di piani di navigazione su X. Il problema di navigazione e' in realta' piu' complesso, perche' e' necessario tener conto anche della cosidetta mappa cinematica, che descrive come la rotazione dei giunti del braccio robotico si traduce nel movimento spaziale dell'attuatore finale (pinza, saldatore,...). E' noto che in genere una mappa cinematica ha singolarita' che corrispondono alle posizioni dei giunti con liberta' di movimento ristretta. Nella conferenza discutero' un'estensione della complessita' topologica di Farber che premette di misurare il grado di instabilita' dei piani di navigazione in presenza di mappe cinematiche con singolarita'.

Interverranno: Prof. Mirko Degli Esposti Dipartimento di Matematica, Università di Bologna Prof. Maurizio Rippa Bonati Dipartimento di Scienze Cardiologiche, Università di Padova Prof. Marcelo Montemurro Faculty of Life Science, University of Manchester, UK Proff. Giuseppe Fallacara e Ubaldo Occhinegro Dipartimento di Scienze dell’Ingegneria Civile e dell’Architettura, Politecnico di Bari

Interverranno: Prof. Mirko Degli Esposti Dipartimento di Matematica, Università di Bologna Prof. Maurizio Rippa Bonati Dipartimento di Scienze Cardiologiche, Università di Padova Prof. Marcelo Montemurro Faculty of Life Science, University of Manchester, UK Proff. Giuseppe Fallacara e Ubaldo Occhinegro Dipartimento di Scienze dell’Ingegneria Civile e dell’Architettura, Politecnico di Bari

Interverranno: Prof. Mirko Degli Esposti Dipartimento di Matematica, Università di Bologna Prof. Maurizio Rippa Bonati Dipartimento di Scienze Cardiologiche, Università di Padova Prof. Marcelo Montemurro Faculty of Life Science, University of Manchester, UK Proff. Giuseppe Fallacara e Ubaldo Occhinegro Dipartimento di Scienze dell’Ingegneria Civile e dell’Architettura, Politecnico di Bari

Let G be a complex reductive group and B a Borel subgroup of G, a subgroup H of G is called spherical if it acts with finitely many orbits on the flag variety G/B. Even though spherical subgroups are classified, the corresponding decompositions of G/B are not yet well understood in general. In this seminar I will consider two special cases, namely that of a solvable spherical subgroup and that of the Levi factor of a parabolic subgroup with abelian unipotent radical. In these cases, I will explain how to attach a root system to every H-orbit in G/B, and how these root systems allow to parametrize the H-orbits in G/B. These parametrizations are compatible with a general action of the Weyl group of G that Knop defined on the set of H-orbits in G/B, and I will explain how to recover the Weyl group action from the parametrization of the orbits. The talk is based on two works in collaboration, with Andrea Maffei and with Guido Pezzini."

Sia G un gruppo di Lie reale, K il suo compatto massimale. G/K e' hermitiano se ammette una struttura complessa invariante per l'azione di G. L'esistenza di tale struttura e' legata alle proprieta' dell'algebra di Lie di G, in particolare all'esistenza di un sistema di radici ammissibile. Tale sistema rende possibile la decomposizione di Harish-Chandra di un aperto di G e la realizzazione dei moduli di Harish-Chandra nello spazio delle sezioni di fibrati vettoriali su G/K. Presteremo una attenzione particolare all'esempio dello spazio di Siegel.

For random dynamical systems, one can distinguish two kinds of limit theorems: annealed results, which refer to the Birkhoff sums seen as functions of both the phase space variable and the choice of the maps composed, and quenched results, which refer to Birkhoff sums for a fixed, but generic, composition of maps. In this talk, I will describe results about the central limit theorem for random dynamical systems consisting of uniformly expanding maps. In particular, I will show that the annealed central limit theorem is valid for such systems, and I will give a necessary and sufficient condition for its quenched version without random centering to hold. This is a joint work with Matthew Nicol and Sandro Vaienti

In this expository talk we will review the theory of multivariate B-splines. These are analogs of the classical univariate B-splines that are built from simplex splines. The presented construction uses the so-called Delaunay configurations. We will survey this area of approximation theory and discuss a number of unresolved issues. We will also sketch an algorithm to compute Delaunay configurations effectively and then apply it to a construction of spline spaces using bivariate B-splines.

In questo seminario vogliamo trattare la classificazione delle forme reali di algebre di Lie semisemplici attraverso i diagrammi di Vogan.

Photometric Stereo refers to the problem of 3D-reconstruction of a scene, given several images taken from the same viewpoint, but under different lightings. Since the early 1980s, researchers have devoted a lot of energy to improving the accuracy of this technique, through the introduction of realistic reflectance, camera, surface and lighting models. In this talk, recent advances in the field of photometric stereo with realistic, physics-based lighting models, will be presented. Clues for numerical resolution will be provided through the study of some energy minimisation approaches of photometric stereo, including matrix factorisation approaches for linear models, optimisation on discrete grids for non-convex functionals, and calculus of variations. It will be shown that, combining such robust optimisation tools with realistic model, 3D-reconstruction through photometric stereo is now almost mature enough to be considered as a real metrological technique.

Abstract. The traditional jump-telegraph processes are based on a Poisson process with alternating intensities. We develop a new model based on an alternating doubly stochastic Poisson process with random intensities of jumps. The market model based on the doubly stochastic jump-telegraph process is studied. A class of complete models is also considered. IL PROF. RATANOV E’ DISPONIBILE AD INCONTRARE I COLLEGHI CHE SONO INTERESSATI A FUTURI PROGETTI DI RICERCA INTERNAZIONALI CON L’AMERICA LATINA.

Ogni forma reale di un'algebra di Lie semisemplice complessa e' caratterizzata dalla propria sottoalgebra compatta massimale che a sua volta corrisponde ad una involuzione di Cartan. Vogliamo ripercorrere i fatti essenziali di queste corrispondenze che portano alla classificazione delle forme reali attraverso i diagrammi di Vogan.

Ogni forma reale di un'algebra di Lie semisemplice complessa e' caratterizzata dalla propria sottoalgebra compatta massimale che a sua volta corrisponde ad una involuzione di Cartan. Vogliamo ripercorrere i fatti essenziali di queste corrispondenze che portano alla classificazione delle forme reali attraverso i diagrammi di Vogan.

Il seminari si inserisce all'interno di un ciclo di seminari di Geometria Algebrica Numerica organizzato congiuntamente da E. Angelini, C. Bocci, A. Calabri, L. Chiantini, M. Mella, G. Ottaviani, E. Rubei. Argomenti di studio saranno: • il metodo di Newton e il metodo di prolungamento omotopico per sistemi polinomiali, monodromia; • applicazioni alla decomposizione di tensori e alle varietà secanti; • applicazioni ad autovettori di tensori; • decomposizione numerica di una varietà algebrica in componenti irriducibili. Il seminario si propone anche di prendere familiarità con il software Bertini (https://bertini.nd.edu/) e il package NumericalAlgebraicGeometry di Macaulay2 (http://www.math.uiuc.edu/Macaulay2/). Tutti gli interessati sono invitati a partecipare. Per ulteriori informazioni sul ciclo di seminari si visiti il link http://web.math.unifi.it/gruppi/algebraic-geometry/GeometriaAlgebricaNumerica20142015.html

We address efficient preconditioning techniques for the inexact second-order methods applied to solve various sparse approximation problems arising in signal/image reconstruction. The preconditioners exploit two features of such problems: (i) sparsity of the solution, and (ii) near-orthogonality of the matrices involved. The latter originates from the restricted isometry properties frequently assumed in such applications. Spectral analysis of the preconditioners and their practical efficiency when solving linear systems in the Newton Conjugate Gradient method will be presented. If time permits then a few comments on the some other problems originating from the "Big Data" buzz will also be given.

ALL FUNCTIONS ARE LOCALLY S-HARMONIC (UP TO A SMALL ERROR) We show that we can approximate every C^k function with a s-harmonic function in a ball that vanishes outside a compact set. That is, s-harmonic functions are locally dense in C^k. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature. This is a joint work with O. Savin and E. Valdinoci.

We shall prove that an stable intrinsic graph of class C^1 in H^1 is a vertical plane. This is joint work with Matteo Galli.

We review some recent results on the regularity of H-minimal boundaries in the Heisenberg group: Lipschitz approximation, height estimate, and harmonic approximation. This is part of a joint research program with D. Vittone.

We consider a class of CR manifolds which are defined as asymptotically Heisenberg, and for these we give a notion of mass. From the solvability of the $\Box_b$ equation in a certain functional class ([Hsiao-Yung]), we prove positivity of the mass under the condition that the Webster curvature is positive and that the manifold is embeddable. We apply this result to the Yamabe problem for compact CR manifolds, assuming positivity of the Webster class and non-negativity of the Paneitz operator. This is joint work with J.H.Cheng and P.Yang.

I plan to discuss the isoperimetric problem, in the Heisenberg space as well as CR 3-manifolds. I begin by reviewing the pseudohermitian geometry, the local geometry of a surface. The problem of Pansu. A new fourth order invariant called the Q-prime curvature, and its relation to the isoperimetric inequality.

I plan to discuss the isoperimetric problem, in the Heisenberg space as well as CR 3-manifolds. I begin by reviewing the pseudohermitian geometry, the local geometry of a surface. The problem of Pansu. A new fourth order invariant called the Q-prime curvature, and its relation to the isoperimetric inequality. Minicorso all'interno dell'evento "Mini-courses and workshop on Geometric Analysis in the Heisenberg group", March 4-6, 2015, Bologna, Italy e dei "Topics in Maths".

We derive variational formulas of the sub-Riemannian area in contact sub-Riemannian manifolds, introduce hypersurfaces with prescribed mean curvature, and discuss the regularity of C^1 surfaces (with prescribed mean curvature) in the 3-dimensional case. Minicorso all'interno dell'evento "Mini-courses and workshop on Geometric Analysis in the Heisenberg group", March 4-6, 2015, Bologna, Italy e dei "Topics in Maths".

Photometric Stereo (PS) is a fundamental problem in Computer Vision aimed at reconstructing the 3D shape given multiple images of the object under different illuminations. It can be expressed as a non-linear system of first order Hamilton Jacobi equations when 3 or more images are given. We talk about realistic PS modeled by PDEs starting from the problem when only 2 images are available. We extend our previous results on this problem, considering the more realistic perspective projection of surfaces during the photographic process and point light source parametrization. We prove that the new differential formulation is well-posed leading to quasi-linear PDEs.

We derive variational formulas of the sub-Riemannian area in contact sub-Riemannian manifolds, introduce hypersurfaces with prescribed mean curvature, and discuss the regularity of C^1 surfaces (with prescribed mean curvature) in the 3-dimensional case. Minicorso all'interno dell'evento "Mini-courses and workshop on Geometric Analysis in the Heisenberg group", March 4-6, 2015, Bologna, Italy e dei "Topics in Maths".

I plan to discuss the isoperimetric problem, in the Heisenberg space as well as CR 3-manifolds. I begin by reviewing the pseudohermitian geometry, the local geometry of a surface. The problem of Pansu. A new fourth order invariant called the Q-prime curvature, and its relation to the isoperimetric inequality. Minicorso all'interno dell'evento "Mini-courses and workshop on Geometric Analysis in the Heisenberg group", March 4-6, 2015, Bologna, Italy e dei "Topics in Maths".

Un biliardo è un sistema meccanico formato da una particella puntiforme che si muove di moto rettilineo all’interno di una regione e viene riflessa quando urta il bordo della regione. La relazione fra le velocità della particella prima e dopo la riflessione è data dalla legge di riflessione. La legge di riflessione comunemente considerata e’ quella speculare: l’angolo di riflessione e l’angolo di incidenza della particella coincidono. E’ noto che i biliardi poligonali con legge di riflessione speculare hanno entropia topologica nulla e perciò non sono sistemi iperbolici. In questo seminario presenterò alcuni risultati recenti che riguardano i biliardi poligonali con un legge di riflessione non-speculare che contrae l’angolo di riflessione verso la normale del bordo del poligono nel punto di riflessione (in collaborazione con P. Duarte, J, Gaivão e J. Lopes Diaz). In particolare, mostrerò che per poligoni senza lati paralleli, i biliardi con legge di riflessione contrattiva hanno attrattori iperbolici e misure invarianti SRB.

Il seminario ha lo scopo di illustrare le attività del Dipartimento di Matematica per l'Industria, di recente istituzione presso la Kyushu University (Fukuoka, Giappone) del quale il prof. Wakayama è stato tra i fondatori ed il primo direttore.

Abstract: We will introduce a construction which produces a family of arithmetic, non-compact hyperbolic 4-manifolds, such that every finite group G is realized as the symmetry group of some member of this family. This allows us to obtain asymptotic bounds which relate the order of a finite group G with the minimal volume of a hyperbolic 4-manifold realizing G as its symmetry group. The construction uses very accessible techniques: basic hyperbolic geometry on one hand, and the combinatorial properties of simplicial complexes on the other. \\------------------------------------------------------\\ Sunto: Introdurrò una costruzione che produce una famiglia di 4-varietà iperboliche tale che ogni gruppo finito G è realizzato come gruppo di simmetrie di un membro di tale famiglia. Questo permette di ottenere limiti asintotici che legano l'ordine di un gruppo finito G col volume minimo di una 4-varietà iperbolica che lo realizza come gruppo di simmetrie. La costruzione usa tecniche molto accessibili: geometria iperbolica di base da un lato, e le proprietà combinatorie dei complessi simpliciali dall'altro.

Reliable estimates for the condition number of a large (sparse) matrix A are important in many applications. To get an approximation for the condition number K(A), an approximation for the smallest singular value is needed. Krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value. Therefore, we study extended Krylov subspaces which turn out to be ideal for the simultaneous approximation of both the smallest and largest singular value of a matrix. First, we develop a new extended Lanczos bidiagonalization method. With this method we obtain a lower bound for the condition number. Moreover, the method also yields probabilistic upper bounds for K(A). The user can select the probability with which the upper bound holds, as well as the ratio of the probabilistic upper bound and the lower bound.

Given the question if two knots inside the 3-dimensional Euclidean space are isotopic, a number of polynomial knot invariants have been constructed in order to distinguish them. We will take a closer look at the Kauffman bracket, the Alexander-Conway polynomial, and the HOMFLY polynomial. To generalize these polynomials to knots in other 3-dimensional manifolds, Skein modules were introduced. To each 3-manifold we can associate infinitely many different Skein modules, the vast majority of which have not yet been computed. We will overview current results and show in what sense Skein modules relate to the classical polynomials.