Archivio 2020

The interest around a sparse representation of data is growing in these years due to its several applications such as image classification, image denoising and image compression. Dictionary learning is one of the most important techniques to address this task. With such an approach, data are represented using a large matrix D and a sparse matrix X. Moreover, to deal with multidimensional data, a tensor formulation of the dictionary learning problem has been recently introduced. Within this framework,we propose a new Tensor-Train based nonlinear optimization algorithm and we compare its performance with well established dictionary learning algorithms such as K-SVD, Ho-SuKro and K-HOSVD.

Plug-and-Play (PnP) is an image restoration framework which leverages on a set of powerful denoisers to induce a priori information on the solution of a general image restoration problem. By using splitting techniques, such as the Half Quadratic Splitting (HQS), the PnP framework solves the classical regularized optimization problem where the regularizer-related step is replaced by a denoiser. In this talk, I will introduce HQS - Deep PnP which is mainly focus on Convolutional Neural Network (CNN) denoisers. I will show how increase the performances of PnP by considering a general framework where the denoiser act on a transformation of the image (e.g. the discrete gradient) and a further handcrafted regularization term is added (e.g. Total Variation). Moreover, I will prove that for the HQS - Deep PnP algorithm a fixed point convergence is guaranteed under certain assumptions. The good performances of the proposed approach for the task of non-blind denoising and deblurring are addressed through several experiments both on synthetic and real data.

The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane Cremona maps of fixed degree. Then, we will address the question of determining which plane Cremona maps of small degree are limits of maps of higher degree. This talk is mainly based on joint works with Jérémy Blanc and with Cinzia Bisi and Massimiliano Mella.

The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this seminar I review the model and discuss its large fluctuations and resulting transport properties, both annealed and quenched, under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity. In particular, by presenting large deviation estimates and the asymptotics of moments for the particle displacement, I show that the motion is superdiffusive in the annealed framework, whereas a normal diffusive behavior characterizes the motion conditional on a typical realization of the scatterers arrangement.

Sia G un gruppo di Lie complesso e semisemplice. Sia P un sottogruppo parabolico di G tale che il radicale unipotente P^u sia abeliano e B ⊆ P un sottogruppo di Borel. In questo caso, se P = LP^u è una decomposizione di Levi di P, diciamo che G/L è una varietà hermitiana simmetrica. Il sottogruppo di Borel B agisce su G/L con un numero finito di orbite. L'insieme di tali orbite può essere ordinato in maniera naturale tramite l’ordine di Bruhat. Per ognuna di queste orbite possiamo considerare l'insieme dei C-sistemi locali di rango 1 B-equivarianti a meno di isomorfismo. Si ottiene così un insieme di coppie orbita-sistema locale a cui possiamo dare un ordine che estende l’ordine di Bruhat e che Lusztig e Vogan chiamano G-ordine di Bruhat In questo seminario presenterò una caratterizzazione combinatorica del G-ordine nel caso in cui il sistema di radici di G sia simply-laced. Questo risultato può essere utile per migliorare l’efficienza nel calcolo dei polinomi di Kazhdan-Lusztig. Mostrerò inoltre un risultato simile per sistemi di radici di tipo B e alcuni risultati parziali per sistemi di radici di tipo C.

Nonlinear differential matrix equations generally stem from the semi-discretization on a rectangular grid of nonlinear partial differential equations (PDEs). The two main challenges related to approximating the solution of such matrix equations includes the high computational cost of time integrating the system when the matrices have large dimensions, as well as the cost related to evaluating the time-dependent nonlinear term at each timestep. In the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to an effective, structure aware low order approximation of the original problem. The nonlinear term is also reduced by means of a fully matricial interpolation using left and right projections onto two distinct reduction spaces, giving rise to a new two-sided version of DEIM. Several numerical experiments based on typical benchmark problems illustrate the effectiveness of the new matrix-oriented setting.

The study of invariants and coinvariants is a classical topic that goes back to the early days of modern algebra. About 25 year ago, Garsia and Haiman initiated the study of diagonal coinvariants of the symmetric group in relation to what has become known as the "n! conjecture". In particular, the combinatorics behind these objects has been intensively studied in recent years, in connection with certain operators defined via the Macdonald symmetric functions. In this talk we review what is known on this subject, and present some of the surprising conjectures that arose in the last two years.

Model theory is a branch of mathematical logic that uses tools from logic in order to explore mathematical structures (models). When those structures are of geometric nature, we tend to call this study "tame geometry". We will give a broad introduction to tame geometry, and survey some topics at the nexus of model theory and: (a) Diophantine geometry, focusing on recovering algebraic curves from "definable" sets with many rational points, (b) group theory, on a definable version of Hilbert's 5th problem, and (c) extremal combinatorics, around the Vapnik-Chervonenkis theory and its links to statistical learning (time permitting).

It is well known that, on a closed Riemannian manifold, the Laplace operator has discrete spectrum. One can wonder if its first positive eigenvalue has some geometric meaning. Cheeger's seminal work, which is now referred to as Cheeger's inequality, asserts that one can give to this first eigenvalue a geometric lower bound, called the Cheeger's constant. A natural question is to find an analogous statement for differential forms of any degree. During the talk we will review Cheeger's theorem and propose a generalization of Cheeger constant for (coexact) 1-forms on closed 3-manifolds. We shall refer to this constant as the open book constant. If time allows it, we will give some elements of the proof of our main theorem which may be thought as a Cheeger inequality for 1-forms on 3-manifolds: the first eigenvalue of the Hodge Laplacian acting on (coexact) 1-form is bounded from below by the open book constant. This is a joint work with Gilles Courtois (IMJ-PRG, Sorbonne Université).

In questo seminario offriro' un panorama dei problemi di cui mi sono occupata nel corso degli anni (problemi al contorno per il laplaciano su domini lipschitziani; diseguaglianze di tipo ``div-curl’’; regolarità in L^p di vari integrali singolari con nucleo olomorfo associati a domini non lisci). Poi mi concentrero' sui miei interessi piu' recenti, tra i quali: la proiezione di Szego associata ai ``worm domains’’ di Diederich e Fornaess; il commutatore della proiezione di Szego associata a domini strettamente pseudoconvessi non lisci; definizone degli spazi di Hardy per una classe di domini singolari e costruzione di nuclei di riproduzione per funzioni olomorfe nei suddetti, con applicazioni al triangolo di Hartogs. Tempo permettendo, descrivero’ brevemente la tesi del mio piu’ recente studente di dottorato (Dicembre 2019).

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining \ell_1-penalized least squares optimization with an appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. In this talk, we are discussing the analytic properties of this class of hypermodels, together with their sparsity promoting effects. The typical benefits of non-convex penalty terms will be coupled with the pleasant convexity guarantees thus making the way for a hybrid solver which allows to have the best of both worlds.

Starting with the early 1990's several outstanding problems of complex analysis were solved by new methods that involved non-homogeneous harmonic analysis, geometric measure theory and random geometric constructions. I mean here the problems of Painlev\'e, Ahlfors, Vitushkin, Denjoy and solved by Tolsa, David--Mattila and Nazarov--Treil--Volberg. This lucky encounter continued for the next 20 years, where those methods were used to solve several free boundary problems of Bishop and Guy David. I will try to give the exposition of ideas behind this development.

The Löwner framework is one of the most successful data-driven model order reduction techniques. Given k right interpolation data and h left interpolation data, the standard layout of this approach is composed of two stages. First, the kh x kh Löwner matrix L and shifted Löwner matrix S are constructed. Then, an SVD of L-ζS, ζ belonging to one of the data sets, provides the projection matrices used to compute the sought reduced model. These two steps become numerically challenging for large k and h in terms of both computational time and storage demand. We show how the structure of L and S can be exploited to reduce the cost of performing (L-ζS)x while avoiding the explicit allocation of L and S.

Let FI be the category of finite sets and injections. For a fixed space X, Church-Ellenberg-Farb demonstrated that packaging the sequence of ordered configuration spaces {PConf_n(X)} as a single object, namely a contravariant functor FI -> Top is very convenient in analyzing the "stable behavior" of the sequence. After reviewing the general framework, I will present general elementary conditions on X (which need not be a manifold) which allows the action of FI to be extended to a larger category. The extra structure yields several improvements in the onset of stabilization.

La didattica a distanza in questo periodo emergenziale è stata l'unica modalità che ha permesso a scuole e università di proseguire le varie attività di formazione.Nel seminario si discuterà come le nuove tecnologie, quali ad esempio un Digital Learning Environment, possano facilitare la didattica online non solo in tempi di emergenza. Particolare attenzione sarà rivolta al loro ruolo chiave che possono svolgere nell'insegnamento e nell'apprendimento della matematica e più in generale delle discipline STEM.

The purpose of the presentation is to introduce a version of a stochastic process known in the physical literature as Lévy-Lorentz gas and derive laws of large numbers and functional limit theorems for it. The model can be described as follows: we consider a point process omega given by an ordered array of points on the real line. We call omega the random medium. The nearest-neighbour distances between the points are i.i.d variables in the domain of attraction of a beta-stable distribution with beta belonging to (0,1) U (1,2). A random walk Y takes place on the medium according to the following rule. Independently of omega there exists a random walk S on the integers with i.i.d increments in the normal domain of attraction of an alfa-stable distribution with alfa belonging to (0,1) U (1,2). The role of S is to drive the dynamics of Y on the point process omega. For example, if a realization of S is (0,3,-1,…), the process Y starts at the origin of the real line, then jumps to the third point of omega to the right of 0, then to the first point of omega to the left of 0, and so on. Our process of interest is Y. We may describe it as a Lévy flight on a one-dimensional Lévy random medium. For all combinations of the parameters alfa and beta, we prove the annealed functional limit theorem for the suitably rescaled process Y, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

An old conjecture of Almgren states that for every convex and coercive potential $g: \mathbb{R}^d\to \mathbb{R}$, every convex and one-homogeneous anisotropy $\Phi : \mathbb{R}^d\to \mathbb{R}^+$ and every volume $V>0$, the minimizers of \[ \min_{|E|=V} \int_{\partial E} \Phi(\nu) d\mathcal{H}^{d-1} + \int_{E} g dx \] are convex. I will review the known results on this problem and present recent progress obtained with G. De Philippis on the connectedness of the minimizers for smooth potentials and anisotropies. Our proof is based on the introduction of a new ``two-point function'' which measures the lack of convexity and which gives rise to a negative second variation of the energy.

Racks and quandles are binary algebraic structures arising in knot theory, representation theory of the braid groups and the study of Hopf algebras. The first part of the talk is an overview on quandle theory and the motivation behind it. The main tools used in the study of racks and quandles are group and module theory. In the second part of the talk we introduce some new ideas coming from universal algebra. In particular, we adapt the commutator theory developed by Freese and McKenzie for arbitrary algebraic structures to racks and quandles. The goal of such theory is to define the notions of abelian and central congruences, extending the familiar definitions in the setting of groups and other classical varieties in order to talk about solvable and nilpotent objects. For racks, the commutator theory has a nice and sharp interpretation in group theoretical language. We also provide some applications, both towards classification problems for finite quandles and knot theory.

We try to describe in mathematical terms a class of models M of languages and of (learning) transformations of M which, when applied to (a collection of texts from) a given library (corpus) L, somehow embedded into M, would result in a model of the language behind L.

We construct a combinatorial abstraction of the Leray spectral sequence associated with the De Concini—Procesi compactification of a complex hyperplane arrangement complement. We show that its homological properties generalise the ones that we expect from topology in the realizable case. Geometric and Hodge-theoretic arguments are replaced by combinatorial blowups, Groebner bases, and sheaves on posets.

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

Isogeometric analysis is an evolution of the finite element method: it employs B-splines or their generalization both to represent the computational domain and to approximate the solution of the considered partial differential equation. The high-continuity of isogeometric basis functions leads to several advantages, e.g. higher accuracy per degree-of-freedom, but it introduces also challenging problems at the computational level: one of the major issues is the efficient solution of linear systems. In this talk, I will focus on the study of an efficient solver for a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, I propose a preconditioner that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. polynomial degree and the time required for the application is almost proportional to the number of degrees-of-freedom. This is based on a joint work with G. Sangalli, M. Tani and G. Loli.

Topological Data Analysis (TDA) is a new and fast-growing branch of math, addressing the need for extracting information from big data sets. One of the methods of TDA is persistence theory, which has proven to be very informative. The standard way to approach persistence theory is using persistent homology. In this seminar, I present a different approach, based on the study of tame parametrised chain complexes. This choice is indeed useful, since, firstly, the study of tame parametrised chain complexes includes the one of persistent homology, and thus is more general. Secondly, there are many powerful theoretical tools available in this setting, and such tools allow us to retrieve invariants, and therefore information, in more general cases. I will provide a quick introduction to these theoretical tools, namely to model category theory, and then focus on some examples to explain in details the constructions and the invariants."

In this talk I will paint a self‐consistent scenario for information processing in shallow neural networks: I will present a minimal reference framework where what is learnt (e.g. via contrastive divergence on restricted Boltzmann machines) is then retrieved (e.g. via standard Hebbian mechanisms à la Hopfield). Then I will generalize this scheme by discussing three variations on theme: the tradeoff between dilution and multitasking capabilities, the tradeoff between storage and resolution and the mechanism of "sleeping and dreaming".

A renowned space of entire functions of one complex variable is the Paley–Wiener space P W A , that is, the space of entire functions of exponential type A whose restriction to the real line is square integrable. After recalling some basic properties of P W A I will present a generalization of this space in several complex variables. In particular, I will consider entire functions which satisfy a suitable exponential growth condition and whose restriction to the boundary of the Siegel half-space satisfy some integrability conditions. Time permitting, I will provide a Paley–Wiener type characterization and a sampling result. This is a joint work in progress with Marco Peloso and Maura Salvatori.

I

The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. In the talk, we present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a strategy for its numerical implementation is also outlined and will be subject of future investigation.

Lecture 2: “Difficult case” 1. Focusing Nonlinear Schr¨odinger equation. Cherednik differential for the Nonlinear Schr¨odinger equation. Characterization of divisors for real solutions in terms of Cherednik differential. Elementary characterization of admissible divisors. Regularity of all real solutions. 2. Sine-Gordon equation. Cherednik differential for the Since-Gordon equation. Characterization of divisors for real solutions in terms of Cherednik differential. Elementary characterization of admissible divisors. Regularity of all real solutions. 3. Number of connected components in the space of solutions for a fixed spectral curve. 4. Real regular Kadomtsev-Perviashvili I solutions.

Subdivision schemes are iterative methods for the construction of curves and surfaces from a set of initial control points. Here we focus on the univariate case (i.e. curves) and on interpolatory subdivision (i.e. schemes with a limit curve that interpolates the initial data). This family of schemes can be subdivided in two classes, primal and dual. An algebraic characterization of both classes in terms of certain Laurent polynomials (called symbols) will be provided, together with some examples.

Lecture 1: “Easy case” 1. The problem of selecting real and real regular solutions. Real algebraic curves. Real ovals. M-curves. 2. “Easy case” and “difficult case”. 3. Real solutions of the Korteweg–de Vries equation. Real regular solutions. Regular and singular real components. 4. Defocusing Nonlinear Schr¨odinger equation. Real regular solutions. Regular and singular real components. 5. Real regular Kadomtsev-Petviashvili II solutions and M-curves.

Una città etrusca dipendeva da un sito esterno con una buona visione sulla città detto auguraculum (A). Nell’auguraculum l’augure dirigeva l’augurazione della città. La diagonale principale della città era fissata da un raggio di sole all’alba o al tramonto di un giorno dell’anno (Natale della città). Su questa diagonale si fissava un punto U (umbilicum=centro del mondo) a distanza R dall’auguraculum. In U si faceva l’inaugurazione della città e si tracciava la diagonale secondaria lungo un raggio di sole al tramonto dello stesso giorno o all’alba di quello successivo. Si fissava il raggio del pomerio, un cerchio di centro U che circondava l’urbe identificata con l’orbe dell’universo. Le intersezioni delle diagonali con il pomerio fissavano i quattro vertici di un quadrilatero che si dimostra essere un rettangolo per simmetria rispetto alle bisettrici degli angoli tra le diagonali. Le stesse bisettrici formano la croce sacrale che indica le strade principali: decumano massimo e cardio maximus. Le bisettrici si trovano tramite le semicorde: i semilati del rettangolo. L’asse dell’universo (Cardine) proveniente dal polo Nord celeste penetrava nel punto U della Terra e ne usciva nel punto S, estremo Sud del pomerio. La proiezione del Cardine sommerso sul piano della città era la parte meridionale del cardine. Si considerano due proiezioni del Cardine sulla verticale in U e in S, rappresentate da una coppia di colonne cosmiche. Le colonne cosmiche sono intese come sostegni del Cielo sulla Terra, o viceversa. A causa del diverso moto di rotazione del Cielo e della Terra queste colonne, intese come plastiche, appaiono a torciglione. Per gli Etruschi erano piuttosto dei perni rigidi detti cardini. A volte erano un sistema articolato di infiniti perni inseriti tra i primi vicini. Ovviamente questo sistema era totalmente instabile. Secondo gli Indiani attorno ad un perno si avvolgeva una corda che veniva tirata agli estremi in una specie di tiro alla fune che faceva girare il perno stesso. Vincenzo Grecchi: Bologna: il mistero delle quattro croci e il rito etrusco (Acacne Editrice, Bologna 2019)

Una città etrusca dipendeva da un sito esterno con una buona visione sulla città detto auguraculum (A). Nell’auguraculum l’augure dirigeva l’augurazione della città. La diagonale principale della città era fissata da un raggio di sole all’alba o al tramonto di un giorno dell’anno (Natale della città). Su questa diagonale si fissava un punto U (umbilicum=centro del mondo) a distanza R dall’auguraculum. In U si faceva l’inaugurazione della città e si tracciava la diagonale secondaria lungo un raggio di sole al tramonto dello stesso giorno o all’alba di quello successivo. Si fissava il raggio del pomerio, un cerchio di centro U che circondava l’urbe identificata con l’orbe dell’universo. Le intersezioni delle diagonali con il pomerio fissavano i quattro vertici di un quadrilatero che si dimostra essere un rettangolo per simmetria rispetto alle bisettrici degli angoli tra le diagonali. Le stesse bisettrici formano la croce sacrale che indica le strade principali: decumano massimo e cardio maximus. Le bisettrici si trovano tramite le semicorde: i semilati del rettangolo. L’asse dell’universo (Cardine) proveniente dal polo Nord celeste penetrava nel punto U della Terra e ne usciva nel punto S, estremo Sud del pomerio. La proiezione del Cardine sommerso sul piano della città era la parte meridionale del cardine. Si considerano due proiezioni del Cardine sulla verticale in U e in S, rappresentate da una coppia di colonne cosmiche. Le colonne cosmiche sono intese come sostegni del Cielo sulla Terra, o viceversa. A causa del diverso moto di rotazione del Cielo e della Terra queste colonne, intese come plastiche, appaiono a torciglione. Per gli Etruschi erano piuttosto dei perni rigidi detti cardini. A volte erano un sistema articolato di infiniti perni inseriti tra i primi vicini. Ovviamente questo sistema era totalmente instabile. Secondo gli Indiani attorno ad un perno si avvolgeva una corda che veniva tirata agli estremi in una specie di tiro alla fune che faceva girare il perno stesso. Vincenzo Grecchi: Bologna: il mistero delle quattro croci e il rito etrusco (Acacne Editrice, Bologna 2019)

There is a new approach that has been proposed recently for the study of certain regular semigroups. This new approach involves the study of finite primitive groups. Via this method, many problems in the context of regular semigroups translate into natural problems in finite primitive groups. We give some details concerning the relation between regular semigroups and finite primitive groups. In particular, we discuss the Road Closure Conjecture on finite primitive groups, which is strictly related to a possible classification of certain idempotent generated regular semigroups.

Lie groups are one of the most fundamental tool in mathematics. They are ubiquitous, starting from mathematical physics, passing through number theory and landing into the world of dynamical systems. In this talk I will give a general overview about the definition of Lie group and I will list some of its property. My attention will be mainly focused on two aspects. I am going to talk about the Iwasawa decomposition of non-compact Lie groups, which can be interpretated as a suitable generalization of QR-factorization. This decomposition result will allow to study some topological properties of Lie groups. Then I will move to some analytical aspects related to measure theory. Studying unimodular lattices on the plane we will "compute" the volume of the quotient SL(2,R)/SL(2,Z) in terms of Haar measure. This will be a very introductory seminar which will be the starting point of a minicourse about Lie groups and representations of lattices.