Topics in Mathematics 2020/2021

We introduce some types of fractional derivatives in one real variable, and try to solve some analogs of ordinary differential equations, in the form \[ D^\alpha u(t) = Au(t) + f(t), \quad t \in [0, T], \] where $A$ is a square $N \times N-$matrix, $f : [0, T] \to \C^N$, and prescribed proper initial conditions are given. Finally, we try to explain how to extend these elementary results to the important case that $A$ is a linear, not necessarily continuous, operator in a Banach space $X$.

The three-dimensional reconstruction of an object is an interesting topic with many applications in different fields and has attracted several researchers. The applications range goes from the biomedical 3D reconstruction of human tissues to the approximation of the surface of astronomical objects, from archeology for the digitization of artistic works to the recent development of 3D printing. The first being interested in this problem were some opticians in the Fifties-Sixties. Afterwards, B.K.P. Horn first formulated the Shape-from-Shading (SfS) problem for a single gray-level image of the object. The goal was to get the 3D surface represented in the input image solving a partial differential equation or a variational problem. This problem gave rise to an expansion in the field of mathematics and some researchers tried to prove the well-posedness in the framework of weak solutions. The first works of Lions, Rouy and Tourin in the early 90s inserted the SfS problem in the context of the viscosity solutions frameworks, hence in a much more theoretical area. In this seminar I will start dealing with the orthographic SfS problem with Lambertian reflectance model, the classical and simplest setup for this ill-posed problem that can be modeled by first order Hamilton-Jacobi equations. During the seminar I will briefly introduce some notions of Hamilton-Jacobi equations, viscosity solutions and other ingredients necessary to understand the problem in a general setting. I will continue exploring some non-Lambertian reflectance models and we will see how it is possible to derive a well-posed problem adding information in a natural way. Finally, I will talk about the more recent Shape-from-Polarization problem and the advantages of it with respect to the SfS.

I numeri reali si differenziano per le loro caratteristiche diofantee, che riflettono diverse proprietà geometrico-dinamiche delle orbite sotto l'azione di PSL(2,Z). La teoria analitica dei numeri studia le proprietà metriche, come misura o dimensione di Hausdorff, dell'insieme dei numeri reali di un certo tipo diofanteo. In particolare, l'insieme dei numeri male approssimabili ha misura zero e dimensione 1 nella retta. Inoltre esiste un' esaustione naturale in sottoinsiemi Bad(c), la cui dimensione converge ad 1 quando c va a zero. D. Hensley ha ottenuto l'asintotico al primo ordine in c della dimensione di Bad(c), attraverso un'analoga stima per l'insieme dei numeri reali la cui frazione continua ammette coefficienti parziali uniformemente limitati. La dimostrazione di questo risultato illustra come la frazione continua permette di introdurre strumenti dinamici, quali l'operatore di trasferimento ed il formalismo termodinamico, che forniscono informazioni metriche molto più precise che un'analisi puramente analitico-geometrica. Tale approccio si estende a diversi contesti geometrico-dinamici. In particolare vedremo una versione generalizzata dell'asintotico di Hensely, in cui la nozione di numeri male approssimabili è riferita non più all'azione di PSL(2,Z) ma a quella di un lattice non-uniforme in PSL(2,R).

I will discuss a theme which, at various levels of complexity, is pervasive in algebra and geometry: The attempt to parameterize algebraic or geometric objects naturally leads to the problem of constructing quotients, almost always by the action of a group. I will start by discussing elementary problems such as parameterizing finite subsets of points in the plane or conjugacy classes of matrices, showing how they lead to the so called geometric invariant theory, which also has a more differential geometric counterpart, called symplectic reduction. In the second half of the talk I will discuss the parameterization of representations of the fundamental group of a surface, and the closely related notion of moduli spaces of vector or Higgs bundles on a Riemann surface, still from the point of view of quotient constructions.

Lo scopo del seminario è quello di esporre ad un vasto pubblico la recente dimostrazione della log-concavità di certi polinomi. Il polinomio cromatico di un grafo conta il numero di colorazioni possibili di un grafo. Negli anni '70 è stato congetturato che i suoi coefficienti $\omega_i$ formino una sequenza log-concava, cioè \[ \omega_i^2\geq \omega_{i-1}\omega_{i+1}.\] L'enunciato della congettura si può dare più in generale per i coefficienti del polinomio caratteristico di un matroide. Queste congetture sono state dimostrate rispettivamente nel 2012 e nel 2018. Le tecniche usate sono sorprendenti e proverò a darne un'idea: costruirò, tramite blow up, una varietà proiettiva e ne studierò l'anello di Chow (che coincide con la coomologia). Infine dal teorema di Hodge-Riemann segue banalmente la disuguaglianza cercata. Nel caso di matroidi la corrispettiva varietà non esiste, ma si può comunque definire un anello con le proprietà desiderate e dimostrare la log-concavità.

Numerical first-order methods are the most suitable choice for solving large-scale nonlinear optimization problems which model many real life applications. Among these approaches, gradient methods have widely proved their effectiveness in solving challenging unconstrained and constrained problems arising in machine learning, compressive sensing, image processing and other areas. These methods became extremely popular since the work by Barzilai and Borwein  (BB) (1988), which showed how a suitable choice of the steplength can significantly accelerate the classical Steepest Descent method. It is well-known that the performance of gradient methods based on the BB steplength does not depend on the decrease of the objective function at each iteration but relies on the relationship between the steplengths used and the eigenvalues of the average Hessian matrix; hence BB based methods are also denoted as Spectral Gradient methods. The first part of this seminar will be devoted to a review of spectral gradient methods for unconstrained optimization while the second part will focus on recent advances on the extension of these methods to the solution of large nonlinear systems of equations, the so-called Spectral Residual methods. These methods are derivative-free, low-cost per iteration and are particularly suitable when the Jacobian matrix of the residual function is not available analytically or its computation is not relatively easy. In this framework, numerical experience will be presented on sequences of nonlinear systems arising from rolling contact models which play a central role in many important applications, such as rolling bearings and wheel-rail interaction.

In the first part of the seminar, we will make a quick introduction to variational problems to motivate the interest in critical points. Then, we will focus on minimax critical points and introduce a useful tool to guarantee their existence: the Mountain Pass Theorem. We will start from the finite-dimensional setting, discussing a result by Courant, and arrive at the infinite-dimensional case, stating the classical Mountain Pass Theorem by Ambrosetti and Rabinowitz. We will try to stress the differences between the two cases and the need for compactness assumptions through counterexamples. In the second part of the seminar, we will show an application of the Mountain Pass Theorem to a semilinear elliptic problem with Neumann boundary conditions and, time permitting, we will try to face the lack of compactness in a particular case.

We present a selection of research topics concerning the study of stochastic differential equations (SDEs) that arise in social, natural and physical sciences. We discuss two macro-classes: (i) jump-diffusion equations with degenerate behavior of their coefficients, and (ii) mean-field (McKean-Vlasov) diffusion equations. For the class (i) we first describe the main features of the models and the general connection with ultra-parabolic differential operators of Kolmogorov type. We then present some recent developments regarding regularity and asymptotic properties of the transition densities for some specific models. For the class (ii) we first describe the interplay between McKean-Vlasov SDEs, mean-field interacting particle systems, and non-linear Fokker-Planck equations. We then discuss some of the problems related to the existence, uniqueness, asymptotic properties of the solutions, as well as to their numerical approximations. We finally present some recent results in particular cases.