Topics in Mathematics 2019/2020

Spiegherò la costruzione dei cosiddetti moduli di Verma ed illustrerò attraverso esempi alcuni problemi, aperti e non, nello studio di questi moduli.

Many real systems in biology, economics, finance, social sciences can be represented as temporal networks, i.e. graphs whose structure is not constant but new links are formed and old ones are destroyed at each time. In the first part of my talk, I will introduce a general class of random models for networks, the exponential random graph (ERG) family, I show the connection with the maximum entropy principle and with the latent variables models, and I describe the inference problem when data are available. In the second part, I show some recent advancements to the use of ERGs to the modeling of temporal networks highlighting different mechanisms which are responsible for the memory in links dynamics. I present some applications to financial problems both for the static and for the dynamic case.

Applying mathematics to real life problems often requires different layers of approximation. For example, to study very complex phenomena we rely on simplified models (e.g. differential equations) that usually do not have explicit solutions. To estimate such solutions we pass trough discretizations and algorithms whose results depend on the amount of invested resources (e.g. time, computational power). In this context, approximation theory takes care of how functions can be approximated using simpler ones and how the approximation error behaves with respect to the properties of the functions involved, exploiting knowledges from different areas of mathemat- ics. In this talk we review fundamental notions and results of constructive approximation and use them to introduce wavelets, frames and subdivision schemes, while showing examples linking also to other topics.

A network is a good representation of a system with many interacting agents and networks with macroscopic structures (communities, hierarchies, cores, ...) naturally emerge from interactions regularities at a microscopic level. In many applications ( financial networks, biological networks, social networks) much of the information hidden in the data can be extracted from the detection of macroscopic structures wich are robust against microscopic noise. Starting from motivations and possible applications I will introduce the inference framework based on the Stochastic Block Model and the statistical mechanics approach to the associated detectability problem. This approach at the same time allows to depict the problem complexity in terms of detectability phase transitions and offers an efficient solution through a Belief Propagation algorithm.

Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.

Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.

Linear inverse problems are typically ill-posed in the sense of Hadamard and they require regularization strategies in order to compute meaningful approximation of the desired solution. Traditional regularization methods solve an optimization problem whose objective function consists of a data-fit term, which measures how well an image matches the observations, and one or more penalty terms, referred to as regularization terms, which promote some desirable properties of the sought-for solution. The quality of the computed solution strongly depends on the choice of the regularization parameters weighting the penalty terms. An advantage of multi-penalty regularization, when compared with one-penalty regularization, is that different features of the solution can be enhanced by using several regularization terms. However, a drawback of multi-penalty regularization is that one has to select the values of several regularization parameters. In this talk, we present spatially adapted multi-penalty regularization for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. In the presented approach, the function to be minimized contains one $\ell_1$ penalty term and several spatially adapted $\ell_2$ penalty terms. An iterative procedure is illustrated for the automatic computation of all the regularization parameters. As a case study, we present the application of adaptive multi-penalty regularization to the inversion of two-dimensional NMR data.

Il seminario, che cercherà di essere introduttivo ed elementare quanto possibile, sarà dedicato a descrivere il problema (largamente aperto) della moltiplicazione nell'anello delle coordinate di una varietà simmetrica. Dopo aver introdotto gli oggetti necessari, descriverò una vecchia congettura di Stanley (1989) sulle funzioni simmetriche di Jack e spiegherò le sue conseguenze nel problema della moltiplicazione.

Simon's conjecture is an interesting open problem in Combinatorics. In the first part of the seminar, we provide all the necessary notions to formulate it. Recently, several authors gave new improvements on this topic. In the second part, we present some of these results, pointing out the main ideas and giving some new questions arising from them.

In questo seminario inizierò parlando brevemente di alcuni meccanismi che stanno alla base di particolari fenomeni caotici, per poi concentrarmi sulla teoria di Melnikov. In particolare analizzerò il caso di un sistema dinamico a tempo continuo autonomo che presenta una traiettoria omoclina (che quindi converge ad un punto critico sia nel passato che nel futuro), soggetto ad una perturbazione non-autonoma. La teoria di Melnikov fornisce condizioni che garantiscono la persistenza dell’omoclina e la nascita di fenomeni caotici. Il modello più noto per questa tipologia di fenomeni è il pendolo (non-lineare) perturbato. Si vedranno brevemente estensioni al caso multidimensionale e a quello discontinuo (piecewise smooth) che trova applicazione nella modellizzazione dei rimbalzi o dell’attrito strisciante.

Nella prima parte del seminario parlerò di varie disuguaglianze geometriche di tipo isoperimetrico. Partendo dalla disuguaglianza isoperimetrica classica (quella che ci dice perché le bolle di sapone all'equilibrio devono essere sferiche), menzionerò alcune sue controparti funzionali e come si possono sfruttare per dimostrare disuguaglianze (di nuovo) geometriche ma definite tramite EDP. Nella seconda parte del seminario ci concentreremo su una delle svariate evoluzioni delle disuguaglianze di riarrangiamento: lo studio della loro rigidità quantitativa. Oltre a spiegare questo concetto, e citare alcuni dei casi più celebri risolti recentemente, vedremo alcune applicazioni (ancor più recenti) di tali risultati.

A quite useful philosophy in mathematics is to use the sharpness of an inequality regarding the "shape" of a topological space in order to detect a precise geometry on it: more precisely, the maximal value of the inequality usually allows to identify a specific geometric structure. Think for instance either to the applications of arithmetic/geometric mean inequality or to the isoperimetric inequality on the plane. Something similar happens in the world of Zimmer's cocycle theory. In this seminar we are going to focus our attention on Zimmer's cocycles associated to the fundamental group a surface S with genus bigger than or equal to 2. If such a measurable cocycle admits a (generalized) boundary map, one can define the notion of Euler number. The latter well behaves along cohomology classes and its absolute value is bounded by the modulus of the Euler characteristic of S. Remarkably the maximal value is attained if and only if the cocycle is cohomologous to a hyperbolization. The first part of the talk will be a gentle introduction to measurable cocycles and boundary theory. Then, we are going to introduce the orientation cocycle on the circle. Finally we will define the Euler number of a measurable cocycle and we will discuss its rigidity property. This is a joint work with Marco Moraschini.

The idea of singularity is found in many parts of mathematics, capturing the idea of a position where some regular behavior breaks down. A standard situation is in linear algebra where for a linear transformation or matrix rank deficiency corresponds to singularity. This provides a basic model for other settings, especially for differentiable functions between Euclidean spaces. Results of Marston Morse (the Morse Lemma) and Hassler Whitney (stable singularities of mappings from the plane to the plane) led to pioneering work in differential topology by René Thom, whose interest in biological morphogenesis gave rise to Elementary Catastrophe Theory and a wide interest in mathematical models of singularity. The foundations of singularity theory were developed, in which the concepts of transversality and stratification played an important role. In the first part of the seminar I will outline some of this history and ideas from singularity theory. In robotics, kinematic mappings relate inputs, outputs and constraints. The impact of singularities on robotic control systems was recognised in the 1960s. Subsequent interest in the variety of ways that singular phenomena occur in robot kinematics has led to a large literature on the subject. In the second part of the seminar, I will discuss some of my research on kinematic singularities.