Chiara Cicolani, University of L'Aquila, November 7, 2024
Asymptotic dynamics of different kinds of opinion formation models with leadership and time-delayed coupling
We present some recent results we obtained in analyzing the convergence to consensus of different types of opinion formation models. We considered the presence of leadership, time-delayed coupling, and non-universal interaction. The first model describes the dynamics of two groups of agents interacting with each other via subsets of individuals, namely the leaders, and natural time delay effects are considered.
In the second example, we deal with a Hegselmann-Krause opinion formation model with non-universal interaction, time delay, and possible lack of connection between the agents. More precisely, we analyze the situation in which the agents involved in the opinion formation do not transmit information to all the other agents of the system. Also, agents linked to each other can suspend their interaction at certain times. We can perform the same analysis on the second-order model, the so-called Cucker-Smale model, and we can establish the exponential asymptotic flocking for it. Moreover, we consider a different kind of structure for a Hegselmann-Krause type opinion formation model in which we assume a property that ensures the presence of a common influencer between two states. Moreover, we explore a case in which this property does not hold, where leaders with independent opinion dynamics appear. Finally, some numerical results are presented.
Ivan Gallo, University of L'Aquila, October 10, 2024
Evaluation of Non-linear Value Adjustments under multiple credit risks
We consider the problem of correctly pricing financial derivatives that might be subject to multiple credit risks, such as default or liquidity risk, that gained importance after the financial crisis of 2008-09. An important achievement in the mathematical modelling of the problem was the representation of the value of such derivatives as solutions of appropriate Backward Stochastic Differential Equations (BSDE), which might be solvable in the samples cases. When various risks are taken into account simultaneously, and correlation is admitted between the processes underlying the price formation, the picture becomes much more complex, and although the BSDE representation still applies, explicit solvability becomes impossible.
Monte Carlo simulations,usually requiring long computational times, are often the only way to get an approximation of the solution, so it might be important to develop alternative approximation techniques that require shorter computational times yet preserving accuracy. Once the BSDE's representation is developed, in a Markovian setting, the derivative's can be rewritten as a deterministic function of the state variables, which verifies a non-linear PDE. Thus, we decided to employ a PDE discretization approach to approximate the PDE solution. By employing an adaptation of the simple method of lines, we were able to construct an approximation method that turned out to be accurate and efficient, thus producing a valid alternative to Monte Carlo simulations.
Roberto Cavassi, University of L'Aquila, June 6, 2024
Problems related to data analysis in non Euclidean spaces
Researchers have to handle spherical data sets in many research areas, like geophysical and astrophysical fields. We can think, for instance, of a planet's gravity maps, or our universe's cosmic microwave background. Since these data sets are non-stationary, it is better to study them through non-stationary techniques, and `Fast Iterative Filtering’ has proven to be an interesting and useful method to achieve this goal, in 1D or 2D cases. However, some problems arise when we want to extend this method to handle spherical data, i.e. non-Euclidean settings, since FIF relies on convolutions that cannot be directly generalized from R^n to the sphere.
In this presentation, after introducing a continuous operator to filter signals defined on the sphere we analysed its discretisation through the Generalised Locally Toeplitz (GLT) sequences of matrices. Using some properties from the GLT theory we show that it is possible to guarantee the convergence of FIF extended to the sphere.In the final part of the presentation, we show a few examples of applications of this newly developed method to real-life geophysical signals and compare them with the ones produced using classical spherical harmonics.
Alessio Barbieri, University of L'Aquila, May 9, 2024
Well-Posedness of Evolution Equations: from Euler to Perturbation Theory for Strongly Continuous Semigroups
In the XVIII century Euler defined the exponential function as the unique solution to a first order linear Cauchy problem. In the following years this idea was developed by many mathematicians giving rise to Semigroup Theory, a branch of Functional Analysis aimed to find abstract tools to deal with evolution processes. In this seminar we retrace the main steps of this route. Then we provide some basic notions of Semigroup Theory in relation to the well--posedness of evolution equations. In particular we present some new results concerning the perturbation of generators of strongly continuous semigroups. Finally, we present some concrete examples on evolution PDEs.
Stefano Abbate, GSSI, February 14, 2024
"Non-local" calculus for beginners
Differential calculus, as developed by Newton and Leibniz, determines the concept of a derivative from the definition of limit. Since this approach depends only “locally” on the point where it is defined, classical differential calculus can be narrow to represent many physical phenomena. Fractional derivatives are one of the oldest and most known approaches to overcome this issue. In the last century, substantial progress has been developed in establishing “non-local” alternatives. The aim of this seminar is to present a rigorous approach to define “nonlocal” operators and a “non-local” vector calculus, introduced by Du et al. in 2011. The goal is to show the analogy with classical differential operators and classical formulas of differential calculus. In this setting, we arrange some models corresponding to the ones known in elementary physics. In particular, we introduce a “non-local” balance law which leads to the study of a basic elliptic equation. In the last part, we give a brief overview on the study of a more complex model, specifically a “non-local” version of the Cahn-Illiard-Oono equation.
Giacomo Vecchiato, GSSI, January 17, 2024
A mathematical description of an ecological behaviour for the mass distribution in climbing plants
Climbing plants exhibit specialized shoots, called “searchers”, to cross spaces and alternate between spatially discontinuous supports in their natural habitats. To achieve this task, searcher shoots combine both primary and secondary growth processes of their stems to support, orientate and explore their extensional growth into the environment. There is an increasing interest in developing models to describe plant growth and posture. In this talk, we investigate an ecological behaviour for the mass distribution in climbing plants. More precisely, we support the thesis that the mass is distributed in such a way as to maximize the length of the shoot, without generating undue stress.
To achieve this goal, we first develop a 2D model that combines the sensing activity (e.g. photo-, gravi-, proprioceptive sensing) and the elastic responses of searcher shoots. Such a model permits the understanding of the role of radial expansion in the shape of the shoot, paving the way for the formulation of an optimal control problem with state constraint. Such a problem is approached from two different perspectives: an analytical one and a computational one. In the analytical approach, we obtain an equation in feedback form that relates the radius, the curvature of the shoot and the stress threshold together. This result supports the computational approach, which uses reinforcement learning techniques to produce a numerical approximation for the optimal distribution of the radius along the searcher shoot.
The numerical simulations are supported by the experimental data on two climbing plant species: Trachelospermum jasminoides (Lindl.) Lem. and Condylocarpon guianense Desf..
Giulia Carigi, University of L'Aquila, December 14, 2023
Long-time behaviour of stochastic geophysical fluid dynamics models
The introduction of random perturbations by noise in partial differential equations has proven extremely useful to understand more about long-time behaviour in complex systems like atmosphere and ocean dynamics or global temperature. Considering additional transport by noise in fluid models has been shown to induce convergence to stationary solutions with enhanced dissipation, under specific conditions. On the other hand, the presence of simple additive forcing by noise helps to find a stationary distribution (invariant measure) for the system and understand how this distribution changes with respect to changes in model parameters (response theory). I will discuss these approaches with a multi-layer quasi-geostrophic model as example.
Andrea Del Prete, University of L'Aquila, December 7, 2023,
Uniqueness of minimal graphs in the Heisenberg group
In this talk we discuss the uniqueness of solutions to the Dirichlet problem for the minimal surface equation both in the Euclidean space and in the Heisenberg group. We will describe the analytical methods employed to establish uniqueness in Euclidean space and, where feasible, adapt these methods to the Heisenberg group.
Jessica Alessandrì, University of L'Aquila, November 9, 2023
Fields generated by torsion points of elliptic curves
In the study of elliptic curves, division fields, that are fields generated by torsion points (i.e. points of finite order) of the curves, play an important role. For example in Galois representation, modularity and the proof of the Mordell-Weil theorem.
In this talk, after an introduction to elliptic curves, we will present some results on torsion points and division fields. In particular, we will see how to compute torsion points, their properties and how to find a "small" set of generators. We conclude by giving some results for 7-division fields of CM elliptic curves (joint work with L. Paladino).
Lorenzo Pescatore, University of L'Aquila, October 19, 2023
Global regularity of strong solutions to the 1D Stochastic Quantum Navier-Stokes equations with density dependent viscosity
In this talk, we consider a stochastically forced quantum viscous fluid on the one-dimensional torus. We will give an overview of the main analytical problems related to the study of quantum fluids, focusing on the description of the stochastic setting and on the differences with the deterministic dynamics.
We prove the global well-posedness of the problem in the framework of strong pathwise solutions, which are strong solutions in both PDEs and probability sense. In particular, we establish a local existence and uniqueness result and then we perform an extension argument by using some a priori estimates that allow us to control the arising of vacuum states of the density. The analysis is performed for a wide class of density dependent viscosity coefficients and as a byproduct of our results we get the global well-posedness also for the deterministic case. This is a joint work with Donatella Donatelli and Stefano Spirito, University of L’Aquila.
Alessandro Camastra, University of Bari, June 14, 2023
Degenerate differential models in Control Theory and generation theorems of semigroups
Degenerate differential problems have been the subject of a large number of papers in the last years. Indeed, many problems coming from Physics, Biology and Mathematical Finance can be described in a natural way in terms of equations whose leading operator present a degeneracy occurring at the boundary or in the interior of the space domain. In a different context degenerate operators have been extensively studied since W. Feller's investigations, where the main motivation was the analysis of transition probabilities.
Moreover, the fields of application of these problems are so wide that it is not surprising that several comprehensive expositions deal with this topic; for example we can think to the applications to controllability theory, which has been considered for parabolic equations over the last forty years. Thus, one of the main purposes of this talk is to illustrate some of the main ideas and techniques of Control Theory, a discipline that has some deep roots.
In the second part we present a suitable notion of degeneracy for parabolic evolution equations and we analyze in detail a degenerate fourth order linear differential operator in one space dimension, proving that it is non negative and self-adjoint in a suitable weighted Hilbert space. As a consequence it generates a cosine family and an analytic semigroup. Thanks to this functional-analytic framework we can obtain the well posedness of the associated degenerate parabolic equation with Dirichlet boundary conditions. In particular these results, that are part of a joint work with Genni Fragnelli (Department of Ecology and Biology, Tuscia University), give the possibility to extend some previous expositions for which the one-dimensional Cahn-Hiliard equation is well posed.
Elisa Continelli, University of L'Aquila, May 25, 2023
Time delays in opinion formation models
Recently, multi-agent systems have caught the attention of many researchers. Among them, there are the celebrated Hegselmann-Krause opinion formation model and its second order version, the Cucker-Smale model, introduced for the description of flocking phenomena. For such systems, the convergence to consensus, in the case of Hegselmann-Krause model, and the exhibition of asymptotic flocking, in the case of the CuckerSmale model, are investigated. It is natural to introduce in the above models time delays. Indeed, in the applications, it could happen that the system’s agents do not receive the information coming from the other agents ”promptly”. The convergence to consensus for the Hegselmann-Krause model in presence of time delays have been studied by many authors. Most of them, require a smallness assumption on the time delay size. However, recently, M. Rodriguez Cartabia has proved the exhibition of asymptotic flocking for the Cucker-Smale model with constant time delay without requiring the time delay to be small. Extending the arguments of M. Rodriguez Cartabia, we are able to prove the exponential convergence to consensus for the Hegselmann-Krause model with time variable time delays, without assuming the smallness of the time delay and the monotonicity of the influence function. Since the estimates we obtain are independent of the number of agents, we can extend the convergence to consensus result to the associate continuum model, obtained as meanfield limit of the particle system. The analysis is then extended to opinion formation models with distributed time delay and with possibly lack of interaction between the system’s agents.
Joint works with C. Pignotti.
Stefano Di Giovacchino, University of L'Aquila, May 11, 2023
Integrazione numerica structure-preserving di problemi differenziali stocastici
In questo intervento, l'attenzione sarà principalmente rivolta verso l'integrazione numerica di problemi di evoluzione stocastici, con particolare enfasi sull'eventuale abilità di tali schemi numerici nel conservare a lungo termine proprietà quantitative e qualitative di suddette equazioni. In particolare, verrà analizzato il comportamento di opportune discretizzazioni numeriche di problemi stocastici dissipativi in media quadrata, così come assumerà particolare rilievo lo studio numerico dell'errore Hamiltoniano derivante da integratori numerici applicati a problemi Hamiltoniani stocastici di Ito e di Stratonovich. Per ottenere accurate stime di errore, verranno largamente usate tecniche di analisi dell'errore all'indietro, che prevedono la visione della soluzione numerica come flusso esatto di opportuni problemi differenziali modificati. Questo seminario è basato su risultati ottenuti in collaborazione con Raffaele D'Ambrosio (Università dell'Aquila).
Lorenzo Campioni, University of L'Aquila, April 20, 2023
Maximal unrefinable partitions into distinct parts
We introduce a new definition for a class of partitions into distinct parts induced by a particular limitation on the number of parts, called refinability. The purpose of the talk is to illustrate how we related to this new object to understand its first basic properties.
In the first part we will focus on how to recognize an unrefinable partition.
From this result we observe a strange sequence in the number of maximal partitions and studying them we discover a relation with a suitable kind of partition into distinct parts.
Filomena De Filippis, University of L'Aquila, March 30, 2023
Regularity in the Calculus of Variations: Lavrentiev Phenomenon
The fundamental problem of the calculus of variations consists of finding minimizer of an integral functional and to describe its properties. A simple example of such a problem is to find the curve of shortest length connecting two points. Therefore, some questions arise: does this minimum always exist? Which space does it belong to? How much regular is it?
After choosing the function space in which we look for the solution of a minimization problem, we wonder whether the infimum value is actually the same when considering different function spaces.
This will lead us to the definition of the so-called ''Lavrentiev Phenomenon'', namely: the infimum over two different spaces may be not the same (even if one space is dense in the other).
Sara Latini, University of L'Aquila, March 9, 2023
Equilibrium problems and electricity markets
In science the term "equilibrium" has been widely used in physics, engineering and economics, among others, within different frameworks. It generally refers to states of a system in which all competing inuences are balanced. Such problems can be modelled through different mathematical models such as optimization, variational inequalities, multiobjective optimization and noncooperative games, among others. All these mathematical models share an underlying common structure that allows to conveniently formulate them in a unique format: the Ky Fan inequality. We will illustrate this problem and show an existence result for finite-dimensional equilibrium problems where the constraint map may not be a self-map. This kind of problems find an interesting application in the electricity market.
Margherita Paolini, University of L'Aquila, February 20, 2023
A continuum analogue of the Braid Group
In the foundational manuscript [1] Emil Artin has introduced the sequence of Braid Group B_n. B_n is a group whose elements are equivalence classes of n-braids up to isotopy. The Braid Group admits different equivalent definitions, in particular, we will introduce the Birman-Ko-Lee presentation [2] whose generators are a_{l,m} (the a_{l.m} braid is the elementary interchange of the l-th and the m_th strand of the braid with all the other strands held fixed). We will use the BKL generators to define a continuum analogue of the Braid group B_n called T (Tapes group). This work is a primary step in order to define an action of T on the continuum Quantum Group introduced by Appel and Sala [3].
[1] E. Artin, Theorie der Zopfe, Amburg Abh. 4, (1925), 47-72.
[2] J. Birman, K.H. Ko and S.J. Lee, A new approach to the word and conugacy problem in the braid group, Adv. Math., 139 (2), (1998), 322-253.[3] A. Appel and F. Sala, Quantization of Continuum Kac Moody Algebras, Pure Appl. Math. Q. 16, (2020), 439-493.
Carmela Scalone, University of L'Aquila, January 30, 2023
The power of dynamical low rank approximation
In this talk, we aim to present how the dynamical low rank approximation has proved to be a powerful approach to solve problems involving matrix differential equations (of big dimensions) with low rank solutions. In particular, we show how it can be successfully used to compute the rightmost eigenpair of some matrix-valued linear operators and to approximate the solutions, preserving the invariants, of particular PDEs.
Alessandro Vannini, University of L'Aquila, December 15, 2022
(Co)Homology: from Algebra and Geometry to Mirror Symmetry in String Theory
What does "the boundary of a boundary is zero" mean? How can we talk about "holes" in a shape? Where vector calculus identities come from? The answer to each of these questions is intimately related to a beautiful and deep concept, namely to a "(co)homology theory".
In the first part of the talk we will understand what a (co)homology theory is through various examples. Then we will apply these ideas to investigate the geometry of higher dimensional manifolds with additional structures. Finally we will see how this language can be used to formulate an outstanding conjecture coming from String Theory which establish an unexpected bridge between complex and symplectic geometry and how this is related to my Phd project.
Luigi Forcella, University of Pisa, December 1, 2022
Finite-time blow-up for a class of non-local NLS equations
In the first part of the talk, we will review the classical topic of determining sufficient conditions which imply the formation of singularities in finite time for the mass-energy power-type intracritical nonlinear Schrödinger equation (NLS) in the energy space. Secondly, we will discuss on some recent results (Proc. AMS, Vol. 150 (2022), no. 12, 5421-5432) for a class of NLS with non-local nonlinearities.
Giovanni Russo, University of L'Aquila, November 10, 2022
GKM actions on cohomogeneity one manifolds
GKM theory is about understanding topological and geometric aspects of spaces with torus-actions in terms of purely combinatorical data, namely graphs.
We first give an introduction to GKM theory via essentials on group theory and explicit examples.
Then we illustrate the above idea on homogeneous and cohomogeneity one spaces.
The results in the cohomogeneity one case are based on a joint work with Oliver Goertsches and Eugenia Loiudice from Philipps-Universitä t Marburg.