Topics in Mathematics 2022/2023

Le algebre di vertice e le pseudoalgebre di Lie hanno definizioni notoriamente ostiche e dure da digerire. Proverò ad illustrarne il significato motivandone l’origine.

This presentation provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaotic trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.

The analysis of linear, time-invariant systems by superposition of modes is a longstanding idea, tracing back to the early works by Daniel Bernoulli on the vibrating string and later formalised by Fourier. Time-invariance and linearity allow to describe systems in terms of eigenfunctions and frequencies, called the modes of the system. Such modal shapes and frequencies may be determined experimentally or from a suitable mathematical model. In the latter case, the model is a system of partial differential equations (PDEs), depending on material and geometric properties, type of excitation, and initial and boundary conditions. Modal equations result after an appropriate projection is applied to the system of PDEs, yielding an eigenvalue problem from which the modal frequencies and shapes are determined. The resulting modal equations depend exclusively on time, and output may be extracted as a suitable combination of the time-dependent modal coordinates. Usually, one is interested in computing a physical output at one or more points of the system via a weighted sum resulting from an inverse projection. Besides being a practical analysis tool, this approach lends itself naturally to the simulation of mechanical vibrations and, thus, to sound synthesis via physics-based modelling. Modal synthesis began in earnest in the 1990s, when frameworks such as Mosaic and Modalys emerged. The early success of modal synthesis was partly due to the ease of implementation, and efficiency, of the modal structure: the orthogonality of the modes yields a bank of parallel damped oscillators. Including complicated loss profiles (necessary for realistic sound synthesis) is also trivial and inexpensive within the modal framework, as is the fine-tuning of the system's resonances. In direct numerical simulation, such as finite differences, distributed nonlinearities can be resolved locally and, in some cases, efficiently via linearly implicit schemes. For the modal approach, the presence of nonlinearities, either lumped or distributed, may become problematic since a coupling occurs between the modes of the associated linear system. In this seminar, an extension of the modal approach, including nonlinearly coupled subsystems, is presented. The enabling idea is the quadratisation of the potential energy in a fashion analogous to that proposed within the SAV (scalar auxiliary variable) approach. It is then possible to derive discrete-time equations whose update remains explicit while guaranteeing pseudo-energy conservation (necessary for the system's stability). Musical examples, including a real-time music plugin developed within this framework, will be offered.

In the first part of this talk I will try to give a (very partial) overview on some of the phenomena of interest in the area of brain diseases, on the kind of contributions mathematical modelling could give in this respect and on which mathematical instruments can be used when it comes to models. In the second part I will present a mathematical model for the onset and progression of Alzheimer’s disease. The synergistic interplay of proteins Amyloid-beta and tau is a subject of considerable interest when it comes to the study of Alzheimer’s disease. The model I will describe is based on transport and reaction-diffusion equations for the two proteins. In the model neurons are treated as nodes of a graph (the connectome) and structured by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid-beta, ii) effects of misfolded tau protein and iii) neuron-to- neuron prion-like transmission of the disease. These processes are modelled by a system of Smoluchowski equations for the Amyloid-beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. I will explain the structure of the model, give a hint of the main analytical results obtained and I will show the output of some numerical simulations

Data assimilation tries to predict the most likely state of a dynamical system by combining information from observations and prior models, often represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches. In particular, three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var), and, if time allows, the Kalman filter. The data assimilation problem usually results in a very large, yet structured, nonlinear optimization problem. The dimension of the latter represents a quite challenging aspect of the entire solution procedure. Indeed, the inclusion of both the time and space dimensions leads to extremely large optimazion problems which need to be carefully handled by designing smart numerical schemes able to fully exploit the structure of the problem at hand. Therefore, the second part of this talk aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches mentioned above.

In the first part of this talk we introduce integer programming models and the two ingredients for their practical solution, namely, the simplex and the branch-and-bound algorithms. Next, we show how to formulate the Traveling Salesman problem, probably the most famous problem in combinatorial optimization, as an integer program having an exponential number of inequalities. Despite the huge size of the resulting formulation, we present a solution approach where only a small subset of inequalities has to be explicitly considered for computing an optimal solution.

International research on tertiary mathematics education has produced till the present days important solid findings which have profoundly influenced research in mathematics education even at other grade levels. In this seminar, after a brief introduction to what research into mathematics education is, results from classical research studies about the teaching/learning of mathematics at the undergraduate level will be presented. Furthermore, classical results will be related to recent research about modern practices including digital resources.

In this seminar, I will overview some applications of logic in mathematics, focusing on a framework recently developed with Bergfalk and Panagiotopoulos. In this context, we apply techniques developed for the study of "definability" notions in mathematical logic to algebraic invariants in homological algebra, topology, and functional analysis. In questo seminario presenterò una panoramica di applicazioni di logica in matematica, concentrandomi su un ambito recentemente sviluppato in collaborazione con Bergfalk e Panagiotopoulos, nel quale tecniche svulippute nel contesto dello studio di nozioni di "definibilità" in logica matematica vengono applicate allo studio di invarianti algebrici in algebra omologica, topologia, e analisi funzionale.

Quantum computing can drastically reduce the minimum number of elementary operations to solve some computational problems that are inaccessible for classical computers. In this seminar I will introduce the principles of quantum computing, present the quantum algorithms for factoring, matrix inversion and combinatorial optimization, and discuss their application potential. No prior knowledge of quantum mechanics is required.

Uno spazio dei moduli è uno spazio topologico (o forse una varietà) i cui punti sono in corrispondenza biunivoca con l'insieme degli oggetti geometrici di un certo tipo fissato. Alcuni esempi: i) lo spazio proiettivo n-dimensionale parametrizza le rette di uno spazio vettoriale di dimensione n+1; ii) le grassmanniane parametrizzano i sottospazi vettoriali di fissata di dimensione di uno spazio vettoriale di dimensione fissata; iii) M_g parametrizza le superfici di Riemann compatte e connesse di genere g. È interessante (e assolutamente centrale nella geometria moderna) studiare la geometria degli spazi dei moduli. Nei primi 45 minuti di questo intervento si darà un'introduzione agli spazi dei moduli accessibile a chiunque abbia una laurea triennale in matematica - nessuna conoscenza di geometria algebrica o geometria differenziale è richiesta. Dopo la pausa e a seconda degli interessi dell'uditorio, si proverà a dare una definizione più precisa degli spazi dei moduli.