Archivio 2024

2024
14 giugno
Kristin DeVleming
Seminario di algebra e geometria
Fano varieties are projective algebraic varieties which are normal and have ample anticanonical bundle. Recently, using K-stability (which originates from Kähler-Einstein metrics in differential geometry), moduli spaces for klt K-polystable Fano varieties have been constructed; these moduli spaces are projective and are called "K-moduli". In this talk, after reviewing some basics of the theory of K-moduli of Fano varieties I will discuss certain explicit examples.
2024
14 giugno
2024
11 giugno
Mirror symmetry predicts a correspondence between the complex geometry (the B side) and the symplectic geometry (the A side) of certain pairs of objects. In this talk I will discuss some aspects of mirror symmetry for Fano varieties, focusing on certain orbifold del Pezzo surfaces falling out of the standard mirror constructions. In particular, motivated by Homological Mirror Symmetry, I will describe the derived category of the surfaces (their B side), and mention some results on the category encoding the A side of their only-known Hodge-theoretic mirrors. This is based on work with A. Corti and work in progress with F. Rota.
Confocal laser-scanning microscopy (CLSM) has long been celebrated in life-science research for its unique blend of spatial and temporal resolution, coupled with its versatile applications. However, recent advancements in detector technology have sparked a transformative shift in CLSM, triggered by the introduction of novel single-photon array detectors. These detectors, poised to supplant single-element detectors (also known as bucket detectors), offer access to previously discarded sample information, reshaping the trajectory of CLSM. In traditional CLSM, images are generated by raster scanning a focused laser beam across the sample, with single-element detectors registering a single-intensity value at each sample position. In contrast, single-photon array detectors capture true temporal images at each scanning position, transitioning CLSM into image scanning microscopy (ISM). Image scanning microscopy transcends traditional CLSM by generating not merely a two-dimensional dataset but a five-dimensional one, incorporating four spatial dimensions and a temporal dimension. This enables the reconstruction of highly informative and super-resolved images of the sample. This seminar will delve into the foundational principles of ISM, starting with the formulation of the forward model underlying the technique. Subsequently, a maximum likelihood approach, considering Poissonian noise, will be presented for reconstructing super-resolved images from the four-dimensional spatial dataset. An extension of this framework will incorporate the temporal dimension, enabling the reconstruction of fluorescence lifetime images that integrate structural and functional sample information. Furthermore, the seminar will explore leveraging the ISM dataset and deep learning techniques to accurately estimate the point-spread function of the optical system. This has the potential to significantly enhance the quality of reconstructed super-resolved images. By elucidating these advancements and future prospects, this seminar aims to inspire researchers to harness the full potential of ISM in pushing the boundaries of biomedical imaging.
2024
09 giugno
Abstract: In this mini-course, we show how various forms of supervised learning can be recast as optimization problems over suitable function spaces, subject to regularity constraints. Our family of regularization functionals has two components: (1) a regularization operator, which can be composed with an optional projection mechanism (Radon transform), and (2) a (semi-)norm, which may be Hilbertian (RKHS) or sparsity-promoting (total variation). By invoking an abstract representer theorem, we obtain an explicit parametrization of the extremal points of the solution set. The latter translates into a concrete neuronal architecture and training procedure. We demonstrate the use of this variational formalism on a variety of examples, including several variants of spline-based regression. We also draw connections with classical kernel-based techniques and modern ReLU neural networks. Finally, we show how our framework is applicable to the learning of non-linearities in deep and not-so-deep networks.
2024
07 giugno
Cecilia Rossi
Seminario di storia della matematica
La storia di Sophie Germain, una delle più brillanti menti matematiche di sempre, vissuta a Parigi tra la fine del Settecento e l’inizio dell’Ottocento, è emblematica del rapporto tra donne e scienza e solleva questioni attuali ancora oggi. Con il suo originale lavoro di ricerca e i suoi importanti contributi scientifici, Sophie Germain si è opposta alla convinzione comune del suo tempo che le donne non fossero capaci di un lavoro scientifico indipendente, cambiando per sempre il concetto di studiosa donna e guadagnandosi il titolo di matematica rivoluzionaria. Nota per gli studi sulle superfici elastiche applicati alla Tour Eiffel, ma soprattutto per i risultati significativi raggiunti nel campo della teoria dei numeri, Sophie Germain si è dovuta in più occasioni fingere uomo per riuscire a coltivare la sua passione per questa disciplina. Dedita interamente alla scienza dall’età di 13 anni, ha dialogato alla pari con i più grandi matematici della sua epoca, ricevendo il plauso, tra gli altri, di Carl Friedrich Gauss, grazie al quale l’Università di Gottinga nel 1830 le riconobbe una laurea honoris causa. Nonostante ciò, i suoi meriti e la sua identità di scienziata sono stati a lungo negati dalla comunità scientifica.
2024
07 giugno
Angelo Carraggi
Seminario di analisi numerica
Anomaly detection is a primary need for industrial/manufacturing applications and Datalogic business, but many challenges persist. The collection and annotation of data is expensive and most of the time unpractical as the deployed systems are usually not remotely accessible. The image resolution and the frame rate require computing-intensive algorithms that often do not fit the real-time constraint on embedded devices. In this study, we propose a novel solution that, inspired by recent advancements in this field obtained by normalizing-flow and patch-based feature distribution, combines unsupervised learning with efficient processing to deploy an optimized solution to reach a high classification accuracy while working with few training samples.
Linear stochastic differential equations: the Doléans-Dade exponential. The martingale property of the stochastic integral.
2024
06 giugno
Over the years since 1970, Magnetic Resonance Imaging (MRI) has developed into as one of the preferred choices for many radiological exams today. It relies primarily on its ability to detect water, which constitutes a significant portion of most tissues (around 70-90%). Changes in the water content and properties within tissues due to diseases or injuries can be substantial, rendering MRI highly effective in diagnosis due to its sensitivity. ESAOTE specializes in designing MRI systems with low-field technology (0.25 T to 0.4 T), offering several advantages including improved patient comfort, cost reduction, less demanding installation requirements, and lower energy consumption. However, the trade-off for using low-field MRI is a decrease in signal strength, often requiring longer scan times to achieve high-quality diagnostic images. Hence, there is a need for techniques to accelerate image acquisition. Specifically, ESAOTE has developed the Speed-Up technique, inspired to compressed-sensing techniques. Recent advances in collaboration with the Amsterdam University Medical Center aim to improve the reconstruction algorithm using artificial intelligence (AI), while maintaining the diagnostic accuracy of traditional, longer scans. This was achieved by optimizing the k-space undersampling scheme and reconstructions, using the Cascades of Independently Recurrent Inference Machines (CIRIM). Promising image quality was observed up to an acceleration factor of at least 2.5.
06/06/2024
07/06/2024
Francesco Esposito
Rigidity and symmetry results for some elliptic problems
Seminario di analisi matematica
In this talk, we investigate qualitative properties of singular solutions to some elliptic problems. In the first part, we will focus our attention on semilinear and quasilinear elliptic problems under zero Dirichlet boundary conditions. In the second part, thanks to the previous analysis, we obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.
06/06/2024
07/06/2024
Valentina Franceschi
Mean value formulas for surfaces in Grushin spaces
Seminario di analisi matematica
In this talk, we consider n-dimensional Grushin spaces, where a Riemannian metric degenerates along a line in the space, resulting in a sub-Riemannian structure. We discuss the validity of (sub-)mean value property for (sub-)harmonic functions on hypersurfaces within Grushin spaces of dimension n>2. Our interest is driven by the classical counterpart: mean value formulas for harmonic functions on surfaces in the Euclidean setting are crucial for establishing the Bombieri-De Giorgi-Miranda gradient bound, which, in turn, plays a central role in the classical regularity theory. We conclude by presenting remarks and open questions about the regularity theory of minimal surfaces within this sub-Riemannian framework, which is yet to be established.
06/06/2024
07/06/2024
Alessandro Cosenza
A Γ-convergence result for 2D type-I superconductors
Seminario di analisi matematica
06/06/2024
07/06/2024
Paolo Luzzini
The Grushin eigenvalue problem: sensitivity, optimization, and blow-up
Seminario di analisi matematica
One of the oldest and most studied problems in the spectral theory of differential operators is the eigenvalue problem for the Dirichlet Laplacian. Classical questions about Laplacian eigenvalues concern their sensitivity analysis, optimization, asymptotic expansions, and many other more properties. On the other hand, similar questions remain open for an important class of degenerate operators, that is the Grushin Laplacians. In this talk I will present some recent results regarding the spectral theory of the Grushin Laplacian and in particular its shape sensitivity analysis, the optimization of the first eigenvalue, and a blow-up analysis.
06/06/2024
07/06/2024
Riccardo Durastanti
Spreading phenomena under singular potentials: statics and dynamics
Seminario di analisi matematica
We look at spreading phenomena under the action of singular potentials modeling repulsion between the liquidgas interface and the substrate. We mainly discuss the static case: depending on the form of the potential, the macroscopic profile of equilibrium configurations can be either droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed. Then we will briefly focus on the dynamics, assuming zero slippage at the contact line. Based on formal analysis arguments, we report that generic travelling-wave solutions exist and have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with equilibrium configurations, travelling-wave solutions have microscopic contact angle equal to π/2 and, for mild singularities, finite energy. This is a joint work with Lorenzo Giacomelli.
06/06/2024
07/06/2024
Serena Guarino Lo Bianco
Aspects of total variation and connections with image processing
Seminario di analisi matematica
06/06/2024
07/06/2024
Michele Marini
The sharp quantitative isocapacitary inequality
Seminario di analisi matematica
The well-known isocapacitary inequality states that balls minimize the capacity among all sets of the same given volume. In the talk, we prove a sharp quantitative form of this classical result. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. We then discuss some possible extensions.
06/06/2024
07/06/2024
Francesca Oronzio
Quantitative Alexandrov theorem and its applications in the volume preserving mean curvature flow
Seminario di analisi matematica
06/06/2024
07/06/2024
Enzo Maria Merlino
Regularity for almost minimizer of a one-phase Bernoulli-type functional in Carnot Groups of step two
Seminario di analisi matematica
The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Generalizations to the nonlinear framework will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).
06/06/2024
07/06/2024
Francesca Corni
An area formula for intrinsic regular graphs in homogeneous groups
Seminario di analisi matematica
2024
05 giugno
Antonella Guidazzoli
Seminario di analisi numerica
This talk will present some case studies on using artificial intelligence (AI) to preserve , study and valorize cultural heritage digital assets. At the heart of the approach is recognizing the need for a human AI frameworks in which data scientists and domain experts collaborate synergistically. Moreover promoting interdisciplinary partnerships is essential touphold ethical principles and serve the public good.
2024
05 giugno
Francesco Gravili
nell'ambito della serie: SEMINARI MAT/08 TEAM
Seminario di analisi numerica
Multilayer networks are a type of complex network that consist of multiple layers, where each layer represents a different type of connection or interaction between the same set of nodes. These networks are used to model systems where entities are connected in multiple ways simultaneously, capturing the complexity of real-world relationships better than traditional single-layer networks. Through a particular interlayer structure, the dynamical evolution of a complex system over time can be represented. Computing the centrality of a temporal network can improve our understanding of how the most important nodes in a network change over time. Our focus is centered on the computation of the centralities of a multilayer temporal network whose modifications over time consist of low-rank updates of the edge adjacency matrix of a transport network. Using Krylov subspace methods for matrix function approximations, we will exploit the particular structure of the problem to gain some computational advantages and modeling insights.
Introduction to stochastic differential equations driven by pure jump processes: the predictable sigma-algebra and a well-posedness result.
2024
04 giugno
Andrea Di Lorenzo
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
Blow-ups are fundamental tools in algebraic geometry, and there are several results (e.g the famous Castelnuovo's theorem) that can be used to determine when a variety is obtained as a blow-up of a smooth variety along a smooth center. Weighted blow-ups play a similar role for stacks. In this talk I will present a criterion for finding out if a smooth DM stack is a weighted blow-up. I will apply this result for showing that certain alternative compactifications of moduli of marked elliptic curves are obtained via weighted blow-ups (and blow-downs). This in turn will prove to be useful in order to compute certain invariants, like Chow rings or Brauer groups. First part of this talk is a joint work with Arena, Inchiostro, Mathur, Obinna and Pernice; the second part of this talk is a joint work with L. Battistella.
2024
04 giugno
Lorenzo Rosasco
Seminario di analisi numerica
L'insegnamento fornisce le nozioni necessarie per la comprensione e l'utilizzo dei principali algoritmi di apprendimento. Verranno introdotte le definizioni fondamentali relative ai problemi di apprendimento supervisionato e non supervisionato. Poi verranno presentati alcuni approcci per l'apprendimento statistico supervisionato, come metodi locali e regularization networks, sia nel caso lineare che nonlineare. Verranno altresì introdotte le reti neurali. Il corso conterrà anche un'introduzione a problemi di apprendimento non supervisionato, come clustering a dimensionality reduction. Gli argomenti trattati dal punto di vista teorico, saranno affrontati anche da un punto di vista numerico durante le lezioni in laboratorio.
Stochastic integral with respect to finite variation processes: integration by parts formula and Itô’s formula.
Ciclo: Optimization based machine learning for computational imaging Abstract: In many scientific and medical settings, we cannot directly observe images of interest, such as a person’s internal organs, the microscopic structure of materials or cells, or distant stars and galaxies. Rather, we use MRI scanners, microscopes, and telescopes to collect indirect data that require sophisticated algorithms to form an image. Historically, these methods have relied on mathematical models of simple image structures to improve the quality and resolution of the resulting images. More recent efforts harness vast collections of images to train computers to learn more complex models of image structure, yielding more accurate and higher-resolution images than ever. These new methods lead to a renaissance in computational imaging and new insights into designing neural networks and other machine learning models in a principled manner, jointly leveraging both training data and physical models of how imaging data is collected. In this course, we will cover some exciting new directions in this emerging area, such as (a) plug-and-play methods; (b) variational networks and deep unrolling; (c) deep equilibrium models; (d) learning regularization functionals; (e) scalable and mini-batch OPML; (f) diffusion models; (g) self-supervised OPML approaches; (h) robustness and domain adaptation.
Abstract: In many scientific and medical settings, we cannot directly observe images of interest, such as a person’s internal organs, the microscopic structure of materials or cells, or distant stars and galaxies. Rather, we use MRI scanners, microscopes, and telescopes to collect indirect data that require sophisticated algorithms to form an image. Historically, these methods have relied on mathematical models of simple image structures to improve the quality and resolution of the resulting images. More recent efforts harness vast collections of images to train computers to learn more complex models of image structure, yielding more accurate and higher-resolution images than ever. These new methods lead to a renaissance in computational imaging and new insights into designing neural networks and other machine learning models in a principled manner, jointly leveraging both training data and physical models of how imaging data is collected. In this course, we will cover some exciting new directions in this emerging area, such as (a) plug-and-play methods; (b) variational networks and deep unrolling;
2024
30 maggio
We present some new results about the concavity (up to a transformation) of positive solutions for some classes of quasi-linear elliptic problems, including nonautonomous cases.
2024
30 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
The Poisson point process and Watanabe theorem. Introduction to stochastic integrals with respect to finite variation processes.
We characterise the exact asymptotic performance of high-dimensional classification and robust regression estimators under convex loss and regularisation assumptions. Using tools from replica theory, our analysis covers a large family of data distribution assumptions, including any power-law tail, and allows us to determine cases where Gaussian data universality breaks. For classification, we characterise the learning of a mixture of clouds by studying the generalisation performance of the obtained estimator, analyse the role of regularisation and analytically derive the data separability transition. For robust regression, we provide an exact asymptotic characterisation of the recovery of a planted estimator under heavy-tailed contamination of covariates and label noise. We show that, unlike in the classical regime of small dimension-to-data sample ratio, regularisation becomes necessary for the Huber loss estimator to achieve optimality under heavy-tailed contamination in the modern high-dimensional regime, and we derive decay rates for the estimation error of ridge regression.
2024
29 maggio
Mohamed Elhamdadi
Seminario di algebra e geometria, interdisciplinare
Protein are linear molecular chains that often fold to function. We will introduce some algebraic structures, called bondles, modeled on projections of proteins. We will discuss colorings of these projections by bondles and construct the coloring invariant which is used to distinguish proteins. We will also discuss an enhancement of this invariant based on introducing some weights at the bonds thus giving the enhanced invariant as a state sum invariant.
2024
29 maggio
Cristian Micheletti
Seminario di algebra e geometria, interdisciplinare
I will report on a series of studies where we used molecular dynamics simulations and various models to study how the properties of DNA and RNAs are affected by the presence of knots and other forms of structural entanglement[1]. I will first consider model DNA plasmids that are both knotted and supercoiled, and discuss how the simultaneous presence of knots and supercoiling creates long-lived multi-strand interlockings that might may be relevant for the simplifying action of topoisomerases. I next consider how entangled nucleic acids behave when driven through narrow pores[2-4], a setting that models translocation through the lumen of enzymes, and discuss the biological implication for a certain class of viral RNAs[4]. Bibliography [1] L. Coronel, A. Suma and C. Micheletti, "Dynamics of supercoiled DNA with complex knots", Nucleic Acids Res. (2018) 46 , 7533 [2] A. Suma, V. Carnevale and C. Micheletti, Nonequilibrium thermodynamics of DNA nanopore unzipping, Phys. Rev. Lett., (2023), 130 048101 [3] A. Suma, A. Rosa and C. Micheletti, Pore translocation of knotted polymer chains: how friction depends on knot complexity, ACS Macro Letters, (2015), 4 , 1420-1424 [4] A. Suma, L. Coronel, G. Bussi and C. Micheletti, "Directional translocation resistance of Zika xrRNA” Nature Communications (2020), 11 , art no. 3749
2024
29 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The last few decades have seen important advances in understanding the consequences of topological constraints in many biological systems. A famous example is the case of knotted proteins, where the presence of entanglement is thought to influence their folding behaviour and mechanisms. More recently, topological data analysis has been providing an effective computational window, aimed at characterising a variety of natural phenomena in terms of their topological features. In this talk, we present techniques and results in computational and applied topology, interpreted broadly, with a focus on applications to biopolymers.
2024
29 maggio
2024
28 maggio
The aim of the talk is to analyze an evolutionary Φ- Laplacian problem, with singular and convective reactions: u_t -A u= f+g .The differential operator A considered is an elliptic operator, driven by a Young function Φ, while the reaction terms are Carathéodory functions obeying suitable growth conditions. The problem possesses three features of interest: • the operator A can be non-homogeneous, and with unbalanced growth; • the reaction term f is singular (i.e., it behaves like u^{−γ} with γ ∈ (0, 1)) and f(x, ·) can be non-monotone); • the reaction term g is convective (i.e., it depends on ∇u). We will first introduce the functional setting of the problem, by recalling the most relevant function spaces involved and their basic properties. Secondly, the main issues concerning both singular and convective terms are highlighted, together with the sub-solution and freezing techniques. Finally, we will briefly sketch the proof of our existence result, based on a priori estimates and a semi-discretization (in time) procedure. The seminar will have an introductory fashion, under the trend proposed by the cycle ASK (Analysis Student Kernel), for young analysis researchers, at the University of Bologna (https://sites.google.com/view/askbologna/home?authuser=1).
2024
28 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
2024
27 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
We show that the natural operation of connected sum for graphs can be used to prove at once most of the universality results from the literature concerning graph homomorphism. In doing so, we significantly improve many existing theorems and solve some natural open problems. Despite its simplicity, our technique unexpectedly leads to applications in quite diverse areas of mathematics, such as category theory, combinatorics, classical descriptive set theory, generalized descriptive set theory, model theory, and theoretical computer science. (Joint work with S. Scamperti)
2024
27 maggio
Introduction to pure jump processes: definitions of (marked) point processes, random measures, and associated counting process.
2024
25 maggio
Francesca Morselli, Università degli Studi di Genova
Seminario interdisciplinare
2024
25 maggio
Giacomo Bormetti, Università degli Studi di Pavia
Seminario interdisciplinare
2024
23 maggio
Sascha Portaro
nell'ambito della serie: SEMINARI MAT/08 TEAM
Seminario di analisi numerica
We introduce a novel procedure for computing an SVD-type approximation of a tall matrix A. Specifically, we propose a randomization-based algorithm that improves the standard Randomized Singular Value Decomposition (RSVD). Most significantly, our approach, the Row-aware RSVD (R-RSVD), explicitly constructs information from the row space of A. This leads to better approximations to Range(A) while maintaining the same computational cost. The efficacy of the R-RSVD is supported by both robust theoretical results and extensive numerical experiments. Furthermore, we present an alternative algorithm inspired by the R-RSVD, capable of achieving comparable accuracy despite utilizing only a subsample of the rows of A, resulting in a significantly reduced computational cost. This method, that we name the Subsample Row-aware RSVD (Rsub-RSVD), is supported by a weaker error bound compared to the ones we derived for the R-RSVD, but still meaningful as it ensures that the error remains under control. Additionally, numerous experiments demonstrate that the Rsub-RSVD trend is akin to the one attained by the R-RSVD when the subsampling parameter is on the order of n, for a m×n A, with m >> n. Finally, we consider the application of our schemes in two very diverse settings which share the need for the computation of singular vectors as an intermediate step: the computation of CUR decompositions by the discrete empirical interpolation method (DEIM) and the construction of reduced-order models in the Loewner framework, a data-driven technique for model reduction of dynamical systems.
2024
23 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
2024
22 maggio
Fabrizio Lillo
nell'ambito della serie: STOCHASTICS AND APPLICATIONS
Seminario di finanza matematica
We consider a model of a simple financial system consisting of a leveraged investor that invests in a risky asset and manages risk by using value-at-risk (VaR). The VaR is estimated by using past data via an adaptive expectation scheme. We show that the leverage dynamics can be described by a dynamical system of slow-fast type associated with a unimodal map on [0,1] with an additive heteroscedastic noise whose variance is related to the portfolio rebalancing frequency to target leverage. In absence of noise the model is purely deterministic and the parameter space splits into two regions: (i) a region with a globally attracting fixed point or a 2-cycle; (ii) a dynamical core region, where the map could exhibit chaotic behavior. Whenever the model is randomly perturbed, we prove the existence of a unique stationary density with bounded variation, the stochastic stability of the process, and the almost certain existence and continuity of the Lyapunov exponent for the stationary measure. We then use deep neural networks to estimate map parameters from a short time series. Using this method, we estimate the model in a large dataset of US commercial banks over the period 2001-2014. We find that the parameters of a substantial fraction of banks lie in the dynamical core, and their leverage time series are consistent with a chaotic behavior. We also present evidence that the time series of the leverage of large banks tend to exhibit chaoticity more frequently than those of small banks.
2024
22 maggio
Lars Halvard Halle
nell'ambito della serie: TOPICS IN MATHEMATICS 2023/2024
Seminario di algebra e geometria
In this talk, I will give a leisurely and subjective introduction to Arithmetic Geometry, focusing on explicit examples and some selected questions. I aim to illustrate how basic questions in Number Theory can naturally lead to rich and interesting topics in Algebra and Geometry. If time permits, I will also indicate how my own research and interests fits into this picture.
We consider two competing companies which generate and sell electricity to the market. The companies aim at maximizing their profit and can increase their level of installed power by irreversible installations of renewable electricity sources, although negatively impacting the price of electricity. We present the model and, for illustrative purposes, we show the Nash equilibrium in the case where the two companies are only allowed to install at time zero. Then we discuss the time continuous case, which is modelled as a singular stochastic game, whose resulting HJB reeds as a system of nonlinear equations with free-boundaries for each player. In particular the free boundaries separate the regions where the two companies should install or wait and uniquely identify the value functions.
2024
21 maggio
'' Nel primo decennio del 1900 si verificò un' improvvisa catalizzazione di idee e metodi che si erano lentamente accumulati nel corso del XIX secolo. Fra i catalizzatori di questo rapido processo vi furono i risultati e le idee presenti nella tesi di dottorato di Lebesgue del 1902''(J. Dieudonné). L'opera di Lebesgue e le ampie generalizzazioni ad essa seguite hanno modificato i concetti di misura e di integrale così profondamente da rendere la teoria dell'integrazione una esclusiva creazione del ventesimo secolo, nonostante la sua idea originaria risalga ad Archimede. L'Analisi Matematica negli spazi funzionali infinito dimensionali, nata quasi contemporaneamente all'integrale di Lebesgue, deve a quest'ultimo i suoi più proficui sviluppi. Fra questi la definitiva dimostrazione del ''Principio di Dirichlet'', dato per scontato - a torto - da Riemann, e inserito da Hilbert nell'elenco dei problemi da lui posti al Congresso internazionale di Matematica del 1900.
2024
21 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
2024
21 maggio
Federico Caucci
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
Tba
2024
20 maggio
Agnese Barbensi
Seminario di algebra e geometria, interdisciplinare
The aim of this course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool, whose aim is to study the “shape” of data via topological techniques. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In this course we will look at the standard PH pipeline, a number of different software programs and computational tools, some real-world applications, and some theoretical generalisations (multiparameter persistence, zig-zag persistence, etc.)
2024
20 maggio
2024
20 maggio
The latest years, machine learning has been one of the main directions in the numerical solution of inverse problems, aiming to face the ill-posed nature of these problems. In this talk, we delve into the solution of inverse problems and specifically inverse eigenvalue and inverse source problems, from a machine learning perspective. In the first part, we focus on the inverse Sturm-Liouville eigenvalue problem for sym- metric potentials and the inverse transmission eigenvalue problem for spherically sym- metric refractive indices. We present the main ideas behind supervised machine learning regression and briefly discuss the basic properties of the algorithms we implement, which are k-Nearest Neighbours (kNN), Random Forests (RF) and Neural Networks (MLP). Afterwards, we numerically solve the direct problems using well known methods, in order to produce the spectral data which in turn are used for training the machine learning models. We consider examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. In the second part, we pose the inverse source problem, to identify the number, posi- tions, and strengths of hidden line sources inside a dielectric cylinder. Using classification Neural Networks, we show that we can predict the unknown number of sources with high accuracy. We complete this talk with a discussion on an ongoing work for the inverse source problem to recover the positions and strengths of the sources. Our experiments validate the efficiency of these machine learning models for numerically tackling such inverse problems, providing a proof-of-concept for their applicability in this field. 1. N. Pallikarakis and A. Ntargaras, Application of machine learning regression models to inverse eigenvalue problems, Computers & Mathematics with Applications, 154 (2024). 2. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Predicting the number of line sources inside a cylinder using classification neural networks, (2024), (to appear in: 2024 IEEE Int. Symp. Antennas Propag. and ITNC-USNC-URSI Radio Sci. Meet.). 3. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Exploring the inverse line- source scattering problem in dielectric cylinders with deep neural networks, (2024), (submitted - under review).
2024
16 maggio
Alberto Parmeggiani
Seminario di analisi matematica
Hypoellipticity is far from being understood, especially that of degenerate operators of the kind sums of squares of complex vector fields. In recent times, J. J. Kohn discovered complex vector fields that, despite their finite-type generation of the complexified tangent space at any given point, nevertheless are losing many derivatives, a phenomenon also discovered by C. Parenti and myself at about the same time for operators with multiple transversal characteristics. In this talk, I will survey Kohn’s results and some extensions of it, that I have given, and through examples give some speculations in order to deepen the understanding of this phenomenon.
2024
16 maggio
Lucio Russo
Seminario di fisica matematica, interdisciplinare, storia della matematica
Si crede spesso che matematica e fisica siano due discipline separate da un chiaro confine rimasto stabile nel corso della storia: è questo il tacito presupposto assunto dagli autori di storie della matematica e della fisica. Il significato dei due termini è invece cambiato profondamente nel tempo insieme allo status epistemologico delle due scienze. Una riflessione sulla storia dei due concetti può essere utile per individuare le caratteristiche essenziali delle scienze esatte, mettere in discussione l’esistenza di un confine invalicabile tra fisica e matematica e immaginare possibili scenari di evoluzione futura.
2024
15 maggio
Francesco Russo
Seminario di probabilità
The notion of weak Dirichlet process is the natural extension of semimartingale. Among the examples we find the following. 1. Irregular Markov processes solutions of SDEs with distributional drift with jumps. 2. Solutions of (even continuous) path-dependent SDEs with distributional drift. 3. (Path-dependent) Bessel processes. Identification problem in BSDEs driven by random measure. The talk puts the emphasis on a BSDE with distributional driver. The presentation covers joint work with E. Issoglio (Torino) and E. Bandini (Bologna).
2024-05-15
Adriano Barra
Networks of neural networks: the more is different
Seminario di fisica matematica, interdisciplinare
By relying upon tools of statistical mechanics of spin glasses, in this talk I will focus on Hebbian neural networks interacting in an heteroassociative manner to show that the overall network as a whole shows computationally capabilities that are lost within a single neural network. In particular I will show how these networks naturally disentangle spurious states recovering the original patterns forming these mixtures, thus providing a novel way of performing challenging pattern recognition tasks. The theory will be developed in the standard random setting then applications will be performed on structured datasets as the harmonic melodies.
2024-05-15
Francesco Guerra
Replica interpolation and Replica Symmetry Breaking
Seminario di fisica matematica, interdisciplinare
The method of Replica Symmetry Breaking is considered in the frame of replica interpolation, where it leads to a kind of phase transition. Applications are given for the Random Energy Model and for The Sherrington Kirkpatrick model. The results show some unexpected surprises.
2024-05-15
Raffaella Burioni
Statistical physics approaches to the social sciences: some applications to the topological and semantic structure of complex historical archives
Seminario di fisica matematica, interdisciplinare
In this talk I will recall some work on the application of statistical physics techniques to social data and discuss some recent perspectives on data from historical archives.
2024-05-15
Cecilia Vernia
A computational approach to spin glasses and beyond
Seminario di fisica matematica, interdisciplinare
I propose a personal overview of my collaboration with Pierluigi since our first meeting in 2003. I’ll review our numerical work on the glassy phase of finite dimensional spin glasses; in particular, overlap equivalence, ultrametricity, clustering property of overlap and monotonicity of the correlation functions will be considered. I’ll also present our research on the inverse problem in some mean field models with applications to the social sciences.
2024-05-15
Francesco Camilli
Breaking identicality: multispecies spin glasses and inhomogeneous inference problems
Seminario di fisica matematica, interdisciplinare
An assumption that typically pervades the study of spin glasses is that of independent and identically distributed random variables. In the celebrated Sherrington-Kirkpatrick model this is manifest in the distribution of the quenched disorder. This homogeneity creates a system whose particles are indistinguishable from one another, namely they can be arbitrarily permuted without changing the thermodynamical features of the model. Breaking identicality in the quenched disorder also breaks this global permutation symmetry, with the possibility of leaving it intact only in smaller subgroups of particles involved. The latter procedure leads to the definition of multispecies spin glasses, which are typically harder to analyse. In my talk I will give an overview of the cases we can solve, with a particular focus on multispecies models on the Nishimori Line, that is a particular region of their phase space where they have a clear correspondence with high dimensional inference problems, and concentration of the order parameters holds despite the presence of quenched disorder.
2024-05-15
Cristian Giardinà
The multifacet Ising model on random graphs
Seminario di fisica matematica, interdisciplinare
The ferromagnetic Ising spin model is often used to model second-order phase transitions and the continuous emergence of order. We consider this model on a random graph, where the additional randomness provided by the graph gives a rich picture with a host of surprises. We identify similarities and differences between the quenched and annealed Ising model. We find that the annealed critical temperature is highly model-dependent, even in the case of graphs that are asymptotically equivalent (such as different versions of the simple Erdös-Rényi random graph). The quenched critical temperature is instead the same for all locally tree-like graphs. Moreover, in the presence of inhomogeneities that produce a fat-tail degree distribution, the difference between quenched and annealed becomes even more substantial, leading in some cases to different universality classes and different critical exponents. The annealed properties depend sensitively on whether the total number of edges of the underlying random graph is fixed, or is allowed to fluctuate. If time allows preliminary results on the annealed Potts model, displaying a first-order phase transition, will also be discussed. [This talk is based on several joint works with Hao Can, Sander Dommers, Claudio Giberti, Remco van der Hofstad and Maria Luisa Prioriello. The preliminary work on Potts models also involves Neeladri Maitra and benefited from discussions with Guido Janssen.]
2024-05-15
Emanuele Mingione
Mean field spin glasses: beyond the i.i.d. setting
Seminario di fisica matematica, interdisciplinare
We review some recent advances in the rigorous analysis of mean field spin glasses. In particular we show how Parisi's theory can be generalized in the case where the spin-spin interaction is not described by i.i.d. random variables but, to some extent, it's of mean field type. We will focus on the multipecies SK model and the multiscale SK model, presenting the variational formulas for the free energy with a sketch of the proofs.
2024-05-15
Jorge Kurchan
Multi-thermalization vs. Parisi scenario, a one to one relation
Seminario di fisica matematica, interdisciplinare
Under the same assumptions and level of rigour as the previous work of Franz, Mezard, Parisi and Peliti, one can show that the ultrametric solution for the equilibrium measure holds if and only if the system's dynamics spontaneously split into widely separated timescales with only one temperature per timescale for all observables.
2024-05-15
Diego Alberici
Ising model on random graphs: a generalisation to many species
Seminario di fisica matematica, interdisciplinare
We discuss a family of multispecies ferromagnetic Ising models on multiregular random graphs. In the large volume limit, thermodynamic quantities are related to the solution of a belief propagation (BP) fixed point equation. A phase transition is identified and the critical region is determined by the spectral radius of a finite-dimensional matrix.
2024-05-15
Silvio Franz
Chaos in Small Field in Spin Glasses
Seminario di fisica matematica, interdisciplinare
Chaotic behavior and Stochastic Stability are two faces of the same RSB coin. In this talk I will discuss the universal properties of chaos against a small magnetic field in spin glasses. The introduction of a small field in a spin-glass modifies the weights of the equilibrium states. Using the fact that the magnetizations form a Gaussian process on the UM tree we can study the progressive decorrelation of the system in the field from the system without the field. We can then provide predictions on chaos that only depend on the Parisi function $P(q)$ in absence of the field. I will discuss in detail the simple case of the 1RSB, where extreme value statistics allow to completely solve the problem. In the full RSB case it is possible in principle to solve the problem through Parisi-like PDE, however, we found it more practical to simulate the infinite-system stochastic process implied by RSB theory. Getting a function $P(q)$ as input, we can generate weighted random trees using the Bolthausen-Snitman coalescent, reweight the states according to the values of their magnetization. We compare the theoretical predictions with direct simulations of Bethe-lattice spin glasses and the 4D Edwards-Anderson model. Work in collaboration with Miguel Aguilar-Janita, Victor Martin-Mayor, Javier Moreno-Gordo, Giorgio Parisi, Federico Ricci-Tersenghi, Juan J. Ruiz-Lorenzo
2024-05-15
Federico Ricci-Tersenghi
Daydreaming Hopfield Networks and their surprising effectiveness on correlated data
Seminario di fisica matematica, interdisciplinare
To improve the storage capacity of the Hopfield model, we develop a version of the dreaming algorithm that is perpetually exposed to data and therefore called Daydreaming. Daydreaming is not destructive and converges asymptotically to a stationary coupling matrix. When trained on random uncorrelated examples, the model shows optimal performance in terms of the size of the basins of attraction of stored examples and the quality of reconstruction. We also train the Daydreaming algorithm on correlated data obtained via the random-features model and argue that it spontaneously exploits the correlations thus increasing even further the storage capacity and the size of the basins of attraction. Moreover, the Daydreaming algorithm is also able to stabilize the features hidden in the data. Finally, we test Daydreaming on the MNIST dataset and show that it still works surprisingly well, producing attractors that are close to unseen examples and class prototypes.
2024
14 maggio
Caterina Campagnolo
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
Promoted by the seminal work of Gromov, bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open. In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group for a big family of coefficient modules. This condition is satisfied by many non-amenable groups of topological, geometric or algebraic origin, and even, surprisingly, by the generic countable group (in a precise sense). In the first part we will present the context and examples, and in the second part we will give some ideas beyond the proof of this result.
2024
14 maggio
Roberta Fulci
Seminario di didattica della matematica, storia della matematica
Cosa ci fa, un laureato in discipline scientifiche, magari anche dottorato in uno dei campi più astratti della matematica, alla radio? Lo scopriremo con Roberta Fulci, che dal 2013 lavora come redattrice e conduttrice della trasmissione Radio 3 Scienza (e non solo).
2024
09 maggio
In this talk, I will present a minimalist approach to the entropic approximations of optimal transport problems. This approach seems to allow some small generalization, in the direction of requiring less regularity of the pointwise transport cost. From a paper in preparation.
We address the control of Partial Differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. The method works as follows, given an estimated parameter configuration, we compute the corresponding control using the State-Dependent Riccati Equation (SDRE) approach. Subsequently, after computing the control, we observe the trajectory and estimate a new parameter configuration using Bayesian Linear Regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. We also focus on the computational cost of the algorithm, since we deal with high dimensional systems. To enhance the efficiency of the method, indeed, we employ model order reduction through the Proper Orthogonal Decomposition (POD) method. The considered problem's dimension is notably large, and POD provides impressive speedups. Further, a detailed description on the coupling between POD and SDRE is also provided. Finally, numerical examples will show the accurateness of our method across.
2024
08 maggio
Francesco Leonetti
Seminario di analisi matematica
It is well known that solutions to elliptic systems may be unbounded. Nevertheless, for some special classes of systems, it can be proved that solutions are bounded. We mention a recent result of this kind and we discuss some examples suggested by double phase functionals.
2024
07 maggio
Carlo Mantegazza
nel ciclo di seminari: MATEMATICI NELLA STORIA
Seminario di algebra e geometria, analisi matematica, storia della matematica
Uno degli eventi più rilevanti della matematica negli scorsi anni è stata la dimostrazione da parte di Grisha Perelman nel 2004, concludendo il lavoro ventennale di Richard Hamilton, della congettura di Poincaré, relativa alla comprensione della struttura degli spazi tridimensionali, che aveva resistito agli sforzi di numerosi matematici per quasi un secolo. Nel seminario verrà illustrato in modo discorsivo la congettura e il cammino che ha portato alla sua dimostrazione, che consiste in una delle pagine più belle e profonde della storia della matematica.
2024
07 maggio
Luca Ratti
nell'ambito della serie: SEMINARI MAT/08 TEAM
Seminario di analisi numerica
An inverse problem is the task of retrieving an unknown quantity from indirect observations. When the model describing the measurement acquisition is linear, this results in the inversion of a linear operator (a matrix, in a discrete formulation) which is usually ill-posed or ill-conditioned. A common strategy to tackle ill-posedness in inverse problems is to use regularizers, which are (families of) operators providing a stable approximation of the inverse map. Model-based regularization techniques often leverage prior knowledge of the exact solution, such as smoothness or sparsity with respect to a suitable representation; on the other side, in recent years many data-driven methods have been developed in the context of machine learning. Those techniques tackle the approximation of the inverse operator in suitable spaces of parametric functions (i.e., neural networks) and rely on large datasets of paired measurements and ground-truth objects. In this talk, I will focus on hybrid strategies, which aim at blending model-based and data-driven approaches, providing both satisfying numerical results and sound theoretical guarantees. I will describe a general framework to comprise many existing techniques in the theory of statistical learning, also reporting some recent theoretical advances (in the direction of generalization guarantees). I will help the discussion by presenting some relevant examples in the context of medical imaging and, specifically, in computed tomography.
2024
07 maggio
Stefano Marini
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
A complex flag manifold F= G /Q decomposes into finitely many real orbits under the action of a real form of G. Their embeddings into F define CR manifold structures on them. We give a complete classification of all closed simple homogeneous CR manifolds that have finitely nondegenerate Levi forms.
After recalling some integrability by compensation results related to conformally invariant Lagrangians in dimension 2, we will present a recent result concerning the upper-semi-continuity of the Morse index plus the nullity of critical points to such variational problems under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse Indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. This is a joint work with Matilde Gianocca and Tristan Rivière
2024
30 aprile
L'abstract del seminario è contenuto nel file allegato.
2024
29 aprile
Salvatore Federico
nell'ambito della serie: TOPICS IN MATHEMATICS 2023/2024
Seminario di analisi matematica, probabilità
The talk is intended to provide a pedagogical introduction to optimal control theory in continuous time and to its connections to PDEs. I will present the main ideas of the Dynamic Programming approach for a family of optimal control problems, both in the deterministic and in the stochastic framework. Some applications in economic growth theory and epidemiological models will be illustrated.
2024
24 aprile
Irene Benedetti
Seminario di analisi matematica
In questa presentazione verranno mostrati risultati sull'esistenza e la localizzazione delle soluzioni per problemi differenziali non locali in spazi astratti. In particolare, verrà illustrato un procedimento basato su teoremi di punto fisso associati alle cosiddette condizioni di trasversalità. Una particolare attenzione verrà dedicata ad alcune tecniche che permettono di indebolire le ipotesi di compattezza classiche spesso presenti in letteratura per lo studio di equazioni differenziali in spazi astratti con metodi topologici. Questo approccio fornisce un metodo unificante per lo studio di modelli che descrivono processi di diffusione in diversi contesti. Permette di considerare condizioni periodiche e condizioni iniziali non locali più generali come ad esempio condizioni multipoint oppure condizioni iniziali di tipo integrale, e di gestire nonlinearità con crescite superlineari, ad esempio polinomi cubici o mappe che dipendono dall'integrale della soluzione, includendo così comportamenti di diffusione non locale.
2024
24 aprile
We are interested in the numerical solution of the matrix least squares problem min_X ∥AXB + CXD-F ∥_F , with A and C full column rank, B, D full row rank, F an n×n matrix of low rank, and ∥•∥_F the Frobenius norm. We derive a matrix-oriented implementation of LSQR, and devise an implementation of the truncation step that exploits the properties of the method. Experimental comparisons with the Conjugate Gradient method applied to the normal matrix equation and with a (new) sketched implementation of matrix LSQR illustrate the competitiveness of the proposed algorithm. We also explore the applicability of our method in the context of Kronecker-based Dictionary Learning, and devise a representation of the data that seems to be promising for classification purposes.
2024
24 aprile
Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.
2024
23 aprile
Dimitrios Vavitsas
nel ciclo di seminari: COMPLEX ANALYSIS LAB
Seminario di analisi matematica
We give a description of the zero set intersection the unit sphere of a zero-free polynomial in the unit ball of Cn. The nice description leads to the formulation of a conjecture regarding the characterization of polynomials that are cyclic in Dirichlet-type spaces in the unit ball of Cn. Furthermore, we answer partially ascertaining whether an arbitrary polynomial is not cyclic.
2024
23 aprile
Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.
2024
22 aprile
Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.
We study the limiting behavior of minimizing p-harmonic maps from a bounded 3d Lipschitz domain O to a compact connected Riemannian manifold without boundary and with finite fundamental group as p goes to 2 from below. We prove that there exists a closed set S of finite length such that minimizing p-harmonic maps converge to a locally minimizing harmonic map in O\S. We prove that locally inside O the singular set S is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in the closure of O the set S is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and O. This is a joint work with Jean Van Schaftingen and Benoît Van Vaerenbergh.
2024
18 aprile
Ermanno Lanconelli
Seminario di analisi matematica
By using a family of harmonic functions introduced by Ulku Kuran in 1972, we define a new harmonic invariant that measures the gap between the perimeter of a domain D and the perimeter of the biggest ball contained in D and centered at a fixed point x_0 of D. From the properties of this harmonic invariant we get new proofs, generalizations and partial improvements of several rigidity and stability Theorems by Lewis and Vogel, by Preiss and Toro, by Fichera and by Aharonov, Schiffer and Zalcman. The complete proofs of all these new results will appear in a paper in collaboration with Giovanni Cupini.
2024
18 aprile
Riccardo Camerlo
Seminario interdisciplinare
The Wadge preorder is a tool to compare the complexity of subsets of topological spaces: if $A,B$ are subsets of the topological spaces $X,Y$, respectively, $A$ \emph{Wadge reduces} to $B$ if there exists a continuous function $f:A\to B$ such that $A=f^{-1}(B)$. While most of the earlier work on the Wadge preorder concerned zero-dimensional Polish spaces, recent investigations have involved more general kinds of spaces. This talk surveys some of the results and presents a few open problems and perspectives in the field.
2024
17 aprile
In this talk I will explain how a certain stochastic pressure equation appears in modelling enhanced geothermal heating (EGS) and how we approach the existence problem. EGS consists of pushing water through crystalline crustal rock at depths of 6-8km, the heat from the rock can then be extracted. Based on empirical observations it seems that the porosity satisfies a log like correlation from at least mm to km scale, the permeability (diffusion coefficient) is assumed to be the exponential of permeability. As a “simple” model, we model the porosity using a Gaussian log-correlated field, the properly normalized exponential thus becomes the so called multiplicative chaos measure. Using Gaussian analysis, we transform the Wick renormalized stochastic problem into a family of weighted elliptic equations, and I will show how regularity of these equations imply existence for the stochastic solution. Joint work with: Benny Avelin (Uppsala), Tuomo Kuusi (Helsinki), Patrik Nummi (Aalto), Eero Saksman (Helsinki), and Lauri Viitasaari (Aalto).
2024
17 aprile
We study a mean field game in continuous time over a finite horizon, T, where the state of each agent is binary and where players base their strategic decisions on two, possibly competing, factors: the willingness to align with the majority (conformism) and the aspiration of sticking with the own type (stubbornness). We also consider a quadratic cost related to the rate at which a change in the state happens: changing opinion may be a costly operation. Depending on the parameters of the model, the game may have more than one Nash equilibrium, even though the corresponding N-player game does not. Moreover, it exhibits a very rich phase diagram, where polarized/unpolarized, coherent/incoherent equilibria may coexist, except for T small, where the equilibrium is always unique. We fully describe such phase diagram in closed form and provide a detailed numerical analysis of the N-player counterpart of the mean field game. Joint work with Paolo Dai Pra (Verona) and Elena Sartori (Padova).
17/04/2024
19/04/2024
Davide Spriano
Curve graphs for CAT(0) spaces
Seminario di algebra e geometria
The curve graph of a surface is a combinatorial object that encodes geometric property of a surface and it is a key ingredient in linking geometric properties and algebraic properties in low-dimensional topology. In this talk I will present an analogue of the curve graph for the class of CAT(0) spaces, and discuss some developments. This is joint work with Harry Petyt and Abdul Zalloum.
17/04/2024
19/04/2024
Maria Beatrice Pozzetti
What are higher rank Teichmüller theories?
Seminario di algebra e geometria
Classical Teichmüller theory can be understood as the study of a connected component in the variety parametrising rapresentations from the fundamental group of a topological surface of genus at least 2 in the group PSL_2(R) of isometries of the hyperbolic space. I will discuss joint work with Beyrer-Guichard-Labourie-Wienhard in which we develop a similar theory for some Lie groups G other than PSL_2(R).
17/04/2024
19/04/2024
Kevini Li
Vanishing of torsion homology growth
Seminario di algebra e geometria
For a residually finite group, we consider the growth of torsion in group homology along a residual chain. It is the analogue of L^2-Betti numbers for torsion. We establish a vanishing criterion that has good inheritance properties. Ongoing work with Clara Löh, Marco Moraschini, Roman Sauer, and Matthias Uschold.
17/04/2024
19/04/2024
George Raptis
Simplicial homotopy theory and bounded cohomology
Seminario di algebra e geometria
17/04/2024
19/04/2024
Monika Kudlinska
Fibering in manifolds and groups
Seminario di algebra e geometria
A group is said to fiber algebraically if it admits a homomorphism onto the infinite cyclic group with finitely generated kernel. Recently, Kielak generalised the work of Agol to show that algebraic fibering is detected by the vanishing of L2-homology in groups which satisfy the so-called RFRS condition. The main focus of this talk is to discuss interesting consequences of admitting algebraic fibrations for groups, with applications ranging from finding exotic subgroups of hyperbolic groups, to analysing the geometry of groups whose (co)homology satisfies a Poincaré–Lefschetz duality.
17/04/2024
19/04/2024
Paula Truöl
3-braid knots with maximal topological 4-genus
Seminario di algebra e geometria
In a joint work with S. Baader, L. Lewark and F. Misev, we classify 3-braid knots whose topological 4-genus coincides with their Seifert genus using McCoy's (un)twisting method and the Xu normal form. We also give upper bounds on the topological 4-genus of positive and strongly quasipositive 3-braid knots. In the talk, we will define the relevant terms and provide some context for our results.
17/04/2024
19/04/2024
Alice Merz
The Alexander and Markov theorems for links with symmetries
Seminario di algebra e geometria
The Alexander theorem (1923) and the Markov theorem (1936) are two classical results in knot theory that show respectively that every link can be represented as the closure of a braid and that braids that have the same clo- sure are related by a finite number of simple operations, namely conjugation and (de-)stabilization. In this talk we will construct an equivariant closure operator that takes in input two braids with a particular symmetry, called palindromic braids, and outputs a link that is preserved by an involution. Links with such symmetry are called strongly involutive, and when we restrict ourselves to knots they form a well-studied class of knots, called strongly invertible. We will hence give analogues of the Alexander and Markov theorems for the equivariant closure operator. In fact we will show that every strongly involutive link is the equivariant closure of two palindromic braids, drawing a parallel to the Alexander theorem. Moreover, we will see that any two pairs of palin- dromic braids yielding the same strongly involutive link are related by some operations akin to conjugation and (de-)stabilization.
17/04/2024
19/04/2024
Pietro Capovilla
Simplicial volume and glueings
Seminario di algebra e geometria
Simplicial volume is a homotopy invariant of manifolds introduced by Gromov to study their metric and rigidity properties. As every good notion of volume, we would expect it to behave nicely with respect to glueings. Unfortunately, this is not always the case. I will discuss under which conditions on the glueing the simplicial volume is additive, with a particular interest for aspherical manifolds.
17/04/2024
19/04/2024
Paolo Cavicchioli
Equivalence of plats in handlebodies
Seminario di algebra e geometria
This seminar elucidates the equivalence between links in handlebodies, depicted by plat closed mixed braids. We introduce an algorithm detailing the braiding process and explore the Hilden subgroup of the mixed braid group. Additionally, a concise overview of the proof of the result will be provided.
17/04/2024
19/04/2024
Martina Jørgensen
A combinatorial higher rank hyperbolicity condition
Seminario di algebra e geometria
We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples related to symmetric spaces of non-compact type and Helly groups. Joint work with Urs Lang.
17/04/2024
19/04/2024
Giorgio Mangioni
Rigidity properties of (random quotients of) mapping class groups
Seminario di algebra e geometria
A theorem of Ivanov states that the mapping class group of a finite-type surface is also the automorphism group of a simplicial complex associated to the surface, the complex of curves. In other words, any automorphism of the complex of curves is somewhat "rigid", since it can only come from a homeomorphism of the surface. This fact, which is the starting point of the geometric group theory of mapping class groups, can then be used to prove other "rigidity" results, such as that every quasi-isometry is within finite Hausdorff distance from the multiplication by some group element, and that every group automorphism is inner. In this talk, we first review the literature on the above results, giving a sketch of how one can see them as "corollaries" of Ivanov's theorem. Then we show that, assuming a forthcoming result of Abbott-Berlyne-Ng-Rasmussen, the same type of properties are enjoyed by random quotients of mapping class groups.
17/04/2024
19/04/2024
Gemma Di Petrillo
Quaternions and isometries of the hyperbolic 5-space
Seminario di algebra e geometria
It is a well-known fact that the group of orientation-preserving isometries of the hyperbolic n-space is isomorphic to the matrix group SO^+(n,1). When n=2 and n=3, these groups have a "friendlier" description as the 2x2 matrix groups PSL(2,R) and PSL(2,C). By identifying R^4 with the quaternion algebra H, we will see that something similar happens in the n=5 case: more precisely, we will show that SO^+(5,1) is isomorphic to PSL(2,H) - the space of 2x2 quaternionic matrices with Dieudonné determinant equal to 1. At the end of the talk, I will give an idea on how these results can be applied to try and understand deformations of complete hyperbolic 3-manifolds (with finite volume) in the 5-dimensional hyperbolic space. This is based on a joint work with Bruno Martelli.
17/04/2024
19/04/2024
Giuseppe Bargagnati
Action of mapping class groups on de Rham quasimorphisms
Seminario di algebra e geometria
The group of automorphisms of a group acts naturally on the space of quasimorphisms by precomposition. In 2023, Fournier-Facio and Wade proved that for a large class of groups there exists an infinite- dimensional space of quasimorphisms invariant for this action. Since their construction is non-explicit, it makes sense to ask whether some interesting subspaces of quasimorphisms admit or not fixed points for the action above. We will focus our attention on de Rham quasimorphisms, which were introduced by Barge and Ghys in the 80s. In this case, the (outer) automorphisms coincide with the (extended) mapping class group. We will prove that there are no non-trivial subspaces of de Rham quasimorphisms which are invariant for this action.
17/04/2024
19/04/2024
Matthias Uschold
Torsion homology growth and cheap rebuilding of inner-amenable groups
Seminario di algebra e geometria
Inner-amenability is a weak form of amenability, which is satisfied e.g. by products where one factor is infinite amenable. Some properties of amenable groups extend to inner-amenable groups, e.g. the vanishing of the first $\ell^2$-Betti number. In this talk, we will treat logarithmic torsion homology growth. One tool for showing vanishing of this invariant is the cheap rebuilding property of Abért, Bergeron, Frączyk and Gaboriau. Certain inner-amenable groups have this property in degree one, thus extending vanishing results that were already known for amenable groups.
17/04/2024
19/04/2024
Jacopo Guoyi Chen
Computing the twisted L2-Euler characteristic
Seminario di algebra e geometria
The twisted $L^2$-Euler characteristic is a homotopy invariant of CW complexes introduced in a 2018 article by Friedl and Lück. Since the invariant agrees with the Thurston norm on a large class of 3-manifolds, it appears quite promising for the study of fibrations over the circle in more general spaces, especially higher dimensional manifolds. We present an algorithm that computes the twisted $L^2$-Euler characteristic, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant of certain matrices. The algorithm needs to run for an extremely long time to certify its outputs, but a truncated, human-assisted version produces very good results in many cases, including hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional examples, such as the fiber of the Ratcliffe-Tschantz 5-manifold.
17/04/2024
19/04/2024
Anna Roig Sanchis
On the length spectrum of random hyperbolic 3-manifolds.
Seminario di algebra e geometria
We are interested in studying the behaviour of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of construction of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model.
17/04/2024
19/04/2024
Edoardo Rizzi
Some cusp-transitive hyperbolic 4-manifolds
Seminario di algebra e geometria
We realize 4 of the 6 closed orientable flat 3-manifolds as a cusp section of an orientable finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps.
2024
16 aprile
Philippe Ellia
nel ciclo di seminari: MATEMATICI NELLA STORIA
Seminario di algebra e geometria, storia della matematica
Un veloce ed informale racconto della vita, dei metodi e dell'eredità matematica (specie in teoria dei numeri) di Pierre de Fermat. Si cercherà anche di sfatare un mito riguardo alla sua famosa congettura.
2024
16 aprile
Pietro Beri
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
2024
15 aprile
Gianmarco Todesco
Seminario di didattica della matematica, interdisciplinare, storia della matematica
Nel 1958 l'artista olandese M. C. Escher fu profondamente colpito da uno schema di triangoli curvilinei presente in un libro del matematico canadese H.S.M. Coxeter. Escher rielaborò la figura a modo suo, creando la splendida serie di incisioni "Limite del Cerchio". La matematica intessuta nelle quattro opere è un buon punto di partenza per esplorare diversi concetti interessanti: il piano iperbolico, le geometrie non euclidee e i rapporti fra curvatura e geometria di una superficie.
15/04/2024
19/04/2024
BARKLEY Grant
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
GAETZ Christian
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
MARIETTI Mario
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
SICONOLFI Viola
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
SENTINELLI Paolo
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
DYER Matthew
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
BOLOGNINI Davide
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
15/04/2024
19/04/2024
ESPOSITO Francesco
Relazione all'interno del convegno: Bruhat order: recent development and open problems
Seminario di algebra e geometria
2024
12 aprile
Vasiliki Liontou
TBA
nel ciclo di seminari: NEUROMATEMATICA
Seminario di analisi matematica, interdisciplinare
2024
11 aprile
Daniela Di Donato
Seminario di analisi matematica
Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize some results proved by Ambrosio, Serra Cassano, Vittone valid in Heisenberg groups to the more general setting of Carnot groups
2024
11 aprile
Peter Friz
Seminario di finanza matematica, probabilità
In this introduction to diamond trees and forests, we focus on their application to computation in stochastic volatility models written in forward variance form, rough volatility models in particular.
2024
10 aprile
Martina Iannacito
nell'ambito della serie: SEMINARI MAT/08 TEAM
Seminario di analisi numerica
Tensors have become widely used in various domains due to their practicality. Tensor factorization techniques are used to solve computationally demanding problems, analyze large datasets, and refine descriptions of complex phenomena. This presentation outlines the development of my research on tensors, including an overview of commonly used tensor methods and their applications in various fields such as remote sensing, multilinear algebra, numerical simulation, and signal processing. Criteria for selecting the most appropriate tensor technique depending on the problem under consideration will be emphasized. The presentation aims to outline the advantages and limitations inherent in these techniques. It explores the challenges and offers insights into current research directions driven by real-world, computational, and applied problems.
2024
10 aprile
Armando Martino
Seminario di algebra e geometria
Seminario specialistico rivolto agli esperti del settore riguardante la cosiddettta twisted conjugacy. Finanziato dall'Unione Europea - NextGenerationEU a valere sul Piano Nazionale di Ripresa e Resilienza (PNRR) – Missione 4 Istruzione e ricerca – Componente 2 Dalla ricerca all’impresa - Investimento 1.1, Avviso Prin 2022 indetto con DD N. 104 del 2/2/2022, dal titolo Geometry and topology of manifolds, codice proposta 2022NMPLT8 - CUP J53D23003820001
2024
09 aprile
Claudio Procesi
nel ciclo di seminari: MATEMATICI NELLA STORIA
Seminario di algebra e geometria, storia della matematica, analisi matematica
Verranno discusse la vita e i lavori di Riemann e si accennerà ai Matematici con cui ha avuto contatti, fra cui molti italiani.
2024
08 aprile
Zoran Škoda
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria, fisica matematica
In Tannakian formalism, groups and generalizations like groupoids and Hopf algebras can be reconstructed from the fiber functor which is a forgetful strict monoidal functor from its category of modules to the category of vector spaces. Can we do something similar for the actions of groups, their properties and the generalizations ? If a Hopf algebra H coacts on an algebra A by a Hopf action (that is, A becomes an H-comodule algebra) then the category of H-modules acts on the category of A-modules, this action strictly lifts the trivial action of the category of vector spaces on A-modules and also H lifts to a comonoid in H-modules inducing a comonad on the category of A-modules. The comodules over this comonad are the analogues of H-equivariant sheaves and a Galois condition can be stated in terms of affinity in the sense of Rosenberg. We propose taking these properties as defining for a general framework allowing for the definition of Galois condition/principal bundles/torsors in a number of geometric situations beyond the cases of Hopf algebras coacting on algebras. We also sketch how many other examples like coalgebra-Galois extensions and locally trivial nonaffine noncommutative torsors fit into this framework.
2024
08 aprile
Piotr M. Hajac
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria, fisica matematica
In topology, principal bundles are often assumed to be locally trivial. While the original Cartan definition of a principal bundle is inherently global, and has been successfully promoted to noncommutative geometry in the compact/unital setting, the standard concept of local triviality is hard to implement for quantum principal bundles. The goal of this talk is to review the state of the art of the concept of locality for quantum principal bundles. We begin with a guided tour concerning the topology of compact Hausdorff principal bundles. In particular, the difference between the piecewise and local triviality shall be explained. In the main part of the talk, the local triviality of compact quantum principal bundles will be compared with the piecewise triviality of principal comodule algebras. (This is a review talk based on joint works with many collaborators.)
2024
04 aprile
Maria Clara Nucci
Seminario interdisciplinare
2024
04 aprile
The functional analytic setting of various variational models in Fracture Mechanics requires the use of classes of functions with set of discontinuities of codimension one. The difficulty of finding good discretization for such classes of functions makes the direct numerical simulation of those variational problems challenging and highly problematic. For this reason, numerous regularizations have been proposed, the most successful of which are phase-field functionals. These elliptic regularizations were first introduced and analyzed in the work of Ambrosio and Tortorelli for the Mumford-Shah energy in image segmentation, inspired by a now classical example in phase transition by Modica and Mortola. Ambrosio and Tortorelli type approximations have become very popular both in the communities of Calculus of Variations and of Computational Mechanics to address a number of problems in applied sciences, especially in brittle fracture. In the talk, we will comment on some of those phase-field models, starting with Ambrosio and Tortorelli's, which eventually led to a useful variant for approximating cohesive energies in Fracture Mechanics.
Sunlight constitutes an abundant and endless natural fuel, available worldwide. In a society where a substantial part of the global energy yield is being directly expended at the city scale, urban areas appear as serious candidates for the production of solar energy. Their intrinsic complexity yet makes it challenging. The morphological heterogeneity between urban geometries and intricacy of their materials optical properties especially contribute together to causing important spatiotemporal variations in the distribution of incident solar radiations. The field of irradiance received by a specific urban region (e.g. façade, building, district) may thus rapidely become the result of complex miscellaneous interactions between many degrees of freedom. Besides, Principal Component Analysis (PCA) has been widely validated as an efficient algorithm to identify the principal behavioural features, or modes of variability, of a high-dimensional phenomenon. An approach is proposed here for analysing the variations in space and time of the solar resource within an urban context by means of PCA. A parametric investigation is conducted on a set of theoretical 100×100 m² urban districts, defined as arrangements of cuboid-like buildings, with various typological indicators (Total Site Coverage, Average Building Height) and surface materials (Lambertian, highly-specular) at three different latitudes. For each configuration, the distribution of irradiance incident on the facets of the central building is modelled via backwards Monte-Carlo ray tracing over a full year and under clear sky conditions, with a 15 min timestep and 1 m spatial resolution. PCA is subsequently applied to the simulated radiative fields to extract dominant modes of variation. First results validate energy-based orthogonal decompositions like PCA as efficient tools for characterising the variability distribution of multivariate phenomena in this context, allowing for the identification of district areas subjected to important spatial and temporal variations of the solar resource. Characteristic time scales are clearly represented across successive orders of decomposition. Information about the district morphology is also obtained, with the contribution of surrounding geometries being portrayed by specific spatial modes. Similar prevalent variables are further repetitively encountered across multiple evaluated surfaces, but at different modal ranks.
2024
26 marzo
Angelo Vistoli
nel ciclo di seminari: MATEMATICI NELLA STORIA
Seminario di algebra e geometria, storia della matematica
Enrico Betti non in realtà non ha mai definito quelli che ora si chiamano numeri di Betti. Durante la conferenza verrà illustrato in modo elementare una connessione veramente notevole tra numeri di Betti e l'aritmetica delle varietà algebriche.
An elementary argument (for sure well-known to the operator theory community) allows to compare the orthogonal projection of a Hilbert space H onto a given closed subspace of H, with (any) bounded non-orthogonal projection acting among the same spaces: this yields an operator identity that is valid in the Hilbert space H. This paradigm has deep implications in analysis, at least in two settings: -in the specific context where the Hilbert space consists of the square-integrable functions along the boundary of a rectifiable domain D in Euclidean space, taken with with respect to, say, induced Lebesgue measure ds (the Lebesgue space L^2(bD)), and the closed subspace is the holomorphic Hardy space H^2(D). In this context the orthogonal projection is the Szego projection, and the non-orthogonal projection is the Cauchy transform (for planar D), or a so-called Cauchy-Fantappie’ transform (for D in C^n with n¥geq 2). -in the specific context where the Hilbert space is the space of square-integrable functions on a domain D in Euclidean space taken with respect to Lebesgue measure dV, and the closed subspace is the Bergman space of functions holomorphic in D that are square-integrable on D. Here the orthogonal projection is the Bergman projection, and the non-orthogonal projection is some ``solid’’ analog of the Cauchy (or Cauchy-Fantappie’) transform. A prototypical problem in both of these settings is the so-called ``L^p-regularity problem’’ for the orthogonal projection where p¥neq 2. This is because the Szego and Bergman projections, which are trivially bounded in L^2 (by orthogonality), are also meaningful in L^p, p¥neq 2 but proving their regularity in L^p is in general a very difficult problem which is of great interest in the theory of singular integral operators (harmonic analysis). Three threads emerge from all this: (1) a link between the (geometric and/or analytic) regularity of the ambient domain and the regularity properties of these projection operators. (2) applications to the numerical solution of a number of boundary value problems on a planar domain D that model phenomena in fluid dynamics. For a few of these problems there can be no representation formula for the solution: numerical methods are all there is. (3) the effect of dimension: for planar D the projection operators are essentially two and can be studied either directly or indirectly via conformal mapping (allowing for a great variety of treatable domains); as is well known, in higher dimensional Euclidean space there is no Riemann mapping theorem: conformal mapping is no longer a useful tool. On the other hand the basic identity in L^2 (see above) is still meaningful but geometric obstructions arise (the notion of pseudoconvexity) that must be reckoned with.
2024
25 marzo
Luca Decembrotto; Giulia De Rocco; Andrea Maffia
Seminario di didattica della matematica, interdisciplinare
La didattica in contesto penitenziario è regolata da una specifica normativa, che prevede un complesso intreccio di responsabilità tra istituzione detentiva e scuola, tra esigenze di sicurezza e diritto allo studio. Comprendere i diritti e i doveri dello studente ristretto è requisito fondamentale per progettare interventi educativi e didattici in contesti penitenziari. Il seminario formativo, organizzato dal Dipartimento di Matematica e dal Dipartimento di Scienze dell’Educazione dell’Università di Bologna all’interno del progetto Learning Math in Prison (LeMP), ha lo scopo di introdurre ricercatori, studenti e altri soggetti interessati a queste tematiche, in particolare al contesto in cui è collocata la scuola in carcere e, quindi, l’insegnamento della matematica.
2024
25 marzo
Francisco Pereira
Seminario di analisi numerica
The Evolutionary and Complex Systems Group (ECOS) is an Artificial Intelligence research group of the Centre for Informatics and Systems of the University of Coimbra (CISUC), Portugal. ECOS research is based on the development of bio-inspired optimization algorithms and on the application of machine learning methods to extract useful knowledge and patterns from different data sources. Our group has a strong tradition for collaborations with national and international research groups, aiming at developing methods that can effectively address real problems arising in different areas. In this presentation we will provide a general overview of several problems that have been addressed in the areas of chemistry, biochemistry, and life sciences.
2024
21 marzo
2024
20 marzo
The many-body Schrödinger equation may be considered a triumph of reductionism: a vast variety of phenomena has been reduced to a single linear partial differential equation. Unfortunately the Schrödinger equation can in general not be solved efficiently, since entanglement suffers from the curse of exponentially growing dimensionality. So this triumph has created a new challenge: revealing the macroscopic behavior from the microscopic equation. I will present recent advances on the emergence of nonlinear equations through the analysis of scaling limits, and then discuss current challenges in the description of interacting fermionic quantum systems. In particular I will discuss the role of non-trivial correlations in the ground state.
We will discuss some recent results concerning weak and strong well-posedness of nonlinear stable driven SDEs with convolution interaction kernel, where the kernel belongs to a suitable Besov space. We will in particular characterize how singular the kernel can be in function of the stability index of the driving noise. In connection with some concrete models, some convergence rates for an approximating particle system will be discussed.
2024
20 marzo
In this paper, we introduce a novel observation-driven model for high-dimensional correlation matrices, wherein the largest conditional eigenvalues are modelled dynamically. We impose equal correlations for any pair of assets from the same sector(s), which facilitates the use of a highly efficient alternative expression of the likelihood of a tν-distributed random vector. This alternative expression utilises the canonical form for block correlation matrices by Archakov and Hansen (2020). The dynamics of the eigenvalues is obtained from the Generalised Autoregressive Score (GAS) framework by Creal et al. (2011). We provide an empirical application by constructing Global Minimum Variance (GMV) portfolios using daily returns of 200 assets. In its simplest form, where just a single eigenvalue is updated, our model is extremely fast to estimate. It surpasses the Dynamic Equicorrelation (DECO) model model by Engle and Kelly (2012) and rivals their Block DECO (BDECO) model’s performance in achieving low variance in GMV portfolio returns. Joint work with: Stan Thijssen and Andre Lucas.
2024
19 marzo
Chenyu Bai
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
The opening segment will explore the conjectural relationships between Hodge structures and Chow groups. The Bloch-Beilinson conjecture suggests a functorial filtration on the Chow groups of smooth projective varieties, underpinned by natural axioms. We anticipate refined structures of Bloch-Beilinson filtrations, particularly within projective hyper-Kähler and Calabi-Yau manifolds, as proposed by Beauville and Voisin. Linking these to the generalized Hodge conjecture allows the formation of explicit conjectures. Verifying these for specific Calabi-Yau manifolds or projective hyper-Kähler manifolds could substantiate both the Bloch-Beilinson and generalized Hodge conjectures. **Part 2 title:** *Voisin's Conjecture and Voisin's Map* Voisin's work, which crafts a series of K-trivial varieties from cubic hype-resurfaces and self-rational maps on them, called the Voisin's map will be the focus here. Notable among these is the Fano variety of lines of a cubic fourfold, a dimension 4 hyper-Kähler manifold. The Voisin's map in this case has been extensively studied. We'll examine higher-dimensional examples, which are all Calabi-Yau manifolds. This session aims to study the geometry of these manifolds and apply their structural insights to the conjectures on algebraic cycles discussed in Part 1, utilizing Voisin's self-rational map as a pivotal analytical tool.
2024
14 marzo
In this talk, we will explore the uniqueness of solutions to variable coefficient Schrödinger equations. We will show that, given appropriate decay assumptions on the coefficients and on the solution at two different times, the corresponding solution must be identically zero. Based on a joint work with Zongyuan Li and Xueying Yu.
2024
14 marzo
Guan Haoran
nell'ambito della serie: SEMINARI MAT/08 TEAM
Seminario di analisi numerica
CholeskyQR is a popular algorithm for QR factorization in both academia and industry. In order to have good orthogonality, CholeskyQR2 is developed by repeating CholeskyQR twice. Shifted CholeskyQR3 introduces a shifted item in order to deal with ill-conditioned matrices with good orthogonality. This talk primarily focuses on deterministric methods. We define a new matrix norm and make improvements to the shifted item and error estimations in CholeskyQR algorithms. We use such a technique and provide an analysis to some sparse matrices in the industry for CholeskyQR. Moreover, we combine CholeskyQR and our new matrix norm with randomized models for probabilistic error analysis and make amelioration to CholeskyQR. A new 3-step algorithm without CholeskyQR2 is also developed with good orthogonality.
In any context of life, human beings aim to achieve the best possible result with minimal effort. In this talk, we discuss how to implement this general principle to the numerical approximation of partial differential equations, where the aim is to obtain accurate approximations at low computational costs. Using the approximation of the Poisson equation by standard finite element methods as a prototypical example, we show how adaptive algorithms based on rigorous a posteriori error estimation lead to approximations that are, in a certain sense, optimal.
2024
12 marzo
Basile Coron
Seminario di algebra e geometria
We will give a brief introduction to classical operadic theory and then define our own operad-like structure, governed by geometric lattices and chains of flats. We will give several examples of such structures and see how some of those examples are related to the Kazhdan-Lusztig theory of geometric lattices.
2024
08 marzo
Michael Hartz
nell'ambito della serie: COMPLEX ANALYSIS LAB
Seminario di analisi matematica
The classical von Neumann inequality shows that for any contraction T on a Hilbert space, the operator norm of $p(T)$ satisfies \[ \|p(T)\| \le \sup_{|z| \le 1} |p(z)|. \] Whereas Ando extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.
In this talk, I will present a renormalization group analysis of the problem of Anderson localization in finite spacial dimensions d and on Regular Random Graphs (RRGs). I will first review and extend the finite-dimensional analysis of Abrahams, Anderson, Licciardello, and Ramakrishnan in terms of spectral observables, and discuss how to take the large-d limit. I will then motivate that the infinite-dimensional case, relevant also in the context of Many-Body Localization, recovers the Anderson model on RRGs. In this case, the renormalization group β-function necessarily involves two parameters, but the one-parameter scaling hypothesis is recovered for sufficiently large system sizes. I will also discuss how to understand this change in behavior in terms of the geometrical properties of the graphs. The talk will be based on arXiv:2306.14965 and ongoing work.
2024
07 marzo
Giovanni Eugenio Comi
Seminario di analisi matematica
The Gauss-Green and integration by parts formulas are of significant relevance in many areas of mathematical analysis and physics, and such applications motivated several investigations to extend these formulas to less regular integration domains and vector fields. These endeavours naturally led to the definition of the divergence-measure fields, which are L^p-summable vector fields whose divergence is a Radon measure. By applying a Leibniz rule between functions of bounded variation and essentially bounded divergence-measure fields, we will prove Gauss--Green formulas for these fields on sets with finite perimeter. It is also of interest to consider as integration domains sets with possibly fractal boundary, such as sets with finite fractional perimeter. To this purpose, we will present a distributional approach to fractional Sobolev spaces and fractional variation, which exploits the notions of fractional Riesz gradient and divergence. This will allow us introduce the fractional divergence-measure fields, which, in perfect analogy with the integer case, are L^p-summable vector fields whose fractional divergence is a Radon measure. Finally, we will provide Leibniz rules involving such fields and suitably regular scalar functions, leading to the fractional version of the Gauss-Green formula. The talk is mainly based on joint works with Kevin R. Payne and Giorgio Stefani.
2024
29 febbraio
Francesca Corni
Seminario di analisi matematica
We present some recent results about a way of defining suitable fractional powers of the sub-Laplacian on an arbitrary Carnot group through an analytic continuation approach introduced by Landkof in Euclidean spaces. Furthermore, we present a stronger outcome in the setting of the Heisenberg group, which is the simplest non-commutative stratified group. Eventually, in this context we propose a geometrical application of our result: we compute the value of suitable momenta with respect to the heat kernel. This is joint work with Fausto Ferrari.
2024
29 febbraio
Yvain Quéau
Seminario di analisi numerica, interdisciplinare
The Bayeux Tapestry is an exceptional Middle Age embroidery, of 70m long and 50cm high. Throughout 55 scenes, it tells the epic of William, Duke of Normandy, who left Normandy with his armada in 1066 to conquer the kingdom of England. However, researchers and scientists interested in the study of this unique artifact are confronted with problems related to temporal or geographical constraints i.e., to accessibility: the number of visitors, the exceptional size of the document, the protective glass, etc. In order to solve these accessibility issues and thus facilitate access to the Tapestry to scientists and the general public, we proposed to create a digital multimodal (daylight, multi-spectral and fine-scale geometry) panorama, which can be explored online in a web interface. This talk will present the mathematical and AI tools which were developed for generating this multimodal panorama, from the spatial and spectral registration to the deep learning-based fine-scale 3D-reconstruction.
2024
23 febbraio
Mauro Di Nasso
Seminario di algebra e geometria, logica
In recent years there has been a growing interest in Ramsey theory and related problems in combinatorics of numbers. Historically, the earliest results in this field are Schur's Theorem ("In every finite coloring of the naturals there exists a monochromatic triple a, b, a+b") and van der Waerden's Theorem ("In every finite coloring of the naturals there exist monochromatic arithmetic progressions of arbitrary length"). A peculiar aspect of this area of research is the wide variety of methods used: in addition to the tools of elementary combinatorics, also methods of discrete Fourier analysis, ergodic theory, and ultrafilter space algebra have been successfully applied. Recently, a further line of research has been undertaken, in which combinatorial properties of sets of integers are studied by methods of nonstandard analysis. In this seminar I will discuss these methods and present some examples of their applications.
2024
22 febbraio
Nicholas Meadows
Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie
Monads and algebraic theories are two categorical approach to universal algebra. In his book Higher Algebra, Jacob Lurie established a relatively comprehensive theory of monads on infinity categories. However, his approach can be difficult in practice to use due to its highly technical nature. In this talk, we will describe a version of generalized algebraic theories in the $\infty$-categorical setting, and show that it is recovers Lurie's theory for nice monads. As an application, we will prove several structural results about monads in the $\infty$-categorical setting. We will also use our result to describe the algebraic theories of E_1, E_2, and E_\infty algebras.
2024
22 febbraio
We discuss the characterization of gauge-symmetric functions in the Heisenberg group via various geometric prescriptions on their level sets. In this talk we mainly focus on a family of overdetermined boundary value problems of Serrin type which exhibit both similarities and differences with respect to the classical symmetry result for the torsion function. We show uniqueness results for gauge balls under suitable partial symmetry assumptions for the class of competitor sets. The main technical tool is a new Bochner-type identity for functions with toric/cylindrical invariances. This is a joint project with V. Martino.
2024
21 febbraio
Aim of the talk is to present an existence result to the anisotropic 1-Laplace problem div [∇_ξ φ(·,∇u)] = μ on Ω with Dirichlet boundary datum u_0 in L^1(∂ Ω) and μ a signed, Radon measure on Ω. Our approach consists in proving the existence of BV-minimizers for the corresponding integral functional Φ_{u_0}. In doing so, we characterize the appropriate assumptions for the measure μ in order to obtain lower-semicontinuity of Φ_{u_0}, and discuss a refined LSC for the related parametric functional. Additionally, we prove the definition of Φ_{u_0} to be consistent with the original anisotropic problem in the Sobolev space W^{1,1}_{u_0}(Ω) and provide some examples. Finally, further research directions will be sketched to include a broader class of functionals with linear growth.
2024
21 febbraio
Markus Fischer
nell'ambito della serie: STOCHASTICS AND APPLICATIONS
Seminario di probabilità
In the context of finite horizon mean field games with continuous time dynamics driven by additive Wiener noise, we introduce a notion of coarse correlated equilibrium in open-loop strategies. For non-cooperative many-player games, a coarse correlated equilibrium can be seen as a lottery on strategy profiles run according to a publicly known mechanism by a moderator who uses the (non-public) lottery outcomes to tell players in private which strategy to play. Players have to decide in advance whether to pre-commit to the mediator's recommendations or to play without seeing them. We justify our definition by showing that any coarse correlated solution of the mean field game induces approximate coarse correlated equilibria for the underlying N-player games. An existence result for coarse correlated mean field game solutions, not relying on the existence of classical solutions, will be given; an explicitly solvable example will be discussed as well. Joint work with Luciano Campi and Federico Cannerozzi (University of Milan "La Statale").
2024
21 febbraio
Modeling traffic dynamics has highlighted some universal properties of emergent phenomena, like the stop and go congestion when the vehicle density overcomes a certain threshold. The congestion formation on a urban road network is one of the main issues for the development of a sustainable mobility in the future smart cities and different models have been proposed. The quantification of the congestion degree for a city has been considered by various authors and data driven models have been develpoed using the large data sets on individual mobility provided by the Information Communication Technologies. However the simulation results suggest the existence of universal features for the transition to global congested states on a road network. We cope with the question if simple transport models on graph can reproduce universal features of congestion formation and the existence of control parameters is still an open problem. We propose a reductionist approach to this problem studying a simple transport model on a homogeneous road network by means of a random process on a graph. Each node represents a location and the links connect the different locations. We assume that each node has a finite transport capacity and it can contain a finite number of particles (vehicles). The dynamics is realized by a random walk on graphs where each node has a finite flow and move particles toward the connected nodes according to given transition rates (link weights). Each displacement is possible if the number of particles in the destination nodes is smaller than their maximal capacity. The graph structure can be very simple, like a uniform grid, but we have also considered random graphs with maximum in and out degree, to simulate more realistic transport networks. We study the properties the stationary distributions of the particles on the graph and the possibility of the applying the entropy concept of Statistical Mechanics to characterize the stationary distributions and to understand the congestion formation.
2024
21 febbraio
TBA
TBA
nell'ambito della serie: STOCHASTICS AND APPLICATIONS
Seminario di finanza matematica
TBA
2024
20 febbraio
Maxim Smirnov
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria, teoria delle categorie
Starting from the pioneering works of Beilinson and Kapranov, derived categories of coherent sheaves on homogeneous varieties G/P have attracted a lot of attention over the last decades. We’ll begin by an introduction into this area and then discuss more recent developments related to Lefschetz exceptional collections, quantum cohomology and homological mirror symmetry.
2024
19 febbraio
The talk is concerned with uniqueness, in weighted lp spaces, of solutions to the Schrödinger equation with a potential V , posed on an infite graph. We distinguish the cases 1 ≤ p < 2 and p ≥ 2. Moreover, we discuss uniqueness of bounded solutions, under relaxed assumptions on V . Such results have been recently obtained jointly with S. Biagi (Politecnico di Milano) and G. Meglioli (Bielefeld University).
Hilbert geometries have been introduced as a generalization of hyperbolic geometry, and provide a family of metric spaces where the Euclidean straight lines are geodesics. A Hilbert geometry is said to be divisible if it admits a group of isometries that acts cocompactly on the space. The aim of this talk is to introduce the class of divisible Hilbert geometries and to look at a characterization of hyperbolicity in this class.
2024
15 febbraio
The fractional p-Laplacian is a nonlinear, nonlocal operator with fractional order and homogeneity exponent p>1, arising in game theory and extending (in some sense) both the classical p-Laplacian and the linear fractional Laplacian. While behaving similarly to its local counterpart from the point of view of variational and topological methods, this operator requires an "ad hoc" approach in regularity theory. We will give an account on some regularity results for elliptic equations driven the fractional p-Laplacian, either free or coupled with nonlocal Dirichlet conditions: in particular we will discuss interior and boundary Hölder continuity, a special form of weighted Hölder regularity, and a recent local clustering lemma. Finally, we will rapidly hint at some applications such as comparison principles, Hopf type lemmas, Harnack inequalities, and an equivalence principle between Sobolev and Hölder minimizers of the associated energy functional. The talk is mainly based on some very recent collaborations with F.G. Düzgün, S. Mosconi, and V. Vespri.
2024
13 febbraio
An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov–Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds famous automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture.
2024
07 febbraio
Aristotelis Panagiotopoulos
Seminario di algebra e geometria, analisi matematica, logica, sistemi dinamici
A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic α-unbalancedness": a turbulence-like obstruction to classification by actions of α-balanced Polish groups. Finally, for each α we will provide an action of an α-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any β-balanced Polish group with β<α. This is joint work with Shaun Allison.
2024
07 febbraio
Kieran O’Grady
Seminario di algebra e geometria
TBA
2024
06 febbraio
Franco Giovenzana
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
TBA
2024
02 febbraio
Anna Miriam Benini
nell'ambito della serie: COMPLEX ANALYSIS LAB
Seminario di analisi matematica
Transcendental Henon maps are a class of automorphisms of $C^2$ with rich dynamical behavior, yet in some sense, tame enough for general theorems to be proven. In this talk we explore some features of the dynamics of such maps, connecting them with some function theoretical properties of entire transcendental functions in one variable.
Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate
2024
30 gennaio
Alexander Kuznetsov
nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
2024
26 gennaio
Antongiulio Fornasiero
Seminario di algebra e geometria, interdisciplinare, logica
Let d be a finite tuple of commuting derivations on a field K. A classical result allows us to associate a numerical polynomial to d (the Kolchin polynomial), measuring the "growth rate" of d. We show that we can abstract from the setting of fields with derivations, and consider instead a matroid with a tuple d of commuting (quasi)-endomorphisms. In this setting too there exists a polynomial measuring the growth rate of d. Joint work with E. Kaplan
2024
25 gennaio
The spectrum of non-selfadjoint operators can be highly unstable even under very small perturbations. This phenomenon is referred to as "pseudospectral effect". Traditionally this pseudosepctral effect was considered a drawback since it can be the source of immense numerical errors, as shown for instance in the works of L. N. Trefethen. However, this pseudospectral effect can also be the source of many new insights. A line of works by Hager, Bordeaux-Montrieux, Sjöstrand, Christiansen and Zworski exploits the pseudospectral effect to show that the (discrete) spectrum of a large class of non-selfadjoint pseudo-differential operators subject to a small random perturbation follows a Weyl law with probability close to one. In this talk we will discuss some recent results on the macroscopic and microscopic distribution of eigenvalues as well as eigenvector localization and delocalization phenomena of non-selfadjoint operators subject to small random perturbations.
2024
24 gennaio
This lecture starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model the introduction of new kinds of expectations and learning mechanisms open new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature.
2024
24 gennaio
We prove a second-order smooth-fit principle for a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone-follower problems and find applications in spatial models of production and climate transition. Let (D, M, μ) be a finite measure space and consider the Hilbert space H := L^2(D, M, μ; R). Let then X be a H-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a linear operator A and affected by a cylindrical Brownian motion. The evolution of X is controlled linearly via a vector-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize an infinite time-horizon, discounted convex cost-functional. By combining properties of semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem V is a C^{1,Lip}(H)-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, allowing the decision maker to choose only the intensity of the control, and requiring that the given direction of control n is an eigenvector of the linear operator A, we establish that the directional derivative V_n is of class C^1(H), hence a second-order smooth-fit principle in the controlled direction holds for V . This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.
24/01/2024
26/01/2024
Chiara Bernardini
Ergodic Mean-Field Games with Aggregation of Choquard-type
Seminario di analisi matematica
We consider second-order ergodic Mean-Field Games systems in RN with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. Equilibria solve a system of PDEs where a Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. In the Hardy-Littlewood-Sobolev-supercritical regime, by means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential term. On the other hand, in the Hardy-Littlewood-Sobolev-subcritical regime, using a fixed point argument, we show existence of classical solutions at least for masses smaller than a given threshold value. In the mass-subcritical regime, we show that actually this threshold can be taken to be +∞. Finally, considering the MFG system with a small parameter ε > 0 in front of the Laplacian, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we obtain existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around the minima of the potential.
24/01/2024
26/01/2024
Yuri Cacchió
On the effect of the Coriolis force on the enstrophy cascade
Seminario di analisi matematica
In this article, we investigate the effects of rotation on the dynamics, by neglecting stratification, in a 2D model where we incorporate the effects of the planetary rotation by adopting the β-plane approximation, which is a simple device used to represent the latitudinal variation in the vertical component of the Coriolis force. We consider the well-known 2D β-plane Navier-Stokes equations (2DβNS) in the statistically forced case. Our problem addresses energy-related phenomena associated with the solution of the equations. To maintain the fluid in a turbulent state, we introduce energy into the system through a stochastic force. In the 2D case, a scaling analysis argument indicates a direct cascade of enstrophy and an inverse cascade of energy. We compare the behaviour of the direct enstrophy cascade with the 2D model lacking the Coriolis force, observing that at small scales, the enstrophy flux from larger to smaller scales remains unaffected by the planetary rotation, confirming experimental and numerical observations. In fact, this is the first mathematically rigorous study of the above equations. In particular, we provide sufficient conditions to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function’s asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. We also prove well-posedness and certain regularity properties necessary to obtain the mentioned results.
24/01/2024
26/01/2024
Athanasios Zacharopoulos
Varopoulos' extensions in domains with Ahlfors-regular boundaries
Seminario di analisi matematica
In this talk we shall describe the construction of Varopoulos' type extensions of L^p and BMO boundary functions in rough  domains. That is, smooth extensions of functions such that the L^p-norms of their non-tangential maximal function and the Carleson  functional of their gradients can be controlled by the norm of the boundary data. After giving the geometric motivation and a brief survey of known results, we will proceed to present a new and more general approach of constructing Varopoulos' extensions in domains with minor geometrical assumptions for the boundaries.
24/01/2024
26/01/2024
Alexandre Arias Junior
3-evolution semilinear equations in projective Gevrey classes
Seminario di analisi matematica
We consider the quasilinear Cauchy problem (CP) P(t,x,u(t,x),D_t,D_x)u(t,x) = f(t,x), with (t,x)∈[0,T]xR, and initial condition u(0,x) = g(x), x∈R, where P(t,x,u,D_t,D_x) = D_t + a_3(t)D_x^3 + a_2(t,x,u)D_x^2 + a_1(t,x,u)D_x + a_0(t,x,u), a_j(t,x,w), (0≤j≤2), are continuous functions of time t, projective Gevrey regular with respect to the space variable x and holomorphic in the complex parameter w. The coefficient a_3(t) is assumed to be a real-valued continuous function which never vanishes. In this talk we shall discuss how to apply the Nash-Moser inversion theorem in order to obtain local in time well-posedness in projective Gevrey classes for the Cauchy problem (CP).
24/01/2024
26/01/2024
Carlo Bellavita
Bounded Truncated Toepliz Operators
Seminario di analisi matematica
I will talk about the Baranov-Bessonov-Kapustin conjecture: "let θ be an inner function. Any bounded truncated Toeplitz operator on the model space Kθ admits a bounded symbol only if θ is a one-component inner function." I will present all the objects involved: the model spaces, the one-component inner functions and finally the truncated Toeplitz operators. Eventually, if there is enough time, I will present a possible (in my opinion promising) approach to tackle this problem.
24/01/2024
26/01/2024
Marcello Malagutti
Asymptotic spectral properties of certain semiregular global systems
Seminario di analisi matematica
In this talk I will be stating some results about spectral analysis of systems of PDEs. Specifically, a Weyl asymptotic is given for a class of systems containing not only certain quantum optics models such as the Jaynes-Cummings model, which is fundamental in Quantum Optics, but models of geometric differential complexes over R^n, too. Moreover, I discuss a quasi-clustering result for this class of positive systems. Finally, a meromorphic continuation of the spectral zeta function for semiregular Non-Commutative Harmonic Oscillators (NCHO) is given. By “semiregular system” we mean a pseudodifferential systems with a step j in the homogeneity of the jth term in the asymptotic expansion of the symbol. The aforementioned results were obtained jointly with Alberto Parmeggiani.
24/01/2024
26/01/2024
Beatrice Andreolli
Spaces of Variable Bandwidth and signal reconstruction
Seminario di analisi matematica
A function f∈L^2(R) is said to have bandwidth Ω>0, if Ω is the maximal frequency contributing to f. The concept of variable bandwidth arises naturally and it is even more intuitive when we think about music. Indeed, the perceived highest frequency, i.e. the note, is obviously time-varying. This observation provides a reasonable argument for the assignment of different local bandwidths to different segments of a signal when representing it mathematically. However, producing a rigorous definition of variable bandwidth is a challenging task, since bandwidth is global by definition and the assignment of a local bandwidth meets an obstruction in the uncertainty principle. We present a new approach to the study of spaces of variable bandwidth based on time-frequency methods. Our idea is to start with a discrete time-frequency representation that allows us to represent any f as a series expansion of time-frequency atoms with a clear localization both in time and frequency. We may then prescribe a time-varying frequency truncation and, in this way, end up with a space of a given variable bandwidth. For these spaces, we study under which sufficient conditions on a set of points a function can be reconstructed completely from the evaluation of the function at these points. Analyzing some MATLAB experiments, we motivate why these new spaces could be useful for the reconstruction of particular classes of functions.
24/01/2024
26/01/2024
Matteo Bonino
Wodzicki residue for pseudo-differential operators on non-compact manifolds
Seminario di analisi matematica
In this seminar I will introduce the notion of Wodzicki residue, also denoted by non-commutative residue, which was first introduced by Wodzicki in 1984 while studying the meromorphic continuation of the ζ-function for elliptic operators on compact manifold with boundary. The Wodzicki residue was independentely defined by Guillemin in 1985, in the equivalent version of Symplectic residue, in order to find a soft proof of the Weyl formula. It turns out to be the unique trace, up to a multiplication by a constant, on the algebra of classical pseudodifferential operators modulo smoothing operators, provided that the manifold has dimension d>1. In the last years, the interest in the study of Wodzicki residue increased due to its applications both in mathematics (non-commutative geometry) and mathematical physics (relations with Dixmier trace). I will discuss the concept of Wodzicki residue on compact manifold with boundary, for SG-calculus on R^d and for the SG-calculus on manifolds with cylindrical ends. Finally, as a joint work with Professor S. Coriasco, I will present an extension of the non-commutative residue on a certain class of non-compact manifolds called scattering manifolds.
24/01/2024
26/01/2024
Francesca Bartolucci
Non-uniqueness in sampled Gabor phase retrieval
Seminario di analisi matematica
Sampled Gabor phase retrieval --- the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice --- is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in L^2(R), but is not equal to the whole of L^2(R) in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.
24/01/2024
26/01/2024
Giacchi Gianluca
Relazione all'interno del convegno: Symposium in Harmonic & Complex Analysis, Microlocal & Geometrical Analysis and Applications, for PhD students (SHaCAMiGA)
24/01/2024
26/01/2024
Guido Drei
Hypoellipticity on compact Lie groups
Seminario di analisi numerica
In this contributed talk we introduce, in a theoretical representation setting, a necessary and sufficient condition, namely the Rockland condition, for a left-invariant differential operator on a compact Lie group G to be globally hypoelliptic. In particular, we focus on the case of a product of two compact Lie groups G=G1×G2 and we show some examples on T^2 and on T^1×SU(2). It is possible to prove the existence of globally hypoelliptic smooth-coefficient operators that are not locally hypoelliptic. In the end, we present a class of pseudodifferential operators on the product G=G1×G2 and the so called bisingular pseudodifferential calculus, as introduced by L. Rodino in 1975.
24/01/2024
26/01/2024
Francesca Corni
An area formula for intrinsic regular graphs in homogeneous groups
Seminario di analisi matematica
We present an explicit area formula to compute the spherical measure of an intrinsic regular graph in an arbitrary homogeneous group. In particular, we assume the intrinsic graph to be intrinsically differentiable at any point with continuous intrinsic differential. This is joint work with V. Magnani.
24/01/2024
26/01/2024
Eugenio Dellepiane
Embedding Model Spaces in Dirichlet spaces
Seminario di analisi matematica
In this talk, we discuss two classes of spaces of holomorphic functions on the unit disk D. First, the Model Spaces Ku, which arise as the invariant subspaces for the backward shift operator S* on the Hardy space H^2(D), given by S* f(z):=(f(z)-f(0))/z (z∈ D). The second class of spaces that we discuss are the harmonically weighted Dirichlet spaces D(m)$. The space D(m) consists of all analytic functions f on D such that D_m(f) :=∫_D |f'(z)|^2( ∫_{∂D} (1-|z|^2)/|z-\zeta|^2 dm(z)) dA(z) <∞. They are a generalization of the classical Dirichlet space D, and they arise naturally when studying the shift-invariant subspaces of D. After a brief introduction, we discuss sufficient and necessary conditions in order for the embedding Ku ↪ D(m) to hold. This work is related to the boundedness of the derivative operator acting on the model space Ku. This talk is based on joint work with Carlo Bellavita.
24/01/2024
26/01/2024
Davide Giovagnoli
Alt-Caffarelli-Friedman monotonicity formulas on Carnot groups
Seminario di analisi matematica
See attached file.
24/01/2024
26/01/2024
Iván Jimenez
Counterexample of normability in Hardy spaces H^p, 0<p<1
Seminario di analisi matematica
It is well-known in the literature on Hardy spaces that the Hardy spaces H^p, 0<p<1, are not normable. However, none of the sources offer proofs of this fact. In 1953, Livingston published an article demonstrating this using a convexity argument based on a theorem by Kolmogorov. In this talk, we will present a direct proof based on a counterexample of the non-normability of the Hardy spaces H^p, 0<p<1. This is a joint work with my thesis advisor Dragan Vukotic.
24/01/2024
26/01/2024
Luigi Pollastro
Approximate symmetry for the Gidas-Ni-Nirenberg result in the unitary ball
Seminario di analisi matematica
In a celebrated paper in 1979, Gidas, Ni & Nirenberg proved a symmetry result for a rigidity problem. With minimal hypotheses, the authors showed that positive solutions of semilinear elliptic equations in the unitary ball are radial and radially decreasing. This result had a big impact on the PDE community and stemmed several generalizations. In a recent work in collaboration with Ciraolo, Cozzi & Perugini this problem was investigated from a quantitative viewpoint, starting with the following question: given that the rigidity condition implies symmetry, is it possible to prove that if said condition is "almost" satisfied the problem is "almost" symmetrical? With the employment of the method of moving planes and quantitative maximum principles we are able to give a positive answer to the question, proving approximate radial symmetry and almost monotonicity for positive solutions of the perturbed problem.
24/01/2024
26/01/2024
Antonio Pedro Ramos
Sharp embeddings between weighted Paley-Wiener spaces
Seminario di analisi matematica
We consider the problem of estimating the operator norm of embeddings between certain weighted Paley-Wiener spaces. We discuss some qualitative properties for the extremal problems considered and provide some asymptotic results. For a few cases, we are able to to provide a precise formula for the sharp constant with techniques from the theory of reproducing kernel Hilbert spaces. As an application, these provide sharp constants to higher order Poincare inequalities via the Fourier transform.
24/01/2024
26/01/2024
Enzo Maria Merlino
Intrinsic Lipschitz regularity for almost minimizer of a one-phase Bernoulli-type functional in Carnot Groups of step two
Seminario di analisi matematica
The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Some generalizations will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University) and will be part of my PhD thesis.
24/01/2024
26/01/2024
Michele Motta
Lyapunov Exponents of Linear Switched System
Seminario di analisi matematica
The principal Lyapunov exponent of a dynamical system is a natural measure of the instability of the system. In our work, we computed the supremum of the principal Lyapunov exponent associated to the system dy/dt = A(t)y, y∈R^2, where the function A ranges in L^∞_loc([0,+∞);{A1,A2}), A1,A2∈R^(2x2). This kind of dynamical systems, where the dynamics can be discontinuous with respect to the time variable, are known in literature as switched systems. This computation is reduced to an optimal control problem. Applying Pontryagin Maximum Principle (PMP) to this problem, we were able to find all controls satisfying necessary conditions prescribed by PMP and then we found among them the optimal one. This is a joint work with A. A. Agrachev.
24/01/2024
26/01/2024
Tommaso Monni
FREEDMAN’S THEOREM FOR UNITARILY INVARIANT STATES ON THE CCR ALGEBRA
Seminario di analisi matematica
The set of states on CCR(H), the CCR algebra of a separable Hilbert space H, is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of CCR(H) that are invariant under the action of all automorphisms induced in second quantization by unitaries of H. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.
2024
23 gennaio
Roberto Frigerio
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
Il volume simpliciale è un invariante per varietà compatte introdotto da Gromov nel 1982. Pur essendo definito solo utilizzando l'omologia singolare, è strettamente correlato alle strutture geometriche che una varietà può supportare: ad esempio, si annulla su varietà che ammettano metriche con curvatura di Ricci non negativa, ed è positivo per varietà di curvatura negativa. In questo seminario confronteremo il volume simpliciale con alcuni invarianti ad esso correlati, come il minimo numero di simplessi in una triangolazione, o il minimo numero di simplessi singolari in un rappresentante della classe fondamentale a coefficienti interi. A tale scopo, introdurremo un nuovo invariante, chiamato "Filling volume", definito sul mapping class group di varietà. Lavoro in collaborazione con Federica Bertolotti.
2024
19 gennaio
Elena Bogliolo
Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie
Bounded cohomology of groups is a variant of group cohomology that, given a group and a Banach coefficient module over such group, gives graded semi-normed vector spaces. A major role in the theory of bounded cohomology is played by amenable groups and amenable actions as they provide vanishing conditions for bounded cohomology. The goal of this talk is to introduce bounded cohomology of groups and look into its realtion with amenability.
2024
19 gennaio
As generative AI technologies are revolutionizing industries and our daily lives, what is going to happen to the role of the mathematician? In this talk, I will highlight recent breakthroughs in deep learning and AI and explore how current and future advancements might alter the way we do mathematics.
2024
17 gennaio
Alexander Kuznetsov
nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria, teoria delle categorie
2024
16 gennaio
Ernesto Mistretta
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
In the first part we will review some notions of positivity and base loci for line bundles and how to generalise them to the case of higher rank vector bundles. In the second part we will discuss some geometric interpretations of semiampleness of the cotangent bundle, and some characterizations of abelian verieties and compact complex parallelizable manifolds.
2024
15 gennaio
Domenico Zambella
Seminario di algebra e geometria, logica
Let L be a first-order 2-sorted language. Let X be some fixed structure. A standard structure is an L-structure of the form ⟨M,X⟩. When X is a compact topological space (and L meets a few additional requirements) it is possible to adapt a significant part of model theory to the class of standard structures. This has been noticed by Henson and Iovino in the case of Banach spaces (and metric structures in general). However, in the last 20 years the most popular approach to the model theory of metric structures uses real-valued logic (Ben Yaacov, Berenstein, Henson, Usvyatsov). Arguably, this is neither natural nor general enough. We show that a few adaptations of Henson and Iovino's approach suffices for a natural and powerful theory. This is based on three facts: - every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent. - in a sufficiently saturated structure, the negation of a positive formula is an infinite disjunction of positive formulas. - there is a pure model theoretic notion that corresponds to Cauchy completeness. To exemplify how this setting applies to model theory we discuss ω-categoricity and (local) stability.
2024
12 gennaio
The classical Waring problem for homogeneous polynomials can be translated into geometric terms, using the notion of defectivity and identifiability for secant varieties. The defectivity problem was completely solved by Alexander-Hirschowitz using classical degeneration techniques. On the other hand identifiability has recently been addressed by Mella and Galuppi. In this talk I will briefly explain the relationship between defectivity and identifiability in a more general setting and give bounds for a generalized Waring problem, introduced by Fröberg, Ottaviani and Shapiro. In particular we will see how the union of classical degeneration techniques combine with techniques borrowed from toric geometry, allowing us to give very sharp bounds on identifiability and defectivity in a much more general context. In the last part of the talk I will show how to generalize the previous approach to singular toric varieties. This is a joint work (in progress) with Elisa Postinghel.
2024
09 gennaio
Tommaso Scognamiglio
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA
Seminario di algebra e geometria
Given a Riemann surface X, character stacks and varieties are geometric objects which parametrize certain representations of the fundamental group of X or, equivalently, local systems with prescribed local monodromies. These objects have a rich geometry and are related, for instance, to the moduli spaces of Higgs bundles through non abelian Hodge correspondence. The cohomology of character stacks and varieties is almost completely understood in the case of a generic choice of monodromies, thanks to the work of Hausel, Letellier and Rodriguez-Villegas and Mellit. In the non-generic case, the geometry of these objects becomes considerably more complicated and their cohomology has not been studied much until recently. In the first part of the talk, I will introduce and define character stacks and varieties and review the known results about the generic case. In the second part, I will focus on the non-generic case and give a sketch of the proof of a formula for the E-series of non-generic character stacks, which is the main result of my PhD thesis.
2024
08 gennaio
Ivan Di Liberti
Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie
2024
08 gennaio