Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 20-03-2023 al 26-03-2023
Lunedì 20 marzo 2023
Ore 11:00, aula "Roberta Dal Passo", Dipartimento di Matematica, Università di Roma "Tor Vergata"
Corso di dottorato
Marco Trevisiol (Università di Roma “La Sapienza”)
Springer Correspondence
This course aims to present one of the first results in geometric representation theory: the Springer Correspondence. The goal of geometric representation theory is to study representations of certain algebraic structures (finite Weyl groups, in our case) via geometric methods. Springer’s result (generalised in several ways and still a central tool nowadays) achieves a construction of all representations of a finite Weyl group (e.g. the symmetric group) in terms of the cohomology of a certain variety. The striking point of Springer’s construction is that the Weyl group action he discovers is highly non trivial and hidden, since the Weyl group does not act on the variety itself. Some of the arguments we are going to address are: representation theory of the symmetric group via tableaux combinatorics; flag variety and Bruhat decomposition; nilpotent cone, the Springer resolution, Springer fibres; construction of the Weyl group action on the cohomology of the Springer sheaf. The discussion will be more or less detailed, depending on the participants’ background and interests.
Per informazioni, rivolgersi a: lanini@mat.uniroma2.it
Lunedì 20 marzo 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Nicola Soave (Università degli Studi di Torino)
Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds
In this talk we present some results obtained jointly with Matteo Muratori (Politecnico di Milano), focusing on qualitative properties for • Extremals for the Sobolev inequality, • Positive radial solutions of the Lane-Emden equation for the p-Laplacian in the critical and supercritical regimes, • Positive radial solutions of the Lane-Emden system in the critical and supercritical regimes, posed on a Cartan-Hadamard manifold \(\mathbb{M}^n\). We are particularly interested in rigidity results, both for the functions themselves, and for the underlying manifold. For instance, we show that if \(\mathbb{M}^n\) supports an optimal function \(u\) for the Sobolev inequality, and the dimension \(n\) is less than or equal to 4, then \(\mathbb{M}^n\) is isometric to \(\mathbb{R}^n\), and \(u\) is an Aubin-Talenti bubble.
Per informazioni, rivolgersi a: azahara.delatorrepedraza@uniroma1.it
Lunedì 20 marzo 2023
Ore 16:00, Aula IV, Dipartimento di Matematica, Sapienza Università di Roma
Seminario "Baby A&G"
Paolo Bravi (Sapienza Università di Roma)
Grafi, arrangiamenti di iperpiani e matroidi
Vedremo come le nozioni di grafo, arrangiamento di iperpiani e matroide sono tra loro collegate. Parleremo della congettura di log-concavità del loro polinomio caratteristico, la cui dimostrazione è valsa, lo scorso anno, a June Huh la medaglia Fields.
Per informazioni: https://sites.google.com/view/sapienza-baby-ag/home
Martedì 21 marzo 2023
Ore 12:00, AULA 1B1, PAL. RM002, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Via A. Scarpa 16
Minicorso Dottorato: Lezioni 5 - 6
PROF. Cornelia Schiebold (Mid Sweden University, Sundsvall, Sweden)
Integrable systems - Methods of mathematical physics in interaction
Integrable systems in infinite dimension refer to an area in mathematical physics which is devoted to the study of a certain group of partial differential equations, many of them soliton equations like the classic Korteweg-de Vries equation and the nonlinear Schrodinger equation. One of the striking features is the existence of solutions with particle character, called solitons, remarkable in view of nonlinearity of the governing equations. Methodologically, integrable systems are a meeting point (melting pan) for methods from very diverse parts of mathematics. The main idea of this mini course is to highlight interactions of some of the main approaches to integrable systems, the inverse scattering method and an operator theoretic approach in the first place, and symmetry methods like Backlund transformations, recursion operators and hierarchies to a minor extent. Throughout we will emphasise the recent topic ofnon-commutative integrable systems, like vector- and matrix soliton equations, where many fundamental questions are still open. Notably, the construction of solutions is not interesting only under the mathematical viewpoint, but also under the physical one. Indeed, very important applications of soliton equations are in nonlinear optics, for instance.
Per informazioni, rivolgersi a: Sandra Carillo
Martedì 21 marzo 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Marta Leocata (Scuola Normale Superiore Pisa)
Modeling an example of green transition
The work presented is part of a collaboration between mathematicians and philosophers of ethics, politics, and society aiming to understand mechanisms of green energy transition where some kind of interaction between many subjects and collective behaviors seem to play a role. We have identified as a first example the case of solar photovoltaic, with the purpose of explaining the time series of Italy and of different regions. We use tools from continuous-time Markov chains, mean field limits, optimal control, and mean field games. Then, we specify our analysis on the case of domestic installations. In order to explain them we propose a Markovian model of interacting particle systems to describe the decision of a significant proportion of the general public in Italy on the installation of solar photovoltaics (PVs) and compare our models to real data. We assume that the adoption decision follows a well precise pattern. First, the general public develops a certain level of sensitivity to climate change and environmental issues. Then it plans to install the PVs because she/he has a sufficient amount of information on the benefits of PVs. Finally, it installs PVs because the economic benefits out-weights the costs. In conclusion, we propose some original policies, not based not on financial incentives. This work is in collaboration with Franco Flandoli (SNS), Giulia Livieri (SNS), Silvia Morlacchi (SNS), Fausto Corvino (University of Gothenburg) and Alberto Pirni (SSSA).
Martedì 21 marzo 2023
Ore 14:30, aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario di Geometria
Salvatore Floccari (Leibniz University Hannover)
Sixfolds of generalized Kummer type and K3 surfaces
The classical Kummer construction associates a K3 surface to any 2-dimensional complex torus. I will present an analogue of this construction in dimension six, which relates the two most studied deformation types of hyper-Kähler manifolds in this dimension. Namely, I will explain that any hyper-Kähler sixfold K of generalized Kummer type has a naturally associated hyper-Kähler manifold of K3^[3]-type. If K is projective then the associated K3^[3]-variety is a moduli space of stable sheaves on a uniquely determined K3 surface, whose Picard rank is that of K plus 15. I will also explain that the Kuga-Satake correspondence for the K3 surfaces so obtained is induced by an algebraic cycle, which gives infinitely many new families of K3 surfaces of general Picard rank 16 for which the Kuga-Satake correspondence is algebraic.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com
Martedì 21 marzo 2023
Ore 15:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Alessandro Alaia (Optimad)
An asymptotic preserving all-Mach scheme for compressible multiphase flow
In this work, we propose a well-balanced Implicit-Explicit Runge-Kutta scheme for the efficient simulation of the Baer-Nunziato model at all-Mach regimes. The numerical method is based on the explicit treatment of non-stiff terms, while stiff terms are kept implicit to remove the time step restriction stemming from acoustic waves. By introducing suitable linearizations in the semi-discrete energy equation obtained by a IMEX-RK time discretization, a predictor-corrector scheme is derived where the implicit contribution of pressure waves is accounted for by solving a system of non-linear elliptic equations in the unknown phasic pressures. A well-balanced discretization of non-conservative terms is also introduced, resulting in a numerical scheme capable of preserving steady-state solutions, including the so-called ”lake-at-rest” state. Finally, we prove that our numerical scheme is asymptotic-preserving. Compared to standard explicit finite volume schemes, the time step of our method is driven by the mean flow velocity and not by the speed of acoustic waves. Moreover, the proposed approach is characterized by a lower computational complexity if compared to fully (monolithic) implicit schemes. Numerical results demonstrate the capabilities of our numerical scheme to correctly capture shocks in the high Mach regime as well as to efficiently simulate flows at low Mach numbers.
Mercoledì 22 marzo 2023
Ore 14:00, Aula Seminari, SBAI
PDE a tutto SBAI
Salvador López Martínez (Universidad Autonoma de Madrid)
Multiple solutions for elliptic problems with subquadratic gradient terms
In the talk we will analyze the structure of (positive, bounded and weak) solutions to a Dirichlet problem which has a nonlinearity depending suquadracally on the gradient. For a quadratic nonlinearty, several recent results show, under very general conditions on the data, that zero is a bifurcation point from infinity. Moreover, if the data are small enough in some sense, then the fact that the solutions bifurcate from infinity at zero implies the existence of at least two solutions near the bifurcation point. Similar results are expected for strictly subquadratic gradient terms, even though there are essentially no results available in the literature. We will present some recent advances in this direction that are part of a joint work with José Carmona and Pedro J. Martínez-Aparicio, both from the University of Almería.
Per informazioni, rivolgersi a: massimo.grossi@uniroma1.it
Mercoledì 22 marzo 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Carlo Scarpa (Université du Québec à Montréal)
Kähler metrics and B-fields
Motivated by constructions appearing in mirror symmetry, we consider the problem of finding canonical representatives for a complexified Kähler class on a compact complex manifold. These are complex cohomology classes whose imaginary part is a Kähler class, while the real part is an arbitrary real (1,1)-class. As is often the case in complex geometry, one way to fix a representative of such a class is to impose an elliptic PDE. In this talk, I will explain why a natural choice of PDE is a coupling of the deformed Hermitian Yang-Mills equation and the constant scalar curvature equation. We will then see how to prove the existence of solutions in some special cases. Based on arXiv:2209.14157, joint work with Jacopo Stoppa.
Per informazioni, rivolgersi a: diverio@mat.uniroma1.it
Mercoledì 22 marzo 2023
Ore 14:30, aula B, Dipartimento di Matematica, Sapienza Università di Roma
Seminario congiunto di Storia della matematica e Fisica Matematica
Alessandro Teta (Università di Roma La Sapienza)
La nascita della Meccanica Quantistica negli articoli di Heisenberg (1925) e di Schroedinger (1926)
La Meccanica Quantistica si afferma negli anni 1925/26 grazie soprattutto ai lavori di Heisenberg e di Schroedinger. Nel seminario, dopo una breve premessa sul contesto storico, si discuteranno le idee fondamentali di questi lavori, limitando al minimo gli aspetti tecnici. Si cercherà in particolare di mettere in evidenza le differenze dei due approcci, basati su modi differenti di concepire la descrizione del mondo fisico. Il seminario sarà in italiano.
Mercoledì 22 marzo 2023
Ore 14:30, Canale Youtube dell'IAC https://www.youtube.com/watch?v=EC9oVJTTtgQ, Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche
Online Seminars on Artificial Intelligence and Mathematics, 2023 Edition https://www.aim.iac.cnr.it/
Boris Hanin (Princeton University)
What is Deep in Deep Learning?
Depth is key to the success of modern neural networks. Mathematically, however, it is not obvious why and when deeper is better. This talk will survey several lines of work, spanning approximation theory, probability, statistical physics, and optimization, that have sought to answer such questions.
Per informazioni, rivolgersi a: roberto.natalini@cnr.it
Giovedì 23 marzo 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Luigi Montoro (Università della Calabria)
Symmetry properties of singular solutions to p-Laplacian systems
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p-Laplacian operators. We prove some fine regularity properties of the solutions, and then we show symmetry and monotonicity properties.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it
Venerdì 24 marzo 2023
Ore 11:00, aula "Roberta Dal Passo", Dipartimento di Matematica, Università di Roma "Tor Vergata"
Corso di dottorato
Martina Lanini (Università di Roma "Tor Vergata")
Springer Correspondence
This course aims to present one of the first results in geometric representation theory: the Springer Correspondence. The goal of geometric representation theory is to study representations of certain algebraic structures (finite Weyl groups, in our case) via geometric methods. Springer’s result (generalised in several ways and still a central tool nowadays) achieves a construction of all representations of a finite Weyl group (e.g. the symmetric group) in terms of the cohomology of a certain variety. The striking point of Springer’s construction is that the Weyl group action he discovers is highly non trivial and hidden, since the Weyl group does not act on the variety itself. Some of the arguments we are going to address are: representation theory of the symmetric group via tableaux combinatorics; flag variety and Bruhat decomposition; nilpotent cone, the Springer resolution, Springer fibres; construction of the Weyl group action on the cohomology of the Springer sheaf. The discussion will be more or less detailed, depending on the participants’ background and interests.
Per informazioni, rivolgersi a: lanini@mat.uniroma2.it
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