Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 20-02-2023 al 26-02-2023
Martedì 21 febbraio 2023
Ore 14:30, aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
seminario di Geometria
Laura Capuano (Università degli Studi Roma Tre)
GCD results on geometric divisibility sequences and a conjecture of Silverman
A divisibility sequence is a sequence of integers d_n such that, if m divides n, then d_m divides d_n. Bugeaud, Corvaja, Zannier showed that independent pairs of divisibility sequences of the form a^n-1 have only limited common factors. From a geometric point of view, this divisibility sequence corresponds to a subgroup of the multiplicative group, and Silverman conjectured that a similar behavior should appear in (a large class of) other algebraic groups. Extending previous works of Silverman and of Ghioca-Hsia-Tucker on elliptic curves over function fields, we will show how to prove the analogue of Silverman’s conjecture over function fields in the case of abelian and split semiabelian varieties and some generalizations. The proof relies on some results of unlikely intersections. This is a joint work with F. Barroero and A. Turchet.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com
Mercoledì 22 febbraio 2023
Ore 10:00, Aula 6, edificio RM018 (aule di Ingegneria, Via del Castro Laurenziano), Dipartimento SBAI, Sapienza Università di Roma
Minicorso
Felix Otto (MPI-MIS, Leipzig)
Stochastic homogenization and large-scale regularity
Lectures: (10/02 14:00, 15/02 10:00, 17/02 14:00,) 22/02 10:00, 24/02 14:00. Recording of previous lectures are expected to be available upon request. In this mini-course, I will introduce the concept of large-scale regularity in case of a linear elliptic equation (or system) with heterogeneous coefficients. It is based on a smallness (on average) of the potentials of the harmonic coordinates, and proceeds via an intrinsic Campanato iteration. I will then apply this to the case of a random heterogeneous coefficient field, sampled from a stationary and ergodic ensemble. I will try to be self-contained and closely follow Theorem 1 and Lemma 1 in Gloria, Neukamm, and Otto ``A regularity theory for random elliptic operators'', Milan J Math 2020.
Per informazioni, rivolgersi a: lorenzo.giacomelli@uniroma1.it
Mercoledì 22 febbraio 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Julie Déserti (Université d’Orléan - CNRS)
Cremona group and regularisable birational maps
This talk deals with the group of birational transformations of the complex projective plane. After some examples, we will see that this group satisfies some (but not all) properties of linear groups. Finally, we will introduce the notion of regularisable birational maps, i.e. those that are conjugate to an automorphism; we will give criteria allowing to determine if a birational map is regularisable.
Mercoledì 22 febbraio 2023
Ore 14:30, Canale Youtube dell'IAC https://www.youtube.com/watch?v=HytZIk2NvGU, Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche
Seminari Generali IAC 2023
Angela Monti (Istituto per le Applicazioni del Calcolo, sede di Bari, Consiglio Nazionale delle Ricerche)
Model Order Reduction for Turing pattern approximation in reaction-diffusion PDE systems.
We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction-diffusion PDE (RD-PDE) systems. We will consider in particular the Proper Orthogonal Decomposition (POD) and the Dynamic Mode Decomposition (DMD). Both techniques present inaccurate approximations, therefore we will introduce two novel algorithms that aim at stabilizing the studied problem. In the first part of the talk we focus on the stabilization of the POD-DEIM technique. We show that solutions of surrogate models built by classical POD-DEIM exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, we add a correction term that provides missing information to the reduced model and we apply the PODDEIM technique to the corrected model. To further improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD-PDE systems, i.e. FitzHugh-Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns. In the second part we show some preliminary results regarding a new adaptive algorithm based on Dynamic Mode Decomposition (DMD). DMD is a data-diven technique that allows to find the best linear fit for a given dataset. We propose to modify the method by splitting the time interval into several subintervals to keep a certain level of accuracy. Numerical results will show the efficiency of the shown method.
Per informazioni, rivolgersi a: maya.briani@cnr.it
Giovedì 23 febbraio 2023
Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre
Seminario di Geometria
Giuseppe Pareschi (Tor Vergata)
Semihomogeneous vector bundles and n-torsion points on theta divisors
I will illustrate a description of torsion points on a theta divisor (of a complex principally polarized abelian variety) making use of certain semihomogeneous vector bundles introduced and studied by Mukai and D. Oprea. As a consequence, I will show an upper bound on the number of n-torsion points on a theta divisor (for a fixed positive integer n). The bound is achieved if and only if the p.p.a.v. is a product of elliptic curves, proving a conjecture of Auffarth, Marcucci, Pirola and Salvati Manni. Partly a joint work with Riccardo Salvati Manni.
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it
Giovedì 23 febbraio 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Francesca De Marchis (Sapienza Università di Roma)
On critical points of the Moser-Trudinger functional
Since the fundamental work by Trudinger from 1967 it is known that in two dimensions Sobolev functions in \( H^1\) satisfy embedding properties of exponential type. In 1971 Moser then obtained a sharp form of the embedding, controlling the integrability of \( F(u):=\int e^{u^2}\) in terms of the Sobolev norm of \( u\). On a closed Riemannian surface, \( F(u) \) is unbounded above for \( \left\|u\right\|_{H^1}>4\pi \). We are however able to find critical points of \( F \) constrained to any sphere \( \left\|u\right\|_{H^1}=R\), with \( R>0 \) arbitrary. The proof combines min-max theory, a monotonicity argument by Struwe, blow-up analysis and compactness estimates. This is joint work with A. Malchiodi, L. Martinazzi and P.D. Thizy.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it
Giovedì 23 febbraio 2023
Ore 15:00, Aula E (e a distanza, tramite la piattaforma Zoom), Dipartimento di Matematica, Sapienza Università di Roma
Seminari di Ricerca in Didattica della Matematica
Nicoletta Lanciano (Sapienza Università di Roma)
Donne e matematica
Chi è interessato a partecipare a distanza può rivolgersi ad Annalisa Cusi (annalisa.cusi@uniroma1.it).
Venerdì 24 febbraio 2023
Ore 12:00, Sala di Consiglio, Dipartimento di Matematica
Seminario di Teoria dei Numeri
Leonardo Carofiglio, Luigi De Filpo (Dipartimento di Matematica, Sapienza Università di Roma)
Valutazione p-adica di somme armoniche
Si vuole presentare una congettura posta da Eswarathasan e Levine sulla distribuzione delle valutazioni \(p\)-adiche dei numeri armonici del tipo \( H(n) = 1 + \frac12 + \cdots + \frac1n \) che afferma che l'insieme \( J_p \) degli interi positivi \( n \) tali che \( p \) divide il numeratore di \( H(n) \) è finito. Partendo da questa si dimostrano due risultati utilizzando un approccio aritmetico: uno per i primi non-Wolstenholme e l'altro per i primi Wolstenholme, che riguardano l'anomalo comportamento asintotico della valutazione \(p\)-adica di \( H(p^{m} n) \) quando la valutazione \(p\)-adica di \( H(n) \) è uguale esattamente a 3.
Venerdì 24 febbraio 2023
Ore 12:00, aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
seminario MoMA
Mirko Degli Esposti ( Università di Bologna)
Entropy, Irreversibility, DNA sequences… and Fake Faces
Entropy and irreversibility, two fundamental concepts underlying physical processes, and now also at the core of digital processes for simulation and evolution of artificial models. We will touch on some of these aspects, taking as our starting point the well-defined and well-explored mathematical context of infinite sequence spaces over finite alphabets. As we will see, universal estimators of typical signal entropy and cross entropy, based on the asymptotics of recurrences and waiting times, play an important role in information theory, and we will discuss some mathematical results. Starting from their construction, we will introduce and discuss universal estimators of typical signals of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts on finite alphabets. Finally, we will discuss some applications to DNA sequences, particularly in understanding the so-called Second Chargaff Rule, a still unexplained family of empirical symmetries present in "most" genetic sequences. We will conclude with some suggestions to show how concepts proper to statistical mechanics and thermodynamics, such as entropy and irreversibility, have become fundamental even in the most recent synthetic image generation models. In collaboration with Giampaolo Cristadoro Vojkan Jakšić, Renaud Raquépas.
Venerdì 24 febbraio 2023
Ore 14:00, Aula 7, edificio RM018 (aule di Ingegneria, Via del Castro Laurenziano), Dipartimento SBAI, Sapienza Università di Roma
Mini-corso
Felix Otto (MPI-MIS, Leipzig)
Stochastic homogenization and large-scale regularity
Lectures: (10/02 14:00, 15/02 10:00, 17/02 14:00, 22/02 10:00,) 24/02 14:00. Recording of previous lectures are expected to be available upon request. In this mini-course, I will introduce the concept of large-scale regularity in case of a linear elliptic equation (or system) with heterogeneous coefficients. It is based on a smallness (on average) of the potentials of the harmonic coordinates, and proceeds via an intrinsic Campanato iteration. I will then apply this to the case of a random heterogeneous coefficient field, sampled from a stationary and ergodic ensemble. I will try to be self-contained and closely follow Theorem 1 and Lemma 1 in Gloria, Neukamm, and Otto ``A regularity theory for random elliptic operators'', Milan J Math 2020.
Per informazioni, rivolgersi a: lorenzo.giacomelli@uniroma1.it
Venerdì 24 febbraio 2023
Ore 16:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
Seminari per i docenti A.A. 2022-2023
Eleonora Colletti con Marta Menghini (Università Roma Sapienza)
manualMENTE - Laboratori di matematica (tassellazioni, gnomoni, trapani, morse, proiettori...)
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