Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 17-05-2021 al 23-05-2021
Martedì 18 maggio 2021
Ore 14:30, Canale Youtube Cnr-Iac https://www.youtube.com/watch?v=h1i3GJLAeMg, Istituto per le Applicazioni del Calcolo - Consiglio Nazionale delle Ricerche
AIM: Artificial Intelligence and Mathematics, fundamentals and beyond
Michele Piana (Dipartimento di Matematica - DIMA Università di Genova)
A data-driven perspective to the forecasting problem
This talk will deal with forecast verification and loss functions design as two intertwined concepts in supervised machine learning. Focusing on binary classification, we will show how this approach impact the forecasting performances of regularization networks in the case of experimental problems in different application domains
Per informazioni, rivolgersi a: roberto.natalini@cnr.it
Mercoledì 19 maggio 2021
Ore 14:00, Seminario telematico via Google Meet all'URL http://meet.google.com/jjt-toji-skw, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Erwan Rousseau (Aix-Marseille Université)
Numerically special varieties
Campana introduced the class of special varieties as the varieties admitting no maps onto an orbifold of general type. They are also characterized by the non-existence of Bogomolov sheaves which are rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana has conjectured that one can replace the Kodaira dimension by the numerical dimension in this characterization. We prove partially this conjecture showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. This is a joint work with J.V. Pereira and F. Touzet.
Per informazioni, rivolgersi a: diverio@mat.uniroma1.it
Mercoledì 19 maggio 2021
Ore 14:30, https://meet.google.com/pip-kvzt-fkz, Dipartimento di Fisica, Sapienza Università di Roma
Seminario delle Meccaniche
Giulia Pisegna (Sapienza Università di Roma)
Renormalization Group approach to the collective behavior of swarms
Biological systems are invariably complex, strongly interacting, and out of equilibrium. The first level of complexity in their world is related to the strong interactions among individuals, which determine the collective behavior of the group characterized by the absence of relevant density fluctuations. Recently, it has been found that the dynamical correlation functions of natural swarms surprisingly exhibit the property of dynamical scaling with an anomalous value of the related critical exponent. We, therefore, investigate the role of the activity in the determination of the dynamical critical properties of out-of-equilibrium systems in absence of density-velocity coupling and heterogenous density structures. Starting from the incompressible hydrodynamic theory of the Vicsek model, we show that increasing the activity leads to a renormalization group (RG) crossover between the equilibrium ferromagnetic fixed point, with dynamical critical exponent z = 2, and the off-equilibrium active fixed point, with z = 1.7 (in d = 3). We run simulations of the classic Vicsek model in the near-ordering regime and find that critical slowing down indeed changes with activity, displaying two exponents that are in remarkable agreement with the RG prediction. The equilibrium-to-off-equilibrium crossover is ruled by a characteristic length scale beyond which active dynamics takes over. Such length scale is smaller the larger the activity, suggesting the existence of a general trade-off between activity and system’s size in determining the dynamical universality class of active matter.
Giovedì 20 maggio 2021
Ore 14:00, Seminario online su questo link, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Analisi Matematica
Giovanni Molica Bisci (Università degli Studi di Urbino)
Nonsmooth functionals in the Calculus of Variations
In the last years, elliptic equations involving a nonsmooth term have attracted several outstanding mathematicians and the interest towards this kind of problems has grown more and more, not only for their intriguing analytical structure, but also in view of their applications in a wide range of contexts. Motivated by this wide interest in the literature, the leading purpose of this talk is to present some recent results on nonsmooth elliptic equations, mainly related to a wide class of functionals defined through multiple integrals of Calculus of Variations; see, among others, the papers [1, 2, 3, 4]. Applications to quasilinear boundary value problems will be presented and some open problems briely discussed; see [5] and [6, Chapter 8] for related topics. References: [1] D. Arcoya and L. Boccardo, A min{max theorem for multiple integrals of the Calculus of Variations and applications, Rend. Mat. Acc. Lincei, 6 (1995) 29-35. [2] D. Arcoya and L. Boccardo, Critical points for multiple integrals of Calculus of Variations, Arch. Rat. Mech. Anal. 134 (1996), 249-274. [3] D. Arcoya and L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, Nonlinear Differential Equations and Applications NoDEA 6 (1999), 79-100. [4] D. Arcoya and J. Carmona, A nondifferentiable extension of a theorem of Pucci and Serrin and applications, J. Differential Equations 235 (2007), 683-700. [5] G. Molica Bisci, Local minima for some functionals in the Calculus of Variations, submitted for publication (2021), 1-53. [6] G. Molica Bisci and P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications 36 (2021), i+vii, 1-266.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
Per informazioni, rivolgersi a: molle@mat.uniroma2.it
Giovedì 20 maggio 2021
Ore 14:30, Sala di Consiglio - https://meet.google.com/ads-dekx-bgm, Dipartimento di Matematica, Sapienza Università di Roma
Seminario P(n)
Lucio Boccardo (Sapienza Università di Roma)
"Quanto" sono positive le soluzioni di alcuni problemi di Dirichlet con termini noti positivi
Le soluzioni deboli di problemi di Dirichlet (dato nullo al bordo) del tipo \( A(u)=f\), nello spazio di Sobolev \( W_0^{1,p}\), \( p>1\), con termini noti \( f(x)\geq 0\), in genere, sono \( u\geq 0\); e la dimostrazione è elementare: basta usare come funzione test \( u^-\). Resta da vedere ``quanto" è positiva \( u\): l’insieme dove \( u=0\) che taglia ha? Risultati classicissimi dicono che \( u>0\) nell’aperto in cui si lavora e \( u=0\) solo al bordo; ormai classici sono le estensioni (Stampacchia, Trudinger, …) al caso di operatori in forma di divergenza (lineari o non lineari). Presenterò recenti risultati concernenti le soluzioni di alcuni problemi non lineari. In alcuni casi si riesce a dire come sopra; più precisamente che, per ogni \( \omega\) contenuto nell’aperto, esiste una costante \( m_\omega>0\) tale che \( u\), su \( \omega\), risulta maggiore di \( m_\omega>0\) (Principio del Massimo Forte). In altri casi si riesce a dire un poco meno: \( u\) può essere nulla solo su insiemi di misura nulla (Principio del Massimo Debole); restando aperta la possibilità che anche in tali casi valga il principio del Massimo Forte. In tutti i casi non è rilevante ``quanto” positivo sia il termine noto: può essere limitato inferiormente da una costante strettamente positiva come può essere ben ampio l’insieme degli zeri, come la funzione caratteristica di un sottoinsieme.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it
Giovedì 20 maggio 2021
Ore 16:00, Streaming via Zoom and Youtube channel, Web site for information: https://www.dinamici.org/dai-seminar/
DinAmicI: Another Internet Seminar (DAI Seminar)
Umberto Zannier (Scuola Normale Superiore di Pisa)
Torsion values of sections, elliptical billiards and diophantine problems in dynamics
After a very brief review of basics on elliptic curves and their families, we shall consider "sections" of such families, and especially their "torsion values". For instance, what can be said of the complex numbers b for which \((2, \sqrt{2(2-b)})\) is torsion on the Legendre curve \(y^2=x(x-1)(x-b)\)? In particular, we shall recall results of "Manin-Mumford type" and focus to illustrate some applications to elliptical billiards. Finally, if time allows we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.) Note: The zoom link to the seminar will be posted on https://www.dinamici.org/dai-seminar/ and on https://mathseminars.org/seminar/DinAmicI. Moreover, it will be streamed live on youtube via the DinAmicI channel: https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it
Venerdì 21 maggio 2021
Ore 15:00, Seminario Telematico, link
Online Representation Theory Seminar
Sachin Gautam (Ohio State University)
R-matrices and Yangians
An R-matrix is a solution to the Yang-Baxter equation (YBE), a central object in Statistical Mechanics, discovered in 1970's. The R-matrix also features prominently in the theory of quantum groups formulated in the eighties. In recent years, many areas of mathematics and physics have found methods to construct R-matrices and solve the associated integrable system. In this talk I will present one such method, which produces meromorphic solutions to (YBE) starting from the representation theory of a family of quantum groups called Yangians. Our techniques give (i) a constructive proof of the existence of the universal R-matrix of Yangians, which was obtained via cohomological methods by Drinfeld in 1983, and (ii) prove that Drinfeld's universal R-matrix is analytically well behaved. This talk is based on joint works with Valerio Toledano Laredo and Curtis Wendlandt.
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