Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 27-02-2023 al 05-03-2023
Lunedì 27 febbraio 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Fabricio Macià (Universidad Politecnica de Madrid)
Reconstruction of potentials from the Dirichlet-to-Neumann map
We address the problem of reconstructing a real potential \( V\) from the Dirichlet-to-Neumann map of a Schrödinger operator \( -\Delta + V\) on the boundary of a domain in Euclidean space (the reconstruction aspect of the Calderón problem). This problem is rather involved in general, from both the analytical and numerical points of view as it enjoys poor stability properties. After reviewing the main tools used in proving that potentials are uniquely determined from the D-t-N map, we introduce an object that is obtained in terms of certain matrix elements of the D-t-N map -- the Born approximation -- which is reminiscent of an approximation for the potential that has been extensively studied in the context of inverse scattering theory. We will show a number of interesting analytical properties of the Born approximation in the case of radial potentials in a ball. Among others, how it can be used to factorize the reconstruction problem into a linear, yet ill-conditioned problem (the Hausdorff moment problem) and a nonlinear step which enjoys Hölder stability. If time permits, we will also present a novel algorithm for numerical reconstruction based on this object. This is based on a series of works in collaboration with J.A. Barceló, C. Castro, T. Daudé, C. Meroño, F. Nicoleau, D. Sánchez-Mendoza.
Per informazioni, rivolgersi a: azahara.delatorrepedraza@uniroma1.it
Mercoledì 01 marzo 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Ursula Ludwig (Universität Münster)
Analytic torsion and the Cheeger-Müller theorem
Analytic torsion is an important secondary spectral invariant of compact Riemannian manifolds. The famous Cheeger-Müller theorem states that for a compact Riemannian manifold equipped with a unitary flat vector bundle the analytic torsion is equal to the topological torsion, and hence a topological invariant. In the first part of this talk I will recall the definition of analytic torsion, the Cheeger-Müller theorem, and how it has been used in the past 10 years e.g. to answer questions motivated from the study of the cohomology of arithmetic groups. In the second part I will speak about the generalisation of analytic torsion and of the Cheeger-Müller theorem to singular spaces.
Mercoledì 01 marzo 2023
Ore 15:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Colloquium di dipartimento
Felix Otto (Max-Planck-Institut, Lipsia)
Optimal matching, optimal transportation, and its regularity theory
The optimal matching of blue and red points is prima facie a combinatorial problem. It turns out that when the position of the points is random, namely distributed according to two independent Poisson point processes in d-dimensional space, the problem depends crucially on dimension, with the two-dimensional case being critical [Ajtai-Komlós-Tusnády]. Optimal matching is a discrete version of optimal transportation between the two empirical measures. While the matching problem was first formulated in its Monge version (p=1), the Wasserstein version (p=2) connects to a powerful continuum theory. This connection to a partial differential equation, the Monge-Ampere equation as the Euler-Lagrange equation of optimal transportation, enabled [Parisi et. al.] to give a finer characterization, made rigorous by [Ambrosio et. al.]. The idea of [Parisi et. al.] was to (formally) linearize the Monge-Ampere equation by the Poisson equation. I present an approach that quantifies this linearization on the level of the optimization problem, locally approximating the Wasserstein distance by an electrostatic energy. This approach (initiated with M. Goldman) amounts to the approximation of the optimal displacement by a harmonic gradient. Incidentally, such a harmonic approximation is analogous to de Giorgi's approach to the regularity theory for minimal surfaces. Because this regularity theory is robust --- measures don't need to have Lebesgue densities --- it allows for sharper statements on the matching problem (work with M. Huesmann and F. Mattesini).
Mercoledì 01 marzo 2023
Ore 17:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Fisica Matematica
Massimo Moscolari (Tübingen University)
General bulk-edge correspondence at positive temperature
By extending the gauge covariant magnetic perturbation theory to operators defined on half planes, we prove that for general 2d random ergodic magnetic Schrödinger operators the celebrated bulk-edge correspondence is just a particular case of a much more general paradigm, which also includes the theory of diamagnetic currents and of Landau diamagnetism. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a "zero temperature" limit. The talk is based on a joint work with H. Cornean and S. Teufel.
Giovedì 02 marzo 2023
Ore 14:00, Aula Seminari RM004, ex Palazzina E, Dipartimento SBAI, Via A. Scarpa 16
Seminario "PDE a tutto SBAI"
Tobias König (Goethe University Frankfurt)
Stability of the Sobolev inequality: best constants and minimizers
Since the ground-breaking inequality of Bianchi and Egnell (1991), which bounds the 'Sobolev deficit' of a function in terms of a constant \(c_{BE} > 0\) times its squared distance to the manifold of optimizers, it has been an open problem to determine the optimal value of \(c_{BE}\) and, if it is achieved, its optimizer. In this talk, I will present some recent partial progress on this problem. The main result is that \(c_{BE}\) admits an optimizer for every dimension \(d \geq 3\). The proof relies on new strict upper bounds on \(c_{BE}\), which exclude that the optimal value \(c_{BE}\) is attained by sequences which are asymptotically equal to one or two Talenti bubbles (i.e. optimizers of the Sobolev inequality)
Per informazioni, rivolgersi a: massimo.grossi@uniroma1
Giovedì 02 marzo 2023
Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre
Seminario di Geometria
Benoît Claudon (Rennes)
Numerical characterization of torus quotients
In this talk I will explain how to recognize complex tori among Kähler klt spaces (smooth in codimension 2) in terms of vanishing of Chern numbers. It requires first to define Chern classes on singular spaces (a rather unstable notion). On the way, we will establish a singular version of the Bogomolov-Gieseker inequality for stable sheaves and study what can be said in the equality case. Joint work with Patrick Graf and Henri Guenancia.
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it
Giovedì 02 marzo 2023
Ore 14:30, Aula Piano Terra, Istituto per le Applicazioni del Calcolo, Cnr, via dei Taurini 19, Roma
INTERNATIONAL PRIZE ''TULLIO LEVI-CIVITA''
Thierry Paul (Cnrs, Sorbonne Université)
Quantum Topologies
Il 2 marzo prossimo, alle ore 14:30, si terrà la consegna del premio internazionale Tullio Levi-Civita 2021 attribuito al Professor Thierry Paul (CNRS, Sorbonne Université). A seguire la Lectio Magistralis del Prof. Paul.
Per informazioni, rivolgersi a: roberto.natalini@cnr.it
Giovedì 02 marzo 2023
Ore 15:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
Seminari di Ricerca in Didattica della Matematica
Alessandro Gambini (Sapienza Università di Roma)
Confronto tra geometria piana e geometria sferica: esperienze sul campo con studenti e docenti
Chi è interessato a partecipare a distanza può rivolgersi ad Annalisa Cusi (annalisa.cusi@uniroma1.it)
Giovedì 02 marzo 2023
Ore 15:00, Aula Seminari RM004, ex Palazzina E, Dipartimento SBAI, Via A. Scarpa 16
Seminario "PDE a tutto SBAI"
Francesco Ferraresso (Università di Roma Sapienza)
Neumann biharmonic eigenvalue problems on singularly perturbed domains
Domain perturbation theory for the eigenvalues of the Neumann Laplace opera- tor on families of bounded, Lipschitz domains of R^N is nowadays a well-understood, yet complicated subject. In general we cannot expect spectral continuity of the eigenvalues when the domain is varying; Courant and Hilbert showed that small perturbations of the unit square can generate an additional zero eigenvalue. For the Neumann biharmonic operator, the situation is more involved, mainly due to two additional hurdles: 1) Neumann boundary conditions are very sensitive to the cur- vature of the boundary; 2) standard techniques, such as the separation of variables, are not available. After a review of the main results and counterexamples for the Laplace operator and the biharmonic operator with Neumann boundary conditions, I will focus on two specific singular perturbations where spectral continuity fails: the dumbbell domain in R^N and thin annuli in R^2. The latter example shows that the limits of the eigenvalues of the Neumann bi- harmonic operator (a scalar operator) on thin annuli are eigenvalues of a system of ordinary differential equations with non-constant coefficients depending on the curvature of the boundary. Based on joint works with J.M. Arrieta, P.D. Lamberti and L. Provenzano.
Per informazioni, rivolgersi a: massimo.grossi@uniroma1.it
Venerdì 03 marzo 2023
Ore 14:30, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Kirill Zainoulline (University of Ottawa)
Oriented cohomology of a linear algebraic group vs localization in 2-monoidal categories
The Chow ring \(CH(G)\) of a split semi-simple linear algebraic group \(G\) is one of the key geometric invariants in the theory of linear algebraic groups, torsors, motives of twisted flag varieties. Starting from pioneering works by Grothendieck and Borel, it has been studied for decades and computed for all simple groups (see e.g. Kac 1985, Duan 2015's). In the present talk we explain how to describe (and, hence, to compute) an oriented cohomology (Borel-Moore homology) functor \(A(G)\) using the localization techniques of Kostant-Kumar and the techniques of 2-monoidal categories: we show that the natural Hopf-algebra structure on \(A(G)\) can be lifted to a 'bi-Hopf' structure on the \(T\)-equivariant cohomology \(A_T(G/B)\) of the complete flag variety. More generally, we prove that the structure algebra of a Bruhat moment graph of a root system is a Hopf algebroid with respect to the right Hecke and left Brion-Knutson-Tymoczko actions. As an application, we obtain an effective combinatorial way to compute the coproduct on \(A(G)\). This is a joint work with Martina Lanini and Rui Xiong.
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