Weekly Bulletin (it)

Notiziario dei seminari di carattere matematico
a cura del Dipartimento 'G. Castelnuovo'
Sapienza Università di Roma

Settimana dal 20 al 26 novembre 2017

Lunedì 20 novembre 2017
Ore 14:15, aula di Consiglio
seminario di Analisi Matematica
Renato Lucà (Università di Basilea)
On the pointwise convergence of solutions of the Schrödinger equation to the initial datum
We consider solutions to the linear Schrödinger equation on Rn with initial data in the Sobolev space Hs. A classic result of Carleson tells us that when n=1 the condition s≥1/4 is sufficient for the solutions in order to converge almost everywhere to their initial data, as time goes to zero. Dahlberg and Kenig have then proved, giving explicit counterexamples, that solutions with less regular data may not converge (this is indeed the case in any dimension). Thus the threshold s≥1/4 has been conjectured to be the correct one even in higher dimensions. However, this conjecture has been recently disproved in a series of paper. We will present the Dahlberg-Kenig counterexample and a more sophisticated one (obtained in collaboration with K. Rogers) which improves it in higher dimensions. Then we will show how to combine both to prove that s≥n/(2n+2) is necessary to guarantee that any solution with data in Hs converges to its datum. This has been recently proved by Bourgain and has been shown to be also sufficient in dimension n=2 by Du, Guth and Li.

Martedì 21 novembre 2017
Ore 14:30, aula 311, Università di Roma Tre, largo san Leonardo Murialdo 1
seminario di Fisica Matematica
D. Monaco (Università di Roma Tre)
Adiabatic currents for interacting electrons on a lattice
Consider a system of interacting fermions on a finite lattice modelled by a gapped many-body Hamiltonian H(t), depending adiabatically on time. I will present an adiabatic theorem for the trace per unit volume of local observables in a state evolving adiabatically from a ground state of H(0). The theorem holds with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for initial data that lie in eigenspaces of degenerate eigenvalues. The adiabatic theorem allows to compute the current density induced by the adiabatic change of the Hamiltonian, and derive rigorously the so-called linear response formula for a system of interacting fermions in a ground state, with error estimates uniform in the system size. I will also discuss the application to quantum Hall systems. This is based on joint work with Stefan Teufel.

Martedì 21 novembre 2017
Ore 15:00, aula di Consiglio
seminario di Modellistica Differenziale Numerica
M. Semplice (Università di Torino)
Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws driven by numerical entropy production
I will present a third order accurate finite volume scheme under Adaptive Mesh Refinement (AMR) on quad-tree type grids. In the scheme, AMR is driven by the, so called, numerical entropy production. This is a residual of the entropy inequality that is computable in each space-time finite volume during the simulation and that is of the same size of the truncation error, which has been successfully exploited to control adaptive behavior of schemes in various ways in one and two space dimensions. Of course, the reconstruction of point values from cell averages requires a procedure that is third order accurate, non-oscillatory, but also versatile enough to handle data on unstructured and non-conforming grids and efficient in computing point data at very many points in each cell: the Central WENO (CWENO) technique is employed here for this task.

Martedì 21 novembre 2017
Ore 15:00, aula D'Antoni, dipartimento di Matematica, Università di Roma Tor Vergata, viale della Ricerca Scientifica 1
seminario di Equazioni Differenziali
Giulio Ciraolo (Università di Palermo)
Stime quantitative per ipersuperifici a curvatura media quasi costante
Discuteremo alcune versioni quantitative del Teorema di Alexandrov della bolla di sapone, che afferma che le sfere sono le sole ipersuperfici chiuse embedded a curvatura media costante. In particolare, considereremo ipersuperfici con curvatura media vicina ad una costante e descriveremo in maniera quantitativa la vicinanza ad una singola sfera o ad una collezione di sfere tangenti di raggio uguale in termini dell'oscillazione della curvatura media. Inoltre considereremo il problema analogo in ambito nonlocale, mostrando come l'effetto nonlocale implichi una maggiore rigidità del problema e prevenga la formazione di più bolle.

Mercoledì 22 novembre 2017
Ore 14:00, aula di Consiglio
seminario di Algebra e Geometria
Claire Voisin (Collège de France)
Segre numbers of tautological bundles on Hilbert schemes
We establish geometric vanishings in certain ranges for top Segre classes of tautological bundles of punctual Hilbert schemes of K3 surfaces and K3 surfaces blown-up at one point. We show how all these Segre numbers for any surface and any polarization are formally determined by these vanishings. Building on these results, Marian-Oprea-Pandharipande and Szenes-Vergne completed the proof of the Lehn conjecture giving a formula for the generating function determined by these numbers.

Mercoledì 22 novembre 2017
Ore 16:00, aula 311, Università di Roma Tre, largo san Leonardo Murialdo 1
seminario di Analisi Matematica
Shidi Zhou (Università di Roma Tre)
An infinite dimensional KAM theorem with application to 2-d completely resonant beam equation
In this talk we shall consider the 2-dimensional completely resonant beam equation with cubic nonlinearity on T2. We prove the existence of the quasi-periodic solutions, which lie in a special subspace of L2(T2). We view the equation as an infinite dimensional Hamiltonian system, and write the Hamiltonian of the equation as an angle-dependent block-diagonal normal form plus a small perturbation with some regularity. By establishing an abstract KAM theorem, we prove the existence of a class of invariant tori of this system, which implies the existence of a class of small-amplitude quasi-periodic solutions of the equation. In the KAM iteration, the measure estimate is reached by making use of the regularity of the nonlinearity.

Venerdì 24 novembre 2017
Ore 12:00, aula di Consiglio
seminari MoMa
Luca Giomi
Active fludis: from liquid crystals to living systems
Colonies of motile microorganisms, the cytoskeleton and its components, cells and tissues have much in common with soft condensed matter systems (i.e. liquid crystals, amphiphiles, colloids etc.), but also exhibit behaviors that do not appear in inanimate matter and that are crucial for biological functions. These unique properties arise when the constituent particles are active: they consume energy from internal and external sources and dissipate it by moving through the medium they inhabit. In this talk I will give a brief introduction to the notion of 'active matter' and present some recent results on the hydrodynamics of active nematics suspensions in two dimensions.

Venerdì 24 novembre 2017
Ore 14:30, aula di Consiglio
seminario di Fisica Matematica
Emanuela L. Giacomelli (Universität Tuebingen)
Surface Superconductivity in Presence of Corners
We consider an extreme type-II superconducting wire with non-smooth cross section, i.e., with one or more corners at the boundary, in the framework of the Ginzburg-Landau theory. We prove the existence of an interval of values of the applied field, where superconductivity is spread uniformly along the boundary of the sample. More precisely the energy is not affected to leading order by the presence of corners and the modulus of the Ginzburg-Landau minimizer is approximately constant along the transversal direction. The critical fields delimiting this surface superconductivity regime coincide with the ones in absence of boundary singularities. We will also discuss some recent results. In particular, we introduce a new effective problem near the corner that allows us to prove a refined asymptotics and to isolate the contributions to the energy density due to the presence of corners. The explicit expression of the effective energy is yet to be found but we formulate a conjecture on it based on the behavior for almost flat angles. Indeed, for corners with angles close to π, we are able to explicitly compute the leading order of the corners effective problem and show that it sums up to the smooth boundary contribution to reconstruct the same asymptotics as in smooth domains. Joint work with Michele Correggi.

Venerdì 24 novembre 2017
Ore 16:00, aula Picone
seminario per insegnanti (Piano Lauree Scientifiche)
Nicoletta Lanciano (Sapienza Università di Roma)
Geometria per l'astronomia

Venerdì 24 novembre 2017
Ore 16:30, aula di Consiglio
seminario di Fisica Matematica
Giulia Basti (Sapienza Università di Roma)
Efimov Effect for a system of two identical fermions and a different particle
In 1970 the physicist V. Efimov pointed out that a system of three different particles, such that the two-particle interactions are short-range and resonant, have an infinite number of bound states. This phenomenon is known as Efimov Effect and it is a paradigmatic example of the so-called universality of low-energy physics. We consider a system composed by two identical fermions of unitary mass and a third particle of mass m. We assume that the interactions are short-range and that the two-particle subsystems do not have bound states. Moreover, we suppose that the subsystems composed by one of the fermions and the third particle have a zero-energy resonance. Under these assumptions we prove the existence of a mass threshold m* such that if m<m* then the number N(z) of eigenvalues of the three-particle Hamiltonian smaller than z<0 is infinite and N(z)≈C(m)|log|z|| as z→0. On the other hand for m>m* we show that the number of negative eigenvalues stays finite.

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