a cura del Dipartimento 'G. Castelnuovo'

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 **R**^{n}
with initial data in the Sobolev space H^{s}. 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 H^{s} 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 T^{2}. We prove the existence
of the quasi-periodic solutions, which lie in a special subspace of
L^{2}(T^{2}). 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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