Notiziario Scientifico
Notiziario dei seminari di carattere matematico
a cura del Dipartimento di Matematica G. Castelnuovo
Sapienza Università di Roma
Settimana dal 21 al 27 maggio 2018
Lunedì 21 maggio 2018
Ore 14:15, aula di Consiglio
seminario di Analisi Matematica
Riccardo Adami (Politecnico di Torino)
Occurrence of negative energy ground states for the critical NLS on graphs
We discuss the existence of ground states for the critical Nonlinear Schroedinger Equation on metric
non-compact graphs. Contrarily to the case of Rn, where the result trivializes, the
topology of the graph can induce a trapping effect that yields ground states, characterized by negative
energy, impossible in Rn due to well-known virial estimates.
The main technical tool is the adaptation to graphs of classical Gagliardo-Nirenberg estimates.
This is a joint work with Enrico Serra and Paolo Tilli.
Lunedì 21 maggio 2018
Ore 14:30, aula D'Antoni, dipartimento di Matematica,
Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario di Analisi Complessa
Andrew Zimmer (William & Mary)
Smoothly bounded domains covering finite volume manifolds
In this talk we will discuss the following result: if a bounded domain with C2
boundary covers a manifold which has finite volume with respect to either the Bergman volume,
the Kähler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is
biholomorphic to the unit ball. The proof uses a variety of tools from Riemannian geometry
and several complex variables including the squeezing function, Busemann functions, estimates
on invariant distances, and a version of E. Cartan's fixed point theorem.
Lunedì 21 maggio 2018
Ore 18:00, Foyer del Teatro Valle, via del Teatro Valle 21
CMTP (Centro per la Matematica e la Fisica Teorica) - Eureka 2018 (promosso da Roma Capitale)
Gianni Jona-Lasinio (Sapienza Università di Roma)
Matematica, filosofia e scienze della natura: un triangolo di relazioni complesse
Le scienze della natura, la fisica in particolare, fanno un uso sempre più raffinato del linguaggio
matematico e si è parlato di irragionevole efficacia della matematica. D'altro canto fisica e
filosofia hanno avuto un rapporto stretto fino alla fine del diciannovesimo secolo. Oggi il linguaggio
filosofico sembra espulso ma una metafisica rimane al di la delle apparenze. Cercheremo di orientarci
in questa relazione triangolare.
Martedì 22 maggio 2018
Ore 14:00, aula Dal Passo, dipartimento di Matematica,
Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario di Equazioni Differenziali
Filippo Giuliani (Università di Roma Tre)
On the integrability and quasi-periodic dynamics of the dispersive Degasperis-Procesi equation
The Degasperis-Procesi equation
Martedì 22 maggio 2018
Ore 14:30, aula 311, Università di Roma Tre,
largo san Leonardo Murialdo 1
seminario di Fisica Matematica
Michele Correggi (Sapienza Università di Roma)
Quasi-Classical Limit for Quantum Particle-Field Systems
We study the quasi-classical limit of a quantum system composed of finitely many non-relativistic
particles coupled to a quantized field in Nelson or Pauli-Fierz-type models. We prove that, as the
field becomes classical and the corresponding degrees of freedom are traced out, the effective
Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schroedinger operator
with a additional potentials, either electric and/or magnetic, depending on the state of the field.
In addition, we prove convergence of the ground state energy of the full system to a suitable
effective variational problem involving the classical state of the field. Joint work with M. Falconi
(Tuebingen) and M. Olivieri (Roma Sapienza).
Mercoledì 23 maggio 2018
Ore 14:00, aula di Consiglio
seminario di Algebra e Geometria
Alessandro Valentino (Università di Zurigo)
Equivariant Factorization Algebras from Abelian Chern-Simons theories
Factorization algebras are a powerful tool to encode observables in classical and quantum field
theory. As suggested by Costello and Gwilliam, to the formal moduli problem describing deformations
of flat G-bundles with connections on a manifold M, one can associate a factorization algebra F
on M which describes the perturbative aspects of classical Chern-Simons theory on M with structure
group G. In the talk I will concentrate on the case of G an abelian group, and show that the
factorization algebra F comes naturally equipped with a (homotopy) action of the gauge group
Maps(M,G), which can be regarded as a genuine nonperturbative aspect of Chern-Simons theory.
Joint work with Corina Keller.
Mercoledì 23 maggio 2018
Ore 16:00, aula F, Università di Roma Tre,
largo san Leonardo Murialdo 1
colloquium di Matematica
Balint Toth (University of Bristol, UK and Renyi Institute, Budapest)
Invariance principle for the random Lorentz gas beyond the [Boltzmann-Grad / Gallavotti-Spohn]
limit
Let hard ball scatterers of radius r be placed in Rd, centred at the points of
a Poisson point process of intensity ρ. The volume fraction rdρ is assumed to
be sufficiently low so that with positive probability the origin is not trapped in a finite domain
fully surrounded by scatterers. The Lorentz process is the trajectory of a point-like particle starting
from the origin with randomly oriented unit velocity subject to elastic collisions with the fixed
(infinite mass) scatterers. The question of diffusive scaling limit of this process is a major open
problem in classical statistical physics.
Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0,
ρ→+∞ so that rd-1ρ→1 and the time scale is fixed, the Lorentz process
(described informally above) converges to a Markovian random flight process, with independent
exponentially distributed free flight times and Markovian scatterings. It is essentially straightforward
to see that taking a second diffusive scaling limit (after the Gallavotti-Spohn limit) yields invariance
principle. I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3:
Letting r→0, ρ→+∞ so that rd-1ρ→1 (as in B-G) and simultaneously
rescaling time by T∼r-2+ε we prove invariance principle (under diffusive scaling)
for the Lorentz trajectory. Note that the B-G limit and diffusive scaling are done simultaneously and
not in sequel. The proof is essentially based on control of the effect of re-collisions by probabilistic
coupling arguments. The main arguments are valid in d=3 but not in d=2. Joint work with Chris Lutsko
(Bristol)
Giovedì 24 maggio 2018
Ore 14:30, aula 211, Università di Roma Tre,
largo san Leonardo Murialdo 1
seminario di Geometria
Ivan Cheltsov (University of Edinburgh)
Burkhardt, Todd, Igusa, Coble, Beauville and rational quartic threefolds
Burkhardt and Igusa quartic threefolds are classically known to be rational. They generate a pencil of
quartics that all admit an action of the symmetric group of degree six. Bondal and Prokhorov asked which
threefolds in this pencil are rational and which are not. All these threefolds are singular, so Iskovskikh
and Manin's result cannot be applied here. Beauiville proved that every quartic threefold in this pencil
is irrational except for Burkhardt and Igusa quartics and possibly two more threefolds. In this talk I
will show how to use two constructions of Todd (dated back to 1933 and 1935) to prove that the remaining
two quartic threefolds in the pencil are also rational. Then I will present a new proof of this and
Beauville's result using birational geometry of Coble fourfold. This is a joint work with Sasha Kuznetsov
and Costya Shramov from Moscow.
Venerdì 25 maggio 2018
Ore 15:00, aula D'Antoni, dipartimento di Matematica,
Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario
Ernesto Spinelli (Sapienza Università di Roma)
Codimension growth and minimal varieties
In characteristic zero an effective way of measuring the polynomial identities satisfied by an
algebra is provided by the sequence of its codimensions introduced by Regev. In this talk we review
some features of the codimension growth of PI algebras, including the deep contribution of Giambruno
and Zaicev on the existence of the PI-exponent, and discuss some recent developments in the framework
of group graded algebras. In particular, a characterisation of minimal supervarieties of fixed
superexponent will be given. The last result is part of a joint work with O. M. Di Vincenzo and
V. da Silva.
Venerdì 25 maggio 2018
Ore 16:30, aula D'Antoni, dipartimento di Matematica,
Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario
Giovanni Cerulli Irelli (Sapienza Università di Roma)
Cellular decomposition of quiver Grassmannians
I will report on a joint project with F. Esposito, H. Franzen and M. Reineke (arXiv:1804.07736).
Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver
representations of a fixed dimension vector. The geometry of such projective varieties can be
studied via the representation theory of quivers (or of finite dimensional algebras). Quiver
Grassmannians appeared in the theory of cluster algebras. As a consequence of the positivity
conjecture of Fomin and Zelevinsky, the Euler characteristic of quiver Grassmannians associated
with rigid quiver representations must be positive; this fact was proved by Nakajima.
We explore the geometry of quiver Grassmannians associated with rigid quiver representations:
we show that they have property (S) meaning that: (1) there is no odd cohomology, (2) the cycle
map is an isomorphism, (3) the Chow ring admits explicit generators defined over any field.
As a consequence, we deduce that they have polynomial point count. If we restrict to quivers
which are of finite or affine type (i.e. orientation of simply-laced extended Dynkin diagrams)
we can prove much more: in this case, every quiver Grassmannian associated with an indecomposable
representation (not necessarily rigid) admits a cellular decomposition.
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