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

ut+c0ux+γuxxx2uxxt =(c2(u2x+uuxx)-2c32 u2)x
has been extensively studied by many authors, especially in its dispersionless form, since it presents interesting phenomena such as breaking waves and existence of peakon-like solutions. Degasperis-Holm-Hone proved the integrability of this equation and they provided an iterative method to compute infinite conserved quantities. Since the Degasperis-Procesi equation is a quasi-linear equation the presence of dispersive terms depends on the chosen frame. In absence of dispersive terms there are no constants of motion even controlling the H1-norm. We show that, in the dispersive case, we can construct infinitely many constants of motion which are analytic and control the Sobolev norms in a neighborhood of the origin. Moreover, thanks to the analysis of the algebraic structure of the quadratic parts of these conserved quantities we show that the (formal) Birkhoff normal form is action-preserving (integrable) at any order. This fact is used to prove the first existence result of quasi-periodic solutions for the Degasperis-Procesi equation on the circle. These results have been obtained in collaboration with R. Feola, S. Pasquali and M. Procesi.

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
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
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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