Notiziario Scientifico
Settimana dal 13 al 19 gennaio 2014
Lunedì 13 gennaio 2014
Martedì 14 gennaio 2014
Martedì 14 gennaio 2014
Martedì 14 gennaio 2014
Martedì 14 gennaio 2014
Martedì 14 gennaio 2014
Mercoledì 15 gennaio 2014
Giovedì 16 gennaio 2014
Venerdì 17 gennaio 2014
Venerdì 17 gennaio 2014
Tutte le informazioni relative a questo notiziario devono pervenire
all'indirizzo di posta elettronica
seminari@mat.uniroma1.it,
o nella casella della posta di Luigi Orsina, entro le ore 9 del venerdì
precedente la settimana di pubblicazione.
Ore 14:30, Aula di Consiglio
Seminario di Analisi Matematica
Given a Lipschitz vector field, the classical Cauchy-Lipschitz theory gives existence, uniqueness
and regularity of the associated ODE flow. In recent years, much attention has been devoted to
extensions of such theory to cases in which the vector field is less regular than Lipschitz, but
still belongs to some "weak differentiability classes". In this talk, I will review the main
points of an approach involving quantitative estimates for flows of Sobolev vector fields (joint
work with Camillo De Lellis) and describe further extensions to a case involving singular integrals
of L^1 functions (joint work with Francois Bouchut) and to a case endowed with a "split structure"
(joint work with Anna Bohun and Francois Bouchut).
Ore 10:30, Aula 34 (quarto piano), Dipartimento di Scienze Statistiche
We consider two types of invariants (geometric and integral), which are connected with the theory of
stochastic differential equations (SDEs). The geometric invariants are the fixed length of a random
chain and the constant radius of a sphere, which is the surface for random walks (for example, in
the case of a turning diffusion). The integral invariants are represented as integrals of kernel
functions of integral invariants on the whole space. These kernel functions are solutions of SDEs,
which were obtained and discussed. Furthermore, kernel functions will be used for deriving the first
global stochastic integral, for constructing the program control (with probability one) for
stochastic dynamical systems on deterministic manifold based on Brownian motion with Poisson jumps,
and for obtaining the Kolmogorov equations.
Ore 14:10, Aula B
We discuss a general approach to understand phase separation and metastability in stochastic
particle systems that exhibit a condensation transition. Condensation occurs when, above some
critical density, a finite fraction of all the particles in the system accumulate on a single
lattice site. We present a detailed analysis of a particular size-dependent zero-range process which
was introduced as a toy model for clustering in granular media. This model also captures all the
relevant details of more generic condensing zero-range processes close to the critical point.
Results on the equivalence of ensembles and metastability are based on large deviation principles
for the maximum of triangular arrays of independent random variables conditioned on their sum. We
derive the saddle point structure of the associated free energy landscape, which implies different
mechanisms for the dynamics of the condensate depending on the system parameters. These results lead
us to an interesting conjecture on the stationary dynamics of the condensate in the thermodynamic
limit.
Ore 14:15, Aula Dal Passo, Università di Roma II
Seminario di Equazioni Differenziali
Given an immersion f of the 2-sphere in a Riemannian manifold (M,g) we study quadratic curvature
functionals depending on the mean curvature, the second fundamental form, and the tracefree second
fundamental form. Minimizers, and more generally critical points of such functionals can be seen
respectively as generalized minimal, totally geodesic and totally umbilical immersions. In the
seminar I will review some results (obtained in collaboration with Kuwert, Riviere and Shygulla)
regarding the existence and the regularity of minimizers of such functionals. An interesting
observation regarding the results obtained with Riviere is that the theory of Willmore surfaces can
be usesful to complete the theory of minimal surfaces (in particular in relation to the existence of
canonical smooth representatives in homotopy classes, a classical program started by Sacks and
Uhlenbeck).
Ore 15:00, Aula di Consiglio
Seminario di Modellistica differenziale numerica
Image restoration refers to the recovery of a clean sharp image from a noisy, and potentially
blurred, observation. Based on the assumption that noise is additive and white, we propose a novel
variational framework in order to enforce whiteness of the residue image. In particular, the
proposed variational model uses Total Variation (TV) regularization (chosen simply for its
popularity, any other regularizers could be substituted as well) and imposes the resemblance of the
residue image to a white noise realization by constraining its autocorrelation function. The
whiteness constraint constitutes the key novelty behind our approach. The restored image is
efficiently computed by the constrained minimization of an energy functional using an Alternating
Directions Methods of Multipliers (ADMM) procedure. Numerical examples show that the novel residue
constraint indeed improves the quality of the computed restorations.
Ore 16:00, Aula di Consiglio
Seminario di Modellistica differenziale numerica
The recent years have seen a significant development in the use of nonuniform grids for the
numerical solution of partial differential equations. This development has given rise to a number of
new problems regarding the analysis of such methods: firstly, on nonuniform grids, many formally
inconsistent schemes converge. We shall report on a numerical study of the properties of
supra-convergence for hyperbolic conservation laws with geometrical source terms, which has
confirmed that the standard consistency condition for the numerical fluxes do not guarantee that the
(local) truncation error vanishes in presence of nonuniform meshes. Nevertheless, the main issue of
an error analysis with optimal rates can be pursued, by virtue of the results obtained on the
supra-convergence phenomenon for numerical approximation of hyperbolic conservation laws. More
clearly, despite the fact that a deterioration of the point-wise consistency is observed in
consequence of the non-uniformity of the mesh, the formal accuracy of the methods is actually
maintained as the global error behaves better than the truncation error would indicate. This
property of enhancement of the numerical error has been widely explored for homogeneous problems,
and we attempt at extending such theory to conservation laws with geometrical source terms that are
discretized by means of well-balanced schemes, as suggested by the classical application to the
Saint-Venant equations for shallow waters. It is worth remarking that the results announced above
cannot affect the case of ordinary differential equation with parameter-dependent (geometrical)
source terms, namely for systems with negligible fluxes. In effects, elementary counter-examples
show that (strong) convergence fails for nonuniform grids, and then some specific approach has to be
designed for recovering the error analysis for finite volume schemes on nonuniform meshes. Precise
comments on the limits and potentiality of these approaches will be done.
Ore 14:30, Aula di Consiglio
Since they were first defined in the early 20th century, Lie algebras have found their place at the
very core of abstract algebra and theoretical physics. Their representation theory was developed
rapidly and is still an area of vibrant interdisciplinary research today - combining algebraic
techniques with those of geometry, combinatorics and category theory. The themes which arose in this
theory have been replicated successfully for many other algebras and so this body of work may be
seen as a guiding paradigm in representation theory. In the late 1970's some glimpses began to
appear of deep relationships between the representations of Lie algebras and nilpotent orbits. These
were mostly understood by a variety of sophisticated methods although, at first, there were very few
unifying themes. In 2002, Alexander Premet defined what is now known as the finite W-algebra. This
is an associative, filtered algebra attached to a complex semisimple Lie algebra and a nilpotent
orbit therein. It has since become clear that the representation theory of these algebras may
explain many of the aforementioned connections between representations and nilpotent orbits of Lie
algebras. As a result, some of the most challenging questions in the representation theory of Lie
algebras are now being answered. I intend to contribute to the theory by initiating two parallel
investigations. The second of these depends upon the first. In type A, the finite W-algebras may be
described by generators and relations, thanks to the work of Brundan and Kleshchev. Finding such a
presentation in other types is perhaps the most fundamental and pressing problem for theorists in
this area. I have conceived of a method to obtain such a presentation, making use of the (geometric)
theory of sheets of adjoint orbits. My second investigation shall reduce this presentation to the
characteristic p realm in order to study the modular representations of Lie algebras.
Ore 14:00, Aula di Consiglio
Seminario P(n): Problemi differenziali non lineari
This paper is motivated by a gauged Schroedinger equation in dimension 2 including the so-called
Chern-Simons term. At low energies, the Maxwell term can be dropped, giving rise to a problem
proposed by Jackiw and Pi in 1990. The study of radially symmetric standig waves leads to a
nonlinear stationary Schroedinger equation involving a nonlocal term. This problem is the
Euler-Lagrange equation of a certain energy functional. In this talk we will be concerned with the
global behavior of such functional. This is joint work with Alessio Pomponio (Politecnico di Bari,
Italy).
Ore 11:00, Aula 34 (quarto piano), Dipartimento di Scienze Statistiche
Latent Markov models (LM) can be seen as a flexible device for taking into account time-varying
subject-specific unobserved heterogeneity. In the basic LM, a random intercept is flexibly allowed
to evolve over time. The available formulations of mixed LM often assume that any additional random
effect is time-constant, with few specific exceptions. In this work we formulate a mixed latent
Markov model in which all random effects may freely evolve over time. The size of the parameter
space is controlled with the possibilities of assuming block independence of random effects, and/or
that groups of random effects may share some or all aspects of their distribution. Parameter
estimation is carried out with a simple expectation maximization strategy, analogous to that used
for the basic latent Markov model, after an adaptation of the usual forward backward recursions and
a parsimonious representation of the expected complete likelihood. Standard errors are derived using
Oakes' identity. Dependence among random effects is summarized using Watanabès total
correlation and described with log-odds ratios and higher-order log-linear interactions. We
illustrate with an original application to the relationship between health literacy and depression
in a panel of adolescents. In this example subjects are clustered in schools, which leads to
high-dimensional multivariate time-varying random effects.
Ore 14:30, Aula 211, Università di Roma III
Seminario di Geometria
On an algebraic curve of genus more than one, there are so-called Weierstrass points which are
rather distinguished points on the given curve. After reviewing some basic notions and several known
results in the Weierstrass point business, we will focus on curves of low genus, say g = 3,4,5.
Especially we are particularly interested in those curves with Weierstrass points of maximal
possible weights.
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