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Seminars 2019
Seminar/Minicourse
Thursday and Friday, February 14-15, 2019
El método de factorización de Wei-Norman para sistemas
de ecuaciones diferenciales ordinarias lineales con
simetrías
Universidad de Jaén
Abstract
En este curso consideraremos ecuaciones diferenciales
de primer orden lineales $y'(t)=H(t)y(t)$,
$y(t_0)=y_0$, donde $y(t)$ es un vector (o función) y
$H(t)$ una matriz (u operador en un espacio de
Hilbert). Esta ecuación puede transformarse en otra
equivalente: $$U'(t,t_0)=H(t)U(t,t_0)\,,$$ donde
$U(t,t_0)$ es una matriz (u operador) tal que
$y(t)=U(t,t_0)y(t_0)$. El método de factorización de
Wei-Norman consiste en que, si $H(t)$ se puede expresar
como combinación lineal de los elementos de un álgebra
de Lie (en una cierta representación), entonces
$U(t,t_0)$ se puede poner como producto de
exponenciales de los elementos del álgebra
multiplicados por funciones de $t$ que verifican un
conjunto de ecuaciones diferenciales no lineales de
primer orden (de tipo Ricatti). Estas ecuaciones no
dependen de la representación del álgebra de Lie
concreta, mientras sea fiel), tan solo de las
constantes de estructura.
En la primera sesión derivaremos las expresiones generales del método. En la segunda sesión, particularizaremos a algunos ejemplos concretos de álgebras de Lie de dimensión 3.
Each seminar will take place from 12:00 to 13:30 in
classroom 1.0.B02 (Edificio Betancourt) Universidad
Carlos III
Seminar
Friday, January 11, 2019
Dynamical criticality in open systems: non-perturbative
physics, microscopic origin and direct observation
Universidad de Granada
Abstract
Driven diffusive systems may undergo phase transitions
to sustain atypical values of the current. This leads
in some cases to symmetry-broken space-time
trajectories which enhance the probability of such
fluctuations. Here, we shed light on both the
macroscopic large deviation properties and the
microscopic origin of such spontaneous symmetry
breaking in the open weakly asymmetric exclusion
process. By studying the joint fluctuations of the
current and a collective order parameter, we uncover
the full dynamical phase diagram for arbitrary boundary
driving, which is reminiscent of a $Z_2$
symmetry-breaking transition. The associated joint
large deviation function becomes nonconvex below the
critical point, where a Maxwell-like violation of the
additivity principle is observed. At the microscopic
level, the dynamical phase transition is linked to an
emerging degeneracy of the ground state of the
microscopic generator, from which the optimal
trajectories in the symmetry-broken phase follow. In
addition, we observe this symmetry-breaking phenomenon
in extensive rare-event simulations, confirming our
macroscopic and microscopic results.
The seminar will take place at 13:00 in classroom
2.0.D14 (Edificio Sabatini) Universidad Carlos III
Seminar
Thursday, January 17, 2019
Frequency-dependent interactions in a computational
model of the primate cortex
University of Amsterdam
Abstract
Cortical areas in the brain are often classified as
sensory areas --those receiving input from the external
world --or cognitive areas --related with association,
memory and more abstract mental processing. It is
thought that interactions between these areas in the
feedforward (sensory to cognitive) or feedback
(cognitive to sensory) direction hold the key to
understand attention, expectations and other brain
functions. However, the underlying circuit mechanism
remains poorly understood and represents a major
challenge in neuroscience. We approached this problem
using a large-scale computational model of the macaque
cortex constrained by novel brain connectivity data. In
our model, the interplay between feedforward and
feedback signals depends on the fine laminar structure
of the cortex, and involves complex dynamics across
multiple scales. The model was tested by reproducing a
wide range of experimental findings about
frequency-dependent interactions between visual
cortical areas. Furthermore, the model replicates the
existence of a functional hierarchy as reported in
recent monkey and human experiments, and suggests a
mechanism for the observed dynamics of such hierarchy.
Together, this work highlights the necessity of
multiscale approaches and provides a modeling platform
for studies of large-scale brain circuit dynamics and
functions.
The seminar will take place at 11:00 in classroom
2.3D04 (Edificio Sabatini) Universidad Carlos III
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