Textos

Presentaciones 2013

“Information, transmission on the complete graph: when Galton, Watson and the couon collector meet”, de René Schott, Francis Comets y François Delarue

“Some applications of dynamic random walks in computer science and quantum, probability”, de René Schott

“A rough-paths type approach to non-commutative stochastic integration”, de René Schott

Últimos textos de la Escuela de Invierno VIII Valparaíso 2010:

“Stochastic Models in Neuroscience.” Textos correspondientes a las charlas dictadas en la Escuela de Invierno VIII Valparaíso 2010.
Michèle THIEULLEN
Laboratoire de Probabilités et Modèles Aléatoires.
Univ. Pierre et Marie Curie, Paris, France.
In the nervous system phenomena with a high variability are frequent: highly irregular signals are observed experimentally, the response of a neuron to a given stimulation may vary from trial to trial. This variability is often referred to as noise. The aim of this course is to present some aspects of a probabilistic study of the neuronal activity of one single neuron considered as a building block of more complex neuronal assemblies. We will focus on the action potential generation and the impact of noise on it. We consider the patch-clamp situation only, which already gives rise to numerous mathematical questions. We will address two types of noise : the external noise which can be considered as resulting from the environment of a given neuron and the intrinsic noise which is the result of the stochastic opening of ion channels present along the membrane. The course will be structured as follows : we will first give a general presentation of possible mathematical models (deterministic and stochastic) for spike generation. Then we will consider questions specifically related to external or intrinsic noise. Both types of models present a time-scale separation. Models with external noise are more classical. One typical question for these models may be the first passage time of a given threshold. Models which take into account channel noise are more recent. They are more biophysically realistic. They provide an access to both the microscopic and the macroscopic levels of the neuron membrane and for instance they enable us to relate the noise with the number of channels involved. In both types of models many mathematical questions remain.

“Course title: An introduction to Quantum Statistics”. Textos correspondientes a las charlas dictadas en la Escuela de Invierno VIII Valparaíso 2010.

Madalin Guta
School of Mathematical Sciences, University of Nottingham, Inglaterra.

Abstract: The recent advances in Quantum Information and Quantum Computation have brought a paradigm shift in the way we think about encoding and manipulating information.
Atoms and photons are carriers of a new type of information and thanks to the modern technology we have reached the point where we can manipulate and measure individual quantum systems. A fundamental implication of these developments is that statistical inference based on data obtained by measuring a limited number of individual systems, will play a much greater role in quantum theory.
These lectures give a short overview of the current status in quantum statistics starting from the first methods developed in the 70′s, and up to the latest theoretical and experimental results. The guiding principle is to adapt and extend well established `classical’ statistical inference techniques to the quantum set-up, and to identify the `purely quantum’ features that need to be explored. In parallel, some recent practical applications will be discussed.


First Lecture: Quantum mechanics as a probabilistic theory.

I will introduce some of the basic notion of quantum mechanics with an emphasis on the parallels and differences with probability theory.

Second lecture: Quantum Statistics: basic examples.

I will define the set-up of quantum statistical problems and discuss a few basic examples such as the estimation of a qubit, Gaussian state, discrimination between two states. Then I will formulate the quantum Cramer-Rao bound and discuss its implications.

Third lecture: Asymptotics in Quantum Statistics

In this lecture I will give an introduction to a new technique in quantum state estimation, called `local asymptotic normality. This tool is an extension of the classical notion of asymptotic normality introduced in Mathematical Statistics by Lucien Le Cam. In the quantum setting, this shows that a large sample of independent, identically prepared quantum systems can be approximated by a quantum Gaussian state uniformly over a range of (unknown) parameters. This reduces the problem of optimal state estimation to that of estimating the mean of a Gaussian state.

Other applications of this technique will be mentioned.

Literature:
Artiles, L, Gill, R., Guta, M., An invitation to quantum tomography, J. Royal Statist. Soc. B, 67, (2005), 109-134.
Barndorff-Nielsen O.E., Gill, R., Jupp, P. E., On quantum statistical inference (with discussion), J. R. Statist. Soc. B, 65, (2003), 775-816.
Guta M., Janssens B., Kahn J., Optimal estimation of qubit states with continuous time measurements, Commun. Math. Phys., 277, (2008), 127-160.
Kahn, J. and Guta, M. Local asymptotic normality for finite dimensional quantum systems, Commun. Math. Phys. 289, 597-652, (2009)
Helstrom C.W., Quantum Detection and Estimation Theory, Academic Press, New York (1976).
Holevo A.S., Probabilistic and Statistical Aspects of Quantum Theory, North-Holland (1982).
Nielsen, M. A. and Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press, (2000)

La Obra Científica de Enrique Cabaña.
Prof. Mario Wschebor. Centro de Matemática; Facultad de Ciencias, Universidad de la República de uruguay.
Texto de la presentación realizada en el IV Encuentro Regional de Probabilidad y Estadística Matemática; Montevideo, noviembre de 2007.

El Azar (Y la manera de calcularlo).
Prof. Mario Wschebor. Centro de Matemática; Facultad de Ciencias, Universidad de la República de uruguay.
Texto publicado en la revista literaria “maldoror” de Montevideo en 2008.

Complete positivity and non commutative Markov semigroups.
Prof. Rolando Rebolledo. Laboratorio de Análisis Estocástico; Pontificia Universidad Católica de Chile. Facultad de Matemáticas.
Presentación realizada en el marco de la VI versión de la Escuela de Invierno de Análisis Estocástico y aplicaciones, efectuada el pasado miércoles 23 de julio.

Stochastic Resonance: mathematical approaches.
Profs. Samuel Herrmann, Peter Imkeller, Ilya Pavlyukevich, Dierk Peithmann.
Presentación efectuada por el Profesor Imkeller en el marco de la VI versión de la Escuela de Invierno de Análisis Estocástico y aplicaciones, llevada a cabo el pasado martes 22 de julio.

Statistical analysis of spatial data.
Alicia Carriquiry. Iowa State University; Pontificia Universidad Católica de Chile.
Presentación realizada en el marco de la VI versión de la Escuela de Invierno de Análisis Estocástico y aplicaciones, el pasado lunes 21 de julio.