Actualización 23 de agosto del 2010
Agosto-Diciembre 2010
Profesor: Víctor M. Pérez Abreu C.
Algunas reflexiones en prefacios e introducciones de libros recientes
y clásicos de Matrices Aleatorias:
Log-Gases and Random Matrices, de Peter
Forrester (2010): Often it is asked what makes a mathematical topic
interesting. Some qualities which come to mind are usefulness, beauty,
depth and fertility. Usefulness is usually measured by the
utility of the topic outside mathematics. Beauty is an alluring quality
of much mathematics, with the caveat that it is often something only a trained
eye can see. Depth comes via the linking together of multiple ideas and
topics, often seemingly removed from the original context. And fertility
means that with a reasonable effort there are new results, some useful, some
with beauty, and a few maybe with depth, still waiting to be found.
An
Introduction to Random Matrices, de Greg
W. Anderson, Alice
Guionnet y Ofer
Zeitouni (2010): The study of random matrices, and in particular the properties of their
eigenvalues, has emerged from the applications, first in data analysis and
later on as statistical models for heavy-nuclei atoms. Thus, the field of
random matrices owes its existence to applications. Over the years, however, it
became clear that models related to random matrices play an important role in
areas of pure mathematics. Moreover, the tools used in the study of random
matrices came themselves from different and seemingly unrelated branches of
mathematics.
Tercera
Edición del libro clásico Random Matrices, de Madan L. Mehta
(2004): In the last
decade following the publication of the second edition of this book the subject
of random matrices found applications in many new fields of knowledge. In
heterogeneous conductors (mesoscopy systems) where the passage of electric
current may be studied by transfer matrices, quantum chromo dynamics
characterized by some Dirac operator, quantum gravity modeled by some random
triangulation of surfaces, traffic and communication networks, zeta function
and L-series in number theory, even stock movements in financial markets,
wherever imprecise matrices occurred, people dreamed of random matrices.
Random
Matrix Theory: Invariant Ensembles and Universality, de Percy Deift y Dimitri Gioev (2009): There has been a great upsurge of
interest in random matrix theory (RMT) in recent years. This upsurge has been
fueled primary by the fact that an extraordinary variety of problems in
physics, pure mathematics, and applied mathematics are now known to be modeled
by RMT. By this we mean the following: Suppose we are investigating some
statistical quantities {ak} in a neighborhood of some point A, say. The ak’s are to be compared with the eigenvalues {lk}, in a neighborhood of some point L, of a matrix
taken from some random matrix ensemble. If the statistics of the {ak}‘s, appropriately scaled, are described by the
statistics of the {lk}‘s, appropriately scaled, then we say that the {ak}‘s are modeled by random matrix theory.