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RANDOM MATRICES, NON-COMMUTATIVE PROBABILITY AND RELATED TOPICS

 

Mexican network of researchers and students of these topics

 

Welcome to the site of a community of Mexican researchers/professors and students that deals with the world in which we work from the perspective of mathematics in the areas of Random Matrices, Non-Commutative Probability and Free Probability, which are modern, multi-thematic, dynamic areas within mathematics. On this site you will find information on these subjects and some of their multiplicity of connections with other branches of mathematics and applications, as well as our activities and lines of mathematical research in these subjects.

Non-commutative probability began to be developed mathematically in the 1970s in the work of Hudson and Parthasarathy, among others, who were inspired by the work of Von Neumann in quantum mechanics. Then, in the 1980s, came free probability theory in the work of Dan-Virgil Voiculescu, whose initial objective was to solve certain problems relating to the structure of Von Neumann’s algebra. Thanks to these efforts, along with a new concept of entropy, in 2004 Voiculescu received the National Academy of Sciences Award in Mathematics.

Now, free probability enjoys a number of relationships with other areas of mathematics, some of them unexpected. Particularly important is its relationship with large dimension random matrix theory and semicircle law, as studied in the 1950s by Eugene Wigner in the context of nuclear physics. This relationship, also demonstrated by Voiculescu, establishes that large dimension independent random matrices behave as free random variables. The paper presented by Roland Speicher at the 2014 International Conference of Mathematicians explains this relationship between random matrices and free probability.

Today, studies are being done on the relationship between non-commutative probability and topological data analysis.

 

[Enlaces:]

Who we are  Our areas of interest        Significance and current developments

Relationship with other branches of mathematics

New notions of independence and conditional expectation

Example of application to wireless communications

Other perspectives in Mexico: Quantum probability, Quantum physics and random matrices

Coordinators: Octavio Arizmendi (octavius@cimat.mx), Víctor Pérez Abreu (pabreu@cimat.mx)