I will give a colloquium talk at the Math Department of the Technion, Israel on next Monday, January 25 – online, of course. They have the nice option of a pre-colloquium talk, which provides students with some background for the material appearing in the colloquium talk. Tobias agreed to give such a preparation for my talk. So he will give tomorrow (on Thursday, January 21) an introduction to free probability and its relation with random matrices. Surely a great opportunity for everyone to learn (more) about the subject.
My talk will, of course, have such material in the background, but I tried to prepare it in such a way that even without knowing about free probability one should be able to get the main ideas. So, it might help to know what free semi-circulars are, but it is not necessary (and not assumed) for my talk.
Below are the titles and abstracts of our talks; and here is a link to the Technion page with access information:
update: actually, next week seems to be a busy week for talks around free probability; don’t forget that the UC Berkeley Probabilistic Operator Algebra Seminar will be starting again, on Monday, January 25, with a talk of Friedrich Goetze; and then there will also be a talk by Serban Belinschi on “The Christoffel-Darboux kernel in noncommutative probability” at the Probability Seminar at Warsaw University of Technology, on Tuesday, January 26.
Tobias Mai: What actually is free probability theory? (Thursday, January 21, 2021)
In my talk, I want to answer this question by giving an introduction to the underlying ideas, basic concepts, and fundamental results of free probability theory. In particular, I will highlight the deep connections of this field with random matrix theory.
Roland Speicher: Singularity of matrices in non-commuting variables and free probability (Monday, January 25, 2021)
The Edmonds’ problem asks to decide about the singularity of a given matrix with linear polynomials in commuting variables as entries, or more general to compute the rank of such a matrix over the field of rational functions. This problem has no known deterministic polynomial time algorithm and it relates to fundamental questions in complexity theory.
Recently, there has been much interest in analyzing a non-commutative variant of the Edmonds’ problem, where the entries are linear polynomials in non-commuting variables and the rank is over the field of non-commutative rational functions (aka free skew field). Garg, Gurvits, Oliveira, and Wigderson showed that for this non-commutative Edmonds’ problem there exists a deterministic polynomial time algorithm. This problem has a remarkable number of diverse origins and motivations and I will present in my talk another such manifestation of the problem, arising from the relation with free probability and random matrix theory. In particular, this approach results also in another, quite analytic, algorithm for calculating the non-commutative rank.
This talk is based on joint work with Johannes Hoffmann, Tobias Mai, and Sheng Yin.