Week 4

The second workshop “Free Probability: the applied perspective”, in the last week of the program, had its focus on more applied directions. In recent years, it has become apparent that free probability, with its tools for calculating the eigenvalue distribution of random matrices or dealing with outliers, might have quite some impact for questions in applied subjects like: physics, statistics, wireless communications, or machine learning. Many of the talks during the second workshop addressed such issues and showed this surprising arch of free probability from the abstract to the very applied: we saw three vignettes on free probability and statistics; heard about a new theory for sketching in linear regression; and learned about free probability for deep learning.

Conclusion

Apart from the exciting and surprising new connections and directions of free probability, there were also a couple of talks which presented new important results per se, creating a lot of discussions and potentially new collaborations. Just to mention a few, the talks “The free field meets free probability theory” by Mai and “The atoms of the free additive convolution of two operator-valued distributions” by Belinschi gave qualitatively new results about one of the hard problems of the analytic theory of free probability, namely the absence or the possible occurrence of atoms for various functions in non-commuting variables. The talk “Traffic independence and freeness over the diagonal” by Cebron highlighted important progress in the theory of traffic independence, by relating the up to now quite combinatorial theory to analytic tools from operator-valued freeness, and thus opening a quite new direction for this important concept. Another impressive advance, on the notoriously difficult task of calculating Brown measure, was presented in the talk “The Brown measure of free multiplicative Brownian
motion” by Kemp.

I think that the whole program on free probability at the CRM was a big success. In particular, the attendance of many young researchers, often for the whole month or a substantial part of it, and the many lively discussions in the seminar rooms and halls of the Pavillon Aisenstadt, following the talks or signalling ongoing or new collaborations, showed clearly that the subject is still very vibrant and full of new ideas. I am quite optimistic about the future of free probability theory.

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for week 3, featuring in particular two of the three Aisenstadt Chair talks of Alice Guionnet. The first of those is intended for a general public, see also the CRM website. The third Aisenstadt Chair talk will be the opening talk of the second workshop.

WEDNESDAY March 20

Pavillon Claire-McNicoll, Université de Montréal, Room Z-220

16:15 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and Random matrices”
Free probability is the natural framework to consider matrices with size going to infinity. Since this key remark was made by Voiculescu in the nineties, these two fields have enriched each other continuously. We will discuss a few of these fruitful crossovers. This talk will only require a general mathematical background.


THURSDAY March 21

Pavillon André-Aisenstadt, Université de Montréal, Room 5340

09:30 – 11:00 Mireille Capitaine
“Introduction to Outliers for Deformed Wigner Matrices”


14:30 – 15:30 Felix Leid
“Maps, Partitioned Permutations, and Free Probability”

16:00 – 17:00 Maxime Fevrier
“From conditional freeness to infinitesimal freeness”

FRIDAY March 22

Pavillon André-Aisenstadt, Université de Montréal, Room 1360

10:30 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and random matrices: Conjugate variables and the Dyson-Schwinger equations”
In this talk I will discuss some uses of integration by parts in free probability, random matrices and related topics. In particular I will show how it can be used to study the large dimension asymptotics in random matrices and tilings.

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for this week; they will be held in room 5340, 5th floor, Andre-Aisenstadt building on the campus of the Universite de Montreal.

Note that the second talk on Friday has changed; Felix Leid’s talk has been moved to next week, Ben Hayes will talk instead.

TUESDAY March 12

9:30 – 12:30 Tobias Mai, Roland Speicher, and Sheng Yin
“Introduction to Regularity and the Free Field”

Lunch break

2:30 – 4:00 Hari Bercovici
“Introduction to outliers”

THURSDAY March 14

9:30 – 12:30 Benson Au, Guillaume Cébron, and Camille Male
“Introduction to Traffic Freeness”

Lunch break

2:30 – 3:30 Laura Maassen
“Group-theoretical quantum groups”

4:00 – 5:00 Simon Schmidt
“Quantum automorphism groups of finite graphs”

FRIDAY March 15

9:30 – 10:30 Josué Váquez Becerra
“Fluctuation moments induced by some asymptotically liberating unitary matrices”

11:00 – 12:00 Ben Hayes
“A random matrix approach to the Peterson-Thom conjecture”

2019 will again be a year with quite a few meetings around free probability.

The main event will be a month-long program on New Developments in Free Probability and Applications at CRM in Montreal in March 2019. There will be two workshops: one, at the beginning of March, on the theory and its extensions and the second, at the end of March, on the applied perspective. In the two weeks in between there will also be quite some activity, in particular, we are aiming at bringing graduate students and postdoctoral fellows quickly to the frontiers of the subject. Furthermore, Alice Guionnet will give the Aisenstadt Chair lecture series between both workshops.

This program is part of the year long celebration of the CRM’s 50th anniversary. It seems very appropriate to have such a meeting on the blossoming of free probability theory, and its promise for the future at the place where the seed was sown. In the spring of 1991 Dan Voiculescu was the holder of the Andre Aisenstadt chair at the CRM in Montreal during the ’91 operator algebra program. At this time, free probability was still in its infancy and only known to a small group of enthusiasts. This was going to change. Voiculescu gave the Aisenstadt Lectures on free probability in Montreal, organizing the material and bringing it with the help of his students Ken Dykema and Alexandru Nica into a publishable form. The resulting book was the first volume in the CRM Monograph Series and was instrumental for making the theory more generally accessible and attracting many, in particular young, researchers to the subject. It is still the most cited literature on free probability. Andu, Dan, and Ken (as well as a couple of other experts) will stay as Simons Scholars-in-Residence for the whole program at CRM

Another month-long program with a substantial free probability component will be the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto, June 10 – July 5, 2019. In particular, one of the workshops of the program, June 17-21, deals with applications of noncommutative functions to random matrices and free probability.

The focus program at the Fields Institute will also include a celebratory banquet on June 14, in honour of the 70th birthday of Dan Voiculescu.