## 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.