# Seminar on Free Probability and Random Matrices Winter-Spring 2021

## Organizer: J. Mingo <!-- // // format date as dd-mmm-yy // example: 12-Jan-99 // function date_ddmmmyy(date) { var d = date.getDate(); var m = date.getMonth() + 1; var y = date.getYear(); // handle different year values // returned by IE and NS in // the year 2000. if(y >= 2000) { y -= 2000; } if(y >= 100) { y -= 100; } // could use splitString() here // but the following method is // more compatible var mmm = ( 1==m)?'Jan':( 2==m)?'Feb':(3==m)?'Mar': ( 4==m)?'Apr':( 5==m)?'May':(6==m)?'Jun': ( 7==m)?'Jul':( 8==m)?'Aug':(9==m)?'Sep': (10==m)?'Oct':(11==m)?'Nov':'Dec'; return "" + (d<10?"0"+d:d) + "-" + mmm + "-" + (y<10?"0"+y:y); } // // get last modified date of the // current document. // function date_lastmodified() { var lmd = document.lastModified; var s = "Unknown"; var d1; // check if we have a valid date // before proceeding if(0 != (d1=Date.parse(lmd))) { s = "" + date_ddmmmyy(new Date(d1)); } return s; } // // finally display the last modified date // as DD-MMM-YY // document.write( "Last modified on " + date_lastmodified() ); // -->

This term will be a learning seminar on Brown measure delivered via Zoom. Send me an email if you wish to join us. We will be following the notes of Flemming Larsen and Chapter 11 in the book of Mingo-Speicher and the 2000 JFA paper of Haagerup and Larsen. Notes for lectures 1 -5, lectures 9 - 11; lecture 12; lecture 13.

Friday, July 16, 10:00 - 11:00, via Zoom
Mihai Popa (San Antonio)
The lecture aims to give a different take on the $S$-transform. We will show that its multiplicative inverse, the so-called $T$-transform, has easier to determine coefficients and allows an intuitive justification of the multiplication property.

Friday, July 9, 10:00 - 11:00, via Zoom
Pei-Lun Tseng (Queen's)
I will conclude the proof of the theorem Haagerup-Larsen [2000] which shows how the $S$-transform can be used to give the Brown measure of a $R$-diagonal operator.
Friday, July 2, 10:00 - 11:00, via Zoom
Pei-Lun Tseng (Queen's)
I will continue the proof of the theorem Haagerup-Larsen [2000] which shows how the $S$-transform can be used to give the Brown measure of a $R$-diagonal operator.
Friday, June 25, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
I will start the proof of the theorem Haagerup-Larsen [2000] which shows how the $S$-transform can be used to give the Brown measure of a $R$-diagonal operator.
Friday, June 18, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
I will review the basic construction of the $R$ and $S$ transforms.
Friday, June 11, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
A crucial part of the calculation of the Brown measure of $R$-diagonal operators is the representation of every $R$-diagonal operator as the product of a freely independent Bernoulli random variable and an even operator.

Friday, June 4, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will free independence and $R$-diagonal operators, which will be the focus for the rest of the lectures.

Friday, May 28, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will review what is meant by the circular operator.

Friday, May 21, 10:00 - 11:00, via Zoom
Ian Charlesworth (Berkeley)
In this lecture I continue to describe the basic properties of the Brown measure.

Friday, May 14, 10:00 - 11:00, via Zoom
Ian Charlesworth (Berkeley)
In this lecture I continue to describe the basic properties of the Brown measure.

Friday, May 7, 10:00 - 11:00, via Zoom
Ian Charlesworth (Berkeley)
In this lecture I describe the basic properties of the Brown measure.

Friday, April 30, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will complete the proof that $\lambda \mapsto L(a - \lambda 1)$ is subharmonic.

Friday, April 23, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will discuss some basic facts about subharmonic functions. Our goal will be to show that $\lambda \mapsto L(a - \lambda 1)$ is subharmonic.

Friday, April 16, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will prove the additivity, $L(ab) = L(a) + L(b)$, of the Fuglede-Kadison logarithm in a finite von-Neumann algebra,

Thursday, April 1, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this lecture I will present the Fuglede-Kadison logarithm, $L(a) = \tau(\log(|a|))$, of a non-normal operator in $(M, \tau)$, a finite von-Neumann algebra, and determine its basic properties.

Thursday, March 25, 10:00 - 11:00, via Zoom
James Mingo (Queen's)
In this initial lecture I will present a broad outline of Brown measure and show that the eigenvalue distribution (normalized to be a probability measure) of a matrix $A$ is the Brown measure of $A$.

Previous Schedules

Getting to Jeffery Hall from the Hotel Belvedere