I will talk about random matrices and free probability and a particular problem which

is spectral deconvolution. So first maybe let me just say a few words about actually what is

random matrices and what is free probability and then we go directly to the problem. So

random matrices is of course many many objects in mathematics are related to linear algebra.

In many cases there is some stochasticity and the classical examples or examples that made the

theory start were coming from statistics, multivariate statistics, the first probably

the first random matrices matrix introduced by Wishart where he considered covariance

matrices and actually today I will talk about this. So it's our own topic but still there is

some questions there and also in physics of course Diego mentioned that the biology is maybe

the new physics but I think physics will be there forever for us providing important problems and

in particular many people in physics studied the random matrices in particular Nobel Prize winner

Wigner studied the model the the spectra of atoms using random matrices because it's very

hard to calculate so you do random and you try to say something and also in the 70s there was

this very nice connection found by Dyson and Montgomery the story is very nice that they were

in a coffee and they realized they were studying the same kind of equations and then this made a

connection between number theory specifically series of the or a specific more specifically

correlation of series of the Riemann-Zitter function and random matrices and actually to

formalize this in the broad sense this is still an open problem but still the connection is there

and it has given a lot of intuition to the people in number theory from random matrices but maybe

now I talk about free probability also in the broad sense so free probability was initiated

by Vykulescu studying hard problems in operator algebras and initially there were like various

developments but maybe I can mention once some of the ones which I think are more important so

if one of the main objects that Vykulescu introduced to tackle this operator algebra problems was

actually that a notion of entropy in this free sense also there was this probabilistic development

initiated by Vykulescu and Vykulescu and maybe one important result was this Vykulescu pattern

where he proved that triangular arrays are actually in total correspondence with the classical case

with the free case and one major also since the very beginning major observation is actually that

many random matrices behave like a free random variables so this is what is called asymptotic

I will talk about this actually today and another major contribution to the editorial was done by

Roland Speicher where he introduced a combinatorial approach to random matrices and this may be more

accessible to many communities which were maybe not so acquainted with operator algebras but maybe

knew the classical approach of moments to probability and then were able to understand free probability

using this approach and also by itself it has a lot gives a lot of insight of what is going on or

why some variables appear and others not by doing an analogy between partitions and necrosyptic partitions

but actually today there is many many applications of random matrices and actually random matrices

using free probability and I will mention some of them and maybe this is these two or three slides

that I will mention is what you can take to your home and then read this kind of papers and maybe

some of your research can be related to this so in for example in graphs graph theory and spectra

graph theory the recently Marcus Spielman and Silvia Steva proved the existence of Ramanujan graphs

of any size and any degree and this was a major achievement compared to the fact that it was only

few examples done by Lubaski-Zarnak but this was done by a random construction and there in the

general more general case they use actually free probability to give some bounds on the second eigenvalue

so if you don't know what is a Ramanujan graph maybe you know what is a sparsifier or expander so

this is basically graphs which have a lot of connectivity but somehow the leading direction

which is given by the main the largest eigenvalue is much larger than the rest of the eigenvalues

okay so this is good because this means that the graph has a lot of connectivity okay so in

wireless communication actually very applied engineering use free probability to understand

for example a capacity of channels in particular a rough muller who is in here at FAU the engineering

department has this contribution together with the group of Spiker also in quantum information

people has found examples again instead of looking at the specific example they can found