 u star then the conditions on things like the normalized trace of a polynomial of x they become conditions on the integral of this polynomial against the the measure the delta masses at the eigenvalues and so more or less you you you're looking I mean you're prescribing the the moments of this measure so more or less you're fixing where these eigenvalues are going to be right and then the question is how to compute the volume of of the set of all matrices whose eigenvalues are approximately the correct thing and that is of course the usual integral which will bring in this this fundamental term lambda i minus lambda j and that what what will tell you that this logarithmic volume will be related to sum of log lambda i minus lambda j and so that will in in the infinity limit will converge to the logarithmic the logarithmic energy plus well there's some constants that are involved in this normalization okay and somehow this computation is can be phrased in a different way this is what was done by in a fundamental paper of Ben Arouss and Alice Guionnay who repackaged this if you like as a large deviation principle so this this chi becomes a rate function in that in that approach this is again the case of one matrix d equals one now for d not equal to one the whole thing is a bit more mysterious but still nice for example you still have minimization or maximization of this relative entropy by semicircular variables I didn't write it but it's still true you still have that freeness is characterized well okay yeah I think it's characterized in this case by equality by the fact that the entropy is the sum of the individual entropies and in the case that it's not minus infinity and in addition you have a very nice thing which is called the change of variables formula so to set this up it says that if I take yj sum well you can take some kind of analytic function of xj so think of this pj is some kind of a convergent power series or something like this a non-commutative power series oh I wrote it wrong sorry it's p p should be of all variables sorry about that I I'm assuming that yj is pj of x1 xn xd and let's assume also that xj is some qj of y1 yd where this p p and q they are some kind of non-commutative power series then there's a very nice formula that tells you that the entropy of the y's so the entropy of the functions of x1 xn are just the entropy of the fun of the variables themselves plus a correction term which has to do with the kind of Jacobian so you take these different quotients you make a matrix out of them by differentiating p i with respect to in direction j this is going to be some matrix in your phenomenon algebra tensor itself you take the absolute value of it you take the log of it and you take traces so this is a kind of a determinant if you like it's a determinant of a derivative so it's kind of a Jacobian term that corrects the whole log of a determinant okay if you want to prove this it's not so complicated I mean you just have to understand how applying such functional calculus distorts volumes on on matrices and that's not so hard so basically this expression will approximately tell you how volume is distorted and then then the rest is just a fairly easy computation okay so the big and only result in the subject really I mean in this in this chi versus chi star subject is this theorem of Philip beyond Ray Capitaine and Alice Guionnay which gives an inequality it says that this chi remembers the thing that is computed with matrices how many matrices do we have that model a given law is bounded from above by chi star so I want to give I mean this is a very non-trivial and it's Jonas paper but I want to give a caricature of the proof I mean it's Friday evening so I can kind of lie to you right so I will lie to you enough to make it simple actually I think it's not so bad lie so the idea is so I just put back the definition of of this chi it's the limit of log volume blah blah blah where this blah blah blah is defined here yeah so all these sets of approximations alright now let's say it in a different way let's call gamma and simply the Lebesgue measure restricted to the microstate space and let's renormalize it to make it a probability measure then actually what happens is that this this free entropy chi is more or less the limit of the classical entropies of these guys okay moreover if you're interested in semicircular perturbations of x1 xn then you can see the entropy of this guy as the limit of classical entropies of Gaussian perturbations of these gamma and t okay so that that formula is not so hard to establish what did I want to say oh yes now the idea is to prove that the two things are equal you want to check two things one when t equals infinity this is true well that's obvious I mean when t equals infinity you have so many semicirculars here that your x's are forgotten so it's just a semicircular n-tuple and here you have added so much Gaussian noise that you forgot where you started and so you just have a ball of some radius you make the computation you're done the next thing that you want to do is you want to check that the fit that the derivatives and t of these quantities are the same or at least there's an inequality between them because then you can conclude so what you want to do is you want to compute d chi of x1 t xn t versus d d t d chi star of x1 t xn t okay and now I'm going to cheat and say that actually this is something to do with the limit as n goes to infinity 1 over n squared and here I will have the Fisher information the classical information of x1 t xn t whereas here I will have phi star of x1 t xn t okay all right fine so entropies go the opposite way with Fisher information so to prove this I think I need to prove that if I'm not mistaken so that's what I'm trying to do all right now for this I have I have a formula in principle what is this thing this is the conditional expectation on to the algebra generated by your x1 t xn t of s n or s j well you have to divide this by you have to sum the other norms of these things sum over j and divided by t to be precise and you have a similar formula for here this is 1 over t there'll be this 1 over n squared whatever sum over j conditional expectation of some Gaussian g j and this conditional expectation will be on what it will be on the algebra generated by the entries of your random matrices which come from the measured gamma n t okay all right but now now we're in business we're in business why because this algebra is so is somehow smaller than this algebra this algebra is more or less the algebra of all polynomials in your matrices these are honest non-commutative polynomials of your matrices some closure of it this algebra is a much bigger algebra it's algebra generated by all the entries so of course when I condition something in a bigger algebra I will get more that's it so that's more or less the proof now the next question you can ask is since there's such a nice argument maybe that chi chi is less recall and chi star and so forth maybe you should actually go for the quality and that's that would be nice of course when d equals 1 there is equality there is unfortunately slight monstrosity that impedes progress and this this monster is called the coin embedding question so the coin embedding question very simply put is whether these sets are always eventually non-empty in other words I give you abstractly a trace on some kind of an algebra you know it's a positive trace these are operators in a Hilbert space as well so there are these variables x1 xd can I actually find finite matrices that approximate my guys in moments up to a certain degree now in fact there is a refinement or you know there are many reformulations of this question you one way to say it is does every separable tracial phenomenon algebra embed into something called the ultra power of matrices and I'm not going to define that what that is but in reality what that what's written there is exactly this but there's another reformulation of it I won't tell you that one let me tell you this one for any u1 ud in some phenomenon algebra m tau and for any epsilon is there an n and capital u1 capital ud which are n by n unitaries with the property that tau of ui star uj is with an epsilon of tau little ui star uj so you just need moments of order to and unbelievably we don't know okay so this this actually this this question has lots of lots of interest and there is now a connection with quantum information theory there's something called some some Cyrilson problem that is associated with that it's also been studied a lot in c-star algebras and phonemian algebras for various reasons it's also related you may have heard let me just say the word about what are sulfate groups you can ask this for a group I mean this is really the maybe the simplest way of asking the question suppose I give you a group gamma which is really given as generators in relations so it's generated by g1 gn subject to some relations r1 rm and even for for the purposes of this discussion even finitely presented so finite number of relations okay and suppose you so can one find so given this g1 gn and a cutoff of rho can one find unitaries say u1 un such that so of size n by n and large enough such that well what I want to say is that things which are non-trivial in the group are non-trivial and things which are trivial in the group are trivial up to this radius row so for all word of length at most row this word of u1 un is approximately identity if so approximately really in two norm is less than epsilon well let me let me just say here delta w equals e so w or w I evaluate where v is w evaluated in g1 gn yeah so I simply want want unitaries which are sorry I'm saying yeah I want to close to identity if if w of g1 gn is identity and far from identity far from identity for not so I just want to be able to embed the group into the unitary group approximately so that all my relations up to some length are satisfied and nothing more okay and the there's a similar question where you replace the unitaries by permutation matrices and and groups that have that property called softening groups and one of the open questions whether every group is suffix so can you embed it in this way into permutations and it's a question seems to be related to the con question certainly if every group were softening it would have to satisfy the con question so that's unfortunately a big problem so at least said that since there is the whole weekend in front of people and there's so many people here that want to work very hard over the weekend I thought I would she wanted me to give a list of impossible questions so I would like to give a list of impossible questions just in case they're possible I mean heaven knows so the first couple are really operator algebra questions and putting them mainly to tell you what motivated free probability theory and continues to motivate it in many ways one the first question is the famous question which probably can be traced to von Neumann in one way or another whether these for no one algebras generated by the D should be and I'm sorry that should be I think that's a perpetual problem I talk today whether the phenomenon to be generated by D semicircular variables remembers D okay and there is a non-trivial result in free probability theory that says that there's a dichotomy either they don't at all in other words if I take two variables or 20 or an infinity it's the same thing or they're all different so it cannot be that the one on five generators is the same as the one on three generators but the one on seven is the same as the one on four that cannot be but but the question still is whether it remembers D and that is that is a famous question and and yeah incidentally there is a similar question for C star algebras which I really like whether you look at the smaller thing the C star algebra generated by D semicircular where that remembers D so the nice thing is there is that in that case the unitary version of it was sold if you take D free unitaries each distributed uniformly on the circle then yes D is remembered and that's a computation in k theory which tells you that in the k one the first k group of this of this guys D minus one so that's remembered but unfortunately the k theory of this C star algebras trivial so actually nobody knows if that remembers D or not so there you go the next question is the con question you know of course there's an upgrade on the con question can you say that kai equals kai star and one kind of one of the stumbling blocks actually to understanding kai versus kai stars to have what's called a change of variables for kai stars so I told you how to change variables there's this form nice formula with the kind of Jacobian for kai prove the same thing for kai star and nobody really has any quick as far as I know now I did not get to talk about this at all but we have this paper with Alisa a few years ago about analogs of transport in the non commutative setting so this is a situation where you try to construct explicitly change of variables that converts one non commutative law into a different non commutative law this is a highly non trivial question in the non commutative case for the following reason you see classically if you look at a measure mu and on some some space and you look at the space of essentially well just look at it as a measure space this measure space remembers absolutely nothing about mu except for atoms so as measure spaces this is always zero one with the big measure except if mu has atoms right so you know there are no big obstructions to transport one probability measure into the other it's just a question of exactly how to do this now in the non commutative case things are really bizarre if you look at two ones of tracial phenomenon algebras which are supposed to be like L infinity of x mu the non commutative replacement for that there's so many of them that they cannot even fit into one there isn't a separable q containing all of them okay so you cannot even put them on the same you know and say one big phenomenon but they they kind of repel you try to do it they fight so much you know it's it's over so so so therefore there are so many different isomorphism classes of these algebras you know that it becomes very non trivial to be able to say that this law has the same phenomenon algebra as that law which of course is what you mean by a change of variables but anyways there are some hints that maybe there is some order in the universe I mean the con question is a kind of a transport question I told you that everything is it's whether everything can be put into this algebra that's not separable so it doesn't contradict what I'm saying and there's another famous question of Neumann whether every non amenable phenomenon algebra contains a free group algebra okay and so for example you can weaken it by saying that if you have a detouple with finite entropy does that imply that this algebra contains a free group factor so that's a very nice question and the last thing that would be very nice to do and you know there's incremental progress on this every time that something happens sometimes it's a big increment sometimes it's small but we would really like to understand what happens in general to limits of random matrix models so you look at things like this with the being arbitrary so not assuming convexity or some some story that it's a small perturbation of a of a quadratic potential or something like this a really very general potential what happens in that case for example even if whether there's always a limit in this case I think that's that's open there are limit points and then there are ways to characterize them but unfortunately nobody knows if there's always that so I think I gave you enough enough stuff to keep you busy for a century and certainly this weekend so it all remains is to thank you and and to say goodbye questions yes they have to be inverses yeah I forgot to say but I'm sure is that what you ask yeah the way it's I mean the the only case where this is known and I mean you can probably extend it a little bit by taking some limits and so forth but it gets a bit murky but yes you are assuming that yj is a power series of x1 through xd so not just one variable but a d tuple and but then you're assuming that it's invertible so there's another power series that inverts this power series and you could go one way that it just needs you need a sufficiently good control of what happens on the matrix level you see for instance if you don't assume that they are invertible or something like this that you get you get into the question of how badly non one-to-one can a non commutative transformation be and that's actually a very interesting question I mean I I'm quite puzzled by this but you take a non commutative you know a bunch of non commutative polynomials and you solve this try to solve the system of equations so here b1 bd are given n by n self-adjusting matrices and you want to solve for a1 ad which are of the same sort how many solutions does this have and it would be nice to know if one over n squared log of the number of solutions goes to zero as n goes to infinity and I have no idea but it there seem to be some mild indications that this may actually be true I mean of course if you simply take algebraic equations in the entries then all you would know is that this is finite it's bounded by d or d times the degree of the polynomials or something so that's you know one of the questions that go in that direction any other questions if not let's thank dima again