 Welcome to the day seven of PCMI summer school. Today we start a new series of lectures that will be about Riemen-Hilbert problems. The Riemen-Hilbert problems had tremendous success, while both in random matrices, improving various delicate asymptotic results and establishing universality theorems, but also very well beyond in different areas. And we are very lucky to have the world's leading specialists on Riemen-Hilbert problems first a dive from the Quarantine Institute in New York who will tell us some parts of the story. Welcome. Okay, first of all I want to thank the organizers for inviting me to participate in this wonderful, quite extraordinary event. Thank you. Okay, so the subject is Riemen-Hilbert problems, R.H.P., and Riemen-Hilbert problems mean different things to different communities, but I'm going to give one particular point of view on what they are. So my goal in these four lectures is to do the following. Is to say what a Riemen-Hilbert problem is, why we are interested in them, and lastly is how do we use them, how do we work with them, going on from that. Okay, these four lectures which I'm giving now are a condensation of a semester course which I gave it to NYU about two years ago. You can find it on AMS open notes. So it's a very nice thing that AMS have done. They ask different people just to put up their notes and you can find a much more extensive version of what I'm speaking about on the AMS open notes. Now there are many references which I've given. I'm not going to write them all down. They're references of two kinds. One's about Riemen-Hilbert problems specifically, and the other's about the general kind of mathematics, particularly from complex function theory that one needs to know to work effectively with Riemen-Hilbert problems. I'm not going to repeat it. You can find them all in the handouts. Now, I want to start off speaking about special functions. And what's important about them is that they give explicitly solvable models for a huge array of phenomena in mathematics and in physics and when I speak about special functions, I mean, for example, the Bessel functions, every function, and so on. Now, of course, let me just say if you haven't met up with these, there is no way that you're going to have any experience in mathematics without coming across them at some point. So a general reference, let me just mention one, is, okay, so this is a book by Abramovitz and Stegen. So it's a general reference for classical special functions. You can also find it on the NIST website in an updated free version which you can just download, so see NIST. So it's a go-to book for finding out anything you want to know about special functions. So how does it work? So let's write down the area equation. Looks like this y double prime of x equals x times y of x. And you look for a solution, so you seek the following, y of x equals an integral over some contour of e to the xs f of s ds. So you want to choose f of s in such a way that you solve this equation. And you begin y double prime of x is equal to s squared over c. And that you observe over here as the following form. If you alter x times y of x, this says x times e to the xs f ds of the contour. And this object here is equal to d by ds of e to the xs. And now you integrate by parts, assuming that you can drop off the boundary terms. And then you see, got a prime prime over here, of course. Then for these two things, for this thing over here and this thing over here to be equal, you see that you have to have that the f prime is minus s squared times f. So in other words, f must equal e to the minus one third of s cubed up to some constant. So what that means is that you can find a solution to the area equation in the following form, e to the xs minus s cubed over three ds over some suitable contour. Maybe with a constant. Okay, now why are you happy about this? Why is this good? Well, if you're an analyst, particularly, what's interesting about special function theory is you can evaluate the asymptotics. In other words, x going to plus or minus infinity. You're very interested in that. And what makes you so happy about this representation is there is a fundamental, all-important mathematical technique known as steepest descent. Descent, sometimes called stationary phase. And this technique applied to that integral allows you to compute with extraordinary precision what the asymptotic says for the solution. In fact, if you didn't have this representation, you would not, I think it's a fair statement to say, you would not ever be able to solve the asymptotic problem. So steepest descent, stationary phase, plays an absolutely central role in mathematical analysis of special functions. Now, if I take this contour, I take my constant c to be one over two pi i. And I take my contour c to be something like this. This goes off at an angle e to the i. And this is e to the minus two pi i over three. That's my contour. Then I introduce the function aix, which is one over two pi i. Integral over this contour e to the xs minus s cubed over three. Yes, and this is called the airy function. And plays a central role in many, many different problems. And you've seen it in earlier lectures, coming up in random matrix theory and different things like that. Now, by choosing the contour appropriately, you can get other solutions of the equation, like something called the bi solution. And I'm going to leave that off. But I want to give you a sense on what the standard is. When I say you can obtain precise information, I want to give you a sense exactly what one means by precision. And you find, again, you can find this in Abramovitz and Stagen or on the NIST website. If you introduce zeta equals two thirds x to the three upon two, then one has ai of x is equal to one over two root pi, x to the minus a quarter, e to the minus zeta. And this is asymptotic expansion. k running from zero to infinity, minus one to the k, ck zeta to the minus k. This is as x goes to plus infinity. And then you've got that ck is equal to the gamma function at three k plus a half upon 54k, k factorial, gamma of k plus a half. That's as x goes to plus infinity. And you also have ai at minus x, again, x going to plus infinity. So minus x goes to minus infinity, one over root pi. And you've got your x to the minus a quarter. And you've got the sine of zeta plus pi by four. Times the sum minus one to the k, zero to infinity. But c of two k, zeta to the minus two k plus cosine, minus cosine zeta plus pi by four, times the sum zero to infinity, minus one to the k, two k plus one zeta to the minus two k, minus one. Now, I'm writing this down because I want to fix the table. Now, this is what one wants and what one needs when one speaks about the pre-precision. You know every single number that you want to know about these functions. Now, what's particularly true is these formula and let me say again you obtain these formulae because you have that integral representation together with this all-powerful instrument called steepest descent. Now, these problems or these formulae solve the connection problem or sometimes it's called the scattering problem. And it says the following, that if you know the asymptotics of the solution as x goes to plus infinity, so you would know this form here, then you automatically can read off what the solution will look like at minus infinity. So it's a scattering problem because you think of x sitting over there, y double prime equals x. You think of that x as a potential, q of x times y of x. And you think you've got some wave coming in looking like the airy function and you want to know precisely what is coming up. And this solves that problem. Now, back to this, the erases are here. Now, what has emerged that was first submerged and it has re-emerged is what one calls modern special function theory. So we spoke a little bit about classical special function theory, Bessel functions, airy functions. But there's something which has emerged in the last 40, 50 years called modern special function. And at the core lie what are called the Pan-Leve functions or equations. Six of them. And just as classical special functions, airy and so on, we're extremely useful in solving the linear problems of the 19th century acoustics in electromagnetism. Now what has emerged is the modern special function theory gives you the language to describe and explain many of the modern phenomena which have arisen in non-linear science. So linear science. Mostly, that's a little bit of a simplification. And modern is non-linear. And as you've seen in the early election, many of the fundamental features of random matrix theory, for example, are expressed in terms of these Pan-Leve functions. Now, let me give an example of how they arise. Let's look at the modified quarterback, the Fritz equation. So he has a PDE, it arises in the theory of water waves, and it's played a very important role in modern non-linear science. He has the non-linearity in the equation. And then you find the following. Big pun. Okay. Find, we find as t goes to plus infinity in the region, in that region, that y of xt is equal to one over three t to the third p of x of three t to the third, plus order one over t to the two thirds. And p solves Pan-Leve two. As I said, there's six of them and I'll say more about them later. And what does it look like? p double prime x equals x times p of x plus pq. Usually there's a two in front there, right? Yeah. So it's a non-linear equation. And if you remember what the area function looks like, it's some kind of non-linearization of the area equation. So you just look at this self-similar solution. For MKDB in this particular region and you see what arises as this function. Now, such a point of view or such a computation would not be of any use according to the standards which we've written down a few minutes ago. What you want to know is as much as you know about the area function, you would now like to know about this Pan-Leve function. Can you, for example, describe the asymptotics as x goes to plus infinity and x goes to minus infinity? And can you solve the connection problem? Okay, so those are the issues. Let me give another example of how they arise. There's a problem which was mentioned. So you look at permutations, for example, two versus, for example, one. Okay, you look at the permutations. SN, permutations of N numbers. For example, if we take N equals six, and we look at the permutation, let's call it pi, is equal to four, one, three, two, six, five. Then these numbers here, one, three, six would be an increasing permutation of size three. So one, three, six is an increasing permutation or subsequence of size three. Another example would be one, three, five. That would be another example of an increasing subsequence of size three. And for a general permutation pi, you let ln pi be the length of the longest increasing subsequence. So in this particular example, L, N equals six for this pi is just three. And then you put uniform distribution on the permutations. There are n factorial of them. So each one has probability, one of n factorial. And you ask the question, what is the probability? That ln is less than or equal to n. And what does this look like as n and n go to infinity? And the answer is this. If you look at the probability that ln minus two root n over n to the sixth, this will converge as n goes to infinity. To e to the minus integral x to infinity t minus x squared, do I have this right? Of y or p of t dt, do I have it right? No, one of them just goes wrong. Sure, where am I up to over here? And again, p is exactly the same function which arises there. On the understanding, so p solves panel of A2 with the boundary condition that p of x looks like the airy function as x goes to infinity. So you see, you begin to get a sense in which the panel of A functions, which are non-linear functions, enable you to express the solution of a problem which arises. In common combinatorics, yeah. This is a very special function. It's the distribution function for Tracy Whitham, which gives you the distribution function for the largest eigenvalue of GUE matrix and that you have seen earlier on. Now, there are many problems in common combinatorics which are expressible in terms of these random matrix theory ideas, but I just want to get the idea across here that these panel of A functions are things which you want to understand. If you want to know now about this probability distribution, for example, when x goes to plus infinity, it's clear that the right-hand side is going to go to one. On the other hand, what does the distribution look like is x goes to minus infinity. Then you're going to have to know a great deal about this function, p, p of t. Okay, so now, as I say, question is this, what do I do with this here? Can we solve a connection problem? So that's the basic issue. If we know the behavior at plus infinity, can we immediately conclude and know what the asymptotic behavior at x goes to minus infinity and vice versa? Now, the difficulty one faces here is that if we knew that there was an integral representation for the pan-panel of A2 function which is similar to the integral representation for the ARI function, we would be in good business because then we can just use a classical STC PIS descent method. But there is no known, and this is where Riemann-Hilbert problems come in. Riemann-Hilbert problems gives you a way of representing these functions which is amenable to as detailed analysis as you could perform on the ARI function using its integral representation. But we do have, and this is one of the many reasons why one is interested in Riemann-Hilbert problems. And there is a non-linear steepest descent method for RHPs. So if there's nothing that you take away from these lectures, except the following, you will have gained something. That when you find that you have a representation for one of these functions, in terms of a Riemann-Hilbert problem, there is a technology there which will enable you to evaluate that problem of any precision that you need. And the non-linear steepest descent method, which is the non-linear analog, was introduced by Zinjo and myself in 1993. So we get to what is a Riemann-Hilbert problem. It involves two objects, a contour, a choir, a contour, which we'll call sigma, which is oriented. So each of the arcs of the contour, which can have points of self-intersection, has a direction associated with it. And by convention, the plus side is always on the left as you travel along the contour. So you have such an oriented contour, and you also have a function V, which is called the jump function. And V goes from the contour to the al-infinity functions. Al-infinity functions on the contour, such that V and V inverse belong to al-infinity. They are the invertible, let me put it this way, GL or KR. So you have a function defined there, a K by K matrix function, and it's in al-infinity and it's in inverse. These two, yeah, big pun, at this stage, any union of arcs, any union of arcs, finite unit of arcs, but I'll say more about that later. We say that the n bar K matrix function, m of z, solves, or is a solution, let me put, is a solution of the Riemann-Hilbert problem, sigma V.