So the purpose of this last lecture is of two types. The first one will be to finish a rigorous proof of the theorem 2 here, at least the point 2 of theorem 2 that I will comment on in a minute. After that, we will go into eigenvectors dynamics and see what we can do for eigenvectors. I will just very briefly give motivations about the study of eigenvectors of my random matrices by the end. So what you can see here on the board is a couple of theorems. The first one telling you that there is some invariance of statistics up to some time. And the second one telling you that after a shorter time, you have already the GOE statistics. Okay, so this is GOE of size n. And in particular in the bulk, what we have give the heuristics for yet on Tuesday, is that for time up to n to the minus one half minus epsilon, so almost up to n to the minus one half, the statistics after running your Dysen-Brahm motion have not changed, okay? At time t and time zero, it's basically the same up to a small error. My capital F here is just a nice function. And for the edge, you can go actually much further. Here I put n to the one over 100, but just for sake of concreteness, but you can go even a bit beyond one. And for considering theorem two, which is much more challenging to prove, you have already relaxation of your dynamics after a shorter time. So just time one over n in the bulk and n to the minus one third at the edge. So I forgot to mention the dynamics I consider, of course, are the matrix Dysen-Brahm motion, okay? Which in use is some spectral dynamics. Theorem one only relies on the matrix point of view, the fact that the matrix satisfies this equation. But theorem two requires the spectral dynamics to be proved, okay? At least the first line, which is Dysen-Brahm motion. So let me comment a little bit about the spectrum dynamics. I wrote the eigenvector evolution here. The eigenvector associated to lambda k is denoted by UK. And this is something we proved as an exercise yesterday. So everyone here knows how to prove this, right? So in particular, what you see is that eigenvectors are very unstable in directions related to eigenvalues that are very close. This is reminiscent of the fact that if these two eigenvalues coincide, you cannot even distinguish eigenvectors, okay? So when you have, for example, your basis on, say, SO3. So this is eigenvector related to lambda 1, lambda 2, lambda 3. If lambda 1 and lambda 2 are very close, then instantaneously, I'm going to rotate very, very fast in the corresponding plane, okay? And I'm going to rotate at a very slow pace, if I'm in a plane related to distant eigenvalues. So it would be important for us, and that would be the core of the class today, to understand the mixing time for these dynamics. And it's very high dimensional, when it's dimension n square. Well, this one is dimension n, so it would be harder in some sense to understand this one.  And here are some heuristics. So remember that it's a coupling argument. But I will just make it one step more rigorous today than it was on Tuesday. So the coupling argument is, on the one hand, you have your dynamics for lambda k, but you also choose y at m0, which is DOE. And the same dynamics for, and the initial condition different from your lambda. And the dynamics for y are just the same. So you do your dyk is equal to 1 over dbkk. And then by looking at the difference between y and x, you get the parabolic equation. So delta k t equal exponential t over 2 yk t satisfies the parabolic equation. Which behaves qualitatively speaking, something like partial derivative in time of your delta k t is not too far from 1 over n delta k minus 1 t plus delta k plus 1 is 2 delta k t divided by gamma k. So here I only keep the nearest neighbor interaction after my subtraction and the parabolic equation I obtained. And it's not too far from discrete laplacian divide speed up by this length. In particular, if you inject what is the difference between typical locations at the edge or in the bulk, you will find out that you get exactly what you expect and to the minus one third speed at the edge and 1 over n in the bulk. Okay, so that was the very first step of heuristics. I will now do a coupling in a slightly different manner, but it's more really speaking the same thing. Now define sk nu at time zero, which is nu at time zero to be nu yk plus 1 minus nu xk. So I do just a linear interpolation in my initial condition. That's nothing more, okay? Remember, all my eigenvalues are ordered. And you run the Dyson-Brahme motion dynamics with this initial condition. And then we will derive the nu to interpolate between both cases. So now you do your dxk of t equal negative, okay? So if I can prove that the derivative in nu at time t of this object is very small, then it will imply by integrating over nu that my value of xk and yk are very close, okay? Because of course remarked that xk nu 1 is just equal to sk 1 t. Just sk t, yk t, and xk 0 t is my lambda k. Just because my initial condition will match with y or x at the extremities of my parameter nu, and then it does the same dynamics as we wrote here for y and lambda. Now, the interpolation is at time zero, but the equation itself. So the interpolation is just at time zero. And it's not true that at time t, xk is a linear interpolation between y and lambda. It's just not true, okay? But still I will prove that it's a small number. Now you can just derive this equation in nu, okay? So you need to be careful when differentiating a differential equation in this parameter. But you can just do it, take a small increment to convince yourself you can do it, okay? So in particular, if I define, this is by notation, let's say, a k at time t to be the differentiate in nu of my xk nu 2. And I multiply by a factor exponential t over 2, it will be to get rid of the arced union back. So you just define this, then by differentiating on both sides here, you get the usual parabolic equation for this, for this guy. So the reason I prefer this interpolation to the one I was doing here by taking directly the difference is that when I was taking the difference for delta k on the denominator, I had xk minus, lambda k minus lambda l times yk minus yl, if you remember from Tuesday. And this is somehow, for algebraic reasons, it would behave better. So you have this equation, okay? So now, there is nothing new so far compared to what I was saying on Tuesday. If you prove that it has been smoothing a lot, it means that a k and a k plus 1 for any nu will be very close. And as a consequence, by integrating over nu, it means that xk and yk are very close. The gap between xk and the gap between yk are very close, okay? So we want to, so for edge universality, say, namely to prove theorem two part two, what we need is the following fact. Is that uniformly in nu between 0 and 1, your ak for t of order much greater than n to the minus 1 third, your ak nu t is actually smaller than its typical size. If it's a small o of n to the minus 2 third, then we already have universality. So in the bulk, I was talking about the gap, so I need to compare ak and ak plus 1. But at the edge, it's not about the gap. Your tracy-widam statistics is just about distribution of the relative position. So if you can prove this one, so this is what we aim at doing, right? And if you can prove this, just integrate in nu now. You integrate between 0 and 1 in nu. This estimate, what you obtain is that the difference between xk and yk, between xk and lambda k, sorry, is going to be a small o of n to the minus 2 third. Sk, so that's my y. So lambda kt is equal to yk t plus a small o of n to the minus 2 third. But this would be enough for tracy-widam. So we want to prove this equation. So the individual eigenvalues after some time, they are on the scale of the natural fluctuations. They just stick together. Okay, the gap is smaller than the typical scale of distribution. So it's enough for universality. If you practice estimating, you'll have fewer. Why nt, lambda nt, or for k plus 2 n? Oh, that's n, yeah. What I meant by k is n, here. Thank you. Actually, it's true for any edge eigenvalue of the first 10 eigenvalues, for example. So we want to understand this. And the reason I focus on the edge here is because there is a particularly simple proof here. It was in particular, particularly, I mean, very difficult to make these types of heuristics correct because your lambda ks may collide. There are some shocks, actually, even though they will never exactly collide, they may get very close together and it happens. And to deal with these shocks, actually, the key is to introduce the following observable. You define ft, let's say ft z to be, so I need an exponential t root minus t root. So from now, I just omit nu in my notation because everything I'm going to say is uniform in nu anyways. Okay, and I want to prove that for 1 fixed nu. This is small. So you define this object. It's a strange object. So this one is the derivative in nu of your xk t divided by xk t minus z. So this can be seen as a derivative of the characteristic polynomial in the parameter nu. But we will not really use this fact. So what's the idea? It's very hard to control ak, but maybe some average of the ak is going to be easier to control. So when you take average of ak, you take a sum of weights times the ak's. And the point is that the weights I'm going to use here is the imaginary part of 1 over xk minus z because we know it's a probability measure, typically. It's the imaginary part. It's a convolution of the spectral, empirical spectral measure on some scale. So we just introduce this guy and this is the sum of all k. And now it's an exercise to prove that this f satisfies a stochastic advection equation. And then it's just a one-line argument to conclude about universal. So on this one you have to trust me, but it's really just a calculation. So let me explain you a bit the idea here. If I take any average of the ak's and I try to understand the evolution of this average with time, in my evolution of the ak's there are shocks. I have this singularity. So it's very hard to control any type of equation that will emerge from an average. On the other hand, if in my average I put the evolution of xk itself, then a miraculous consolidation will occur which gives an equation with no shocks. So this equation has no shocks or no singularity. Now how should we understand this equation? We should understand that the second line is just error term and the first one is the dominant term. So st is a still-gest transform. s always stands for the still-gest transform. Thanks. So st here, that's my sum of 1 over xk t minus z with a 1 over n. That's my still-gest transform. And what we will see is that all of this is just error term. And st is a still-gest transform of somehow a Wigner matrix. So we expect that it's very close to the limiting still-gest transform of by the local law. So st is supposed to be about m of z but m of z is now is remember your m of z is minus z over 2 plus square root of z square minus 4. That's the still-gest transform of the semicircle. So we expect st of z to be about this. So that's an equation. You add z over 2 and you only get the square root of z square minus 4. And this is error term. So let's understand why this is error term. So this is error term for z in the mesoscopic scale. So everything here is in the mesoscopic scale. So my eta is of scale greater than the typical gap between eigenvalues below. So in part, think about this. Here I have, this is an object of order 1 because it's just a still-gest transform. It's just a finite formula. This is a first-order derivative. Here I take a second-order derivative. So it's a bigger, it's a bigger size typically. However, I have an extra 1 over n. So if I am on a scale n to the minus 1 plus epsilon, it would be a smaller size. Okay. And this you can also control and prove it's a smaller order. So this equation, let's call it 1. 1 is well approximated and this one is easy to justify. So it's well approximated by dft is just 1 half of the square root of z square minus 4 dz. Everyone follows at least the chain of equalities here. So what type of equation is this? It's an advection equation, meaning that think about what happens. You start with at some z and the derivative in time of the value here is equal to square root of z square minus 4. But this is typically, if you are in the bulk for example between minus 2 and 2, this is basically purely imaginary because it's at first order. So it means that it's something like i times a constant dz. If you have i times a constant dz, it's exactly the most standard type of stochastic advection equation. It means that the value at time t here is equal to the value at time 0 much higher. Okay. So I will go into more details after. So it really tells you, because you have the good sign here, it's a plus, it tells you that it's regularizing. This ft at time t for very small scales is going to be given by f0 at larger scales. So, but it happens that this equation can be just solved exactly because the characteristics, so this type of advection equation is solved by the method of characteristics. But for this factor here, there is an explicit formula for the characteristics. So the derivative in time of ft is equal to, thank you. So this is an advection equation with solutions of following one. ft at z is equal to f0 at a deterministic point zt, where zt is the following. Where's my zt? Okay, so it's a bit formula, but not so bad. And so this you can just check. Okay. Of course, it's not for any function here that you would be able to find the characteristics, but here you have a formula. So what does it look like? What do the characteristics look like? So as I told you in the bulk, if you start at z0 somewhere here, this is minus 2, 2. If you start here, what you will find out is that the characteristics take you almost straight up. And so this is, sorry, this is z0. But when you add the edge, it's a bit different. You will go on a transverse all the way. You will actually, at first order, it will be like a parabola like this. So z0 is just my initial point here on any, the mesoscopic scale you prefer, any of zt. So in particular, what it means is that the average of the uk's, of the ak's that I introduce in this observable, after a long time, it's just like taking the average on a bigger scale before. Okay. So it's an automatic call-smoothing effect. It's just that this is a good observable that makes it completely transparent. So now let's understand in particular what happens at the edge in terms of scales. So if z0, let me take z0 to be just one point on the mesoscopic scale at the edge, which is just 2 plus i times n to the minus 2 third. Okay. Remember that my gap between two again values at the edge is n to the minus 2 third. So I just consider z scale. Actually, I need to be in a mesoscopic scale for my theorem to apply. So let's, let's add some epsilon. It's fine. And here I can apply any types of local laws and my approximation of this by the deterministic advection equation is correct. And now what you will find out is that zT is the following type. Then zT, if you just tailor-expand this, it is at first order. So it's going to be 2 plus t squared plus, so that's my zT minus z0. Plus i n to the minus 1 third at second order in t. That's an expansion from here. So what it means is really that you have this parabolic shape at the edge. Yes, because I removed the zero. Thank you. Fortunately, you're around. So what it means is that after time t, what I observe here is going to be like what I observe when I'm actually quite far away from the spectrum. It's far from itself. But when I am far away here, the average of what I see just below is basically zero. It's very small. So it means that my AT, the quantity I was interested in from the start here becomes a very small naturally by this equation. So this implies, this implies that for t, much greater than n to the minus 1 third. If you're much greater than n to the minus 1 third, then this term is going to win over this one. So it means that you are really far away from the spectrum. You don't even see it anymore in the average I consider. And as a consequence, what you end up with is the fact that f t of z, zero, is equal to f zero of z t. This we already know at first order. But we now know thanks to this that this is actually you can just check. It's a small o of n to the minus 2 third. So many calculations have to be checked here. I'm sorry about it. I don't have time to put full details about them. But I just want you to remember that if you consider this type of observable, then all types of shocks you may be annoyed by, they just disappear. And you have an automatical smoothing effect which gives you some type of regularization for the things you are interested in. Any questions about this? The cancellation between the shocks is just because you compare the equation for a k and for x k. Yeah, so apply it to your f t of z. Just apply it to, of course, you have to look at the the usual derivative in time of a k because a k is smoothing time. A k has no martingale term. So the evolution of this guy makes some shocks appear.  Is the proof clear? So I mentioned it at the edge because it's simpler at the edge than in the bulk. You just have this characteristic taking you away, so it becomes small and that's it. And also I believe that for the purpose of proving tracy-widam in a model you like, no matter which model you like, introducing this just makes it very, very transparent. All right, so if you don't have more questions, we will go to eigenvectors. Okay, so what do we want to prove for eigenvectors? You have let's say h, so yeah, h-vigner matrix. And remember that my notation for eigenvectors, they are here. This is the u k's. And for any sequence q, q n, so q n is in our n and it's normalized. And it's a deterministic sequence of unit vectors. You have u k projected on q n. So this u k depends on n implicitly. And you multiply by square root n. This converges to a standard couch. So this is the first part of the statement. So what do you expect for your eigenvectors? That they are uniform on the sphere somehow, right? Because this is a case for GOE. For GOE it's completely transparent because the model itself was designed so that you have invariance by orthogonal conjugacy. Here it's something you probably expect. And you can choose your favorite direction. Of course you have a representation of the uniform measure on the sphere as a sequence of independent Gaussians that are normalized. So then this theorem for the uniform measure on the sphere would be obvious. So we want to understand this here. So that's the first thing. And the second one is the fact that the eigenvectors are flat. And this is something that we may call a probabilistic version of quantum unique algorithm. I mean that for any i, so you pick your favorite interval deterministic in one n. And you look at the mass, the l2 mass given by your eigenvector on i. So you look at the sum of uk of alpha square on i, all coordinates in i. And you wonder how much does it deviate from what I expect. If it was completely flat, all of this uk alpha square would be of order 1 over n. Okay, because each coordinate has size 1 over square root n. So you want to compare it with what it is when you remove, when you're recenter. Okay, and you consider the probability that this object is large or small or things like this. And depending on the i, so you want to prove that the probability that it's large goes to 0 as n goes to infinity. So this is for any positive epsilon. But if i is a macroscopic proportion of the coordinates, then this is a good, some good scale because it's an object of order 1 typically. And you compare, we want to prove that it's a small probability to be greater than any constant. But you may choose i which is just a small fraction of the coordinates. It also works, you just need to multiply by a factor n over cardinality of i. So you are interested in the probability that this quantity is large. Now if i is just one coordinate, this is a random variable divided by n. I multiply by n, so it's just a random variable all together. So if there is no chance, it's true. So I need to add just 1 over the cardinality of i. Okay, and now when I was correct. So I will not come back to, I will not mention the origins of this name. It was introduced by Ronik and Sarnak in the context of manifolds, some context very close to the Boygasian and Anish Mith conjecture I mentioned. For generic manifolds, you expect that in the semi-classical limits, the eigen states occupy space very well in terms of their L2 norm. This is a modest probabilistic analog here. How do you prove these types of things based on the dynamics? So theorem one, you're just going to work exactly the same way, proving that you have invariance of your eigenvectors up to some time. Thanks to the local law as an input, which is a fundamental object here. You can just prove theorem one in the same way. Remember that for theorem one, we were just using the fact that the entries of the Green function were bounded. So we were not using dynamics at all for theorem one. But we want to understand theorem two, the analog of theorem two, exactly for eigenvectors now. This is true for all k. So you choose a deterministic, so things are a bit non-precise here, thank you. So your i is a deterministic subset of one n, which may vary with n. But it's a deterministic one. And k, same thing. It's a sequence of numbers, all of them between one and n, but the deterministic one. And then it's correct. The probability that this is true for one k or for all of them together? This one? Yes. This one is for one k. It's a challenge to prove it for more, for all together, and it's a challenge to have the absolute optimal error term here. As I mentioned by the end of the lecture, this is an important thing actually. If you don't prove it for all ks, then I think quantum munic is unique. Unique munic means that all eigenfunctions are flat. It means that you're, okay, yeah. So let's call it quantum munic, quantum ergodicity if you want, but quantum ergodicity on the other hand is not that either, because it's a bigger average. But that's perfectly fine. The first part, you take k also, the domestic sequence also. Yes, yes. So what you have is... So theorem one is still true for the observable, which is, for example, a function of this, or a function of square root n. So with the same scales. It's going to be correct. So we don't really care about theorem one here. For theorem two, we want to analyze these dynamics and understand the mix. So you see, as I mentioned at the beginning of the class, what is complicated is that some directions are going to move very, very fast, some others will not move fast. How does it all together combine to get uniform on the orthogonal group? So that's the question. So I would love to have a proof based on arguments like this, just identifying this as a diffusion with a very good convexity type of constant, but we don't have such a proof. And what you can do is to introduce again a set of observables which behave pretty well under these dynamics. So theorem two is just going to be true on the same scales. So is still true for the same f. And here is a sketch of proof. So for eigenvalues, we were proceeding by coupling. And here I have no idea how to couple. This is dimension n square. Since I could not find any way to make it parabolic. And on the other hand, what we want to prove convergence to is much simpler. It's just a Gaussian. So maybe just by taking moments, we can prove that they converge to the moments of a Gaussian. So for this, let's look at the very first moment. Let's try to understand where it goes to zero and then the second one. And let's try to understand the heuristics here. Okay, first moment. So first, I just make my life easier. I will not take any qn. I will just take the first coordinate of the canonical basis. Okay? So take qn, go to one. Let's prove it in that case. We'll be convincing enough. Okay? So what do we get here? I just project on E1. This is projected on E1. This is projected on E1. So it's an equation involving all projections of my injectors on E1. And if I take expectation, this term goes away. So I just end up with an exponential decay to zero. So the d over dt of the expectation of my ukt. And I condition on my path lambda from zero to infinity. I will comment a little bit more on this. What you obtain is that this is minus one over n, minus one over two n, sum of one over lambda k minus lambda l square, l equal to one over two n, l different from k, sorry. Multiply by the same thing. So the reason I can do this conditioning and take my lambda k out is because these b's here, and that's very important fact, are completely independent. The noise driving the evolution of eigenvalues is independent of the noise driving the ones of eigenvectors. So you can see these couple dynamics. And that's a key point here. It's actually a very nice fact for these dynamics. Imagine you are first given the eigenvalue path and on the top of it, conditionally on this eigenvalue path, you run the eigenvector dynamics. These dynamics that we proved yesterday were first proved in the context of covariance matrices by Marie-François Brue in the 80s. And it's even simpler to prove it for the Hermitian case. I think they require even more attention. Those are really interesting. So now what is the size of this here?  So this will be of order n. And this will be n to the 1-th third on the edge. So this is like n in the bulk and n to the 1-th third on the edge. Okay. So obviously here it's extremely simple to prove that the expectation goes to what you want on the good time scales and at an exponential rate. It's a very fast, very strong convergence. But this is a bit too easy. Let's look at the second moment and let's give it a name. Let's call it gt of k, the expectation of my uk of 1 square conditionally on my path. Now it's an itto formula on the square and you will find out that you get the exact same parabolic equation as the one we are used to. Okay. So again, exercise. So the uk of 1 after projecting on u1, this equation of dimension n square because the equation of dimension n takes a square. You apply it to the square, take expectation to make the martingale terms go away and this is what you're putting. But for this equation, we just know that it regularizes on these time scales too. We just, this is basically everything I told you about eigenvalues. That it is a very same equation. So regularization on the same scale. So we know the first and second moment converges to the Gaussian ones but this is not very surprising because if you believe in some symmetry between the uk's, of course all expectations because it's the same for uk square in the limit because it's normalized in l2. What's more difficult to understand is the higher moments. If you take uk 1 to the 4, for example, then it will not be true that the equation becomes autonomous in these observables. The uk 1 to the 4 will also include some mixed terms like uk 1 to the square times ul 1 to the square. So you need to enlarge your space here to get the good point of view. So to enlarge your state space. So the good way to do this is as follows. So it's really a random work in a random environment perspective here. It's the random environment is given by your eigenvalues trajectory which is random and on the top of it you run a random work. So for general moment, here is how it goes. I need to first talk about something completely different. You look at configurations on the set of n points. So you start at 1, 2, 3, at 2n and you look at some configuration let's call it eta in the particle systems point of view that's a traditional way. So let's say for example they have three particles at i, 2 at i and 2 and 1 at side j. And this is one configuration eta of my three particles. I can decide. So these are indistinguishable and I just decided about the configuration how to put these three somewhere in between 1 and 2 and 1 and n. So this is my eta and here I have three particles. So this number of particles let's call it p. And to this eta I associate an observable gt of eta which will be the expectation of a mixed moments depending on the position normalized by the corresponding outend moments. So you take gt of eta to be the expectation of the product of uk product over all k uk1 to the power 2 eta i 2 eta k. And this is an expectation conditionally on the trajectory again where my eta k denotes the number of particles of eta at side k. And I normalize by what I expect it to be in the limit. I expect these guys to be independent gaussians, right? Each coordinates correspond to any independent gaussians. So I just normalize by the corresponding thing for gaussians. So let's normalize with a square root n. These are independent standard gaussians in this n case. And now the good algebraic fact is that the gt of eta satisfies the power brick equation just as the space is not just one particle it's not just k it's not a function of k it's just that these particles will move. Then it's a fact and please make sure you know your eta for proving it. The fact is that the derivative in time of f t at side eta over gt at side eta this is basically a sum of my gt at the side eta ij where eta ij means I take a particle at side i and I bring it to j minus that I will gt of eta divided by lambda k, lambda i minus lambda j. And I have a factor here which is twice the number of particles at i times one plus twice. That's sum overall. Is this readable? So eta ij so remember eta k is a number of eta subscript k is the number of particles at k eta superscript ij is a configuration obtained after taking a particle at i bringing it to j. Of course you cannot bring a particle from i to j if there is no particle at i but this is fine because I have a factor two eta i here. So anyways it's an equation and again these guys are positive so it's a parabolic type so all of them will become equal. So in the in the large n limit and large time limit all of these observables become equal which means that the the moments converge to converge to the Gaussian ones. Once you know they are equal it's easy to justify that they are equal to one actually. Okay we're just a normalization and if they are equal to one because of the ratio is with the Gaussians you just have whatever you expected. Okay I think the message here about the two examples I gave you for one good observable observable for edge universality and one good one for bulk universality is for for proving that you have this statistics go down or or or Tracy Whidham or Gaussian QE and so on. You cannot rely on any explicit integrability obviously but you can rely on somehow integrability of the dynamics. Your your exhibit observables that satisfy nice dynamics. Okay so are there questions about about this? I just gave the main ideas of course technically from there proving that it becomes equal it's it's a technical idea. This is true for any fixed eta. So the equation is true for any fixed eta.  Okay and they will jump to to converge to the equilibrium measure through these dynamics and the rates depends on the inverse of the distance square between the eigenvalues. So a particle will be very likely to jump to its nearest neighbor and not very likely to go far on one just by one jump. And when you say they jump you mean that GT you interpret GT as the density of particles? No what I mean by they jump is that this is a generator of a process on this on the space of all particles eta and this process corresponds to jumps. Okay it means that GT is a is a sort of a density density function and you look at the density or the probability of being at the at the configuration eta? Oh yeah okay. So the GTs will all will all become become flat. I mean yes you can you can consider it as just what you said but if you start with a Dirac for example it would just completely split out. This kind of the analysis of this equation can be done only if you know the equilibrium measure pretty well because you need tools like Dirichlet forms and so on. You need to know the equilibrium measure to to talk about Dirichlet form use reversibility things like this. So you need to have an explicit equilibrium measure for this. It happens that if you consider the GOE the GUE dynamics the equilibrium measure on the space of configuration eta is just the uniform measure. Any eta is as likely as the other. It's not obvious from the equation but it's true. But if you consider GOE it's a slightly different I mean this is GOE GUE is a bit simpler and you get equilibrium but if you consider GOE it's a more complicated equilibrium measure but still explicit. So once again it's a weird fact that GUE is simpler than GOE from a calculation perspective. But I don't have much time to just write what the equilibrium measure is but it's it's not a complicated one even for GOE. What is the equilibrium measure? Is it given by gt of eta or t of infinity? No no no. No let me So imagine you start with the with the configuration eta at m0 eta at m0 and then you put a you put a clock on each particle of eta which will ring at a rate and this clock will ring for example for this link you put a clock on each link between one particle and empty sites and the clock will ring at this rate and when the clock rings the particle jumps. Okay so that's what that's what happens. So your eta is zero it's it's eta itself will not converge to something but the probability distribution of eta will converge to the equilibrium measure. If it's clear maybe let's let's go to motivations for eigenvectors study. Okay so is this clear? So I hear what you need to do is to and about the spectral gap uniformly the lambda So you cannot redo that. There is no you know the problem is again that some directions are good some are bad some lambda keys are closed then mix very fast but when it's far further so what you what we are using technically speaking here is the maximum principle. You take the maximum over all configurations so you pick the configuration in eta so that zt of eta is maximum and we prove that the derivative in time goes at least at some rate by maximum principle and for this as an input we need the local law from the very first class and so it's a grand val argument and it proves that gt of eta goes very fast to one so it's a more classical PD argument here any other question? So let me tell you a little bit about the the problem of going non-min fields okay and it will be related to to a question about how strong is our quantum megacity is it is it unique or not and things like this so let's consider band matrices but before going into that I just want to mention this is far from a closed subject in particular what we prove here is that some observables which are these projections on the eigenvectors contrast to equilibrium very fast we don't know that the measure itself is close to r like in total variation okay the total measure of the group being close to r in total variation we just not know it it's it's way too strong as a if you find an argument for this it would be very interesting okay so it's a it's a way to make linked with what was last week general mixing type arguments and so on but here we are in dimension divergent so so for band matrices so so it's good I still have these statements here now here is a classical problem which is also related to the so-called understand transition that was probably mentioned by Simone last week you now consider in don't don't consider a vigner matrix but you consider a matrix with entries being only close to the diagonal so in other words you you take some bandwidth here which will be of order actually for convenience I will call it 4w-1 and where my matrix is still of time size n by n and you wonder about what are the special properties of this matrix where my hij's are independent of finite moments of order orders and but now if w is very small compared to n it's basically a sparse matrix not only a sparse matrix but a sparse matrix where you impose the geometry so we have the expectation of hij square here is 1 over let's take it 1 over 4w-1 this way the sum of variances is constant on each line so we have the semi-circle distribution is the limit in the same scale we can prove so you still have a semi-circle distribution between minus 2 and 2 moments are finite of order orders and you wonder what happens about eigenvectors for example or eigenvalue statistics for this matrix so the extreme cases we know when it's w of size complete size so that it's a full matrix you end up with obviously the GOE and the delocalization of eigenvectors this is what we just proved for Wigner matrices but if the matrix is completely diagonal then of course the eigenvectors are completely localized and the statistics typically if your distribution is smooth for the entries the statistics would be Poisson okay so is there a phase transition so it's really a question of transition here okay so the conjecture by Fyodorov and Myelin and it was there were numerics in the 80s and 90s about it is that the transition occurs exactly for W of order square root n W is much smaller than square root n but you have this Poisson plus localization and for W much greater you have go down in the bulk say plus delocalization so I must say that at the edge things are pretty well understood thanks to work of Soudin and the reason the edge can be done is that the moment method applies no matter that the eigenvalue that matrix is full or not you can take the moments and if you do very smart combinatoric tricks like Soudin did you can calculate the moments of even for the band matrix and in the large moments give you access to the greatest eigenvalue it just singles it out in the bulk there is no such argument and so this is a dimension one band matrix but this is really not to be it's good to keep in mind that it's not restricted to the analog with a random Schrodinger in dimension one because you could look at the band matrix in dimension two namely you take a box in Z2 imagine you make it periodic boundary condition and you decide that each vertex is going to interact not only with its nearest neighbor but with neighbors up to distance w and this is n so this this works in any dimension and it's a difficult problem to prove that for this model in dimension two actually for any w of type n to the epsilon you are supposed to have the Godin delocalization regime so there is no transition for one for a polynomial n this transition is supposed to occur for a logarithmic n in dimension two but let's stick to dimension one so of course this model was introduced a supposedly an easier model for the understand transition so what is known about this on the Poisson and localization side there is a work of Schenker who proves that w up to n to the 1 8 you have indeed delocalization and on this so Schenker okay for w of order up to n to the 1 8 and on the delocalization side there are basically two approaches the first approach is by supersymmetry so this is a term you heard in Christoph's talk you heard about in Christoph's talk already supersymmetry and Bérezain integrals so what the type of results that I known is that if your matrix has a specific variance profile and the entries are Gaussian then you have a representation of the Green function in terms of very high dimensional integrals I don't understand everything about and you can perform some rigorous asymptotic analysis and it was proved by Charbina that if w is greater than a small constant times n and the entries are Gaussian then indeed you have good n and delocalization so this is the work of Charbina so Tatiana Charbina and Tatiana and Maria Charbina have proved that actually you really have the transition at square root n but not for the local statistics for the characteristic polynomials in some sense which is for technical reasons an easier thing to handle so for the local statistics it's just not known for any model that there is a transition with the supersymmetry method there is also work of Erdosch of Bauer and Erdosch who prove delocalization in some regime I think w up to n to the 6 over 7 something like this but this is always in the sense of either Gaussian entries or entries that match Gaussian random variables up to fourth moment because then you can do some moment matching and what I want to just mention in the next 15 minutes is how is the QUE picture or QE picture and the dynamics can help also to understand this problem in the delocalization side and this allows us to do something quite modest w greater than just a small constant times n so the constant I could actually be like one over double log n if you want but we are not in the polynomial scales and but the entries don't have to be Gaussian just any entries is fine okay so it's a it's a famous problem this conjecture was was given by Fyodorov and Mierlin and what fails here so remember the global approach we have here so the second point tells you that after some time you already have relaxation actually theorem 2 may work just you start with your Wigner spectrum with not Wigner with your band matrix spectrum you run the dynamics a little bit actually proving that after a little bit of time you you have a GOE is fine not easy but fine but theorem 1 has absolutely no chance to be true remember that theorem 1 relied on the fact that the variance was constant along the dynamics if you start with the matrix where the geometry is imposed it's completely wrong so for the proof theorem 1 fails so you need to do something different so what can you do so what I just want to explain in the remaining time is the technique of mean field reduction I think it's interesting to remember I just say as a general technique if you have a model where some geometry is imposed you can maybe somehow come back to a mean field problem so here is one way to think about it you have your matrix so let's let's call this matrix H let's call it AB V star B and D where I'm going to take this block of size 2W and this one of size N minus 2W so one way to think about it is that my A is this one and trace the complete one so now you're going to write down what the eigenvalue equation means by going to short complement in terms of A so you have your AB star BD times let's say you write your eigenvector WK and PK and this eigenvector is just the block decomposition for UK typical eigenvector for H and you just assume that this is lambda K times WK so this implies that you have the following fact some matrix I'm going to define Q evaluated at lambda K at WK is equal to lambda K and this Q, QE is A, it's just a short complement it's A minus B star D minus E to the minus 1 B it's actually a B so you define this one for a fixed parameter E this is your matrix now when you do the short complement E becomes random itself it's lambda K so it's a complicated equation because the dependence in lambda K you cannot say I have a fixed matrix and I look for the eigenvalues because the matrix itself depends on the eigenvalue but let's make a graph of the eigenvalues of QE as a function of E and then what we are interested in will be the intersection with the x equal y axis so here is some big graph I want to do here is the parameter E and for each fixed E I want to put the eigenvalues of QE so I will have two W points for each parameter E and they will vary in a continuous way typically in E except when I cross an eigenvalue of D itself because I have a singularity so let's single out the eigenvalues of D here which will be the singularities so let's call them mu 1 mu 2 mu so it's N minus W minus 2 W so this here are the eigenvalues of D and then my eigenvalues of QE will depend on E and I know that I will have some vertical abscissa at each mu i so typically for one given E I will have my two W eigenvalues somewhere here they will evolve and this one will just drop at the first mu 1 this one will drop at the first mu 2 at the second mu and so on but when this one drops another one appears here okay is this kind of graph so you so you get a graph of this type and then some others appear they may drop again anyways it's a complete mess if you want it's fine okay it is something like this and but the eigenvalues you are interested in is is going to be the intersection because Q at lambda needs to have eigenvalue lambda so you are interested in these numbers here now how can you really understand this so let's let's make a zooming so so the first time we consider these graphs it was not clear at all whether there is some structure say but it's very simple you if you zoom in you take n extremely large and you zoom around some one given axis and what you will observe after this zooming is that the the blue lines are basically parallel in the limit they they they never intersect there is a the fact that they never intersect it's qualitatively easy to to proof but but then it really looks like they are parallel not necessarily 45 degrees this way but just parallel if is a really our what it means is that for one given e say here for one given e you consider the eigenvalues of q e but now q e is a matrix of type some some some complicated object plus a mean field and a is independent of this one so for fixed e it's it's a one deterministic matrix plus a mean field so it's a mean field matrix and the techniques I just mentioned in this class they apply there is a lot of work but they apply okay so here what it means is that we actually know that for fixed e the statistics of these gaps are g o e but now if the line is if the lines really are parallel you just make a projection and it means that this one will also be g o e okay so that's how you want to proceed so of course I'm just pushing the problem further here I want to prove the lines are parallel so how do you prove the pros the lines are parallel and it's it's the fact that these slopes are exactly related to quantum unique type Ergolisity types of problems so and just one formula to write what is so let's call let's call these slopes c 1 c 2 maybe for fixed e you can you can order your your eigenvalues and your these are just some lines depending on e and one formula is that the derivative in e of your c um k of e is basically 1 minus 1 over the sum between 1 and 2 w of your uk that's the k so it says that's your k uk of alpha square and alpha equal to 1 so it's it's not completely true you need to put a small perturbation but whose effect is negligible but this is really related to this object so you look at your original matrix it has a negative vector remember your h has a negative vector uk the big matrix uk was decomposed in blocks but you may wonder about what is the mass given by the first two w coordinates so if this mass happens to be very close to a deterministic constant then this is independent of k and the slopes are the same so it's independent of your k so imagine you have the strongest form of q e statement you want okay then this mass is nothing but 2 w over n it's a it's a it's a constant and it does not depend on k these are just the components of w uk yes yes yes so the question is a relative mass of w k compared to p k um so um so so now you you understand why knowing q u e for the band matrix implies somehow university so but again i'm just pushing the problem one step further i want to prove q u e for my band matrix but this is an easier problem because um imagine that for this mean field type of matrices you know that the eigenvectors are flat you you know that q u e is true for each one such matrix eigenvectors give have a well-spread out mass then you can do some patching namely your so you have your band and you what what we just did is to look at this short complement related to this box and we know that the mass of the eigenvector is what you expect say but now you can do it for another box say this one and you know that the mass of the eigenvector here in this box is going to be what you expect okay but because these are after a short complement these are mean field problems so you can actually prove q u e okay and you do it for another box so if each of these boxes the corresponding coordinates of your eigenvectors so the l2 mass here is going to be well-spread out the l2 mass here as well and so on then you can just patch and it means that the whole the whole vector has the mass you expect in any smaller box and so i think the moral of the story is that the l2 mass of eigenvectors is a more extensive quantity so that you can patch and after learning something about eigenvectors you can deduce universality thanks to the fact that the lines are parallel okay so that's the idea how far can you go so so far we cannot go very far we can go to here because there are technical issues everywhere but let's come back to your question if you knew if we knew the very strongest form of q u e for these eigenvectors for example if the basis was really hard in the sense of if for any such matrix q e so here's a problem for if for any such matrix u e the eigen basis was a very very close to hard okay say in total variation optimal rates and so on then we could push very far here and and it actually gives heuristics for the square root n transition because lines keep parallel exactly up to when you put all indices together with the optimal q u e up to square root n you you can still out that the the parallelism keeps enough and in some sense it's also give heuristics for the Poisson regime because these lines oscillate too much after when you go beyond square root n but this is this is a very hard very hard problem but but to make progress here it would be wonderful if someone could find a way to use mixing type arguments for the dynamics I consider for eigenvectors okay so that we get a stronger stronger form of q u e all right so that's the end of this lecture and I just want you to remember that this dynamic thing which looks a bit artificial in some sense for the types of problems we we consider it happens to be not so difficult you have you had this zero one and two which summarize the general idea but oh they're not here anymore but but there is still a lot of room for improvement in the technical ideas okay so that's it just want to make sure the dynamic for the dynamic resist do you run also run the dynamic for the zeros or no so we never run the dynamics here with the starting point we first perform a short complement formula you obtain matrices of this type and on this one you run the dynamics because on this one you have a chance to keep variance constant so that's the idea of mean field reduction you first reduce your problem and then you run the usual things the usual types of arguments and but if you run dynamics here directly there are two natural choices if you run it everywhere then the eigenvector and eigenvalue equation is integrable we have a very nice equation but the theorem one will fail because variance is not constant on the other hand if you only run the dynamics on the basis where there is initially randomness the dynamics are not integrable anymore it's a very it becomes a very complicated dynamics involving eigenvalues and eigenvectors together which I have no idea how to analyze question here when you show complement formula there's the the fact that a b and d are independent very important yes it's completely fundamental because I can see this matrix as I have a deterministic one plus something independent I'm adding it's um I mean maybe you can relax a little bit this hypothesis but the heart of the argument it is important yes yeah so we are interested in local things right we are looking at what happens in a box like this and we we want to prove universities so we are interested in what happens around somewhere where there is an intersection and we can once the box is centered around an intersection you can you there is an argument which says there is no singularity in that very small box and here you say that you use qe for this partial partial eigenstates but you need all of them together yes all the equations so this is why um oh no no you need you need qe for one uk but for different boxes one uk but different boxes and and you don't need all your case if you want universality because you if you want universality you just need to know that two lines are parallel so two uk's will be enough and but what you what you point at the fact that the strongest forms of qe are very important it's absolutely clear here


I must say that in this age of the celebrity, I was a wee bit surprised and maybe a wee bit flattened to actually be asked to come and take part in this ceremony today. But I must say I'm very pleased to be here because I was brought up in Aberlawer and I have a lot of very pleasant memories playing along the Riverbank here. And especially by the bridge here because we knew it was a swing bridge. And of course Davy Crookshank did sit at the other side over there and collect it all. And for a few extra pennies he would give you a haircut. And as I said I started fishing here and my first contact with the salmon was just below the bridge here. And I was fishing with a rod made up of a lanswood butt, a green heart centrepiece and a hazel tip. The hazel was cut from a tree just above the football pitch here. It was just whittled the splice with my knife. It was tied on with a bit of string. The eyes were made from the end of safety pins. And I was fishing with what we called a penny line, just a piece of string dyed brown. It was just a discarded bass that I found in the river here. I remember I was wading out on my bare feet in short breaks, trying hard to reach a fish, rising just by the stones down here. And eventually the fish took it. First run it broke the rotten cast in the front. And I can remember pretty well sitting by the bank here with a tear in my eye. Now I don't think really there's anything better than being in the river having a cast. To go down there and to be there on your own, surrounded by nature here in this beautiful countryside, and just be there on your own with your own thoughts is one of the best things you can possibly do. But if sometime in the future you do manage to hook into lively large salmon in the river here. And after a struggle or maybe an epic battle, you manage to haul them back up onto the chingle. Bend down carefully, remove the hook out of its jaw, give it a gentle part in the head, and slip it back into the river, because you never know it might just be me. Everybody, every one of you will have a very successful season, and you'll have as many happier memories as I have of the River Scroop.


I am Tom Schwedhelm, Mayor of the City of Santa Rosa. I want to start by thanking everyone who is doing their part to stop the spread of the coronavirus and save lives. I've noticed so many in our community who are embracing the new health order to wear face coverings when leaving home for essential purposes. And true to the spirit of this community, I've seen some very creative face coverings out there. The purpose of this video is to outline some of the specifics of the new face order and answer some common questions that we are hearing. The order was issued by the Sonoma County Health Officer and is in effect now and will remain in place for the foreseeable future. It requires that everyone cover their face before entering any building or enclosed open space other than their home. You must also cover your face when you're outdoors, if you can't stay at least six feet away from other people. It's okay to exercise in your neighborhood without covering your face, but you must stay at least six feet away from others. There are a couple of exceptions for people who should not wear face coverings. These include children under the age of two, anyone who has trouble breathing, and anyone who is unable to remove the face covering without assistance. If you're wondering about the term face covering, and if it's the same as a mask, the answer is they are similar, but different.  You can make it from household items like scars, bandanas, or t-shirts. Just make sure it includes multiple layers of fabric. Mass, including surgical and N95s, are for use by healthcare workers. Members of the public should not buy these. However, you may use them if you already have them at home. What's the purpose of a face covering? Simply put, it reduces the risk of spreading the virus if you have it. Not everyone with COVID-19 knows they are sick, so we all need to cover our faces so we don't unknowingly affect others. Keep in mind that wearing a face covering is in addition to all other requirements of the COVID-19 health orders. We all still need to stay home other than for essential purposes as outlined in the shelter in place order and maintain social distancing and safe hygiene. Speaking of hygiene, there are a few important things to do to properly use and maintain your face covering.  Make sure it covers your nose and mouth and that it fits properly so you don't have to keep adjusting it. Each person should have their own face covering and it should be washed at least once a day. Throw it out if it gets stretched or damaged. These times are very difficult but our community's efforts are making a difference. Please continue to follow the direction of health experts by staying home only going out for essential reasons, maintaining your social distance and when you do leave home wearing your face covering. By doing this together, we are saving lives. Thank you and be well. For more information, visit srcity.org slash prevent the spread or call 2-1-1. Watch local coronavirus updates from the city of Santa Rosa on Comcast Channel 28 and AT&T Uverse Channel 99.


So I'm here with my niece Zaria, and she is a big fan of the snack who unboxing episodes So I wanted to let her try a few of the most recent Japanese snacks Are you excited to try snacks?  So this one is a pastry, but it's a baked potato pastry If you like it, you can probably have the whole thing honestly Um, so have you ever had a baked potato? I don't think so Have you ever had potatoes kind of like just the potato with skin on it? No, well, this is what it kind of tastes like, but it's a little sweet So go ahead and try it and see what you think What do you think? This isn't looking enthusiastic about it What do you think? Do you like it? Do you not like it? What does it taste like to you? It's very starchy. Yeah, she's gonna need some water Don't like it Okay, I have a feeling maybe you wouldn't like it, but We're trying we'll let other people try some of it later. You want to let's do this mochi now This is a gold leaf mochi. So there's actually there's actually pieces of gold in there. Do you see that? Little pieces of gold. Do we eat them? Yes.   It's like you're gummy Yum, you like it more More of the mochi. Does it taste like a marshmallow to you? Yeah, that's what it kind of tasted like to me. Can you taste the gold? You can eat the whole thing then Not right now though. After try this one and then and then you can eat the rest of it, okay? Couldn't taste the gold though I couldn't either because I don't know what gold tastes like. Yeah Like you know, I don't know. It's a good good point. I don't really know what gold would taste like either We don't know what gold tastes like because we're not Japanese because we don't live in Japanese We don't live in Japanese. Yeah, because we're not Japanese people Okay, so the last little snack. This is a um brown sugar cracker with I believe it's like milk sugar on the outside As sugary So I thought it kind of tasted like a donut. Does it taste like a donut to you? Yeah Yeah, and you like donuts I guess you're just gonna can we share a little bit with a few other people or this is all for you No, we need to share. Okay, we're gonna share. Okay. So what did you think of all the snacks? Are you? Mainly likes too All right.  She was trying to share with you. Oh, you're trying to share with me Thank you. Thank you very much


I'd like to thank Robert for the introduction and I'm not imagining a thousand people somewhere else watching this. I'm imagining a much smaller number. I'd like to acknowledge the traditional owners of the land with meaning on the number of people. Don't give me a title, I kind of got to do what I thought was a big thing. I left this page blank for a week because I couldn't think of a cool catchy title and this is what I came up with. The titles I came up with, apart from that, were things like, well they had the vibe of the thing in it. So I can't really use vibe in the title page. I don't know how to use the technology. How do I change the screen? You're still trying to say next? Next. I'm giving you a Jack's perspective on this. It's not the only Jack's perspective. Jack's is a broad portfolio and there are many people with different ideas and perspectives. So how do you tell people about Government Block? The approach a lot of people take is to write to ministers to come to the director, write to a director general, come to a community consultation. And people come in and they say, we have the best idea ever. Really, really. It's not a light bulb, it's a fluorescent light bulb. It is so good, it is awesome. If we're really lucky they'll say, and what's more it's so good and we believe in it so much we actually grafted the legislation forward. So we don't always give effect to those kinds of things straight away really this conversation is around why we don't do that, why we might not do that and to give you some of the perspectives that we have on things. So I guess I have to start off by saying we value the community's views and part of the reasons for that. Apart from the fact that there is value that you have good ideas that we don't have a monopoly on good ideas. There's some other reasons around the size of the ACT government that lead themselves to that. We no longer have an expert per issue. Certainly when I started in the public service last millennium we seem to have a much more depth and there would be so two of my areas were actually two different branches and there were people there who had spent their whole lives on a particular topic area. We just don't have that anymore so we're really spreading much thinner next. So one of the areas I'm in charge of is copyright which is why you're getting drawn pictures. So things to remember about the context is we are an island in the middle of New South Wales so things that New South Wales does impact on us. People go from living in New South Wales and work here or live here and get working in New South Wales so we try to be consistent, that's important. And really when I go to my minister or one of my ministers and I say there's this idea one of the first questions I get is what does New South Wales do? And quite often the second question is what does the Northern Territory of Tasmania do? And the reason we get that is the second question is there are the small jurisdictions so there are some things that we wouldn't do in the ACT because we don't have the population. New South Wales has some really excellent schemes and you kill for them but we don't have the population density to support those sorts of things. The other thing, I couldn't draw the whole of Australia but you could imagine it. We are an island but we are part of Australia, we recognise that. So that has a couple of implications. One is harmonised legislation. Australia has regular national meetings where legislative proposals will be considered and there are some things that really benefit from having the same laws across Australia. Some people would like the same laws across Australia for everything. I can't see it taking off while states have particular sectional interests. So anyway we certainly look at, we get on the bandwagon for harmonised legislation. Things like road rules, it makes sense that wherever you are driving in Australia you have the same rules apply to you. There are small exceptions on that though. We are also a territory subject to Commonwealth law. So territories are recognised in the Australian constitution but really the Australian constitution is an agreement between the federal government and the states really so it talks about states powers. We are actually, the territory is a creation of the Commonwealth so we are established by a Commonwealth Act and that sets out a lot of the laws, sets out a lot of the rules around how we operate. So for example because we are subordinate entity until 2011 our laws could actually be disallowed by an executive act of the federal government so they could say to the Governor-General no disallow that and that could work within a certain timeframe. Can't do it now. The other thing is because of the self-government act there are some things we can't do so we can't execute people. If we were going to go there we can't, I'm sorry. And we can't make euthanasia laws, things like that. Another way that the ACT is different from the states is that we only have two layers of government. So we've got the state powers and the local government powers rolled into one. So this can make things more complicated and really it kind of means that we're covering more ground than in other jurisdictions like local legislatures, like the New South Wales government doesn't do the things like look after the roads and things quite as much as we do so it just means we cover more. A while back there was a report called it's the Hawke Report, governing the city-state and really what it said, it looked at the way we operate and it said actually guys, it didn't quite say it in these terms but it said you're a mile wide and an inch thick and really what you're trying to do is a lot of things and when you try and do a lot of things and there's not a lot of you you don't really hit the mark as much. What you should do is work out what your priorities are and focus on that which makes good sense. The issue is I think that along with having priorities as a government we also need to meet the needs of the community and if you have a problem and you have the problem today if I say to you when you bring me your lightbulb no I'm sorry this isn't on my priority list that doesn't make you happy so what we need to do is work out a way to be responsive focus on our priorities and still hit as many marks as we can so when Robert said you can ask questions you can ask questions I did put a slide at the end that said ask questions but if I'm not making sense I'm sorry next if I'm not making sense talk to me it does happen I have teenagers I'm used to feedback this is a picture of the Westminster system a couple of things so your traditional Westminster system has two houses we're a unicameral system so the legislative assembly is the single house you would think that would make it easier to get legislation through but I don't think so so really in the time since self-government there has only been one single party majority government so I think it was three or four years I think it might have been three years so what that means is that there's a lot of negotiating to get any sort of legislation through because you need to have two parties agree to it usually so at the moment there is a Labor Greens government previously there have been independents who have held the balance of power and there have been minority governments okay Westminster system we have a cabinet process so ministers can't go off there are some things that the executive can do let's take a step back there are three parts of government there's the executive legislature and judiciary, thank you the judiciary of mine so I shouldn't have forgotten them okay really what it's about is about balance so the executive can take certain actions so there are things like making regulations which are subordinate laws but they can't make the big laws and the big decisions they can come to the legislative assembly through a cabinet process and put laws through that way but that has to get the agreement of the legislative assembly the reason that there's a separation of powers like a Westminster system is really about balance and it's about making sure that processes are followed and that not all the power rests in one place because of the Westminster system separation of powers judiciary has independence we have independent courts so there's independence around judgements it also ties in with independence of prosecutors so we get a lot of letters and a lot of feedback around particular court cases and things like that the legislature shouldn't be involved in that the legislature should set up the system but really to stop the legislature getting the power to actually say how its laws operate apart from in broad terms the judiciary judges makes judgments on that and says how it will operate so it's the balancing again and we really don't make laws for individuals we make them for the system as a whole okay, next so this gets, yes, good this gets into the balancing again you've come up with your light bulb it's beautiful, it's awesome and it's so inescapably right and you can't understand why I'm not immediately doing something to legislate it the issue is that rights affect other rights and this is kind of a well-done thing but there are very few things in life that are completely right one way so I was at a meeting last week and we were proposing something to a group of informed stakeholders they had a particular, it was around courts and they all knew about the area and we said we're looking at doing this and somebody actually said well, you know, that's obvious why don't you just do it, why are you even talking to us and the person next to us turned around and said because I think it's the wrong thing to do so it's, even in areas where the stakeholders know each other and know the areas there are divergent views and really it's our part as the public service to balance rights and to take views and to make an informed decision and really this is where the community is important because we don't have the lived experience of how things operate we don't have the lived experience of what it's like to go through systems but we do have a broader picture and we do have contacts with other parts of the community rights affecting other rights if you look at the Human Rights Act pretty much so there's really this there's two rights that you can't you can't change or detract from but all the rest you can and that's because there's a recognition that one person's rights will affect another person's rights and the Human Rights Act again is around balance the Human Rights Act is a decision making tool so it won't stop us from making a decision doing something that we need to do what it will do is make sure that where that decision will affect impact on particular rights personal rights we've gone through a good process and that's around evidence that's around balance that's around appropriateness and it's around doing the thing that's least restrictive so really what the Human Rights Act does is it helps us decide whether we really need to do something that we want to do and really that goes back to having an evidence base we really need an evidence base we don't do things because they're the right thing to do we do them because we have an evidence base because it's not clear what the right thing to do is and what that will come through is probably the best way to describe that is what's the problem will what we're proposing fix it how often does it happen how proportioned it is it are we using something that's quite large to solve quite a small problem will there be collateral damage, that sort of thing so really it's going through it's really going through good governance and good decision making processes I guess the other thing that is always we certainly always think about is decisions that we make around what to propose to the minister are going to have an impact on tax usually quite often are going to either impact the community or cost the community money and it's taxpayers money and we have to think about that so we get quite repeated calls for things that might cost a lot of money and really we're not going to go for a big solution straight up if there are alternatives we'll go through them and that might take time to get us an evidence base next please so this is one of these things it's not like the other really I just wanted to give you two examples to sort of demonstrate the real life application of this one of the best one of the best indicators of whether something will work in the ACT is whether it's already working in somewhere else this is not infallible one of the things we put in back in the early 2000s was a right to trial by judge alone and that came in because there was a perception that there were quite complicated cases around sort of finances like fraud where perhaps the jury wouldn't understand and it didn't happen in all the jurisdictions across Australia but certainly there was I think we adopted the South Australian model and that worked really well and it reduced the impact on meant that there weren't retrials where juries came up with basically couldn't decide and things like that what happened in the ACT to this system that had been operating quite successfully in South Australia forever was we went to a situation where in South Australia and in other jurisdictions there was between 10 and 20% of trials were going to judge alone in the ACT it was more than double that and what we were getting was a skewing of the way people used the courts because of that so we can't always predict what the effect of what we're doing will be even if we've got a live example in Australia where you think it should apply the other thing I wanted to other example I wanted to give you was around bail reform so this is another one where it was obvious what the problem was so we had a problem with bail what happened was people were coming to the Magistrates Court for a bail application getting knocked back, appealing to the Supreme Court getting bail and this was leading to the Supreme Court because people didn't want to go to the Magistrates Court because you've always got a rubbish result and people wanted to go to the Supreme Court so it looked like a disconnect between the Magistrates Court and the Supreme Court when we actually looked at it the solution was to make people have a second application at the Magistrates Court stage there were two applications at that as a right because previously you could only apply once the issue was that when you get arrested the first thing you want to do is get out of jail so people would straight away apply for bail but they wouldn't have all their information they wouldn't have a good case that it was almost a gut reaction by making them have a second application at the Magistrates Court stage the stage that they had previously been going to the Supreme Court the Magistrates Court got to decide on what the Supreme Court had previously been deciding on and there were a lot more bail applications being granted I think the reason I wanted to share that one with you is if somebody had said we've got too many bail appeals to the Supreme Court let's make people go to the Magistrates Court again I don't think that that doesn't seem intuitive but in fact it's worked really well and when you look at the numbers, it's been in for about two years three years and the numbers are looking really good and it's led to people applying for bail at the right time at the right place I guess there are only a couple of other things around the ACT about us being different special so one is we have a population profile we have a lot of younger people and that will have implications for some things the other thing that sometimes we don't think about and certainly we do think about in crime prevention space is indicators of poverty socioeconomic indicators in the ACT are quite masked so because of a policy way back you don't tend to get poor suburbs you tend to get suburbs with a variety of people across them and that makes it harder to do some things that in other jurisdictions have been easier where you've actually targeted particular areas and we have a lot of hidden disadvantage so that's really I think next that's really what I wanted to say I hope that's kind of given you an indication of why things are not always as obvious as they should be and that there are checks and balances through the system that something that I think might be a good idea or you might think might be a good idea we might not have enough of the picture to be able to make an assessment Thank you very much Julie While we've got it here does anyone have any questions of Julie? I did them Sorry Yeah Yeah And totally hear you Really what other jurisdictions do is a factor so that when we make a decision about going forward if no one else has done it at least we know that we're trailblazing and that our mistakes are our own Again it's about balancing what the political appetite is at a particular time and what resources we have to put into things but no I totally agree and we do do things that other jurisdictions don't do and we are looking at Scandinavian countries are really excellent around law reform and that sort of thing So yeah Is there any sort of connection with the New South Wales departments around similar work? We do, we probably So the question was do we liaise with New South Wales around proposals and initiatives because we're an island within New South Wales So the answer is that we do a bit of it we probably don't do as much as we could but we could always do more consultation on everything Certainly because New South Wales has been around a lot longer than the ACT it has a lot of really good programs and we're really not averse to stealing ideas sorry, borrowing using So certainly there's a big initiative we're working on at the moment that we've pretty much talked to New South Wales from day one and they're our kind of mentor or learned friend on that So yeah I was wondering if organisations if in their work they come across areas where they do think that there could be improvements in the law how would you advise them to go about just having those discussions with government and calling us about how we should do that So the question was any advice on if you find areas for improvement for law you can get that message through to government you can write to ministers you can write to directorates there's a time to talk website there's a whole lot of community consultations around so feel free to put stuff in through that if you write to directorates that's probably a useful way if it's something that's sort of more low-key and what we actually find is that it's the small things that make a big difference to people's lives so things that just aren't working right and it might just be something small that we could sort of pop into a bill or something like that so yeah right, talk to us ring us even I think there's another question coming back there No, I must be Yes there is When you're weighing the evidence is the financial It's a tricky question it kind of depends whether it's a proposal or an initiative that already has funding so that ties into budget proposals so later this year there'll be the ACT government will put up an invitation for people to make budget proposals and that sort of thing but it's not a proposal that has a budget implication certainly use the community consultation which Geoffrey will talk about sorry look it's really hard to tell we don't go for a Rolls Royce model of anything because we don't have the funding we really don't have the size to do that I don't it's always a consideration but it will actually probably depend on the particular case so sorry I can't be more helpful on that I don't believe so I think it's the Commonwealth has responsibility for Jarvis Bay and we I think we currently provide services but Geoffrey might have a better idea on that one sorry Geoffrey Rodley to push even this it's Jarvis Bay is a Commonwealth territory of which they flee the service to the ACT government to do or deliver some services so we deliver care and protection services the courts go down but if and registration driver's licenses stuff like that but place is done by a local government very unique situation which we bore you endless we'll we'll sell it to you if you're interested we don't know can anyone want to any other questions for Julie no yes I guess it's all about which perspective you have I'm really happy to hear that we're the Jam in the Donut for new supports thank you


The computer is a valuable tool. Once I have access to the information, the computer makes me more independent. When I'm new here, I come through now. I am like any other student. The computer is a vital tool for education and employment. There's really nothing that does the variety of tasks that the computer can do, but it's not perfect. That's because not everyone can use the standard computer. And that's where Adaptive Technology comes in. With a little extra hardware or software, computers and the Internet can be accessible to people with a wide range of abilities and disabilities.  The cost of Adaptive Technology really is quite small. Particularly when you look at the impact that can have for a person able to do their job more effectively, more efficiently. Adaptive Technology addresses the challenges imposed by specific disabilities. For example, low vision. I use a screen in larger, which allows me to see everything on the computer screen that a normal person would see with regular type on the computer. For someone who is sensitive to light, software can reverse the screen from dark on light to light on dark. Large print keytop labels may also be useful for people with visual impairments, especially if they're just learning to type. The most common adaptation for people who are blind is speech output. The lab is well equipped with Adaptive Technology. Tours and demonstrations can be arranged by appointment. Really, it helps me out a lot on the Internet. I have voice output that reads everything that comes up on the screen, on the computer screen. So I'm able to access anything that I want to on the computer. And it's really helped me out a lot. A scanner combined with speech output allows people who are blind to read printed materials. A tall fellow in a Batman costume comes soaring out of the wind for a sky. Other adaptations include Braille displays and Braille printouts. People with speech or hearing impairments can use their computers to communicate with friends, teachers or coworkers. I really like using the Internet because it's easier to communicate with people rather than using the telephone. I can read it instead of listening and it's easier for me to read it than to hear. Yeah, the Internet is helpful. It allows me to communicate more easily because of my voice. It makes me, it allows me to say more, express myself more easily. How are we playing this again? People who can't speak can use communication devices to participate in group discussions and one-on-one interactions. That's exactly right. Those who can't hear require visual alternatives to sound output. When the computer speaks, they have ways of capturing that. The computer system I use uses visual output rather than sound output, which means instead of making a chime or a ring, it blinks the screen. People with learning disabilities can use a variety of software to help with reading and with writing papers. Adaptive technology ranges from spell check and grammar check to speech input and output. Washington is leading the world in global health. Schoolwork, it helps me because when me and my mom try to work to do it, we usually fight and so it usually ends up being a bad consequence. So if I can do it on my own, it's way better. Washington phase two, schedule August 7th to 13th. I basically just use standard word processors with a grammar checker and a spell checker and dictionaries on the computer. Just using the word processor alone reduces the amount of time that it takes to write things. I have a voice box that it will read it to me so I understand what I'm reading. Was Helen Keller the first deafblind person in the United States to be educated? And then when I have to like read books, I just scan those so they can read the books to me so I don't have to spend two hours reading one page or something. Things that I found really helpful have been speech to text programs. You know, you talk to your computer and it writes. At the University of Washington, a variety of hardware and software. I've written papers at college in a quarter of the time that it would have taken me to type them by hand. With speech to text, I just say the word and it shows up on the screen. People with mobility impairments have a wide range of adaptive technology options. For some, flexibility in the positioning of tabletops, monitors and keyboards is helpful. I use a mouthpiece that I type with. I can do at least 30 words per minute when the words are going from my head to the keyboard. I actually write lots of things and we're not for computers and word processing and spell checking and things like that. It would take me ages. I have a trackball which I roll around and I use sticky keys to hold down control and shift. The computer helps me type reports better and it's easier on my arm. I don't have to wear my prosthetic. One thing that I use is a keyboard where the keys are enlarged and there's more space between because when I use regular keyboards, I get double letters. For people who need to type with only one hand, left and right-handed keyboards are available.  Word prediction software can increase speed and accuracy. I have an on-screen keyboard and it also has word prediction to where I throw in a letter. Let's say I throw in a T and five words that start with T will pop up. The most common ones that I use, they'll pop up and I'll click on it and it'll just print it out. I got fairly fast. Some people may choose to bypass the keyboard by using Morse code. A sip and puff switch registers dot with a sip and dash with a puff. Special hardware and software translate Morse code into a form that computers understand. I'm a junior this year. Other people may choose a speech input system to replace the keyboard. I use a program that helps me type. Whatever I say or type, I talk into the microphone and it types it out on the computer screen. It makes me feel a lot more independent and I don't have to rely on somebody for so much and I can do it myself. The internet can be accessed from almost any location at any time that a person wants to use it. This is a real benefit for people with health impairments. I think that computer can help hospitalized kids and I was in the hospital for like one month. I talked to other kids and I could like socialize with them and people send me meals, creating meals to get well. In the past year I lived in a hospital and an internet connection there allowed me to communicate with teachers. I think that the internet is really helpful for students who miss school due to medical reasons because it allows them access to teachers, resources and fellow students which can be very helpful. For anyone with a disability, adaptive computer technology is a vital link to success in school and in work. It's more fun. I've had people try to type for me.  I took a poetry class and I felt weird trying to tell some friends of my age what I'm thinking and what I'm trying to put on paper. I like using my computer because it helped me to be independent. It helped me and my ideas. I virtually live on computers. On the computer people are more on an even keel. On the electronic field we're all equal. For more information about IT accessibility consult www.uw.edu. The content of this presentation is based on work supported by the National Science Foundation under grant number 9800324. Any opinions, findings and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation. Copyright 2015, University of Washington. Information is granted to copy these materials for educational non-commercial purposes provided the source is acknowledged.


Ladies and gentlemen, I wonder if you would take your seat, please, right after the President. But Jay is here, and I'd like to introduce Jay to you, who is the Special Assistant to the President on Private Sector Initiative. Jay? Thank you very much, Bill. I'd just like to take a couple of brief minutes to explain this administration and something that you all ought to point out to those people that you talk to, is that this is the first administration that created in the White House the Office of Private Sector Initiatives, which I now currently head, with 44 of the best individuals around the country, people who were leaders and leaders in for the next several months to define those 43 outstanding people. From December 2nd, up through the end of this year, the White House, the President, and this administration, how they can best mobilize this effort. And we hope that at the end of the year, when the Task Force submits its final list of Private Sector Initiatives, becomes a little bit more important and receives a little bit more focus because it will be that office's responsibility, let me say that from all of my indications that he is not going to let this initiative die. He plans to continue to speak out as often as he has in the past. He plans to continue to visit cities around the country who have been Private Sector Initiatives' effort would follow very closely along the President's major domestic initiatives that he has put forth over the last several years, as all of you know, and we have every indication to believe that in 1983, 1984, and beyond that Private Sector Initiatives will also join those previous two initiatives. We take two examples out of our project bank and we send those to newspapers all over the country and radio stations, and it's called The Brighter Side of the News. These are two examples of how a community is in a very creative way. Ladies and gentlemen, the President of the United States. Thank you very much. Well, seeing so many Association leaders here today through me for a moment, I thought that I was supposed to lobby you on a budget bill, but it really is a pleasure to welcome you here and to tell you the truth, I'm going to do a little lobbying after all, and that's on behalf of our Private Sector Initiatives program. Now, I know you must have been briefed very thoroughly and know all about that, and I'll try not to plow the same ground, but I heard a story the other day about a third grade teacher who was explaining the importance of good penmanship to her class. She told the students, if you can't write your name when you grow up, you'll have to pay for everything with cash. Well, we know there are certain things you can't buy with cash or even credit, and that's involvement, the kind that historians describe as the American spirit. You've been hearing about that spirit and involvement in the briefing today, and I'd like to just expand a little. I know that voluntarism didn't start with this administration. Americans have been helping their neighbors ever since the country was settled. It's a grand tradition that unfortunately hasn't gotten the attention that it's deserved in recent years. What we're trying to do is make sure that these good deeds receive their fair recognition, hoping that in turn this will spur others to participate, and that's why I asked Bill Verity to chair the Private Sector Initiative Task Force, and let me tell you he's been keeping me busy. We've had six major meetings here at the White House. I've talked to the nation's governors, asking them to form statewide task forces in over 40 states have now done so. I've asked the cabinet to encourage voluntarism and to promote public-private partnerships in their departments. I've met with independent broadcasters from around the country, and now I'm glad to report that we're finally seeing more stories about the good things that people are doing. To give you an idea, at least 40 TV stations around the country are running programs that show how communities are solving their problems through partnerships. I've also made visits to cities all across the country in order to spotlight private sector initiatives. And now before I wear out my arm padding, Bill and myself in the back, let me say I mention all this to show you how deeply we believe in private sector initiatives. The push for private sector initiatives is as central of this administration as cutting spending and taxes. But what's even more important than what we're doing is what the American people are doing. In Las Vegas, the Independent Youth Athletic Association provided volunteer labor and equipment to light up city and county baseball fields, saving taxpayers more than $270,000 in lighting costs. Then working with local agencies and private foundations, they built an athletic complex that is now valued at over $3 million. The old Newsboys Goodfellow Association sounds like something I should belong to, doesn't it? In Toledo provides more than 2,000 food baskets and other essentials at Christmas time for school kids. In the association's words, our project will last forever. Our membership's pleasures are realized when they see the smiles on the recipient's faces. An American Medical Association auxiliary in Flint, Michigan operates a free spinal screening program for youth in the area. In November, more than 13,000 children were screened with the help of 300 volunteers. And the list goes on and on. And many of them stories of almost happening by accident, as a matter of fact, in Santa Barbara, California. Just a couple of years ago, a young girl in school needed to have the total kind of bone marrow transplant for the catarious condition that had afflicted her. But her family, there was no way they could send her to Mayo's, where this was to be done at that time, and fund this. And some of her schoolmates just in her class began knocking on neighborhood doors and telling their story and asking for money to help their schoolmate. And from that, suddenly, as adults began to hear, a committee was formed and they raised money. Oh, they did it with a symphony concert. They did it with all kinds of things and they raised the money to send this little girl to Mayo's. The operation was a success, but then tragically, a staff infection took her life. And the group that had formed and raised the money for this particular thing, then suddenly, faced with this tragedy, said, wait a minute, they had money left over. They said, let's stay together. Why don't we make this a permanent part of our community? Because there are going to be other cases like this and we'll have a head start and we'll be prepared to take care of those other cases. And it just started with some school children knocking on a door. You know, I wish all those who claim our greatest days or past could grasp the energy and vitality in grassroots America today. Henry Luce, the founder of Time Magazine, saw this. He spent a lifetime watching and recording it. And in 1962, he observed, we're the country of the endless frontier of the big sky, of manifest destiny, of unlimited resources, of go west young men, of opportunity for all, of rags to riches, mass production, nothing to fear but fear itself, technical know-how, a chicken in every pot, gung-ho and can-do. Well, it may sound a little corny, but Luce was absolutely right. And that spirit is still out there. And you here today are in touch with it through your memberships. You can stimulate and activate the manufacturers, the small businessmen, the doctors, the nurses, the lawyers, the realtors, the public officials and concerned citizens who are your members. They're the people who make this country go and they're the ones who can make the difference. And you might point out to them that the quality of their community is key to the well-being of their plants, stores, practices and neighborhoods. We need you here today to get the message out that involvement is not only in the public good but in your members' own interest, best interests as well. We need you to encourage them to put their talents to work directly on some problems that we've let sit for too long. I urge you to take on a private sector initiative and put the full resource of your organizations behind it. Show your members how it's done and what they themselves can do. I know the task force will give you all sorts of help and ideas to get your projects rolling. Although the task force is going out of business at the end of the year, we'll still have the White House Private Sector Initiatives Office and I'll still be speaking out as I have been. And I hope that your organizations will be out there carrying the ball for us also. I guarantee you that you will have plenty of support from here. When I was inaugurated, I said that our people have a potential for greatness and they've proven it every time it counted. I believe we're in one of those periods of our history when it counts more than ever. And I ask you to join us in releasing that potential and to share in the satisfaction of making America all she can be. I thank you again, all of you, for being here. I will turn you back to the gentleman here in the platform and the continuation of your meeting. And as the little girl told me, an 11-year-old girl who wrote me a letter shortly after I got here, she told me how many disappointments I'd have and how busy I would be. And she said, you'll feel just be glad that you're not God. And then she ended with a PS and said, PS, now get back to the Oval Office and get to work. So that's what I'll do. But thank you again for coming here.