Thanks very much for the opportunity to give these lectures. So we have three hours today and one tomorrow, which means you have to bear with me. You might get tired of me already by the end of the day. So I'm going to cover some topics in quantum field theory that are interwoven. There are some intricate relations between these different ideas. So let me quickly go through about the plan for these lectures. So first, I'll start with some basic introduction into how anomalies creep into quantum field theory, mostly in two and four dimensions. I'll discuss three examples. One is for current algebra and other four central charges in two and four dimensions. This results in various applications for a condensed matter physics, statistical physics, particle physics. But in these lectures, I would not have much time to emphasize the applications. I mean, for example, this has played a major role in the quantum hall effect in the past, but I'm not going to emphasize the applications very much because I want to emphasize more the ideas and the formalities of how these things arise. So I'll be mostly talking about the theory, not about the applications. If somebody wants to know about the applications, we can maybe discuss it later in the day or during some break. And so that would be the first part, some very simple anomalies and how they creep into quantum field theory. The second part is going to be more advanced. I'm going to try to define a geometry in the space of theories. And then I'll apply these ideas for cyber-witten theory in which there is some space of models and one can try to compute the geometry in the space of models. And that would be related to some other anomaly, which I will call B type rather than A type. So their first part would be about this A type. The second part would be about B type and there would be an application to cyber-witten theory in which I will show you that you can compute exactly some properties about theory space. So these ideas are a combination of new and old ideas, but this is completely new. And then the last lecture I'll devote to a topic which is completely different about the S matrix of QCD. It has some interesting ideas and it's more like a teaser to I think some progress that can be made in this field rather than a lecture in which I'll present some complete results. Okay, now because we have like three hours today, instead of me speaking, it would be very nice if you ask lots of questions and it would be a more interactive session. Otherwise what I'll end up doing is to basically reciting my notes which I'll put online today. So there are 70 pages of notes. They're somewhat disorganized, but the gist is there. So if you don't ask questions, what I'll end up doing is to recite my notes which would not be very interesting. So, okay, so please ask questions. So let me just start with a little bit of motivations. So obviously the framework of quantum filtering, the framework of quantum filtering has singled itself out as a universal framework, a mathematically consistent framework that describes lots of phenomena in physics, in many, many different branches of physics. It's even hard to enumerate in cosmology, in condensed matter, in statistical physics, and of course in high energy physics, and even in stochastic processes. So even people who study stock market, they need to know a little bit of quantum filtering. And in each of these instances, the reason that quantum filtering emerges is somewhat different. So in condensed matter physics, of course quantum filtering emerges because you can have particles generated from the vacuum and things like that in the same way. And you have sometimes emergent relativistic invariance so you even get to study relativistic quantum filters in statistical physics. It's the old story that statistical systems near a critical point, near a phase transition, they have long range correlations and so you can forget about the lattice structure and you can pass to some continuum description. That's how people understand nowadays critical phenomena, which you must have heard about from Slava. In high energy physics, obviously we have Lorentz invariance and so we can have particles generated from the vacuum so we need to use this framework and so on. So this is a very, very universal framework that can be used to describe many different physical ideas. And so it's obvious that anything that we can say about quantum filtering, any general statement that we can make about quantum filtering is going to be useful for many, many people, many branches of physics and we're still exploring the basic structure of quantum filter even though it's four decades old, the framework we're still trying to explore to what extent we understand the intricacies of the theory, we understand the basic results in the theory and so on and so forth. So the general picture that, the general understanding of quantum filter comes from Wilson's ideas. What Wilson said is that basically you can think about quantum filtering as an evolution process of some sort. So if you look at very, very short distances, so this is going to be let's say short distances. So inconsistent quantum filtering, okay? In consistent quantum filters, the short distance physics is described by some scaling variant model and also the very long distance physics is described by other scaling variant model. So there is some scaling variant model here and in the middle there is some crossover behavior which is characterized by some scale. So there will be some scale which in particle physics conventions you can think about it as an inverse mass scale. So there is some crossover from one scaling variant theory short distances to another scaling variant theory at long distances and the critical exponents here so we characterize scaling variant theories by the correlation functions and the correlation functions in scaling variant theories are some power laws and these power laws have some exponents gamma N in the infrared and here there are some other exponents gamma N in the UV and the crossover interpolates between one set of exponents to another set of exponents. So that's the general framework that Wilson gave us for quantum filter. So what is our job in quantum filter? So this kind of scenario is realized of course in many instances in particle physics, statistical physics, and that's matter and so on. For example, one of the first examples was for instance a renormally flow from the tri-critical model to the Ising model and this is something that has been observed experimentally long time ago and there are lots of dozens or hundreds or thousands of other examples where we've seen this picture happening in nature. So what are the interesting questions that we can ask? So the interesting questions could be structural which is what is the connection between these exponents and these exponents? This is one sort of question that you can ask. Can we go from any scaling variant theory to any other scaling variant theory or are there constraints? Are there constraints that if we specify the scaling variant theory here there is no crossover that would lead us to some given scaling variant theory here. You can ask the question of given a scaling variant theory can it be the bottom of this crossover behavior or maybe it doesn't have any way of uplifting it into what we call a flow. So this is called an RG renormalization group flow, a renormalization group flow. So these are sort of structural questions that you can ask of when can a given model be understood as sitting here or here when there is a crossover behavior. You can ask, this is one set of questions. Another set of questions is about what's happening at the crossover scale. So the crossover physics could also be very interesting. For example, deep in elastic scattering which is an interesting experiment in quantum chromodynamics is more about the cross, what happens here, very near the fixed point. This is called the fixed point the scaling variant theory in the UV. This is more about the physics in the crossover regime but not very far from the fixed point. QCD, the hadronic physics is more about the crossover scale. In statistical physics, critical behavior, these critical exponents that you read about in the books in statistical physics are about this regime. So not quite the scaling variant point but very close. That's how you read out critical exponents in statistical physics. So there are different applications of setting the crossover regime, the scaling variant theory, the scaling variant theory. There are also lots of applications in higher physics. Yes, yes, the picture could be more complicated. It could be that for example, you start from some scaling variant theory, there is some complicated crossover regime which actually hovers near some other approximate scaling variant theory and then it continues to some other point. The diagram of possible renormalization group flows could be very intricate and well understanding that global structure is one of the things that you may want to do. Since the framework is so widespread, understanding the global structure of these possible flows is a very important question. So different people have different tastes for what are the interesting questions. In condensed metaphysics nowadays, in fact they've gone to the extreme because what they are studying are this topological insulators. That's one very big theme in condensed metaphysics nowadays. In topological insulators, this scaling variant model in the infrared is empty. So there are no interesting correlation functions. And when this is empty, we call it a gap phase. And what they're studying are some non-local observables in these gap phases. So they're interested in trivial scaling variant theories in which there are some residual non-local observables that are still sort of interesting topological observables. And in statistical physics, of course, people are much more interested in the critical exponents. That's what they measure. They measure local correlation functions. So different communities have different interests and but it all falls under this umbrella which is very nice. There is one picture that encompasses lots of lots of physics. And so different people have different tastes for what is interesting. So let me tell you what I'm going to describe. Of course I made a very personal selection of the things that I'm going to describe. So I'm going to describe a little bit. So the first thing that I would like to describe is the question of when are the scaling variant points actually conformal? So we've observed experimentally, it's an experimental observation or an empirical observation. We've observed experimentally that sometimes these theories are more constrained than just being power laws. They have some additional symmetry and I would like to give you a quick argument that explains at least four two-dimensional models why you have these extra symmetries. That's the first thing that I'm going to discuss and then we'll discuss some very basic properties of the relations between these critical exponents and these critical exponents. We'll discuss some very simple applications of that idea and some simple results. And then I'll try to describe more collectively the geometry of all the possible theories. So there's some geometric manifold of all the possible theories and all the possible renormalization group flows and you can define an interesting geometry and study it and you can even make exact computations in some supersymmetric theories which allow you to learn about the space of theories. And then in the last lecture, I'll be discussing really the crossover shift where I'll be studying the S matrix of a large N QCD. So I'll cover some topics of various sorts. Are there any questions about this introduction and before I start, the first topic which is to explain why the scale invariant models are now to be conformal invariant in two dimensions at least. Okay, so let's start by reminding you about the Poincare group that will be very important to remember. So the Poincare group in D dimensions. I'm going to discuss for now the Poincare group of RAD. So the space is going to be RAD Euclidean. The Poincare group of RAD consists of translations. So we have D translations and then we have some rotations which are in SOD. So this is the Poincare group consisting of translations and rotations. Now what is the scale invariant model? So there is a formal definition. A formal definition is that the symmetry of the scale invariant model consists of the Poincare group plus an additional conserved quantity which we can call delta. That's a conserved charge. So that's a new conserved charge which acts on the coordinates by multiplication. So that allows you to rescale. And once you add this conserved charge to the theory then everything has to be a power law because when you measure, when you study let's say now correlation functions of two local operators phi one and phi two, this would be at x and this would be at zero without loss of generality. If this has scaling dimension delta one and this has scaling dimension delta two which I'm sure it's that makes talk to you about then we have to write the following formula. Delta one plus delta two. Okay, so everything has to be a power law because this is a symmetry. So now what for example, Lando did when Lando sort of described for the first time the theory of phase transitions. Lando realized that the fact that there is scaling variance allows you to derive many interesting relations between critical exponents. And everything that essentially was done back then follows from this idea that two point functions are given by power laws. This is the so called hyperscaling relations that you might have heard about but even if not it's not important. Hyperscaling relations. These are some relations in the statistical physics that just follow from the fact that there is scaling variance at the critical point. Okay, so this is the naive description of the scaling variance theory but experimentally, so an experimental fact, an experimental fact is that what we end up seeing in many of these examples is a bigger symmetry. So what we end up seeing is the full conformal group. So we have Poincare plus this new conserved charge which corresponds to rescaling of space but we also end up seeing the special transformations. Special conformal transformations and all of this together forms the group SOB plus one comma one. Okay, so this, the generators combined into a big group which is bigger than Poincare and bigger than this group the scale group, it's SOB plus one comma one and this group has lots of interesting consequences. So there are interesting consequences. One of the measurable consequences of having this extended symmetry is that two point functions are actually diagonal in dimensions. So a scaling variance allows you to have a nonzero two point function between any two insertions who is well defined eigenvalues under scaling but one of the consequences of the bigger symmetry is that for what's called primary operators you get a diagonal two point function for primary operators and then you have one over X to the power two delta one. Okay, and this has measurable consequences and in many cases we know that this is true. This is a consequence of this bigger symmetry and there are also lots of consequences for three point functions but I'm pretty sure you've seen some of that too. Okay, so the first thing that I would like to do is to explain why there is this symmetry enhancement. So there is the question of, it's a general question in quantum filtering which is why a Poincare plus delta often gets enhanced into a SOD plus one comma one. This is Poincare D. So it's a very general question that pertains to many applications in quantum filtering and we don't know. The short answer is that we don't know why this is true but in some cases we can say a lot. We can more or less give a proof in D equals two. This is what I'm going to do now. I'll give you a proof. In D equals four there's almost a proof. Almost proof meaning that it's not mathematically rigorous. It's a physical argument. So it's almost a proof and all the other cases, for example, three dimensions which is most relevant to statistical physics, there has been no progress essentially. And of course, you know that there are also interesting quantum filters in five and six dimensions and also in those cases there has been not much progress. Okay, so the question is why does this happen? Why does this symmetry enhancement happen? You might think, you might actually recall when you see this formula, that in the hydrogen atom there is a symmetry enhancement from rotations to the Runge-Lenz Laplace group which includes an additional generator. So remember that in the hydrogen atom you have an enhancement from SO3 to SO4. You might think that understanding the symmetry enhancement is very important because for example, the symmetry enhancement plays a huge role in the solution of the hydrogen atom. So understanding the symmetry enhancements is a kind of a basic question. So I would like to give an explanation. So our journey into all of these topics will begin from starting the two point function of the energy momentum tensor. Of the energy momentum tensor. I'll start by doing it in any number of dimensions and then we'll specialize to two dimensions and we'll see something special happening. So that's the object that I would like to study. I'm going to do it in momentum space because it's a slightly more convenient and one doesn't have to worry about various issues that arise in position space. So I'm going to study this fundamental object which is the correlation function of two energy momentum tensors with momenta q.  So in statistical physics this object you would call the stress tensor in higher energy physics it generates energy momentum so we call it the energy momentum tensor but in both cases it's a very fundamental operator that creates some interesting local excitations. So we're going to try to fix this two point function but what are we going to assume? So what I would like to assume is just Poincare. Okay and we'll do it in steps. We'll first assume Poincare, then we'll assume scale invariance, then we'll see that you can prove that conformal invariance emerges. So this bigger group emerges in two dimensions at least and then we will go further to prove various interesting structural theorems about these two point function. So we'll just use Poincare. So if you use Poincare you just have to satisfy the various symmetry conditions. So it has to be symmetric in mu nu, it has to be symmetric in rho sigma and it has to be conserved. The conservation is the big deal that we'll worry about a lot. So remember that there is this classical equation whereby this is conserved and we'll worry a lot about this conservation equations. You'll see that sometimes we cannot satisfy it, sometimes we can't satisfy it. We'll see various interesting applications but I want to make a small digression to explain what does this equation mean. You've probably seen this equation many, many times but what does it actually mean? So when we write a general equation that some operator, some quantum operator is equal to zero, it has a precise mathematical meaning. The mathematical meaning is that at separated points when you insert this operator you get zero but at coincident points you might need to be careful. So this is an equation that always should be interpreted at separated points. So what it means is that if we insert this operator at some point x1 and then there are lots of other operators and x1 is different from all the other xi which sit here then this vanishes. This is the mathematical meaning of a vanishing operator that at separated points it gives rise to vanishing correlation functions but sometimes you cannot uphold these equations at coincident points. This is where anomalies arise and various other issues creep in. Now in momentum space, remember that when you do a Fourier transform you integrate over all the possible positions with some weight and therefore imposing conservation equations in momentum space is always subtle and this will be a major theme that we'll discuss. You cannot quite impose these conservation equations right away in momentum space, you have to think what corresponds to separated points and what corresponds to coincident points. If you try to impose conservation equations you get wrong inconsistent results as you will see. Okay, so let's start by just writing all the possible point-curring variant pieces and then we'll discuss the conservation equation. So there are two possible tensors you can write. One in which there are constructions across the two energy momentum tensors and the other is when there are only constructions in pairs as you'll see. I wanted just to remind those so that the energy momentum tensor is symmetric in its indices so this is how these structures are constrained. So one possible contraction is q mu q rho minus eta mu rho q squared times q mu q sigma minus eta mu sigma q squared a plus rho interchanged with sigma and this would multiply any function of q squared. So let's call this function f and then there is another possible contraction which looks like q mu q nu minus eta mu nu q squared times q rho q sigma minus eta rho sigma q squared and there is g of q squared. So these are two tensors that you can write down so there are two functions f and g and I've chosen the tensors in such a way that the conservation equation is obeyed but you'll see that it's a little bit subtle. We'll discuss more why did I choose this particular tensors? One is really obeyed, one is not really obeyed and so on. This is just a naive first attempt that implements this equation. So in momentum space this equation can be written as q mu nu q mu nu equals zero. So in general there are two functions that appear here f and g. The first claim that I want to make, these functions would in general be functions of the crossover scale of the momentum over the crossover scale. So in scaling variant models where there is no external crossover scale, this would be some homogeneous function that we'll try to fix but in general it could be a function of the momentum over the crossover scale. So one interesting claim that you can make, you'll see I'll derive this claim later from another point of view but you can already, you can try to prove it also from this point of view is that if the number of dimensions is two, then f and g are linearly dependent. So they're not different functions. They're not different tensors. So the claim is that this tensor and this tensor in two dimensions are actually the same. Not independent, so they're not independent. This is a small claim that I would make. We'll prove it from another point of view but it's important to appreciate it already at this stage. Okay, so now I'd like to talk about what happens if we impose scaling variants. So we impose that if we impose that there is no external scale in the problem. So everything is just some power law of various momenta. Okay, so let's discuss three different cases. Two dimensions, three dimensions and four dimensions. What happens if we impose scaling variants in these three different cases? Now, you have to appreciate that the energy momentum tensor in position space has dimension D and therefore it's Fourier transform has dimension zero and because there is always a delta function for momentum conservation which I've omitted, the right hand side has to be of dimension D where D is the number of space time dimensions. Now we already have four powers of momenta here and here and so in two dimensions it has to be one over Q squared. So F and G, the F to B, some coefficient over Q squared. It won't be important to fix these coefficients. It just follows from dimensional analysis. Now you could ask what about logarithms? So we can discuss logarithms a little bit later. You could add various logarithms here and would still be naively consistent with scaling variants but we'll discuss that later. This is the most straightforward solution. This is the most straightforward choice. Now in three dimensions F and G are going to scale like one over the square root of Q squared and in four dimensions they have to be logarithms of Q squared. These are the three cases, two, three and four dimensions. Are there any questions about why this is true? You could choose either a logarithm or a constant function. It would still be okay but you'll see later that the only consistent choice is a logarithm, not a constant function. Okay, so these are three cases that you could discuss but then there is an interesting computation that you can do. You can try to take the trace over the indices mu nu and rho sigma, both of them at the same time. So let's trace over mu nu and rho sigma. So we're going to trace over mu nu and rho sigma. Let's write down the answer. We get to study T mu mu of Q and then T rho of minus Q and we'll do it again in two, three and four dimensions up to this coefficients that I haven't fixed. So let's start from two dimensions. Two dimensions F and G are one over Q squared. So if we trace over mu nu and rho sigma this would be Q to the four. This would be Q to the four. They would cancel two powers from the denominator and you will get Q squared. This is in two dimensions. Three dimensions, we will get something like Q to the four over the square root of Q squared. So that's going to be Q squared to the power three halves. Yes, so if you take TQ, P, that's what you're asking, right? Then in general, this would be a delta function in the dimensions of Q plus P. Why? Because if you create an excitation of momentum P it has to be absorbed in the vacuum with momentum P. So this creates an excitation from the vacuum with momentum P and then it gets read. What this describes in the language of statistical physics is that you create an excitation of some momentum P and then it is absorbed sometime later in time. And so it has to be absorbed with the same momentum because we're doing it in the vacuum. Of course, if you do this kind of, if you do this correlation functions in the presence of some non-trivial medium then some momentum can dissipate into the medium or some momentum can be extracted from the medium and then you don't have a delta function. But I'm doing these computations now in the vacuum where no momentum can be absorbed or extracted from the vacuum. Okay, so I get Q to the four. In four dimensions we get Q to the four times logarithm of Q squared. Okay, these are the three cases that we've got. Now, the most striking fact about this result is that this is a pure, this is the only case in which you get a pure polynomial. So this is a pure polynomial. This is not a polynomial. This is a fractional power of Q squared. And this is also not a polynomial because there is a branch cut due to the logarithm. There is only one case in which you get a pure polynomial. What does it mean in position space? If you have a polynomial in Fourier space in position space it's a delta function. The Fourier transform of a polynomial is always a delta function. That's very important to remember. So what it means is that in two dimensions the correlation function of t mu mu at x and t mu mu at y, where y is different from x is vanishing or more precisely the Fourier transform of this polynomial is a laplacian of a delta function of the two-dimensional delta function. So the two-point function of t mu mu in two dimensions vanishes at separated points. In three dimensions it's not a polynomial. It has a non-trivial inverse Fourier transform and you get some interesting correlation function. In four dimensions you have a logarithm. So again you get an interesting correlation function at separated points. It's not just a pure delta function. Now you could have asked, it's important to think, why did I choose a logarithm? If I had chosen a constant which is also consistent with the naive scaling then this would be also a polynomial and then I would also get something like that in four dimensions. But let me explain why this is not the right choice. If I had chosen here, if I had chosen here a constant then all the components of the energy momentum tensor with all the other components of the energy momentum tensor would be pure polynomials because this is a polynomial and this is a polynomial so this would be a constant, you would get pure polynomials and therefore all the correlation functions of the energy momentum tensor would not have support at separated points and that would contradict unitarity and various other things. It would mean that the energy momentum tensor is a trivial operator which is not good. In two dimensions this choice does not trivialize the two point functions. It only trivializes one component which is the trace of the energy momentum tensor and so it's okay. But in four dimensions this is the right choice and therefore it's not a polynomial. Now if you have, yes, which more general situations? Oh okay so there is a great question of why didn't I add a log of q squared? That's the question. Okay let's discuss that. If I had chosen to multiply it with a log of q squared then indeed this would not be true and the conclusion would not follow. But let me explain physically why this is not allowed. So you see what I'm imposing here is scale invariance. Like any other symmetry or operator equation this should be upheld at separated points. So when we do the Fourier transforms back to position space the correlation functions have to be pure power loss even at separate points, sorry, at separate points. They have to be pure power loss. Now if you had the log q squared you would need to divide by some scale, mu squared. Okay just for dimensional analysis. And if you change the scale from mu to mu prime the difference is going to be a over q squared and when you plug it here you get something that is not a polynomial. So you violate scale invariance even at separated points. So if I had put a log here that would mean that there is no scale invariance even at separated points in this model. Okay so there are no critical exponents. This log is different. Now you can start seeing when can we add a log and when we cannot add a log. In this log if we change the scale from mu to mu prime the difference between the two logs is constant. And therefore it corresponds to a pure polynomial in momentum which only has supported coincident points in position space. And therefore it's okay. So this log does not violate scale invariance which is what we're assuming but this log does violate scale invariance. So we are not allowed to put it here. And therefore this result is inevitable. Okay now one final general comment that I wanted to make before I proceed is that in quantum field theory if we have an operator and it's here mission conjugate at let's say x and zero. If this vanishes for all x, sorry. Yeah if this vanishes then for x not equal to zero then the operator is trivial. This is in good unitary theories where there are no imaginary parts in the action. This is like a norm of a state. You can think about it as a norm and therefore if this vanishes then the operator itself vanishes. When it acts on the vacuum it doesn't create anything. For if it created something you would have a nonzero result. You just imagine inserting a resolution of the unity in the middle and you would get that this is true. So if the two point function vanishes the operator vanishes at separate points. And so we conclude that in two dimensions T mu mu has to vanish. We just proved that. It follows from scale invariance. I've not assumed that there is conformal invariance. It follows from scale invariance that T mu mu vanishes. Now when if T mu mu vanishes in conjunction with this condition you get far reaching consequences. Namely you get the full conformal symmetry of two dimensional models. Let me review why this is true. So since we're in two dimensions let's choose light-con coordinates. They will be very convenient in what follows. So I'll choose the coordinates x plus and x minus and correspondingly there's p plus and p minus for the two components of the momentum. So we can write this conservation equation in light-con coordinates and p plus minus coordinates.   These are the two conservation equations in momentum space, in the light-con space. T plus minus is just T mu mu. So if T mu mu vanishes, which is what I've proven, then these two go away and we remain with these conservation equations that q plus times t plus plus and q minus times t minus vanishes. Or in position space, that means that the d minus of t plus plus vanishes and d plus of t minus minus vanishes. This is something that we've proven based on scaling variance. But if this equation is true, I can multiply t plus plus with any function of x plus. So therefore it's also true that d minus of any function of x plus times t plus plus vanishes. And d plus of any function of x minus of t minus minus vanishes. So you get infinitely many conserved charges if T mu mu vanishes because I can multiply t plus plus with any function that I want and that generates the full bero soror algebra. So that's the infinite dimensional conformal algebra that you certainly heard about in two dimensions. So you see that we proved in 15 minutes a very far-reaching result in quantum filtering which is that in two dimensions, if you just assume the scaling variance which is just saying that you have power laws, you get the full bero soror algebra. You get the full bero soror symmetry. And the only ingredients that went into the proof were unitary, meaning that I assumed that the resolution of the identity is present and also that all the states have positive norms and one current variance with scaling variance in the usual sense. So this is a very strong result that we know is true in two dimensions and it explains why in quantum filtering, the scaling variance points that appear at long distances at short distances are actually bero soror invariant theories which are much more constrained and there are many experimental consequences of having this bero soror group. So unfortunately you see that this argument does not carry through to three dimensions or four dimensions because we get two point functions which are non-vanishing at separated points of T mu mu. So there doesn't seem to be an easy way of that sort to prove that scaling variance implies conformal invariance. You can assume that conformal invariance is true and then T mu mu vanishes and then you can proceed but you can't prove it in this way from first principles. You have to do something much more elaborate, presumably. Okay, are there any questions about this part? Yes, right, this is like a local operator. This is just zero as an operator, as a local operator. Typically you mean when there is conformal invariance or what? Right, right, indeed. So yeah, I'll repeat the question. So suppose, so in four dimensions, we can't prove that scaling variance implies conformal invariance. Let me just repeat the question. The question was, what happens if you don't have conformal invariance? That's my interpretation of the question. Then this is not true but maybe some weaker form of this equation is true, okay? So indeed, indeed. Let me quickly tell you what do we know if there is no conformal invariance. So if there is no conformal invariance, then T mu mu does not vanish as an operator. Rather, T mu mu, we know, is equal to the gradient of some other operator. So it's not a completely general operator, it's just the gradient of something. I mean the divergence of something. How do we see that? There is this charge that I defined, which exists in scale invariant models. This is the integral over d, d minus one x over some space like slice of some current, which we can call a, let's call, let's denote this current by b. And this current b is given by x mu T mu mu plus v mu. This is a general form for this current, which is conserved and whose integral gives the scale charge. Now, for this current to be conserved, you need to satisfy this equation. Just hit it with nabla mu. Use the fact that the energy momentum tensor is conserved and you'll get the T is the gradient of v. Now, if T vanishes, then the gradient of v vanishes and then you get many more conserved charges. But if you don't have conformal symmetry, the only thing you know is that the integral of b naught is conserved and that's it. So the integral of b naught is very close to what you said. It's a v naught plus x i, x mu T zero mu. So if there is no conformal invariance, this is all we can say. That there is some operator v whose gradient is, whose divergence is d and that's it. But if you have conformal invariance, you have a much stronger result because you know that this vanishes and this vanishes. You can throw it away and you have lots of new conserved charges. So conformal invariance is indeed much stronger than the scale invariance, but it's not always easy to prove that fixed points have conformal invariance. In fact, Landau did not anticipate it. Landau never used conformal invariance. He just used scale invariance in his theory of second order transitions. But I think in the late 70s or maybe mid 70s, it already became apparent that all the known models do happen to have conformal invariance. And many people wrote early papers on trying to explain that, but still the only case in which it's completely well understood is two dimensions and to some extent four dimensions. Okay, now I'm switching to the new topic. Well, it's not completely different. I'm just going to talk about gravitation anomalies, gauge anomalies and all that and how they evolve on the renormalization group transformations. So let's have this picture in mind. Now we are smarter. We know that this is a conformal filter. This is some other conformal filter. I'm going to do two dimensions. I'm going to discuss two dimensions for now. So we're a little smarter. We know that this is the picture and here there is some crossover in the middle between the two theories. So the question that I'm going to ask now is are there interesting quantities that have to be the same here and here? So maybe the critical exponents may change. Many things can change, but some things may be preserved under this evolution. Or maybe there are some inequalities along this evolution. So I'm going to study two examples of things that are preserved or have a controlled evolution. And you'll see what kind of ideas go into that. So I'll start with something simple, which is to suppose that this whole evolution preserves a U1 symmetry. So there is some global symmetry and then we'll generalize it to central charges and so on. So I'm going to assume that there is some global symmetry. So we have some current, which is preserved. Everywhere, it's preserved everywhere, not just at the fixed points. We just have a complete global symmetry, a complete honest to God global symmetry that's preserved along the flow. And there are many examples of that, sorry. For example, take the O2 model, which is an interesting, which has interesting applications in statistical physics. So we have a honest to God conserved current and I'm going to study its correlation function in the same way as we did with the energy momentum tensor and prove some general facts about it. So we will use light one coordinates in which there are three interesting correlation functions to study and I'm going to use momentum space again because in momentum space anomalies in general are much easier to discuss in momentum space. I'm going to write down the most general possible answer that follows from one current variance and then we'll use the conservation equation. We'll discover that sometimes it's anomalous, sometimes it's not anomalous and we'll try to see how does this story go. So just from one current variance, you have to put a P plus squared over P squared times some function, which is a dimensionless function of P squared over the crossover scale. How did I get to these sunsets? So the fact that you have to have two powers of P plus is obvious. That's just boost invariance. Now the fact that A is dimensionless follows from the fact that A current has dimension one into dimensions. So the dimension of the current is one and therefore its dimension in Fourier space is minus one and we have a delta function so this thing has to be dimensionless. This is given by some function B of P squared over M squared which is some dimensional function, a dimensionless function and this is P minus squared over P squared times some function C of P squared over M squared. So far I only used dimensional analysis, I did not assume scaling variance, just dimensional analysis and boost invariance and of course the full point correct group. That's what I used so far. Okay, now, so now there is a very interesting and subtle story about what does the conservation equation imply? Okay, so this is a very interesting and subtle story. Perhaps if you haven't seen these kind of things before that would be extremely confusing and you'll have to think about it but I'll try to explain it to the best of my ability. So suppose you just impose the conservation equation. Suppose you just impose the conservation equation.   So that's the conservation equation in the momentum space and you can just impose this equation on these three possible functions. So the conclusion, the main conclusion that you find is that these three functions have to be identical. Okay, now if you impose this condition you'll end up getting lots of paradoxes. For example, you won't be able to reproduce these results by Feynman diagrams. There will be some contradictions with unitarity and so on and so forth. So the name application of the conservation equation fails and the reason is that you have to be very slightly more careful. What the conservation equations actually imply is a slightly weaker equation. They imply that P squared of A squared over M squared is equal to P squared of B and equal to P squared of C. So if you allow yourself to divide by P squared then you get this equation. But there may be some subtlety with P squared at the origin, okay? Another way to try to describe where the subtlety is is that when we impose the conservation equation then we basically want to demand that this would be true as an operator equation. Now operator equations cannot be imposed at coincident points. That sometimes may lead to inconsistencies and this is one example where it would lead to inconsistencies. So what you actually have to require is that this equation is true at separate points and you should allow yourself some freedom in what's happening at coincident points. So the fact that there is a P squared here means that you should only satisfy these equations up to things that would correspond to polynomials in momentum space after you hit them with the derivative. So the actual general solution of the word identity is not that A is equal to B equal to C but it's the following. So the plus plus correlation function, I'll just call this plus plus plus minus minus minus to make the notation simpler. So the actual solution is P plus squared over P squared and then there is A, which is some function of P squared over M squared plus some constant which I will call K lift. Why is this constant allowed? Because when we try to study conservation equations because we hit everything with P squared that corresponds to a polynomial in momentum space which is a contact term. So it doesn't actually change the fact that this equation is upheld because it should be only upheld at separated points, not at coincident points. Now plus minus, for plus minus we get B of P squared over M squared plus some other coefficient that we call K. We don't know what K is, we just put it there because we may. And minus minus is given by, sorry, this is the same function A. And minus minus is given by P minus squared over P squared, A P squared over M squared plus K right. So this is the most general solution to the worded entities. I've included three new constants beyond the naive solution. So the classical solution would be A equals B equals C but in fact we may not be able to impose this equation at coincident points and therefore I allowed three new constants, K lift, K lift and K right. Okay, and without loss of generality and without loss of generality, we may assume, I'll be done in a second, we may assume that the function A of P squared over M squared goes to zero at very high momentum. Because if it doesn't go to zero then we can absorb the constant in this case. So without loss of generality we can assume that this is true. So what I claim is that these three constants are necessary, they're physical and you can't determine them from the conservation equation. Okay, they are completely free. Now one comment before I'm done. This K here is a pure number. It doesn't multiply any function of momentum. So this is a contact term already before you hit it with a conservation equation. So this is just some delta function, a completely uninteresting delta function. While this guy only becomes a contact term after you hit it with a derivative. But before you hit it by a derivative it's a non-trivial function P plus squared over P squared. P squared is just P plus times P minus in light cone coordinates. So this is basically P plus over P minus. It's a non-trivial function and it's Fourier transform is not a contact term. So this actually does have some implications for separate points physics. So does this. So these two K left and K right are completely physical measurable quantities because they have some effect on current correlation functions at separated points. This is completely ambiguous. This you can play with whatever you can play with to your delight. You can try to set it to be one, zero. Maybe you can try to set it to be equal to this or to this or some geometric or some average of these two quantities. So this is completely ambiguous but these are completely physical. And the fact that they can be different is in violation with this naive classical conservation equation. So if K left and K right are not the same the wider than D, D mu, J mu cannot be satisfied at a coincident points. And that's where an anomaly arises. So what am I going to do next? What I'm going to do next is I will prove that if we have some RG flow or normalization group flow of that sort and we know something about K left and K right here. Then we know something about K left and K right here. So these are completely different physical regimes but you will see that some information about this case is preserved. The difference is preserved and the sum gets decreased. And we can prove it rigorously using some ideas that will take me 15 minutes to develop. But maybe I'll finish by asking if there are any questions about that. This is a very subtle business and you should be confused. So, right, so that's a great question. This is at the core of how an anomaly is creeped into quantum filtering. The conservation equations, these conservation equations classically just mean that like charge is preserved or the amount of charge that enters some region is equal to the amount that came from somewhere else. So basically this just classically means that charge is preserved. But in quantum filtering, this is promoted to a local quantum operator, a Hermitian operator in this case. And so when you write this equation, it's an equation about some operators. Hermitian operator is detecting some Hilbert space. And you have to understand what does it mean mathematically to write this equation. What it means mathematically is that when you study correlation functions of this operator, this quantum operator with other operators, this will indeed be true, but only as long as this operator does not hit an operator that sits at a coincident point. Why is that? Good, so I'll give you, I'll answer the question of why is that, which is a great question by the examples, okay? I'll give you a quick example, which is the Klein Gordon field, a free Klein Gordon field. So the equation of motion of a free Klein Gordon field is that, agreed? Classically, this equation is true. It's a set in stone, it's a true equation. But quantum mechanically, we have to interpret it as an operator equation because there may be some examples where we would not be able to satisfy this equation at coincident points. So let's see why. Remember the propagator. In four dimensions, this propagator is one over X, oh, sorry, I forgot. Is one over X minus Y squared, right? Now let's try to hit it. This are some quantum operators. Let's try to hit this equation with a box. So if this is a true operator equation, then this better vanish when X is not equal to Y. But if you take the depletion of this function, you get the delta function. This is the green function in four dimensions. So what you get is a delta of X minus Y. So you see that quantum operator equations, sometimes you just can't satisfy them at coincident points. You can satisfy them at separate points, but sometimes it's just forced to new that they have to be violated. You see, this is the simplest example one can give. Or it's just impossible to make sure that this equation would be true at coincident points. Now, usually these coincident points are dismissed as unphysical, but you see that in these cases, if I'm really careful about the meaning of this equation of imposing it only at separate points, you get this new coefficients K left and K right, which correspond to separate points physics. Only when you hit it with another derivative, they become coincident point physics and uninteresting. But before you hit it with a derivative, that's a genuine non-zero correlation function at separate points. And it was measured in the quantum Hall effect. Okay, so in the quantum Hall effect, there is such a K left is equal to one, K right is equal to zero in some sense. Okay, so this coincident point physics may some, most of the time, it comes from some other correlation function, which has the support at separate points. Like in this example, the fact that this is non-zero at coincident points can be traced to the fact that there is a propagator, which is physical. So one has to be really careful with this operator equations, because if you were not careful, you would miss the fact that there is K left and K right. And in some models, like a right moving fermion, you would not be able to write consistent correlation functions. Okay, so just to summarize, are there any other questions about that? Okay.


from the Sands Convention Center, Las Vegas, Nevada. Extracting the signal from the noise. It's theCUBE, covering AWS re-invent 2015. Now your host, Stu Miniman. Welcome back to theCUBE, SiliconANGLE TV's live production of AWS re-invent 2013. I'm Stu Miniman with wikibon.com. Excited to have on the program, Mylon Thompson-Bukovec, who's the general manager of Amazon S3. Mylon, thank you so much for joining us. Stu, it's a pleasure to be here. Thanks for having me. All right, so Mylon, there were, as always, a lot of announcements here at the show. Really good excitement, things that people are kind of cheering for. And, you know, storage had a pretty prominent role in Andy Jassy's keynote and some of the announcements. Some of them, actually we've been before the show and at the show. So can you kind of walk us through it's the S3 infrequent access, snowball and the rest of the announcement suite? That's right. So it's a big show for us. There's a lot that we do in storage services and one of the things that we're pretty careful about doing as with all of the AWS services is listening pretty carefully to what customers ask for and doing whatever we can to deliver that as fast as we can. So here I have with me one of the new things that we launched. I'm very excited to introduce you, Stu, to a snowball. This is the Amazon AWS import-export snowball. This was one of the things that we launched a couple of days ago. And it really addresses a need that we heard from customers where they had hundreds of petabytes all locked up in older data centers. They didn't have the networking in place to make it easy to get the data out and into a cloud environment where it's so much cheaper to store. So those customers didn't want to incur that one-time cost to go amp up the network for a data center that they didn't really need to have it for anyway. And they asked us to help. So we built the snowball, which is a physical appliance. As you can tell, we built it to be very rugged, tamper-proof. And this device, which is about 50 pounds, holds 50 terabytes of storage. And so you would go up into our management console and you would essentially order snowballs. And the snowballs would arrive at your facility. You would go ahead and plug it in. We provide, as you can see, all the cables and all the cords for you. You would plug it into your network. You would turn it on and you would start the process of loading the storage that you have on your on-premise over your existing network into that snowball. So this is pretty exciting for customers. Makes it very real. You can order multiple snowballs at a time. There's a 10 gigabit connection network connection on this so you can daisy chain as many of these up. If you wanted to get, say, a petabyte of storage off, out of your data center, into a snowball in under a day, you would get maybe 20 of these snowballs. You would line them up and you'd have that petabyte of storage on your collection of snowballs and send it off. I mean, Maylon, bring up a really good point because one of the challenges is moving data is difficult. We have the laws of physics and, right, that very expensive if we wanted to have a really big pipe to be able to do that, so. Yeah. So I mean, Stu, one of the most exciting things about this is I feel like we've leveraged all the learning that we've had for Amazon retail to make the best shipping experience possible. So once you get done with the data transition or the data transfer, you just power down, you close up the device and we have an e-ink shipping label that automatically updates with the address to send back to the Amazon facilities where we'll go take all the data off that snowball, load it into either S3 or Glacier and then erase the snowball on your behalf. So what's really exciting about this is all you have to do is set it out with your UPS pickup and your UPS pickup now. We'll just take that snowball and just take it away to Amazon and there goes your petabyte of storage. Yeah, that's impressive.  Now your data center people are going to be excited. When this box shows up, you got a Kindle sitting on front of it so that label with the e-ink technology is pretty cool and it tracks, you know, where your information is. Can you talk a little bit about just kind of the ruggedness and security of it because, you know, the physical, you know, we're worried about as well as the data itself that's sensitive information. That's right. We built this case to be very tough. It's very rugged. It's tamper proof. We actually have a hardware module inside the snowball that tracks if anybody tries to tamper with it. When the data is coming out of your data storage and going onto the snowball, it's encrypted with 256-bit encryption before it goes into the snowball. And all of the keys, the keys used for that encryption are never on the device itself. They're actually stored in Amazon's key management service in a very secure way and used with the encryption and decryption process. So this device that you see is both tough. I have to say our engineering team had a lot of fun with the testing of it because we got to drop it from very high places and see what happens. So this device we think is just going to unlock a lot of storage. And it's really the result of customers, a customer saying, I have a problem and can you help me? And, you know, the other two things that we launched this week. If I can, a couple of questions on the snowball. First of all, logistically, how long does it take to get it? How much does it cost to use this service? So the snowball itself takes a couple of days to arrive. By default, we've set it up for its two-day shipping. Just like everything else on Amazon, right? You can see how that makes sense. Now you can change that shipping at any time. The shipping is paid by you. So you can overnight, you can make it longer. The pricing of this is $200 per job. And then we won't charge you for the day that you receive the snowball. You get 10 days to go and transfer the storage onto the snowball and then ship it back to us. So we don't charge you the day it arrives and we don't charge you for the day that it leaves. You have 10 days to do it. If you need it for longer, it's only $15 a day to keep the snowball onsite a little bit longer. And one thing I did want to say is that the snowball today at launch, it supports the movement of data from on-premise into Amazon S3 or Glacier. It does not support the move of Glacier for data out of S3 and Glacier just yet, but that is going to be coming soon. We want you to be able to set up a job where we take the storage that you have on Amazon Web Services and be able to put it in a snowball and then ship it to wherever you would like. Yeah, and thank you for clarifying, because that's a point I've gotten a lot from the community is a lot of these services, it's easy or free to necessarily get it in, but if I want to get it out, there's a penalty on that. Is that, do you get feedback from your customers on that? Is there concern about making it easier to get my data out of Amazon? Yeah, the thing we hear the most from customers is it's basically surprised at how easy it is to get out. So we don't charge for deletes, for example. That's very core to us. If you have your data on Amazon S3 or Glacier, you should be able to delete it, you should be able to get it out. And in fact, the snowball makes it even easier to do that. All right, great. Thank you, snowball, really interesting. A lot of people, I'm sure, touching it, getting to know it, I lifted it, it's not that heavy. And yeah, some really, it's nice to see kind of some of the various pieces of Amazon working together, which sometimes we want to talk about how you work with the various services. But first, finish out kind of the announcements that you have, IA is a big one. Talk about how pricing of S3, IA and Glacier is working these days. I'm pretty excited about standard infrequent access. Again, just going back to how we think about storage. We've had standard storage out there for a while and customers love it, but what they told us is that there's a lot of storage that they have and they know that it's very infrequently accessed and they want a special solution for that. So what we did is we engineered a new storage class that we call infrequent access that is 55% cheaper than standard and really purpose built for data that you don't access very often. So in this storage class, we introduce a data retrieval charge. So if you don't retrieve that data very often, you just get all the benefits of that 55% storage discount and if you do it to retrieve it occasionally, it still ends up being cheaper. Now one of the really interesting things about infrequent access, it's the same durability, it's the same performance as Amazon S3 standard. So our idea is that even if you don't need your data very often, you can get it and you can get it just as fast as you can get it from standard. So when we think about that, think of it sort of as a warm archive. If you have data that you know you're not going to access very often, for example, a lot of our customers in the enterprise, they have data that they know is going to be susceptible to legal holds or HR holds of some sort. That's a great candidate to put in standard infrequent access because when the legal department or the HR department needs access to the data, they need it right away, but you don't want to pay the same prices as standard. In all of our data tiers, we think of the data tiering as just core to the business. So infrequent access, you can tier data from standard to infrequent access using policies and from infrequent access over to Glacier and you can do it all automatically based on things like the date of the object being created. That is so important for customers who have all their data on Amazon storage services. I had the opportunity to actually talk to a couple of customers here at the show that are using this. We're pretty excited about it because they understand when they understand their data, it's like, all right, we know our hot data that I'm going to reasonably access that pretty short term, definitely the first couple of days, then over the first 90 days, depending on I talked to one that said, oh well, I know a certain class of data actually might be in some kind of illegal information, as you said, therefore I can put it in this class, save myself a lot of money. And unlike traditional tiering was kind of tough, I had all these knobs and settings and moving things back and forth and estimating it was difficult as opposed to moving between these tiers. You know, it seems a lot simpler when it comes. That's very central. Just like it's easy for us to get data in and take data out. We want to create the notion of data movement by policy and by automation to be super easy for our customers to do. We had a chat with Andy Jassy earlier and the question was, when you're building these services, does the simplicity and the frictionless nature of it kind of get built into it? I think it comes in the name of your product line that when you're offering it, but yeah. Yeah, that's, I mean, every time we do a design review, every time we look at a new feature idea, one of the things that we always look for is, how easy is it for a customer to understand and to use and this data tiering and data visibility and all of these constructs around a data lake, which is what most customers, how most customers treat S3 are just central to how we built. All right, so the other couple of announcements around really leveraging the data, like the Kinesis fire hose, can you talk about how your storage group interacts with kind of the whole portfolio of Amazon products? Well, I think one of the biggest benefits of just moving to AWS is that you get the benefits of the entire platform. So if you think about Amazon S3 and you think about Glacier, we integrate with CloudTrail audit logging. So you have the benefits of a central place CloudTrail to look at all of our audit logs. We have CloudWatch metrics for size of storage. This is kind of in our DNA because we think this is what customers want. They don't want silos of functionality. They want this integrated solution where they can get the best of security and they can get the best of auditability and they can get the best of monitoring. So what fire hose really did, again, listening to what customers said, which is Kinesis Streams, super powerful. You can build amazing applications. We have a lot of customers like AdRoll that have on Kinesis Streams. But I want to build essentially a fire hose of small objects, any object of the size of one megabyte to come into Redshift, S3, or both Redshift and S3 at the same time. And I want that service to take care of the batch, compression, and encryption for me. So all I do is I set it up and the data just flows. So you're going to see a lot more of this thinking out of storage, which is how can we make it really easy for customers to get the benefits of a solution that take bits and pieces like fire hose, providing the data ingestion for S3 and Redshift and just make it seamless for the customer to just interact with that storage. Yeah, when you look at the analytics solutions, the new IoT announcement today, storage is an underpinning that is just critical to be able to make these solutions not only work, but affordable. Yeah, we're very excited about this. It's, you know, it makes us very proud on storage services when we know that when people talk about IoT and those hundreds of thousands of sensors, it's just all coming back into S3. And that's what the value of our storage services are is that we have this low cost, high durability, high availability. Customers don't even think about it. They just create what, you know, essentially our data lakes that are there today and then they act on it in all the different ways that they want after. Yeah, so last question I have for you. When we look at kind of the storage industry, you know, outside of cloud in general, one of the things we kind of came out of this week saying is not only is Amazon probably the largest growing storage company in the industry right now, but, you know, it's been a long time since there's been a company that said, oh, they're doing over a billion dollars in storage and, you know, our estimates are that you're doing more than that in storage. So any comment on kind of, you know, the role of storage in the IT ecosystem in general and where you play? Yeah, I think the role of storage plays so closely with the idea of innovation. If storage is expensive, if storage is hard to use, then IT departments and developers have to think about it. And we don't want them to have to think about it. We think the reason why we're here providing a storage platform is to make it so cheap and so easy to use that the developers can focus on the innovation that they want to do to go create these amazing things that you've been hearing about all for the last couple of days that customers are doing to transform their different industries of healthcare, financial services, you know, startups doing amazing things and innovation and enterprises just taking off right now. So storage for us, it is absolutely part of the foundation for how we make that innovation happen. All right, well my lawn, thank you so much for joining us. Thanks for showing off the snowball and I look forward to catching up with you more in the future. Thanks for watching, we'll be right back here with more coverage at AWS ReInvent 2015. This is theCUBE.


guy that I sketched with using sine and cosine and trig functions and stuff like that in Mathematica. He's obviously more detailed than this, but you can just go with me. So this is illustrating what he would have done. So he would have, actually, this is the opposite direction that he walked, but again, bear with me. So he'd go forward into the blue room, turn 90 degrees to the left, and in real life, he should have turned 90 degrees to the left. Then he'd walk into the pale green room, turn 90 degrees to the left, walked into the rainbow sherbet ice cream colored room, turned 90 degrees to the left, walked into the sort of salmon colored room, turned 90 degrees to the left. This is where he should have gotten back to where he started, but he didn't. So he has to finish his tour. He walks into the dark red room, turns 90 degrees to the left, walks into the dark green room, turns 90 degrees to the left. And now he's back where he started. And in real life, he did, in fact, make six right angles up here. So what's going on is a phenomenon that's called holinomy. So this is something that we're familiar with if we think about the surface of a sphere. So imagine I'm standing at the equator and my left hand is pointing along the, I guess, red arrow to the east. And my right arm is pointing along the blue arrow to the north. And instead of actually turning, I'm going to always face in the same direction and I'm going to walk in a path around like this. So first I go around the equator a quarter of the wave and up to the north pole and back down to the equator. And when I've completed this circuit and I get back to the place I started, I'm going to find that my arms have now rotated. So my left arm is now pointing north and my right arm is now pointing towards the west. So this phenomenon called holinomy is something that is sort of a signature of living in curved space is when you go around a closed loop, you come back as having been rotated. And this is something if you take differential geometry like towards the end of it, you work out Christoffel symbols and it's really nasty and you find this out using like rank three tensors and it's really not fun. But this is a really easy way to figure this out. So this is what Henry would have done had he not been turning. So imagine he's facing along the blue arrow and his right arm is pointing along the red arrow. So first he walks forward, then to the left, then backwards, then to the right, then forwards, then to the left. And you can see that when he comes back to where he started, he's rotated by 180 degrees. So this is sort of equivalent to him starting facing you and ended up facing 180 degrees from you. And so this is a phenomenon that we're going to explore a bit more in our other views of hyperbolic space. So what we're going to do is look at fully 3D hyperbolic space. And so this is going to be made up of cubes again, but instead of cubes meeting six around every vertical edge and four around every horizontal edge, they're going to meet six around every edge. And we can expand our version of Schlafly symbols to take into account three-dimensional tilings as well. So we start out with this tiling of the plan. This is 4, 4. We have squares that meet four around every vertex. When we move up a dimension, the first two numbers stay the same. So 4, 3 would be what shape cube? So these are squares that meet three around every vertex. So you can look at sort of the corner of the room to see that. And then the next number here is how many of those meet around every edge. So in the Minecraft tiling of space, you have four cubes that meet around every edge. In hyperbolic space, keep forgetting this doesn't work. In hyperbolic space, we can play around with these numbers again. So this is what would happen if you had cubes that meet five around every edge. And this is in the Poincaré ball model. And we could do this again for cubes that meet six around every edge. So this is a cube inside the Poincaré ball model. And we're going to start adding cubes around each of these edges. We're going to complete at the rest of the five around them. And we get this. And then we're going to add five around every edge there. And we get this. And this just doesn't make any sense to me. I mean, I do have a picture of something from this type of hyperbolic space tattooed on my shoulder, but I still can't make any sense of what's going on here. And so this makes a lot of sense as to why we want to move to virtual reality. So we don't need to look at this sort of mess of sort of mess of sort of squished cubes. And so this is just a model of what you're actually going to see in this space. This was designed by Royce Nelson. So this is again these truncated cubes. You can imagine if you put your eye right at the origin here and look radially outward. This is kind of like what you would see if you were actually in the virtual reality, but you don't get to walk around on this. Okay, so Henry, hopefully the headset works. We might have to keep you closer to the corner over here. I think it was just losing tracking. Okay, so before, I guess before you put this on, bear with me and the rest of the audience while I teach you the moves to the hall and only dance. I know you already know them, but it's way more fun for them if you pretend you don't. So okay, so take off the headset because otherwise they're going to know what happens in advance. Okay, so this is for all of you dance aficionados in the audience. This is like bad middle school dancing. So you're going to keep your feet planted and move your head around night at the Roxbury style. Okay, only the old people in the audience got that, sorry. So okay, so the first one is going to be the hula hoop.  Okay, you think you got that? Okay, and the next one is the TV set. So again, feet planted, hands up. Okay, you're going to go over to the left, down over to the right, up and keep doing this. Okay, and then the last one is the bicycle.  So now we're going to go forward, down, back, up, forward, down, back, up. Okay, I think you got those. Okay, so let's put you in the headset now.  And I'm going to switch everything over so they can see it. Okay, can you, are you good? Okay, can you turn like 45 degrees to either side? Keep going. Okay, perfect. Okay, so I'm going to change his view so that it actually is, it looks more like cubes that you're seeing. So he's looking straightforward into a magenta cube, look up a little bit. That's a red cube, down is a cyan cube, to your right is a blue cube, and to the left is a yellow cube.   Okay, so now pick your favorite one of those moves. Okay, so let's start with the Hula Hoop.  Okay, and stop and face straightforward. Okay, and what you'll notice is that the blue cube that was on his right side has now sort of started migrating towards the front.  Okay, stop. Do that once more. Okay, stop. Perfect. Okay, so now he's facing square onto the blue cube. Okay, so pick another move. No, let's try the TV. Okay, so let's try that. So let's see what happens. All right, and stop. Okay, so the green cube that was on his right has now rotated up so that it's sort of at a 45-degree angle up here. And let's try the last move. But first, I'm going to ask the audience, what do you think is going to happen when he does this move? I mean, I see vertically stuff. Is he going to rotate? I guess so if this is the axis, is he going to rotate this way or is he going to rotate this way? So I guess from your point of view, this is clockwise and this is anti-clockwise. Am I right about that? So which way do you think clockwise or anti-clockwise? So you think he's going to go this way? Okay, let's... There is not consensus in the audience, but guess what? This is experimental math and we can actually test it out. So let's try it out and see what happens. So in fact, he is taking the stuff from the top and dragging it down towards the front so that is going for you clockwise. Okay, so what I'm going to do now is... This is going to involve me having to direct your head because I'm going to stick you inside. Yeah, you know the story. Okay, so what I'm going to do is change his view back to the sort of original view of the cubes, the truncated cubes here. And then what I'm going to do next is I'm going to remove the pillars that are connecting the triangular windows to one another. Okay, so this kind of looks like now he's facing sort of in outer space filled with all of these jewels. And these things look kind of like Eicosahedra, right?   So, okay, so walk this way and put your head down and down. Perfect, okay. So now he's inside one of these. Thank you, Henry. Now he's inside one of these and it really doesn't look like a closed polyhedron from the inside. So it turns out that what he's looking at from the inside, this is actually an entire Euclidean plane.  So the idea here... Oh yeah, perfect to do that. If you can actually sit down a little bit. So what you can see here is that each of these is an equilateral triangle. So we've got a cube and we truncate the cube corner that gives us an equilateral triangle. There are six cubes that meet around every edge. That means six equilateral triangles meet around every vertex. So this is a three, six tiling of space and we showed at the beginning that that is a tiling of the Euclidean plane. So we can see this here. So he's looking at one of the vertices and there are indeed six different triangles that meet around these vertices. So I guess there's one last question and normally I ask other people this, but I don't have to ask you. I'm going to put you in here with an infinite number of monkeys. So normally I check to make sure that no one has like a monkey phobia because that would be pretty not fun. So you're not going to be able to see anything.  So normally when this happens just by the pure fact that there are infinite number of monkeys, someone is inevitably staring straight at a monkey's butt. So infinite number of monkeys means an infinite number of monkey butt. And it turns out that these monkeys, what they're doing is they are assembled into sort of an infinite structure. So they're all doing some sort of crazy yoga move like this, where their arms are pointing through two sides of the cube. Their head is at the top. One foot is at the bottom. Their leg is at this way and their tail is out the back. And this tiling, you go forward and then rotate by 90 degrees about that axis and continue going. And this is something, this is a manifestation of the quaternions. And this is something that you can't do in Euclidean space. So I apologize for all of the technical problems here. And if anyone has a very old version of iPhone software, there is a version that works on your phone. There's a current problem with it not talking to the gyroscope in the newer versions. But if you happen to have an older version, or if you happen to have a computer browser, you can go to h3.hypernom.com or h2xe.hypernom.com. And you can play around with this software yourself. So thank you so much for putting up with all of these technical issues. And I'd be happy to take any questions. Thank you very much, Sabata. So before I take any questions, I just want to remind something I forgot to say at the beginning. So at 5.30 today, there's going to be an informal presentation from a representative from the NSF who will speak about the types of grants that are out there. And I think those would be interesting to many of you. So it will be in here at 5.30. Okay, so any questions for Sabata? I was so clear there's nothing you'd want to ask. Okay, well, so she'll be here. And again, please come to the 3D printing workshop this evening and you can ask questions informally. And to our interpretive dancer, Henry, give us a hand.


Welcome traders to another Tick Mill earnings season preview with me, Patrick Munley. Before we jump into today's report, as always, want to hear that risk disclaimer. Most pertinence of today's presentation is the fact that the views and opinions expressed by me are solely mine, they're not addictive or representative of those held by Tick Mill UK or Tick Mill Europe Limited. Okay, so let's jump into today's report. We are looking at Nvidia. Nvidia set to announce earnings after the close of trade in New York today. We are looking for an earnings per share of 0.811 on a revenue estimate of 6.015 billion. A lot is at stake in this release, in its previous quarterly release, Nvidia missed analyst estimates and guided for a week fourth quarter. If the company beats estimates this time around, then its stock is a chance of climbing out of its current slump. Beating estimates will be a tall order for Nvidia at this release. The company faces a number of headwinds, most recently including the weak RTX 4080 sales, a declining video game industry, and an ailing cryptocurrency market. Nvidia makes a lot of its money off gaming PCs and crypto mining. Its bread and butter clients suffering will of course lead to its own sales taking a hit. It's pretty much a foregone conclusion that Nvidia sales and earnings will ultimately decline in the fourth quarter. The company guided for severe declines in both, so any positive growth in either metric is extremely unlikely. The question really is whether Nvidia will be looking at any buyback program. If it does another three billion dollars in buyback, then the EPS will ultimately make analysts consensus range. If it does no buyback, then it will miss by about 0.01 cents. All of this assumes of course that Nvidia's guidance was accurate at its previous announcements. If the guidance was way off, then the numbers could be totally different from what we're looking at. But I don't really see any indication at this stage that the revenue figure specifically should be too far out of sync with analysts estimates. In terms of the statistical trading patterns around the earnings release, Nvidia shares have ultimately moved higher in the immediate aftermath of earnings 8 out of 12 previous reports. On average, stock moved up 1.1 percent in the first day of trading after earnings. Based on the previous 12 earnings releases and videos more likely to trade higher one day after earnings for an average gain of 0.7 percent. On average, stock has moved higher by 0.8 percent one week after earnings. In terms of the the analyst community, let's check back there. Out of the 44 analysts giving a rating over the past three months, pretty much the consensus is that the stock remains a buy. In terms of estimates over the next 12 months for the stock price, we have an average of 212. On the upside, we have a 325 being the the max estimate there along the minimum downside would be a 110. In terms of the flow and sentiment from the options market around this earnings release, options traders are pricing in an 8 percent move on earnings and the stock has averaged a 4.2 percent move in recent quarters. The options market overestimated video stock earnings move 75 percent of the time really in the last 12 quarters. Notable buying though in terms of the option market contracts 7374 for a 215 call which expires on Friday and options order flow in general has been bullish. 70 percent of analyst coverage is also bullish on the stock. Let's pull up a chart here and see if we can identify any near term trading opportunities in terms of Nvidia stock. Last time out when we covered the earnings, we were looking for a pullback into the 140 to give us an upside objective of 229 and we pretty much traded right into that and then we've seen this pullback. Now, bear in mind this pullback that we're seeing has coincided with the general pullback in terms of markets at the moment. What I'd be looking for technically now would be any three-wave corrective move back into these prior highs. So 188, 187 is the yearly pivot. So any pullback into that area, I'd be watching for bullish reversal patterns on the four-hour, the daily timeframe to reengage on the long side and I'd be looking for a move up now. The next technical target for me is going to be a 242 that coincides with the ascending trend channel resistance as well. So any weakness into the late 180s, we're watching for bullish reversal patterns to take this on the long side to target a 240 test. Now, equally, any loss of that 180 support zone, I think we can see a much deeper pullback and we have monthly projected range support down into the early 160, 164 specifically there. Again, I would be looking for bullish reversal patterns there to reengage on the long side, same upside objectives. However, any daily close through that trend channel support will be a bearish development and we could be looking at retest the pivot down to 138. As always, trade as planned the trade, trade the plan and most importantly, manage your risk. Until next time, thanks very much.


the president of the United States thank you thank you very much to David Rockefeller and Ambassador George Landau and Robert Helander and all of you welcome to the White House complex I think they call this the White House complex because nothing in Washington could ever be simple except maybe Congress is compulsion to spend and it's a it is a pleasure to have you the members of the Council of Americas here today in 1933 Franklin Roosevelt said common ideals and a community of interests together with a spirit of cooperation have led to a realization that the well-being of one nation depends in large measure upon the well-being of its neighbors it is upon these foundations he said that Pan-Americanism has been built well FDR could easily have been speaking for you for nearly 30 years the Council of the Americas has promoted understanding and cooperation between the United States and the nations and peoples of Latin America the Council has brought leaders to our private sector together with business and government leaders throughout Latin America it's helped strengthen the old and enduring friendship between the United States and Latin America it's been helping to make our nation's great engine of hope and opportunity the private enterprise system the engine of hope and opportunity for Latin America as well and let me just say that the entire hemisphere owes its gratitude to the Council and in particular to your chairman one of the great citizens of the Americas David Rockefeller you know when some people talk they make it sound as though development is some kind of a magic process not the hard work of real men and women reminds me of the story a lot of things remind me of stories these days and if I've told you all this one before just forget it and pretend I didn't it comes in handy every once in a while this particular one is about an old fellow that lived down by the creek and he had some creek bottom land and he went to work on it one day it was all brush covered in rocks everything there and he worked and he got rid of the brush and then he hauled the rocks away and he cultivated and he fertilized and planted and finally he had a real garden spot there and one Sunday morning after church he asked the minister why didn't he after lunch come on out and see what he'd accomplished well the minister came out that afternoon and he took him down there and the minister said I've never seen calls so corn so tall he said my goodness the Lord has blessed this land and then he went on he said melons he said look praise the Lord look at the size of those melons and he went on about everything else and all of that and praising the Lord the old boy was getting pretty restive and finally he said Reverend I wish you could have seen this place when the Lord was doing it by himself well today two great winds are sweeping across Latin America the wind of free enterprise and the wind of democracy they're warm and nurturing winds that carry within the general reins of hope for Latin America's future country after country has seen the disaster of state controlled and dominated economies in both small and large steps nations are beginning on the difficult path away from stateism and toward freer economies as Brazilian President Jose Sarnes said recently private initiative is the engine of economic development in Brazil we have learned that every time the state's penetration of the economy increases our liberty decreases I want to diminish the state's presence well from one end of Latin America to the other the message of reform is on everyone's lips and despite problems progress is being made in the South Argentina for example has reformed its tax code liberalized trade and moved to privatize and reform publicly owned companies between 1984 and 86 it reduced its budget deficit from over 12% of gross domestic product to 4% it has cut inflation dramatically and once again the economy is growing not sluggishly but at a robust five and a half percent last year meanwhile on the American border Mexico has also reduced the number of peristatal companies and is moving toward tax reform and more market-oriented pricing and has begun to make trade more open debt remains a burden for too many countries but the final and best way to lighten that burden is not by jeopardizing access to the international financial markets but with freer trade sounder monetary and fiscal policies and greater economic growth to take one example Colombia is following this path and last year was able to make its first voluntary euro financing since 1982 and other nations have eased their debt loads with debt equity swaps all the countries of Latin America need to rely less on borrowing by one course or another and more on investment from abroad the United States is determined to help the countries of Latin America grow as a young tree grows toward the Sun pushing the boulder of debt out of the way as they do we've encouraged continuing private lending we're encouraging Latin American leaders to trust more in the energies of their people and less on government just as important we've provided the market that Latin America needs if it is to pay off its foreign debt we buy nearly half of Latin America's exports while Europe and Japan together by less than 10% if our trade balance with Latin America had been the same over the last five years as it was in 1981 our overall trade deficit would have averaged 25 billion dollars less a year put another way one quarter of the trade deficit during our administration came about as a result of the debt crisis we're convinced that if Latin America commits itself to sound policies for economic growth it's going to bounce back then our sales to them will rebound as well which will be good for everyone so you see this is an investment in the future of our entire hemisphere and investment in our future as well as Latin America's that's the best investment the United States can make but the economic growth of the next century in Latin America is to be as powerful and relentless as the Amazon then democracy in that vast region must become as towering as the Andes just as the Amazon rises in those magnificent mountains so too does the river of opportunity rise in the highlands of freedom today it's possible for the first time in our history to see approaching the moment when the entire Western hemisphere from the Canadian Arctic to Tierra del Fuego is composed of democracies as the Peruvian novelist Mario Vargas Yosa said recently for the first time democracy or incipient democratic forms of government are being established in the countries of our hemisphere with clear popular support and with equally clear rejection of Marxist revolution or military dictatorship today anti-democratic alternatives are running against the will of the people supported only by economic and intellectual elites you know there's a thundering sound that echoes from the peaks and across the plains of Latin America it's the sound of people marching not in uniforms not behind red banners but rather marching one by one in simple everyday working clothes marching to the polls ten years ago 33% of the people of Latin America and the Caribbean lived in democracies or in countries that were turning to democracy today over 90% do several of these new democracies have faced crises in the past few years but unlike earlier times every crisis has ended with democracies forces still in control only a few countries resist the democratic tide and among these the most dangerous are Cuba and Nicaragua as president Kennedy told us more than a quarter century ago in his words the forces of communism are not to be underestimated in Cuba or anywhere else in the world the advantages of a police state its use of mass terror and arrest to prevent the spread of free descent cannot be overlooked by those who expect the fall of every fanatical tyrant if the self-discipline of the free cannot match the iron discipline of the male fist then the peril to freedom will continue to rise we must remember that in Nicaragua the freedom fighters fight is our fight our goal is democracy in Nicaragua and throughout the hemisphere the 19th century Europe emerged as the first great industrial continent of the earth in the 20th century North America joined it in the 21st century Latin America will also enter that company for the sake of our own peace and freedom it must be a democratic region when it does or as the Argentine poet Jose Hernandez wrote more than a century ago the Americas have a great destiny to achieve in the fate of mankind an American alliance will undoubtedly be achieved and the American alliance will bring world peace the Americas must be the cradle of the great principles which will bring a complete change in the political and social organization of other nations so to all of us and all of you who are helping build the future of this hemisphere of hope I thank you for what you're doing I can't resist this comparison we all have the common heritage from the Arctic down there to that southern tip and the pioneer heritage and to those of us that turn to freedom as the basis for all that we did we can see what has resulted just those others that still have yielded to statism I have become a collector of stories that I can prove are told by the Soviet people among themselves showing a certain growing cynicism about their heritage and this last one had to do with a young man buying an automobile it's not an exaggeration it takes 10 years to get delivery in the Soviet Union but you have to pay for the automobile right at the first not when you get it so this young fellow was going from agency to agency and getting permits here and it's there and stamps he was collecting and finally at the final place in the final stamp was put on and then he laid out the cash and the man said come back in 10 years and get your car the young fellow started to turn away and he turned back and he says morning or afternoon and the fellow behind the counter said what difference does it make well he said the plumbers coming in the morning thank you all and God bless you all thank you very much Mr. President for what was really a very inspiring and I think most significant speech it's a great privilege for all of us to be able to welcome you here again to the Washington Conference of the Council on the Americas and also to the America's Society which the Council is an affiliate we're particularly grateful to you also Mr. President for having participated in four of these conferences since you've been in office and more importantly you have demonstrated through your presidency that you were an ardent champion of freedom and democracy through the nations of this hemisphere as you yourself pointed out today over 90% of the people of our hemisphere live in countries where democracy reigns reigns at this present time but I would have to add to that that you have never swerved for a moment in your effort to make sure that the figure is 100% countries such as Argentina Brazil and Uruguay Haiti have to take satisfaction in the active moral support that you gave them in their successful struggles to regain the cherished blessings of freedom and pluralistic society. You've also been a champion of the virtues of the market economy and a strong private sector as a means of promoting growth in this hemisphere and this too has given courage to nations struggling to rid themselves of the crushing debt burden which have come to see that the inward looking government dominated economies which they had encouraged in the past were not producing the desired result. Many of them are now promoting greater export orientation with far more private sector participation. Mr. President for these and many other reasons the Pan-American society which is also an affiliate of the America's society has designated you to be the recipient of our highest recognition the gold insignia. With your permission I'd like to say just a few words about its origin and history. Founded in 1912 the Pan- American society has dedicated to fostering better relations among the hemisphere nations. It's become a tradition to encourage this objective by awarding the society's gold insignia to outstanding leaders of the democratic nations of our hemisphere. As we celebrate our 75th anniversary this year we're delighted that the president of the United States is being so honored. The first gold insignia was given in 1932 when the Pan-American society celebrated its 20th anniversary. Numerous heads of state from the Americas have been recognized with it since that time. Needless to say we're deeply honored that you could spare the time to be with us today Mr. President and we're pleased that you will join the list of distinguished leaders of this hemisphere in accepting the gold insignia. Since you were sworn into office for example Mr. President 14 democratically elected Latin American heads of state have honored us by accepting this award including among others such distinguished people as President Duarte of El Salvador, President Sarnay of Brazil, President Alfonsina of Argentina, President Lucinchia of Venezuela, President Febrez Cordero of Ecuador and President de la Madrid of Mexico. Among the US presidents who have been honored over the years include presidents Hoover, Roosevelt Truman, Eisenhower, Kennedy and Johnson. You're accepting the gold insignia today reestablishes an important link between the Pan-American society and the White House for which we of course are especially grateful. So at this time Mr. President may I ask you and our President Ambassador George Landau and the vice chairman of the Pan-American society Mr. Robert Hielander to come forward to join me in presenting this award to you. Ladies and gentlemen that concludes