Thank you very much. Thanks for organizing this great program, for, well, having this stimulating summer school in this nice environment. I'm glad to be here. I have to say right away that my talk is somewhat different from the talks that we've heard before. In particular, there will be no time and there will be no waves. Now that is, I have to say, not because I don't like time, but simply because I can't do it in a time-dependent way. And I think it would be great if I could somehow motivate people here to work on this problem and to do some time-dependent version of what we do here. And if you think somehow historically, there has been about 10 years ago a lot of work. This was to a great deal, motivated or initiated by Erdersch, Schlein and Jao about how some microscopic quantum dynamics give rise to an effective macroscopic dynamics. And also for their work, I think it's fair to say that the understanding of the stationary theory is basic. It was the first step and once this is understood, then one can do the time-dependent step. So therefore, I'm trying to make, I mean, it's kind of analogous to what they did except we do things at positive temperature. It's a different regime, but the setup is similar. And so I try to explain this stationary model. But keep in mind that there are time-dependent Ginsburg-Landau equations and would be a big problem to understand those and derive those from these microscopic equations. So the title, as Funk already mentioned, that's Microscopic Derivation of Ginsburg-Landau theory. Most of the results that I will present were obtained in joint work with Christian Heinzel, Robert Saringer and Jan-Philippe Soloway. And especially today's lecture is also based on joint work with Marius Lem from Caltech. And if you want to read something about this background, so we've been working on this for at least five years. Now there's a review of Heinzel and Saringer. So this is a review in Journal of Mathematical Physics 2016 where you can get some background information and some, I mean, references to related results. Okay, so there are two buzzwords that I want to mention that are really important for what's going on. The first word is that of a phase transition. So phase transition, that's a word from statistical mechanics. This happens that if you vary some parameter in the system continuously, then some property of the system changes abruptly. Changes abruptly means that the quantity or derivatives of it are discontinuous. Okay, now that's a rather surprising phenomenon when you think about it because usually these things, they are finitely many particles and usually things behave analytically in this parameter. And then somehow when we pass to this infinite volume limit, we lose this analyticity and we get these jumps. There's one phase transition that everybody knows that's that of water, right? We know water in its solid phase, its liquid phase and its gas phase. And you see the phase transition, that's not a random phenomenon. It's rather the contrary, I mean our thermometers, right? When we talk about Celsius, right? That's zero Celsius, that's where the phase transition between solid and liquid happens. And when we talk about 100 degrees centigrade, that's where we go from liquid to the gas phase, right? So it's really, we've made our thermometer based on these phase transitions of water. Now, I will also be speaking about the phase transition and its so-called superconducting phase transition. That's a transition that happens at much, much lower temperatures between either 1 and 20 Kelvin. And it's a phenomenon that certain materials lose, completely lose their electrical resistance when you pass a certain critical temperature, okay? It's a phenomenon that was discovered, I think, in 1911 by Kammerling Ons. And it's a quantum mechanical phenomenon. It's a phenomenon that cannot be explained by classical physics, classical mechanics. And it's a macroscopic phenomenon. By this, I mean you don't need a microscope or anything to look at this, but you can really watch it with your bare eyes. And it's one of the few quantum mechanical phenomenons that we have like this, okay? And that brings me, this scale thing, that brings me to the second big word that we have here. The second thing is that of effective equations or effective theories. I should draw the same. By this, I mean when you drink water, when you do something with water, you don't want to think about H2O molecules, right? You just want to have your water and you think of it and you have a piece of water, a size, and that's how you want to think of it. You want to forget all the atomic structure that's below it. And the same thing happens about superconductivity. When you want to make an experiment or use it for something, then you want to forget the atomic level and you want to think of this really on the scale of the thing that you look at. And that put differently when you say some are in this superconductivity phenomenon, they are somehow 10 to the 23 particles involved. You don't want to work with a function of 10 to the 23 variables, right? That's what quantum mechanics would tell you to do. But that's just an insane thing. You cannot do this, right? But you rather want to describe this thing simply by a single function of three variables. And so what an effective theory does is somehow it takes this huge wave function of 10 to the 23 variables and somehow cooks up an effective object, which just depends on three variables. And everything, or at least the important things, should be describable by this new thing of three variables, okay? And so I told you that this phenomenon was discovered in 1911 by Kamaling Ons. It took quite some time until 1950 until Ginsberg and Landau came up with an effective theory. And let me write down this functional. It's not so important yet. There will be some constants and we'll talk about those later. It looks very much like a Schrodinger energy functional. A is a magnetic field. Psi is this function, which depends on three variables. It lives on some open set omega. That's where your superconducting sample. Let me put some w, psi squared, perhaps another parameter. I'll speak a lot more about these things, what it means. I just want to have it on the board here. So I want to have a minus and then plus lambda 3 psi 4, okay? So essentially this looks like this ordinary Schrodinger energy functional, except that you've added a nonlinearity. And it's the simplest nonlinearity that you can think of. It's psi to the power 4. And then what Ginsberg-Landau said was, well, you have here four parameters, lambda. The lambdas, you go measure those. And then once you have this, then you can use this functional and make predictions. And somehow it works. That's kind of a miracle because there was not much understanding why you should have those things. What these terms mean, it just is a surprising thing that using this functional, you can describe superconductivity to very high precision. It's a thing that's also numerically very convenient. And also in the mathematics literature, this has been studied a lot because it's somehow prototypical for many interesting phenomena. And then, so this was 1950, in 1957, BCS, 1957, Bardin, Cooper and Schriefer proposed a microscopic model. Now really a model for superconductivity that really works on the atomic level. And then, of course, so now you have two different models and somehow how are these two things related. And that was clarified, at least in the physics literature, in 1959 by Gorkov. And there's an alternative argument later, later Duchen. And these two, they showed how BCS theory leads in a certain limit to Ginsberg-Landau theory. And so what I'm interested in is making this statement mathematically rigorous. I mean, these papers are kind of mathematical, but not to the extent that a math that can be made rigorous. I mean, this is not hand-waving, is what I want to say. But it's not like we quantify error terms in these approaches, but we have to do something completely different. And this is actually a rather important aspect of our work. We have to really come up first of all with a model where we put length scales in and really get some well-defined limit that we can analyze mathematically. Anyway, if you want to give a name to the field that this is about, or this talk is about, you can call it non-commutative, qualitative calculus of variations. Calculus of variations, therefore, what do people do in calculus of variations? They want to minimize something. And that's actually a fundamental idea in our work as compared to these physics papers. We do not want to derive the Ginsberg-Landau equations from the BCS equations, but rather we want to derive the ground state, the minimal energy state in Ginsberg-Landau theory from the minimal energy state in BCS theory. It's one motivation why we want to do this is because we say, well, perhaps BCS is not the whole truth either, so I should backtrack a little. We do not only do it for the ground state here, but for approximate ground states. And so we're saying that our work is therefore more general because even if BCS theory does not describe the whole truth and if there's a more complicated model which has more effects, then what we would hope is that these more complicated ground states are approximate ground states for the BCS theory. So there's a stability statement involved. And these almost ground states of BCS are almost ground states of Ginsberg-Landau. That's what we prove. It's yet another question to really do something on the level of equations. And by this I mean solutions to the equation which are not ground state solutions. For instance, such vortex solutions that you might have heard about these are precursor of lattices. They are probably not ground state solutions, but they are higher up in the spectrum. And so there would be some correspondence between solutions that we do not have. That was the calculus of variations part of this name. And if you look at this functional, what I will talk about today is that to a logic's extent this part here, that's what's often called the Mexican head. You see if you look at this, this is a psi to the power 4. This goes up and then it goes down. There's a certain minimum and at zero it goes like that. You think of psi as a complex valued function. This part, this can be explained by a classical calculus of variations theory. That's what I will do today. Now the other part here, in particular that one, that's a lot harder. I hope I can tell you a little bit about this in the two following lectures. This is done using semi-classical analysis. This thing really comes from the fact that if you have a multiplication operator and a differential operator they do not commute. And that gives you commutators and these commutators they're really relevant for these terms. And that's somehow this non-commutative part that will play a big role. But in order to just explain you what we want to do and you know what the difficulties are I just want to start today with a commutative version, just ordinary calculus of variation problem. It's something that I guess could have been solved like 20 years ago or something. Nevertheless, I think it's good if we go through this slowly and get used to this mechanism. So that's now really the first part here. And that's translation invariant ECS theory. And this follows to a great extent this paper with a lemma. Okay, and so the goal for today is that we will see mathematically rigorous a phase transition arise and we will see how this Mexican head is related to this phase transition. And so we have a certain class of admissible states. There will actually be pairs of functions, gamma and alpha. Okay, and this alpha, this is what first is called the Cooper pair function or Cooper pair wave function. And that's the object that describes superconductivity. I should, by the way, just a small remark. I will use the word superconductivity. There's a similar phenomenon which is called superfluidity. I will use them interchangeably and always mean superconductivity. Okay, even though this might be not completely physically correct. Anyway, so we have a functional F which depends on T. T is the temperature and we'll minimize this over all such admissible gamma and alpha. And the question is simply if we minimize this, does the minimizer have alpha equal to zero or not? Having alpha equal to zero means we're in the normal state. Having an alpha which is different from zero means we're in the superconducting state. And what we hope is that if we change the temperature, then there's a transition. Namely for low temperatures, we're in a superconducting state. So this minimization problem should have a minimizer alpha different from zero. And if T goes above something, then we want to have an alpha which is non-zero. Okay, that's the basic question that we will ask. All right, so let me do some assumptions. I think we have to start on this. Okay, thank you. So first of all, well, there's an underlying dimension and we'll always assume that this is one, two, three. This is something to do with solar left inequalities and the power of psi to the four. But that's certainly fine for all physical applications. Now, the admissible states, these are functions, gamma and alpha. They are both functions which are defined on Rd and take values in the complex numbers. And such that there are two conditions that we want to have and they will be nice functions. So I can Fourier transform them. And so the Fourier transform of alpha is point wise. I mean, it's square is point wise bounded by the Fourier transform of gamma hat times one minus gamma hat. Okay. And the second assumption is that's a regularity assumption. That's that one plus psi squared gamma hat of psi. So this in particular means that gamma hat has values between zero and one. And here we have that this integral d psi is less than infinity. Okay. This is kind of H1 assumption, right, except that somehow the gamma hat is kind of the square of something. So it enters here linearly. This implies you see that alpha hat squared is less than gamma hat. Right. And so therefore this implies in particular that alpha is an H1 function. Okay. Now what we will often do is we will write this first condition. So we'll write, we will often write capital gamma. This is a two by two matrix, which has the little gamma as its one one entry. The alpha hat as its one two entry. Then we make it self-adjoint by putting the complex conjugate down here. And there we put a one minus gamma hat. Okay. Now it's a little computation, right? This is a two by two matrix. We can all compute its eigenvalues. And we can say that this assumption that the diagonal entry squared is less than the product of the diagonal entries. That means exactly that this matrix in the sense of matrices lies between zero and one. And observe. All right. Good. And then one more thing. This I just say this for completeness. One could also assume that this function alpha is an even function. That's what we will need in the second and the third lecture. Therefore I'm writing it on the board, but the theory goes through exactly without this assumption. So don't worry about this. All right. So these are the admissible states. Okay. So the first one is gamma and the alphabet that sits in the off diagonal. Now I'm going towards defining this energy functional. And I will have the, so that's often called the free energy. And it will depend on a couple of parameters. And the first one is, well, the obvious one is this parameter T, which is the temperature. The parameter mu, which is a real number, which is called the chemical potential. The interesting case is mostly when mu is positive, but you can just treat this as a parameter. And then there is an interaction potential V, which belongs to some LP on our D. It's a real valued function. And so this is called the potential. And in order to have a minimization problem that is bounded from below, we have certain, we have to assume certain regularity conditions. So that's P greater than one in dimension one P greater than one in dimension two and P greater than three over two in dimension three. Don't worry too much about these things. This is really just if you want to push it to rough potentials. But everything is also interesting if you assume that V is a bounded compactly supported function. Okay, good. Now what's our energy functional? So f of T depends on these two parameters gamma and alpha. There's a first term that corresponds to a type of kinetic energy. XI squared minus mu gamma hat of XI D XI. Now there's another term, which I'll write later. And the last term is V of X alpha of X squared DX. Okay, that's like a potential energy if this would be a Schrodinger functional. Now you see these terms do not interact. So this is just involves a gamma. This just involves an alpha. So now the missing term in the middle, that's where they compete with each other. Okay, and that's an entropy term. And that's the following. So that's T. This is how the temperature enters in the game. Let me write it like this. So I've introduced for you these two by two matrix. And now we just take this matrix times its logarithm. Okay, that's still a two by two matrix. You do this for every fixed XI. And while you compute this trace, the C2 trace, and that's the coupling term. Okay, yes? Can you briefly touch on the physical interpretation of gamma? Yes, I mean these are the electrons. So to say, or the particles, the usual particles, that would... I don't know when this is your question. I dropped here certain interaction terms of the gamma. So there is an interaction of the gamma with itself and some exchange term. Those I've dropped in order to make the model simple. Okay, so therefore you don't see them. Therefore it looks very simplistic at this point. But I think this captures the difficulties. Okay, we can talk more about it. Let me just make this a little bit more explicit, right? If you don't like somehow traces and this notation. Well, we can just compute the eigenvalues of this gamma, right? For every fixed XI. And we'll see that the eigenvalues of gamma hat of XI are 1 half plus minus gamma hat of XI minus 1 half squared plus alpha hat of XI squared. Okay, these are the eigenvalues. So therefore what is this trace?  Okay, that's all there is. Okay, that's how the gamma and the alpha are coupled. So, and as I said, right, the goal is to, I've written it up there is to minimize this. And more precisely, we're not really so much interested in minimizing. I mean getting the number, but really determining whether we want to have alpha equal to zero or not when there's a coupling or not. And therefore I think the natural first step to do is to see what the enemy is that we are competing with, right? So let's see if alpha is equal to zero, can we compute what gamma wants to be? Okay, and that's called the normal state. So what we want to do is let's compute, let's solve this minimization. Now I just minimize over gamma. Well, and we'll turn out that we can can compute this explicitly. Here it is. Perhaps I should actually write it down if you want. Okay, so this is the integral. We have a xi square minus mu, a gamma hat of xi, and then we have a plus from over there t times. And now we have gamma hat xi log gamma hat of xi plus one minus gamma hat of xi log one minus gamma hat of xi. Yes. And you see there is no, well there might be one parenthesis missing, you see there is no global constraint on on gamma hat. So we can just minimize this thing point wise. Okay, so let's do this. I've written this down as a little lemma. So for all h in R, t positive. So we have to do this point wise. So there's a h rho plus t times rho log rho plus one minus rho log one minus rho. So I've just called gamma hat of xi. I called this rho. That's what I've done. Well, and now we can minimize this simple exercise. You see, if you abstractly, this is computing a Legendre transform. Okay. And so the answer is equal to minus one over minus t, sorry, minus t logarithm of one plus e to the one minus one over t times h. Right. And the inf is attained if and only if rho is equal to one over one plus e to the h over t. Okay. This might be familiar to some of you. That's what's called the Fermi Dirac distribution physics. Okay. So what we found here, if we go back now to this, it was the minimization problem that we solve, that we wanted to solve. We do this. So h is equal to xi square minus mu. We plug this in. So we find, so the minimizer. And it's in fact the unique minimizer of the problem with alpha identically zero is gamma zero hat xi equal to one over one plus e to the xi square minus mu divided by t. So that's a definition and that's a normal state. That's the one that we have to beat. If you want to have superconductivity, we have to get a better energy, a smaller energy, a free energy than this one. Good. So I don't know what I should draw a picture. I mean this, so this describes the occupation of momentum. This is the xi axis. Somewhere here is xi equal, I mean xi squared equal to mu. Let me draw it like this. Okay. So this function, it's very, very close to one and then it goes down and then it's almost equal to zero for large values of xi. Okay. And so the temperature is the parameter that the slope how fast it goes down. Okay. Good. So now let's talk about phase transitions. I need to introduce one more thing. It's a key player for the next talks. So one more, once you have this thing, so one over one plus e to some things, there will be a lot of cautious and tanges and so on appearing. Okay. That's just, if you differentiate this thing, play with this thing, you get these things. They're not particularly nice, but I try to emphasize their main features. So here's the operator that will play a crucial role. I call it KT. And this is minus Laplacian minus mu. Well, that would not be so frightening. But now I divide by Tange minus Laplacian minus mu divided by two T. Okay. So by this, as you all know, I mean a multiplier in Fourier space. And the crucial features is that for large moment of this thing, right, the Tange down there is like one. So this just behaves like the ordinary Laplacian. Okay. And of course I have to tell you why this thing arises. We will see it soon. Let me state one little lemma before that. There's a unique TC. Okay. Greater or equal than zero, but it's finite. So that, so we look now at the, it's asserting a type operator. So you take the KT plus V, you take this operator and add this V as a potential. And now you look at, it's a self-adjoint operator. We can look at the lowest point in its spectrum. Okay. And now this thing, the infimum of the spectrum is less than zero. If T is below TC and it's greater or equal than zero, if T is greater or equal than TC. That's not a difficult lemma, right? Because this symbol for each fixed Xi is a monotone function in T. This means if you think about the variational principle that the eigenvalues also are monotone functions of T. Okay. So it's a monotone function and now this TC is just the first point where you hit zero. Okay. So there's nothing deep in here. Okay. But now the important thing is what this TC stands for. And that's the first main theorem here. This one I should start with. Okay. So this is a theorem. And that says that if we're below the critical temperature, you know, the critical temperature might be equal to zero, then the first statement is an empty statement. Okay. That's included. I'm not claiming that it's a positive number. In the same sense, I mean this might be an empty statement if TC is equal to zero. Then we have superconductivity. Okay. For small temperatures, this is less than, and now I can write because we know what the solution is if alpha is equal to zero. It's just FT of gamma naught zero. Okay. Below TC, we have superconductivity and above TC, well, we do not have superconductivity. And we have this even in a strong sense, namely the free energy is strictly greater than the free energy of the normal state for every admissible state, which of course is different from the normal state. Okay. So the nice thing is really there is a fixed T temperature where it goes from superconducting to normal. And what's particularly nice is that we can characterize this number in a linear way, right? This is a linear theory. So now at this point, we can, and that's in a lot of these references that I mentioned up there, we can really do computations now, right? We can add parameters. So this TC depends, of course, on V, it depends on mu. And now we can, in certain regimes, see how it depends. And that has been done both in the math and in the physics literature. It really is a computationally very, very useful thing. Now, there is one part of this theorem, which is very simple. That's the one I'll show you. And that's the part that answers how this KT operator comes in. And this is the obvious thing that you want to do, right? You want to create superconductivity. So how do you do it? You ask yourself, is this normal state stable or not? Okay. It's a local question. So what does this mean? Well, you just compute. So compute is your normal state.  And I'm going to differentiate twice with respect to this T. Why do I do it twice? You see here, this thing just depends on alpha squared. And over there in this formula, right, this is what I didn't write out. This is also alpha squared only. Okay. So I can, the first derivative is clearly zero. And what I get is, well, that's obviously a quadratic form in A. And that's exactly where this KT plus V operator rises. Okay. And therefore, obviously, if this guy has a negative eigenvalue, then I can make this thing smaller than its value at T equal to zero. Okay. So that's a simple thing. The thing that's not trivial in this theorem is that this local condition of stability actually is a global criterion. That if you cannot get to superconductivity in this by perturbing slightly around the normal state, then you cannot get to superconductivity at all. That requires a proof. It's not very difficult, but I want to skip it just to get ahead and do some other things. Okay. So let me just say this proofs, this proofs the first part for the second part. See these notes. Okay. Perhaps it's helpful if I draw a picture. We'll have some board space here. Picture that I would like to draw is the energy. As a function of the temperature. I don't know. I mean, there's no nothing special about zero. This is a convex function. There's something like that. Now over here, there's a TC. That's a critical temperature. This thing I've drawn here, that's the energy of the normal state. And here I leave the curve of the normal state. I know it's a simple exercise to see that the energy is continuous. Okay. And it's also a K function. So the question is somehow, how does this bifurcation here work? And as I said, by phase transition, we understand that there is a non continuity, at least in some derivative. And that's what the second theorem will be about. And which is really the part that gets us close to the Skinsburg-Landau theory, the Mexican hat. Okay. And what I will show you in the next theorem, that's what this picture says, is that this leaves here quadratically. Okay. So that is quadratic. And that's what's called a second order phase transition in physics. So the first derivatives are continuous. The second are not continuous. That's really the infimum of Ft, gamma alpha. Okay. And we will compute the coefficient here. And later on, to tell you what will happen in the second and the third lecture, this is where the Skinsburg-Landau arises, just in a non commutative version of this talk today. Okay. So here's the theorem. Obviously, there is only something to prove if the critical temperature is positive. So let's assume this. And so I'm looking at the free energy for T very close, but below the critical temperature. Well, we said that it's continuous. So I put here the energy of the normal state. By the way, I put here a T, right? Not a Tc. That's slightly more convenient, but you can just expand that. And then what you'll find is that this is Tc minus T over Tc to the power 2, as I told you. And there is a number E here plus and the remainder, which is power 3, as T approaches Tc from below. And I want to tell you what the E is. Well, the E is again the solution of a variational problem, some problem script E, some functional that I will tell you in a minute. And what we've seen here in this computation, really the interesting states are those that make this KT plus V operator equal to zero, right? So we're close to the critical point. So this thing is about zero. So therefore we will minimize here a certain functional of all A in the kernel of KTc plus V. And that's unfortunately a little bit lengthy. I don't know whether I will write everything down. Minus 1 over 2 Tc integral kosh. So please don't copy that. It's really going to be horrible. Divided by 2 Tc times xi squared minus mu divided by 10. I just want to show you that we can do these things explicitly that while you get formulas particularly nice. But the important thing here is destruction. So that's the first term, right? It depends on the Fourier transform of a squared and then multiplied by some positive function in Fourier space. And that comes with a minus sign. And now there's another term which comes with a plus and well g1 xi squared minus mu divided by 2 Tc divided by xi squared minus mu. And now this thing here to the power 4 a head of xi to the power 4 d xi. So this thing while I didn't tell you what g1 was it's again something related to exponential function. We will get some idea next time how these things come up. Z squared 1 plus e to the z squared. It's terrible I know. Anyway the structure of the second term here is again the Fourier transform of a now this time raised to the power 4 and again multiplied by a certain function and the important thing is that this function is positive. So we have minus a quadratic plus a quadratic term. And so the formula becomes much nicer in particular if the dimension in particular if we're in the non degenerate situation that this a kernel is one dimensional and I pick an a, should I call it a star yes, then e of right then you can write everybody in the kernel as some number psi, psi is a complex number times this guy that spans the kernel and what you get here is minus lambda 2 psi squared plus lambda 3 psi to the power 4. Now obviously we can minimize this. I mean that's not the point I mean. So e is equal I guess minus a quarter lambda 2 squared divided by lambda 3. That's not the point that we can compute the number. The point is that we see here arising the psi squared plus the psi to the 4 term. It's just right I don't have anything that breaks the translation variance so I don't have a gradient up there on the first board the first thing I in the line down there. Nothing breaks the translation variance size a constant but that's exactly how this thing comes about. And somehow the point is, well I guess I explain this later in the proof, you see here but it's a quadratic expansion around that point. Good. And these are explicit formulas for these lambda 2 and lambda 3 and such things I mean that's what physicists really care about and really I mean they plug in numerical values and they mean something even for mathematicians often they're just formulas. Alright let me tell you a little bit on how one proves something like that. And so first it's convenient to introduce a notation. I call it H. It has nothing to do with a blanks constant but it's just a usual notation for a small constant and will be even more important in the next lectures where really there are commutators involved. Okay and it turns out that the natural thing is the square root of this temperature difference. Good. And so what's the, how does one prove such a theorem? Well we want to prove an asymptotic equality by proving two different inequalities. So there's an upper bound and a lower bound. What does the upper bound say? Well the upper bound says that find somehow a pair gamma and alpha such that the energy of this pair is less than well the energy of the normal state plus now there's an H to the power 4, right? If I look at the square that's H to the power 4 E plus a capital O of H to the power 6. So that means we're guessing something. I mean of course we should guess the right thing but this is somehow at our disposal we have to write down some gamma and some alpha and that proves the upper bound. Now the lower bound we have to prove something for every gamma and every alpha and we have to show that f of t of gamma alpha is greater than this. And so usually this means that well you guessed something in the first step and now you really have to ask yourself why is this guess really correct? So in some sense you want to reduce the second thing always to the first thing. By saying well you didn't really lose the generality by picking some particular alpha and gamma. That's a general strategy for such minimization problems. Now I want to write down a somewhat stronger statement which is actually what we'll prove and then I will stop for today. Let me write this down here which really says that in a certain energy region this E problem there is really the effective problem that you should care about. So here's what we really prove. In this paper I don't think I will prove all of this. In this course here in fact we show so the upper bound for every A so you give me an A in this kernel. Remember this kernel of the KTC plus V operator that's over which we have to minimize E script E. Now you give me any such A and then I can produce for you a gamma and an alpha so that the energy of this pair alpha and gamma well of course to leading order I want to get the energy of the normal state otherwise there is nothing to be said I want to get the script E energy of the A that you gave me plus perhaps some small remainder. Conversely I'll show that for every gamma and alpha there exists an A in the kernel of KTC plus V such that the H1 norm of A is bounded by 1 such that alpha is given now I have to construct an A for you which such that H times A is close to alpha it's actually an H squared what I'm saying here is alpha really is close to is of order H plus some small remainder of A order H squared and such that the energy of this thing that's given is the normal state of course plus H to the power 4 E of the A that you constructed here plus a capital of order H to the power 6 so now obviously this thing implies the lower bound because the largest this can be is just order A it's just E and for the upper bound well you just choose as A the minimizer in this optimization problem so once you have proved these two things then you get this theorem what I really want to emphasize here what's important is where you see the subtlety of the problem is that the off diagonal entry alpha that's only of order H whereas the energy difference is order H to the power 4 so there are lots of computations here that's here that's somehow doable here but that will really create problems once we have to do this all semi-classically anyway that's where I want to stop for today let's see whether there are any questions everybody's hungry I have a question concerning BCS actually Gorkov and the genes did they also provide a framework for a time-dependent problem? there's a paper by Gorkov and and then there's another paper by Schmitt German mathematician Schmitt and both of them derive a time-dependent Gainsburg-Landau equation from BCS theory there are some sort of disputes it's not somehow as universally valid as what I'm telling here so there are some additional assumptions and it's not completely clear to us what they mathematically mean but there's certainly work that's accepted in the physics community good what kind of evolution equation? it's very very interesting because it's so I don't know I'm running out of my lambda letters so perhaps there's a lambda 4 plus I lambda 5 dT psi okay so there is both a dissipative part and a dispersive part and in principle one in principle of course somehow this part should win right the the dissipative part it should go what we have so far managed is rather seeing the dispersive part so there is a paper by Heinzel well myself I guess Heinzel, Schlein and Seringer appeared I think this year where we look at an equation like this and where we get a dispersive equation but I think first of all it's rather believed in the dissipative effect so it depends maybe you have to wait more that's right I mean there are lots of questions exactly right I mean this is really an interesting question thanks other questions for Mark? this one so below the critical temperature can you say anything about whether the free energy is actually minimized and can you say anything about the minimizer is such a minimizer? yes yes it is both in this problem and in the more complicated non-commutative version one can prove existence of a minimizer and one can actually compute the Euler equation and I think I'll say a little bit about that but somehow it's kind of an energy subcritical problem so ordinary tools like Rayleigh or Rayleigh-Kondratsov give you existence of a minimizer okay thanks do you think that this can be seen for excited states this kind of transition? it would be very interesting to do something really with non-ground state solutions and somehow establish a correspondence between these microscopic solutions excited solutions and Ginsberg-Lander solutions also excited exactly I mean that would be very interesting I think physically it should be possible there's nothing that contradicts it at least in certain regimes but I don't know any books on that thank you


Our next speaker will be Dorothy Deasy. Dorothy is a design researcher with experience in both qualitative and quantitative research methods. She excels on strategic qualitative projects, which allow her to explore the intersections of meaning, culture, and technology. She holds a BS degree in industrial organizational psychology from Loyola University in Chicago and a master's of applied theology from Merrill Hearst University in Portland. The emphasis of her theological focus is on spirituality and transhumanism. And I'd like to add that Dorothy is on the board of directors for the Mormon Transhumanist Association. She is herself a Christian transhumanist and not a Mormon transhumanist. And she has made spectacular contributions to our organization. I'm happy to have her speaking today. Thank you, Dor. Thank you. It is an honor and a privilege to be a part of the Mormon Transhumanist Association. The transhuman age, like the kingdom of God, is already present and still unfolding. There are those who feel that these two paradigms are at odds, that they represent opposite and rival views of humanity, perhaps on par with good and evil. In this paper, I will argue, however, that the perceived threats and comparisons to Nazi Germany stem not from science and technology, but rather from an ideology of total domination and the commercialization of humanity. In contrast, if transhuman philosophy embraces spirituality, ecosystem, and symbiotic thinking, thereby co-creating with God, then both humankind and society may be transfigured to bring about the promises of the kingdom. I will draw on the work of Jewish philosopher and historian Hannah Arndt, who wrote extensively on totalitarianism and the Holocaust and apply her work to the subject of transhumanism. Where possible, I've edited quotes to be gender neutral. What is the hope of spiritual transhumanism? It is the transfiguration of society and ourselves to live in unison with God. For me, it is synonymous with the realization of the kingdom of God, and it has three components, non-violence, neither physical nor economic, codification of the rights of man, and co-creating in a way that reflects an understanding of the sacredness of creation. Jesus referred often some 50 sayings and parables to the kingdom of God. From these sayings, Jesus' followers learn how to behave in the here and now. The present change in behavior is part of what allows God's vision for life on earth to evolve. The kingdom is realized when humankind is working in concert with divine intent for social justice and non-violence. For me, the kingdom of God is not theocracy, especially a theocracy that lays claim to know the mind of God. Historian Gary Wills quotes a passage from the parable of the rich man in Matthew. Many of the first will be last and the last first. And goes on to say that anti-hierarchical last sentence shows that the symbolic prophetic meaning of the 12 has nothing to do with church governance below. In the 21st century and beyond, the kingdom of God represents acknowledgement of and flourishing for all of humankind. It is inverting how we think about economics and realization that true capital comes from human ingenuity. Eliminating poverty ensures that the resources of all voices may contribute to the whole. I emphasize human aspects, not to deny or belittle the emergence of artificial intelligence, but to underscore that the kingdom of God is based on humans working in cooperation with God. The kingdom of God is directly related to the Edenic allegory. What we experience as the fall from grace, which is knowing good and evil, is the loss of paradox. We, with our human nature, believe in our own convictions, sometimes at the exclusion of respect for the wisdom of others who do not share our perspectives. The kingdom of God is present when we realize that God, and perhaps God alone, is able to weave what we see as disparate parts into a unified whole. The disobedience of the fall is believing our own egos rather than seeing God active in the world. This vision may seem inconsistent with transhumanism because transhumanism often brings to mind images of technology from science fiction. Yet I look at the amazing world unfolding before us and see God active in the world. There are daily breakthroughs in research and execution, such as the use of genomics to cure cancer, creating crops resistant to blight, saving lives with artificial and animal grown organs, helping infertile couples conceive, implants that can short-circuit Parkinson's tremors, cars that do not need drivers, and the news that Voyager has crossed over into interstellar space. Already, emerging technologies have helped to reduce global poverty, and this year, while polio in India was eradicated. We are all transhumanists because the very context of industrialized modernity is to push beyond the boundaries of nature to become supra-natural. But does the inevitability of the progression of history necessarily mean that it's evidence of God's work in the world? Those who find transhuman-age technologies threatening sometimes make comparisons to Nazi Germany and the abuses of the Third Reich. They see an emphasis on eliminating disease as further marginalizing those with physical impairments. They see genetic engineering as eugenics designed to create a master race. Such concerns are more than an example of Godwin's law. There are legitimate desire to learn from the past. They serve, however, to obscure the fact that the abuses of fascism were political and socioeconomic rather than technological. Transhumanism is possible because of the convergence of technologies, not conspiracy. Individuals that reach transhuman goals do not have the power to recreate Nazi horrors, only when people are stripped of their political and judicial personhood, as is the case of a totalitarian government or the dissolution of government can such a system reemerge. Hannah Aren't a Jewish philosopher and historian from Germany writing in the wake of the Holocaust. She is most well-known as the person who coined the term the banality of evil. She wrote, the impetus in what is more important, the silent consent to such unprecedented conditions are the products of those events which in a period of political disintegration, suddenly and unexpectedly made hundreds of thousands of human beings homeless, stateless, outlawed and unwanted while millions of human beings were made economically superfluous and socially burdensome by unemployment. This in turn could only happen because the rights of man which had never been politically secured but merely proclaimed, have in their traditional form lost all validity. It was the political ideology that perverted the nascent science of genetics. There is only one thing that seems to be discernible, she said, we may say that radical evil has emerged in connection with a system in which all humans have become equally superfluous. What happened under the Third Reich was a creation of a system of total domination which sought to destroy the human nature of all citizens, not just those killed in the concentration camps on several levels, judicial, social and moral. The government enacted laws that were arbitrary, laws that classified some citizens as non-human, created a penal system that allowed for detention without regard to criminal action and used arbitrary punishment to turn neighbor against neighbor, friend against friend and family members against one another. Some citizens were made stateless and thereby lacked protection under the law even at the global level. The world found nothing sacred in the abstract nakedness of being human, aren't said. Eugenics were certainly an abuse of the regime but that abuse followed the creation of a system of total domination. In short, fascism was at the root of the Holocaust. And now we stand at a unique time in human history. Increasingly climate change and severe drought may be harbingers of threats to our food production system and the introduction of a combination of machine intelligence and robotics may displace millions of workers leading to another wave of massive unemployment. We see imperfection in our economic structures such that the gap between the wealthy and the poor is increasing. The standard of living for many is in decline. Michael Lewis in his book Flash Boys notes that flash trading has rigged the stock market. This should be a warning that AI is not morally neutral. In a market-driven economy, innovators seek to meet the needs of their consumers.  How do we rethink product valuation such that profit is only one measure of ROI not the only measure? How do we ensure that the needs of all are taken into account? How do we employ these technologies for beneficence and avoid creating a system of total domination? How do we accept the promise of machine emergence while still valuing humans not as consumers but as inherently deserving the rights of man? Arnt wrote, a conception of law which identifies what is right with the notion of what is good for the individual or the family or the people or the largest number becomes inevitable once the absolute and transcendent measurements of religion or the law of nature have lost their authority. That is precisely why this organization is important. We stand for the division, I'm sorry, we stand for the vision of the divine. We are part of a more secular ecosystem including the presidential commission for the study of bioethical issues, IEET, Center for Genetics and Society, American Society for Bioethics and Humanities and of course Google's AI Safety and Ethics Review Board. Together we can help shape the unfolding of the transhuman age to respect human flourishing and work towards avoiding the creation of a biological caste system. Together we can lobby and avoid political abuses that would pervert the intentions of emerging technologies. Preventing political abuse is also why gay rights and a reinvigoration of the rights for the disabled are essential. Not simply for the protection of these minority groups but for the stance that the guarantee of protection of law applies to all humans. The essence of spirituality in general and Christianity in particular is not to adhere to any given set of beliefs but rather to become and remain committed to a lifestyle of action dominant among these being the universal acceptance of and reverence for humanity itself. To become transhuman is not to endeavor to replace or move beyond humanity. To become transhuman is to remove the barriers and obstacles which might exist preventing each individual from reaching the apex of their potential. Transhumanism which seeks to replace humanity with a new species of technology based life forms is at best in denial and at worst leading down a path to a total domination system. Spiritual transhumanism is rather concerned with human flourishing itself. It is to transfigure the human so that every opportunity is given for self-actualization. Moreover, spiritual transhumanism is concerned not with the individual alone. It is about the individual or person. It is not about individual or personal self-actualization but rather with the flourishing of humanity as a whole. To allow some to transmute at the expense of others is to cede to selfishness and greed. Rather flourishing or godhood occurs only at the communal level only when the welfare and wellbeing of all are taken into account and improved. Here is the distinction between commercial transhumanism versus spiritual transhumanism.  We recognize that humans have the ability and the right to be co-creators with God but we also recognize that there is this distinction between being co-creators and rivals. The respect for godhood is the recognition of ecosystem, that we live in a universe of symbiotic relationships. If we seek to establish one right way, whether that benefits a church, an ideology, a political party, a gender, or a species, then we are placing ourselves outside of spirituality in general and Christianity in particular. The notions of ecosystem and symbiosis need to pervade all our institutions from healthcare to economic structures. Writer Andrew Solomon points out, the exceptional is ubiquitous, to be entirely typical is the rare and lonely state. In his book, Far From the Tree, he explores the complex relationship between identity and illness and makes the observation, the problem is to change how we assess the value of individuals and of lives to reach for a more ecumenical take on healthy. We need to get good at paradox. While emerging technologies may seek to mitigate disability, we must come from a place of compassion and not stigmatization. Rather than seeking one view of bioethics, either purely secular or purely faith-based, how do we develop eclectic options allowing individuals to decide for themselves into which paradigm they fit? Symbiosis applies to economic solutions as well, seeking economic reforms that rebuild our business ecosystem. This may mean replacing GDP with an alternative measure that supports income for individuals across a wide spectrum of fields. There is no question that STEM fields are important, but small business, social work, pastoral care, and nonprofits are equally as important. Corporations that realize excessive profits from the deployment of AI should be responsible for funding non-technology startups and subsidizing main street ventures. We must also pay attention to our legal system with such steps as reform of NSA surveillance practices, extending postal privacy rights to email, moving toward a system of direct referendum at the federal level, the restoration of voting rights to former convicts who have completed their sentences, constitutional change to eliminate the death penalty and codifying the right to vote, congressional hearings on aspects of the Patriot Act that subvert protections of due process, elimination or stronger citizen oversight of privately run prisons, and read only safeguards and databases that hold forensic data such as fingerprints and DNA. Transhumanism can lead to the coming of the kingdom, but only if we are intentional about creating our society to be on earth as it is in heaven. Thank you.