Thank you for inviting me to give a talk here. I'm pleased. It's nice to be back at the Azure S after a number of years, so I'm happy about that. So the title here is discrete minimal surface algebras and I will try to describe a little bit what we've been doing over the last maybe 10 years or more related to finding solutions to equations that somehow relate to minimal surface equations. I will tell you exactly what I mean by that. So we have taken a number of different approaches in different contexts and so on and I will try to outline these and also you'll see hopefully some equations that you will see again and have already seen during this workshop. So what do we want to do? So we want to do discrete minimal surfaces, quantum minimal surfaces, non-commuted minimal surfaces, whatever you want to call it and well there are several motivations. First of all, as we've already seen today, I mean some equations that you may enough have realized that they were minimal surface equations appeared and these kind of equations appear in physics like the IKKT model we heard about and also in membrane theory in the matrix regularization. But also there is a general mathematical interest I think to see what extent can we do non-commutable minimal surfaces. Is there some kind of nice theory or is there nothing like the classical situation or do we have some results? So what I'm talking about today is I'm joined work with a number of authors like Jaegyang Choi in Korea and Jens Hoppe here, Gerd Hysken and also at the end with Maxim Konsevic. So some outline what I will do. First of all I will sort of tell you something about that you can formulate Kehle geometry in terms of Poisson brackets. A little bit like the previous talk you've seen and how that leads to equations for minimal surfaces written as Poisson bracket expressions. Then I will go to the paper which I borrowed a title to this talk which is the discrete minimal surface algebras trying to solve these equations that you have in front of you. I will then do something slightly different but related. I will try to solve these equations in the vial algebra. So u and v are operators or algebra elements which commute to one and the x i's are the elements in the vial algebra you're trying to find to solve these equations. And then also I will end by saying something about a non-commutative cadenoid and solving these equations which you now recognize from the last talk and connect that a little bit to the other two approaches. So these type of equations have also been studied earlier in particular the top one and the bottom one. My friends by Kohn and Michel Divoire Villette and they call them Young Mills Algebras and this for the Young Mills Algebras and then some inhomogeneous Young Mills Algebras. So they've been studied. I mean in this talk I will not focus so much on mathematical aspects of these algebras and so on but I wanted to give you an overview of how we try to solve these equations. So in order to motivate this why these equations pop up I would like to quickly recall how we may formulate Kelly geometry in terms of the Poisson algebra generated by isometric embedding coordinates into some ambient space. So this is part of a you can do this more generally we wrote a number of papers here. If you have an arbitrary n dimension Riemannian manifold you can rewrite the geometry in terms of an n bracket a multilinear bracket instead but for Kelly manifolds or Parakeller and so on almost Kelly and so on you can do with the Poisson brackets. So how do we do this? Right so on a Kelly manifold we know that the symplectic form of the Poisson structures is intimately related with the metric and this compatibility you can write in many different ways. I've chosen to pick a particular way of writing it in local coordinates so g is the metric tensor gab inverse and then you have the Poisson bi-vector and then the metric here and this equation holds on a Kelly manifold where gamma is equal to one but I will keep gamma here because that's a degree of freedom that is convenient to have as we continue. Right gamma is for I will gamma is for the moment one I will show you on the next slide what why I put this gamma in although for a Kelly manifold is one okay it has to do with the choice of Poisson bracket on your manifold. So the claim is here that if the Poisson structure is compatible with the metric in this way with gamma possibly being non-zero you can reformulate all of Jumanian geometry in terms of Poisson brackets of embedding coordinates into some ambient space so that's the statement we and we do it explicitly in these papers. Now so let's go to surfaces and embedded surfaces and for simplicity we will choose the ambient space in which these surfaces are embedded to be RM because that's simple and of course we know we can always do it we are Nash embedding theorem and it will be the case in our examples. Surface means two surface. Surface really means a two-dimensional real dimension surface right yeah and so it's a surface which gets an induced metric from the Euclidean metric and now for surfaces if we take an arbitrary density row we can introduce a Poisson bracket on the space of functions like this. Now there is of course a natural choice or canonical choice of this density which is simply the square root of the determinant of the metric maybe the Kepler choice one could call it and now we introduce this gamma which is the ratio of the square root of the determinant of the metric and this row you choose for the Poisson bracket so it tells you how much your choice deviates from the Kepler choice or whatever you want to say so and if you have this Poisson bracket in theta here you really realize that you have this relation again but now gamma doesn't have to be one it could be something else and now for surfaces this is sort of just one another way of writing the cofactor expansion of the inverse of the metric is just a this anti-symmetric sum of terms of the matrix elements of the matrix so that's just an identity. Now to come to minimal surfaces let us apply this to the Laplace operator on the surface so the Laplace Beltrami operator we know how to write in local coordinates like this and we can write it in different ways so I chose two different ways here so the first way of writing it is with a Poisson bracket with the embedding coordinates x i's a double Poisson bracket and here you see you have to introduce these gammas here if gamma is not one you get these extra gammas now you can also write it in terms of the local coordinates on your surface so u1 and u2 or u and v as they will also be called and you can write the Laplace operator like this of course you have sums over a and b here yeah okay so some particular examples when gamma is equal to one it becomes particularly simple right then it's just this double Poisson bracket equation for the Laplace operator in terms of the embedding coordinates now if you have a conformal metric on your surface just being proportional to the delta ab the metric and you choose rho to be equal to one that's the choice where the Poisson brackets of the local coordinates u and v is equal to one and you also get that gamma is square root of g is equal to this e then the second formula on the previous slide the formula which wrote the Laplace operator in terms of the embedding sorry the local coordinates it simply becomes this so it's just proportional to the sum of two double Poisson brackets with one of the coordinates here and the other coordinate there yeah assuming a conformal parametrication of your surface right so these are two equations that we will use to motivate what we introduce as non-commutative minimal surfaces so just a few remarks before we go on just to see how it looks like so I claim that everything can be written now in terms of Poisson brackets of the embedding coordinates how does it actually look like in practice right well for instance if you have a surface you can compute the Gaussian curvature in terms of the embedding coordinates and the formula for the Gaussian curvature is this huh so it's just a sum of products and iterated Poisson brackets so that's how all these formulas look like just to give you an idea and of course in terms of trying to find that's one of the original motivations here trying to find sort of how can we quantize these kind of geometrical systems well we sort of know how to quantize them if we can see things in terms of Poisson brackets we can just replace them by commutators naively so that was one of the original motivations for doing this and but of course it has an independent interest as well so just one more remark not so much related to this talk but one could of course now turn the question around when you see that you have all these formulas for Emanian geometry written in terms of Poisson brackets of some generators Xi some embedding coordinates you can ask well now if I start with a Poisson algebra can I introduce Emanian geometry in a consistent way in a Poisson algebra without referring to any underlying manifold and I mean we did this I did this with a student and yes you can find some simple conditions for a Poisson algebra and that allows you to do Emanian geometry in a Poisson algebra so that's just a remark but you can turn this this question around which is quite nice yes so with the help of these three formulations of the Laplace operator and we would like to formulate the equations for a minimal surface so it's well known that for a minimal surface in Euclidean space the equation so that you have for the embedding coordinates is simply that the Laplace operator on them is equal to zero so it's a it's a harmonic embedding and in terms of our Poisson bracket formulation this is simply this equation of course should say for i equals one two to n here that this should be zero but we can formulate it in this way coming very close to the equation we saw in the previous talk or we can choose another Poisson bracket where we the density was equal to one and not square root of g and then we see that well the conformal factor we had in front one over e doesn't matter if you want it to be zero so the so it's actually not equal here but proportional to this one and you can formulate the condition that the surface is minimal in terms of the parameters u and v like this and this is of course assuming that the metric is conformal that these tangent vectors here are orthogonal and have the same length so now also we have been interested in equations that look similar to minimal equations describing minimal surfaces in spheres in these these type of equations here from a physical point of view arose if you if you try to do regularization of membrane theory time dependent equations you choose a particular gauge and you work and you end up with equations similar to this one that you have to solve now these equations here Laplace of xi equal to minus 2 xi these are equations that now you assume that the the length of the vectors is one so you're in s d minus one should be d minus one up here as well and if you're all on top of that so these equations you know you have a minimal surface embedded in s d minus one also now written in terms of Poisson brackets so now we want to take these equations we want to replace Poisson brackets by commutators and we want to study whether or not we can find solutions right as and by solutions I mean anything algebras the algebras matrices operators whatever try to find solutions and what do we expect well from these equations here we expect something very rich already when d is equal to four because when d is equal to four these equations are minimal surface equations for minimal surface in s3 now by well-known paper by Lawson we know that surfaces of arbitrary genus minimal surfaces of arbitrary genus exist in s3 of course this equation should have all these kind of surfaces as solutions and of course if you do some non-commutative version of this you also expect somehow that you probably have a rich solution space so already d equals four should show a rich structure and you might also say that it even for d equals four it might be impossible to solve the non-commutative equations completely because we will have so many objects lying around there yeah but that's good because we want many objects to find examples so now with this introduction how you can rewrite the Laplace operator in terms of Poisson brackets and look at the different types of minimal surface equations and let us now go to non-commutative or discrete versions by using of course this correspondence to define our objects what y minus two that's what you get that's what you get when you when you when you when you derive the minimal surface equation in it's the dimension of the surface two right yeah you have a Lagrange multiplier and then you vary the equations and you get something like that also yeah and so this is an overview of one paper we wrote which I borrowed the title of this talk discrete minimal surface algebras and those equations correspond to minimal surfaces in spheres right now we allow for this eigenvalue or here whatever you call it it should be an i here sorry xi of course xi this eigenvalue to be arbitrary I mean it was minus two on the last slide but we we just to have some more room and flexibility we allow for arbitrary values and we call this collection of these mu i's the spectrum just to have something to say and we look at this equation as in different ways so of course these equations make sense in a Lie algebra right you let x i's be elements of a Lie algebra and you can make sense of this equation so that's one way of looking at this equation that given a Lie algebra and given a subset of elements which I call curly x you want this equation to be solved so really crucial or complex equation um doesn't matter I mean you mean over the complex real numbers Lie algebra yeah it doesn't matter so much now it doesn't right of course when we later on maybe want to find representations with the x i's our Hermitian matrices um it it matters but here it doesn't matter mu i can be complex just from the start right it will of course be real um if you have Hermitian matrices or so on and you take that take the conjugate of both sides yeah but for now we just leave it complex and also the multiplication the multiplication of of elements is normal multiplication or can you have a star product right now there is no multiplication right now it's just a Lie algebra so the only multiplication you have is the Lie bracket yeah however of course now the next thing to say is of course you can consider I mean sort of an associative algebra the associative free algebra x1 to xm you generate an ideal by this relation and you quote out and you get an algebra an associative algebra right so that's of course like the enveloping algebra of your Lie algebra that's one other way of looking at these equations maybe if you want to find some kind of basis and so on to use the diamond lem and all these things it's quite useful to do that right so I was thinking of a cube bracket over there and well we wanted to find this really wanted to find operators so then it's not so but of course you could consider these equations you could have any bracket it doesn't even have to be maybe Jacobi identity or I don't know but if that's interesting or not I don't know right so so you can do several algebraic things with these objects and of course con and juba we let for for the case where this is zero on the right hand side they put a lot of properties like causal and so on and homological properties of these algebras as I said I will focus on solutions to these equations so what kind of solutions we have well we have for instance Clifford algebra solutions now going away from the Lie algebra solutions I just mentioned you will see them later on so if I have a Clifford algebra satisfying these relations here I can easily see that and check that well if you do this sum here you will end up with something proportional to EI again now this this commutator is really the commutator of course not the anticommutator then you would just have zero right so it's really the commutator so representations of Clifford algebras if you want or just Clifford algebras as they are and solve these equations so Lie algebras maybe perhaps more natural as written well if you take an orthonormal basis with respect to the Keeling form of a semi-simple Lie algebra then you can easily see that the x-sides are solutions to the equations simply because the double commutator is the product of two structure constants and they will just be the Keeling form if they're orthonormal so you have a solution now of course let us look just a little bit more in detail on some other solutions you get from the algebras and let us look at sln here just to have something concrete to look at so sln you have alpha 1 to alpha n minus 1 being simple roots and we have to choose some kind of of of basis here so we choose elements e alpha e minus alpha and h alpha to satisfy these commutation relations quite a normal choice of basis and where these h alphas are part of the Carton subalgebra fulfilling that this linear functional and h is just this inner product with respect to the Keeling form and in sln all roots have the same length so let us call this l squared the length of our root and now having these roots we can just take the plus and minus combinations of the negative and positive root like this and these elements will now fulfill nice double commutator relations okay so you actually get this maybe not so important exactly what you get here but you see if I take one of these e alphas commuted with e beta twice and I get something back which is proportional to e alpha plus again now this is of course alpha plus or minus beta has to be a root here so what happens of course is that e alpha with e beta gives you the root e alpha plus beta then you get the term when you subtract beta again and you go back to alpha so there is no mystery here and so you can compute all these not so interesting so what can we do here so there are many ways now of choosing subsets of these things to get solutions to the equations for instance you can choose positive roots like this I think even you can choose the the signs to be independent and in this case every commutator like this would be proportional to xi again for all these elements here that's one way of constructing solutions there are other ways for instance here you include some elements of the carton subalgebra and take every root with a plus and a minus like this and now it's no longer true that every double commutator is proportional to to the element again but things cancel and work out here so you get solutions to the double commutator equation where it acts on the carton subalgebra as being zero and not on the carton subalgebra it has some other eigenvalues yeah so just to give some examples that there are many ways of of choosing these elements to get solutions so now let us consider the case when d is equal to four remember that dx equal to four you have four operators or four algebra elements and that there were equations corresponding to minimal surfaces in s3 which we expect to be rich already so and then we do some simplifications here so we assume that at least two of these eigenvalues are equal and the other two are also equal we call them u and rho here and we complexify these matrices Hermitian matrices xi as lambda and t now of course you don't have to talk about matrices you could equal or well talk about star algebra or so on but we talk about matrices to be concrete that was also an original motivation for for doing this now you just rewrite the double commutator equation in terms of these new variables and you get some expression like this and what you can note here is that things drastically simplify if lambda and t are normal operators now normal operators means that lambda lambda dagger and t t dagger commute so this term goes away and these two terms here are actually equal and these two terms are also equal and this goes away so you just have one term on the right hand side that's all can make things simpler so what do we do well when lambda is normal we don't assume t to be normal for now we assume lambda to be normal because it's unitary diagonalizable so we can find a basis such as and always find a basis where lambda is diagonal and then let me show you how we think of the solutions these equations in terms of a directed graph which turns out to be useful so what do I mean well it's like the adjacency matrix of a graph right and you let g with vertex set v and edge set e be a directed graph and the vertices are the eigenvalues of lambda okay and there is an edge between these eigenvalues if and only if the corresponding matrix element of t is non-zero okay so the graph encodes the eigenvalues of lambda that's drawing the the the graph in the complex plane with the eigenvalues are the vertices and then you just look at t and you draw the edges from the respective eigenvalues to get this graph so although of course this graph as depicted here does not encode the value of the matrix elements of t it sort of encodes the structure which turns out to be useful so this is just an example we'll see later so this I mean we use this simple representation of solutions to matrix equations in a number of different situations where it was very useful in some papers here for instance in this paper here it was crucial to to sort of classify all possible finite dimensional Hermitian representations of some cubic algebras right so which was much more tricky to do without graphs than with graphs so if you you can ask questions like if I have some equations and I have a graph of this type what kind of restrictions do I have on my graph coming from the equations can I derive rules for my graph to be a solution and so on yeah and so the fuzzy sphere solve these equations right so this up here is just a concrete representation of SU2 here with some choice of normalization which you see here and this is solution which actually these are equal to two here and for lambda is equal to that which is already diagonal and t is equal to this matrix and now if we just depict it in our graphical way it will be a simple string and like this with the eigenvalues of lambda are just these real values on the real line and then we just connect them one by one like that now these equations also have a rotational symmetry which I didn't talk so much about so you can if you want you can rotate this eigenvalues from the real line to go out in the complex plane if you want but you can get other solutions from SU2 right so for instance you can for arbitrary complex numbers z and w you can construct these two matrices which have different eigenvalues related to these of course and a different type of representation graph here and they're not equivalent these representations so they're different now the fuzzy torus is also a solution so it's generated by matrices g and h which have this commutation relation that those are the clock and shift matrices we know and now this is a rational root of unity here so q to the sum n is equal to 1 and you get a solution by in principle taking g and h here just to be lambda and t and g is the diagonal clock matrix and h is a shift matrix and this gives you a solution to the equations with this eigenvalue here depending on q and it looks like the graph looks like this those are just of course the n different roots of unity here now of course now you start wondering here now these equations they contain should contain solutions which correspond to surfaces of arbitrary genus in s3 of course the torus here has genus 1 and we saw the fuzzy sphere has genus 0 of course you would like to find solutions maybe explicit solutions of arbitrary genus right this is in general a tricky problem to find explicit things of higher genus but they should be here somewhere and we don't understand this completely at all and i mean maybe it's also very difficult because there are so many solutions in here but let us try something else here so let us take sl3 and let us take the this t1 to be an element of the carton subalgebra typically the ones we choose to be diagonal matrices that's also lambda is phi of t1 where phi is a representation of sl3 by anti-harmission matrices now if you take these these two elements and you just compute that you get a solution of the equations with these eigenvalues here and of course you can write down for instance if phi is the n comma 0 highest weight representation you recognize the weight diagram of course of of this kind of representation and this is n equals 3 perhaps and then you have to look at this matrix here to see how they are all connected with directed arrows so not so important but and of course you can go on you can take now this n to infinity limit and you get a sequence of of matrices somehow inside sl3 with perhaps some limit this i don't know yeah so here so to conclude here it says it is not it is not so difficult to find explicit solutions that you can actually work with for these equations which might be useful in i don't know in physics or not and but it's surely interesting that you have the structure in mathematics and you can say you can actually say a lot more here uh general about the representation theory and the connection between graphs and representations and irreducible not irreducible and and so on which i've chosen not to present so you're assuming q is not just one because t q q because mu is equal to 0 if q is less so on no q is not equal to 1 that's right no better way when what you construct here is a projection of cp2 okay yeah okay yeah good interesting yeah so what happens have you considered the case of bps kind of solutions meaning that you also have these num equations with the extra terms sort of stabilizing no i suppose solutions that also satisfy the num equation meaning that they have this commutator equal to epsilon ijk times you know right equal to the commutator the sum of commutator xi xi xj or something like that squared yeah multiplied by epsilon is the solution no i haven't looked at i mean the fuzzy sphere yes i think right but but this i don't know i don't yeah okay so now there was one set of equations to solve now let me proceed to another set of equations which are maybe not directly relevant for physics as equations but still very interesting and you can actually prove more for these equations and those are the the non commutative minimal surfaces which we look for in the vial algebra um when you wrote the Laplace operator in terms of local coordinates so we return to the equations defining a minimal surfaces in rm and we think of having a parameterized minimal surface in r3 and we know that the minimal surface equations can be written like this if the metric is conformal and and recall here that the common this choice of Poisson bracket was u with v is equal to 1 and the question we had is what happens if we naively translate these equations into non commutative algebras do we get something non trivial do we sometimes come in simple or not can we find solutions and so how do we think about this well we know we can find a lot of explicit parameterized non commutative and minimal surface in in r3 that's a classical subject so we thought maybe we can actually find analogues of these surfaces um right that's what i said we in terms of physics we may not solve the more relevant physics equation but maybe this will help us to understand what to expect from the other equations and also it turns out to be a nice um nice theory in itself here so what do we do since since the classical Poisson bracket is one we start with an algebra generated by u and v satisfying that the converters equal proportional to one this is the vile algebra we denote it by a h bar and we also for more technical reasons you can say we need this fraction field i mean where you consider also all the inverses of elements polynomials in u and v and now we start from a extremely naive um definition not very elegant but very naive and very close to the classical situation now what do we do well we take some tuple of elements in the vile algebra or if you want we take an element of the free module of rank n over the fraction field remember the fraction field is not so important we will later see that we can find solutions in the vile algebra itself but to make the general statements we need fraction field so we call it a non commutable minimal surfaces if they are all um all uh emission really in some sense they satisfy the minimal surface equation but we remember that this minimal surface equation only holds if we have a conformal parametrization so with the help of um derivatives here we write down these conditions very naively so what's du and dv well they are simply the commutators with v and u which in the classical situations where you have the Poisson bracket are exactly the derivatives with respect to u and v but here we have them as inner derivations in the algebra we write down this fact that the length of x prime u and x prime v are the same it's and the fact that the off diagonal terms they are orthogonal so you have this equal to zero somewhat symmetrized to be a real condition these two f's are same the two f's are the same thank you yes let me just different typeset right so these are the equations we want to solve you want to find x i's and elements of the vile algebra for which this is true you could imagine doing a lot of fancy algebraic things here you could define oh this is the algebra generated by these x i's completed to to include all the possible derivatives of the x i's and so on yes but this is not the route i want to take i want to show you theorems about solutions so this is actually what you get here a non-commutative vice stress theorem so recall for those of you who don't remember vice stress representation theorem tells you how you can write every minimal surface in r3 okay so it goes two ways so first of all you say here well you choose two elements f and g now they need to be holomorphic and here we we write that so d bar f and d bar g is equal to zero where d and d bar are just a complex combinations of d u and d v as usual and so you take such and so if you think about this these are quotients of polynomials in lambda only and not lambda dagger you take these and you write down x1, x2, x3 like this now f now this integration is just the anti-derivative of polynomials right so there is nothing fancy going on here if so if you can integrate these polynomials here you know that these elements satisfy this Laplace equation and also this these restriction that you need the parametrization to be conformal so this gives an a way to generate infinitely many explicit solutions to the equations in the algebra okay and thank you i didn't write lambda lambda is u plus iv let me just write that on the board so thanks should of course have been here thank you so lambda is the complex combination of u and v and we write everything in terms of lambda and the real part so that's where you get a mix of all the lambdas and u's and v's and so on when you write it out okay so it's nice that you can generalize the Weierstrass representation without change more or less to the non-commutative setting so one more thing so of course the the non-trivial part is not to show that this is a solution perhaps but the other way around that all solutions can be written this way in the vial algebra yeah there is another classical representation theorem which tells you how to obtain minimal surfaces and this is the following and this representation formula just depends on choosing one polynomial or quotient of polynomials f in lambda like this and construct this one then if you sort of integrate these phi 1, phi 2, phi 3 as x i's you get a minimal surface just another way also usually in a in a book about minimal surfaces you find this and so given any polynomial you find a minimal surface here and for instance now let me go to some examples you can find a non-commutative versions of the NEPA surface that this one is constructed this way right so we choose in principle we choose f to be just lambda to some power right and then we just write down what will this be well you can integrate that explicitly and you just get the real part of these things of course as usual I didn't say but the real part is just I mean one half the element plus it's of course a emission transpose yeah and now n equals 3 corresponds to the NEPA surface right and in a parametric yes and now you see when you write this out you recognize sort of if you recognize the parametricization of the NEPA surface as you find it in Wikipedia or in some book you will of course recognize this part here this but without the h bars and this part here now of course what this theorem provides you is the sort of non-trivial correction terms that you have to do in order to compensate for the non-commutativity so this theorem gives you these correction terms automatically if you want to look at it that way so unless these corrections are here it will not satisfy the equations of course you can go on and produce a lot more examples so it's really you can produce an infinite number of explicit examples if you like okay so let's okay just so now before going to the next part you can wonder a little bit here now if I have my non-commutative minimal surface in the setting right or if I have it in the other setting with the commutators with the x's I mean of course classically they are equivalent you can solve the equations in local coordinates you can write them in the in the embedding coordinates and they are equivalent and we have many many solutions to this setting right can we use them to produce solutions to the other setting which perhaps contain the more relevant physical equations now that we have so many examples here yeah of course classically is a change of coordinates but non-commutative in the non-commutative word is not as simple but we tried we have tried and we have succeeded to some extent so and that's this next part here together with Jens and Maxime which we are writing up now and we did the following so let me be a little bit more precise about the things I just said so if we have this Poisson bracket here with the sort of natural one over square root of g factor we get the minimal surface equations to be this and if we have this Poisson bracket with just a one here we get this now the two are related by a change of coordinates right so this comment is what I just said we want to produce solutions to the top equation from the bottom equation in the non-commutative setting as well so assume that we have a parametric minimal surface and assume that we sort of reparameterize these in terms of u tilde and v tilde which has a Jacobian which is just equal to the square root of g here now for a Poisson bracket where the u tilde and v tilde the Poisson bracket is one and you get that these x's now solve this equation right so in terms of u and v it didn't solve it solved the other equation in terms of the parameters but if we reparameterize this way you solve this equation which we want to solve so can we make use of this in the non-commutative setting and let's consider the case of the catenoid embedded in r3 how can we do this okay so let us recall what's the catenoid well we parametrized the catenoid in r3 for instance in the following way with caution hyperbolic functions and trigonometric functions and we introduced w to be the complex combination of these and it's just being equal to that now we can explicitly do the reparameterization we talked about on the previous slide by setting u tilde to be u and v tilde to be these this function of v alone you easily check that you get the the Jacobian to be the correct one and now you have these w's and z being functions of u's and u tilde with this these functions inside here and of course one would have to invert this up here to get this now so what have we obtained here well what we have obtained is sort of a parametrization in terms of the variables u and u tilde which have Poisson bracket one that's the main difference right in the in the other case we didn't know what the Poisson bracket of u and v was or maybe one of a square root of g and we don't know maybe we don't know what to do with that in the non-commutative setting but in this setting u and u tilde v tilde have Poisson bracket one which sort of make them easy to quantize in some sense so now we want to take these expressions and just replace u tilde and v tilde by operators that commute to one more or less right so so in analogy with sort of standard representations of operators which commute to one and we let them in this case we let them act on sorry for the four for your modes on the circle like this this is really just a motivation for how we now make the ansatz to solve the equations right we see that this e to the i u tilde shifts just one step and this v tilde is diagonal this way right so we make the following ansatz for w and z and w is just something whatever shifting one step z is a diagonal operator and it corresponds to these ones the functions right so this one the e to the i u tilde would just shift one step and whatever this is will be a just diagonal so that's the right ansatz and this z will just be diagonal now remember that w is x one plus i x two so and z is x three so we want these three operators which satisfy these double commutator equations now we just plug things in and you notice that these equations now in terms of this just becomes these two equations here as written and then you plug in the ansatz right and as you expect these the equations you get will be equations for these w n and the z n which defined our operators now we write it in terms of the modular square of w n which we call r n and then z n this is z n on the previous page so if we think of the w's and z to be operators in sort of the the doubly infinite fox space just the states labeled by an integer going from minus infinity to infinity these are the recursion relations we get for the the action on the different states of of w and z and we can solve these recursion relations yeah so we immediately note here that you see this is just this equation shifted one step right so this combination here will be constant for all n's and that's what we call c here so we immediately see that and now you can just solve the recursion relation by by picking some initial values and then doing the recursion in both directions now the only slightly non-trivial thing is that of course we have to keep our n positive in order to solve for w n at the end so we have to prove that these recursion relations they preserve positivity and they do so for a specific set of initial conditions so r 0 is positive you choose some c which was this combination of which was constant we noted on the last slide and you choose r 1 to be between r 0 and r 0 plus something constructed out of c and r 0 so you have a range in which you're allowed to choose r 1 to get a solution and this is to guarantee positivity of r and then you just simply you take this you do the forward recursion for n bigger than equal to 2 or the backwards recursion for n less than equal to minus one and you just get everything in terms of r 0 and r 1 and you can do the same thing for z here and you get all the z's so given the initial conditions you get a unique solution and this solution satisfies that the r n's are positive and you can define the following operators you can always include a phase here but it's not important you can you can buy unitary transformation you can remove it so there is nothing here so in principle you get these operators now so remember from our general ansatz here to the double commutator equations remember this was just a motivation to get this ansatz so this ansatz has really from the look of it nothing to do with the cateenoid really it's just a some ansatz motivated by the cateenoid and we get solutions which we have down here now how do these solutions look like do they have anything to do with with the classical functions defining the cateenoid well um they do somehow so you can just plot this recursion relation for some initial values right and this is r n you plot here which is more or less the the square of w n and here is the z n going like this and if you plot the corresponding functions i didn't do that here and you have a quite nice match of the behavior so really r n and z m seems to be more or less discretations of the functions of the cateenoid although that was not really how they were constructed it turns out that the solutions are exactly like that and in the things we are writing up now there are more interesting relations between the initial conditions and the parameters when you define the cateenoid you can you can shift the face a little bit you can multiply everything by a constant you can do all these kind of things and they seem to have a corresponding freedom sorry a corresponding um notion in the non-community world and there's also interesting to note that there is some choice here of r1 between r0 and this value which sort seems to be some sort of non-community effect here that you're allowed to choose some non-community activity in this thing um because in in in the classical case this is a symmetric function here which is it's not here it's not symmetric but for certain values of the initial conditions you get something symmetric okay so okay before i summarize so what did we do well we looked at um a cateenoid in some parameters which was not really correspond to the equations we want to solve we looked at how to do a coordinate transformation in the classical case and we thought of can we implement this in the non-community case by this procedure and the cateenoid is not the only example it seems like we can do the same kind of reasoning to obtain solutions to the double commutator equations even for even for other surfaces minimal surfaces so it seems like to some extent this would help us to define solutions so just to summarize um what i i try to give an overview of what we've done different approaches to non-commutative discrete quantum minimal surfaces here um and they are these equations are motivated both from physics as we've seen and but also from mathematics because i think it's an interesting question to what extent these objects exist and i have nice properties in the non-commutative world um and in particular presented these two different ways one in terms of the vial algebra and the other one in terms of just the x's to obtain equations for minimal surfaces and one may find solutions many solutions and in the vial algebra one even can sort of in some cases classify all solutions like with the vial stress representation so you have infinitely many explicit non-commutative minimal surfaces that you can write down and at the end i try to present an idea how to sort of connect these a little bit how to go from one to the other is it possible in the non-commutative setting it seems like yes in some cases you can actually do this to obtain non-commutative minimal surfaces which solve the more relevant physical equations and that was it thank you very much yeah if you kind of go back to this slide when you choose the complex structure w no no after that after that yeah is it the reminiscent of self-duality in the kind of Youngville's case means is it the can we interpret this z w w dagger as some self-dual equation and the other one is because it's young I'm I'm sorry I'm not a physicist so I don't have a good question answer but after choice of this w equal to x1 plus ix2 it is after choice of the complex structure sorry I don't know no no the answer is no this is just rewriting the equation it's no self-dual it's not something yes the question here so in the case of three emission matrices precisely this equation so how close are you to having a complete understanding of all solutions of these things all solutions of these equations or the catanoid case no with three matrices and maybe with a mass on the right hand side I don't know um with three matrices um this equation um no not the general I mean for the corresponding equation in the vial algebra some sort of a complete understanding but here no yeah it I mean all these morally speaking all these solutions should be here somehow lurking around right all these minimal surfaces yeah yes I have two questions so you said you were taking a field a vial field I'm not an expert so is this just vial algebra localized in with respect to some multiplicative sets or localized I mean it's a total fraction field for the vial algebra you can construct a total fraction field yeah yeah so you're just quotient of polynomials right you get you can show that these or conditions are satisfied for if you take the complete algebra to localize at the complete algebra and you can show that these conditions are fulfilled and you can construct by this or localization you can construct the quotient field yeah have you found second question have you thought of taking say other nice converging sub-algebra of star sub for instance on on on the sphere you can there's a explicit star products on the upper half plane there are explicit star products which are converging on some class of functions two construct solutions that's what you mean yeah I mean of course the vial algebra is the simplest choice simple yeah right um well you would I mean if you if you work like this you would like really you um oh you mean having an algebra with still keeping u commutator v is equal to one but allowing for more more functions in there apart from polynomials no that was not your you commutator would sometimes be the vial right right yeah some other I mean you think of the sphere as well you have three functions and right patient identities like yeah I mean yes I mean maybe you would have to go to s3 I mean for the ordinary sphere maybe you don't get so much interesting like because they already got two dimensional manifold so you would have minimal surfaces in the two sphere which is yeah I don't know really what that would mean but you can you can imagine having introducing more elements in the vial algebra for instance I mean I know for instance Stefan Waldmann constructed a sort of a much larger algebra including a lot of more smooth elements of the vial not just the polynomials of course you can work in such an algebra also in this case if you want to define the catanoid in the vial algebra of course you cannot directly do it because you don't have access to the cosh of you right but in this this much larger algebra you could imagine doing it yeah for my culture if I'm interested in the quantum string is there a link between quantum solutions of the string and what you would do in Lorentzian manifold for the string worksheet connection with the minimal yes I mean the string version of classical equation for string is a minimal right right yeah if I'm interested in quantum strings real quantum strings right yeah so you should not action you should not really ask me I I'm not a physicist but one of the point was of course to see to try to provide explicit solutions to these classical equations to try to understand maybe perhaps more the quantum version of the strings and so on but this I don't know much about thank you thank you


Livestreaming will help you get to that 4,000 watch hour mark that everyone needs to get by February 20th in order to stay monetized and partnered with Freedom. Livestreaming is available on mobile and desktop for YouTube and I've seen research that shows a channel grows four times faster by Livestreaming. Look, everyone has a cell phone. You can just Livestream while you're out and about to your channel. Every time one person is watching your Livestream for one minute, that counts like a minute watched as if they watched one of your videos for one minute. That's exactly how you should be growing your watch time to get to that 4K limit. How many of you Livestream already on YouTube? Tell me down there in the comments below. I will show you just how easy it is. It is literally built into the YouTube mobile app, just the one you use to watch YouTube videos. And here I'm about to start my first Livestream. Okay, we're live. Have you ever used this to Livestream to your channel? Have you ever used this to Livestream to your channel? Now, every minute watched counts. You need to get 4,000 watch hours in the last 12 months in order to stay monetized and partnered with Freedom as of February 20th. Click that I to learn all about that rule where I first introduced it. We had a few videos about this and I'm going to stop my Livestream now. What are the requirements to stream with your mobile device? Very simply, you install the standard Windows mobile app. You have to have your account verified. This doesn't mean the verified tick. This just means you've used your phone to verify your account. It's in the channel settings. And that you have no Livestream restrictions in the last 90 days, meaning you don't have a copyright strike or something that would disable Livestreaming. You don't have to be partnered. You don't have to be part of a YouTube network. That's the whole point. You can raise your watch minutes by Livestreaming even if you can't monetize your videos. This YouTube help article, which I've linked down there in the description below, says that you can Livestream using mobile and webcam. Now, when you look at that page and you go to the webcam section, it says visit youtube.com. To start Livestreaming without any software. So when you go there, you get this page is not available. Sadly, YouTube has not updated their help article because when I googled the issue, apparently the webcam feature is not available on YouTube. Previously, it was there, but it was turned off because it is rarely used and built on technology that is no longer supported. Good job, YouTube. How about you update your help article so that you say that here instead of getting everyone's hopes up. This is video number four in our 4K 1K series. We started with this video where I announced the 4000 watch hour rule. Then we talked about let's beat the 4K 1K rule together as a family because that is the spirit of freedom. We help each other grow together as a family. And then we confirmed that YouTube will unlink every channel from every MCN that does not meet the 4K 1K rule on February 20th. And I have to add a little caveat to that in an affiliate CMS. Channels in a managed CMS will not be unlinked on February 20th that don't meet the 4K 1K rule, but they will be demonetized. So you will still not be able to make money, even if you are in a managed CMS. Now let's read some of your deserving comments. Fine gaming connoisseur wrote, George, you should record videos of you working out in the gym like you did climbing the flights of stairs in the Philippines. I'm sure many will enjoy watching that. Did you really enjoy watching that? Of course I'm happy to do it, but I didn't think many of you did. Tell me in the comments below. Now on to my terse but heartfelt opinion piece on the subject of YouTube's latest ruling, the 4K 1K rule. YouTube is going to hurt the small channels the most, especially those who have spent years building up the channel in a legitimate way and have been rewarded with a few dollars each month for their hard work. Now I'm going to mention the C thing again. This is our project that will help you replace these few dollars a month that you have been getting if you get demonetized after February 20th. The C thing will actually let you make as much, maybe more, maybe less, but about the same as you're currently making on YouTube, even though you won't be making money on YouTube. I can't tell you more yet, and I know I keep saying this, but the C thing. You'll understand when I can tell you more. Now they will get nothing until they reach the new milestones, which make absolutely no sense as far as I'm concerned. So that's the part we're going to fix where you will get something. Then he quoted some financial figures saying, since 99% of the small channels that are making less than $100 a year and 90% of them are making less than $250 a month, they're just insignificant collateral damage in our financial statements. Nothing feels more like having salt rubbed in your wound than that. Anyways, George, it's so relieving that you took the time to tell us that you're going to do what you can to help us feel valued, something YouTube isn't doing and never has, in my opinion. Your words in this video are like music to my ears. Sincerely thank you, and I'm sure many of us, Freedom Partners, would feel the same. This is the Freedom Family. We all help each other grow together as a family. That's always been true from day one, and that will be true for as long as there is freedom. That's my promise to you, all of you. Vigilant wrote, Hey George, big message for you, please don't miss it. I didn't. I've been a long supporter of freedom since 2015, and I decided as a New Year's resolution I wanted to make videos, get partnered, and start building my business from there. You know the usual, better PC parts, but YouTube just implemented this, and by the looks of it, you've done all you can to negotiate with them, and they're not dropping down. Actually, we can't really negotiate with YouTube. We can tell them what we think, but that's not really negotiation. As you know, YouTube does what YouTube will do, because it's their platform, and I have no problem with that. Like I said in a previous video, I really appreciate the YouTube platform. I love that we have this thing called YouTube, so we just do our best to adapt as YouTube changes the rules. That's, I think, the best kind of attitude we can have about this. I've got a list of ideas for you. Some you may have already thought of, but hopefully this can help you. I'm also willing to upload some videos based on my expertise, technology, and gaming for freedom on my channel. Things such as best budget PCs for streaming. How to react, how to read analytics on YouTube if you'd like that. Here's my list. Game selling sites. Getting partnered with freedom partners. In his video descriptions, he puts a link to Kinguin, a website that sells CSGO skins, knives, and game keys at really low rates. Other sites include G2A, CD Keys, Green Man Gaming. How many of you use this technique to make additional money by putting affiliate links to sites like that beneath your videos? What other sites do you use? Let me know down there in the comments below. A streaming site like Twitch or supporting Hitbox or whatever it's called now. It's called Smashcast. I know they've renamed it since they merged with Azurius. Now it's smashcast.tv. But if you go to hitbox.tv, it redirects to the new site. It's no lie that freedom and Hitbox partnership worked really well. People really enjoyed it, even though Twitch is more popular. I'm happy to hear that. I know there are other streaming sites good for this, but it seems like whatever Hitbox we named itself to just ruined their system. Do you agree? I'd love to know. If you could talk with their developers and maybe you guys could negotiate a deal like the one we had before with their promise of making an actually good streaming site that could be a good way of doing it. Actually now we have to encourage everyone to stream on YouTube, hence this video, because the first goal should be to reach the 4K watch hours because gaining subscribers is, in my opinion, a lot easier than gaining watch hours and live streaming is the best way to do that. So here's my best idea. Don't miss it! Don't worry, we're not. Introducing a Steam tipping service like Lutz.com. Lutz basically puts this thing where people can send a message on the stream and it costs nothing to the person that requests it, but it will give money based on how many viewers are present to see the tiny ad that displays with it. That's an interesting way to do advertiser supported live streaming. How many of you like this concept? Tell me. And if we have enough support, we'll do our best to get such a stream tipping service like Lutz.com. Hopefully these help. If there's anything I can help with, let me know. I think you guys at Freedom are genuinely good people. Thank you. And don't deserve to get hit as hard as YouTube is hitting you. This reminds me of a very important quote. It's from Rocky. Nothing will hit you as hard as life, but it's not about how hard you can hit. It's about how hard you can get hit and keep going forward and keep moving forward. That's what I live my life under that concept doesn't matter how hard I get hit it's important that I keep moving forward. In other words, hashtag never give up. It's another nice way of saying never give up. Do you have any famous quotes or popular quotes that you really remember and enjoy? Share them down there in the comments below. Dr. Dunk wrote a tip for you all. Be active on other YouTube channels. I talked a bit about the new IPP on videos related to them the positive points and I gained 11 subs in two days and counting. This also applies on being active on forums etc. It's a very good tip because collaboration is the fastest way to grow on YouTube and the more places you can put your videos and your brand the better. Like forums. www.community.tm Those are our freedom forums. I hope you check them out. Create an account and post an introduction. It's a good way to meet new friends and get a few more subscribers. Haka Ren wrote, Okay, this is a huge problem to the majority of the smaller channels on YouTube but after what YouTube is doing to us I'm not giving up. Very good. And I advise other small channels out there to do the same. Me too. You may think YouTube doesn't care about us smaller channels and that's understandable but that's not the case. How do you think many of the biggest YouTubers have gotten to where they are today? Everyone of course started with zero subscribers and zero views at one point in their life even PewDiePie. Well, maybe not PewDiePie. Did he start with like 2 million subscribers? I don't remember. YouTube today is a lot easier compared to what it was a few years ago. It used to be a lot harder to earn money on YouTube so really we've been given more opportunities and more chances to earn money today compared to back then. I completely agree. There's now super chat, sponsorships and many other ways that you can make money on YouTube like merchandise and affiliate deals. Some of these of course are not built into the YouTube platform you have to go get them yourself but that's the whole point of this video and why I chose some of these comments to remind you that there's more to revenue than just YouTube. AdSense. And there's that C thing.  We've been together for how many years? One, two, three, four years. We started four years and two months ago as a YouTube MCN. Back then we had zero channels zero subscribers and zero views. I'm missing one more hand. And of course we grew because of you of how we all helped each other. That hasn't changed. Just because YouTube throws us some new challenges does not mean we should give up. Does not mean we should change any of our culture. This is just a test. We're being tested. This is like math quiz. Of course we're going to pass. Of course we're going to recover the lost revenue from all this. And we're going to do it. Say it with me. Together! As a family. Who am I? I am Papa George. George Sunpai. In many other names, you, the beautiful Freedom Family, has given me. I am George, CEO and founder of Freedom! For Freedom and the Freedom Family. And all the amazing people that are commenting down there. We're getting like 500 plus comments up to a thousand comments beneath each of these videos. Where did that come from? I mean obviously it's an emotional reaction. These are, this is an important topic but I'm so happy that all of you are giving great advice, great motivating speeches, some of you walls of text, to keep going. The messages never give up and the messages were here to help each other grow together. And with that, I am George and you've been watching Click that I to partner with Freedom and join the Freedom Family so we can all grow together. You get many perks like position music, you also get epidemic sound, a lot of other access to royalty free videos, sponsorships, and many things to help you grow. Just click the links down there in the description below to get involved in our community, our forums, our discord chat servers, meet our graphics team, meet our community team, all of that on discord and the forums. What are you waiting for? Get started. And we will grow together as a family because this is the Freedom Family. You are part of it. We are all part of it and we're all growing together. To get more George click that big F that will subscribe you to Freedom Central home of the George show and PewDiePie gave one of you Freedom Family a big shout out. Click that video to see the shout out and to see our new 3D sets for you and click that video to see what YouTube recommends you watch next.


and to ask for eight percent cut in pay on the backs of these employees when management and the poa negotiated a far less and when we have so many employees that have recently left i just don't see the numbers of $68,000 and i just feel that as a council you should look at the entire picture of what the cost savings of these recent vacancies and consider that for these employees because so many of these employees that they're just barely making it and i just feel greatly that we should consider that thank you for your time thank you for ever we appreciate your comments i believe ali is the next person with their hander hi can you guys hear me okay i had actually emailed twice but i don't know how that never made it to larry anyways so my name is ali cliford i'm a recreation coordinator with the city of capital and my last day of work is tomorrow i have one of those three positions being laid off i have been with the city for just shy of nine years excuse me in a recreation coordinator for almost two i'm a three-quarter time employee i was responsible for all of our recreation classes independent contractor instructors our recreation brochure building rentals events marketing and social media i voted no on the furlough knowing my job was at stake because myself as well as my fellow as union employees are already so overworked and underpaid i just couldn't in good faith ask my fellow co-workers to take a pay cut to save my job the salary study done last year was proof of how underpaid many of us from within the city are my position in particular is already 14.6 below 14.6 percent below the market median if you were to take a look at a paycheck for myself or any other our union members you would see that the take-home pay is barely if even at all enough to be able to afford to live in scenic creek county i will soon be on unemployment and with the temporary extra six hundred dollars a week of federal stimulus i'll actually make more unemployment than i do now the proposed 8.1 percent furlough would have been too much too huge of a financial hit for many of my fellow co-workers we already struggle enough as it is we shouldn't have to struggle more than we already do i personally don't feel the city has been diligent enough or transparent enough in finding other alternatives besides these layoffs thank you thank you ali all right we're going to bring it back to the council uh councilmember story has his hand well thank you mayor um you know after uh hearing barbara and ali speak um i couldn't hold back and not acknowledge their pain and how difficult of a situation that is this is um for everyone concerned um and we certainly hate to see that it came to layoffs um and the ali in particular and any of the staff but you know everyone's in a dire strait um and all the other staff have agreed to reductions um and um you know it's unfortunate that you know we couldn't get other reductions from the a staff i understand their viewpoint um but you know i think it's important that everyone um kind of move together uh in that um and um i'm hoping um you know maybe it didn't need to be eight percent but i didn't i didn't hear or understand that there was any counteroffer made um but and so this is where we've ended up um if there's some way of ability to salvage um these layoffs i would certainly be open to it um but i'm really um you know i'm sorry that we have to lay out positions at this time um but um i think that we felt that we were kind of pushed into that position so um i just want to acknowledge and thank bob and ali for speaking up and thank you man thank you council member story are there any other comments from members of the council okay um i echo what council member story said in acknowledging the pain and the hurt that is clearly heard here tonight is understandable and justifiable that they there is that you are experiencing this pain and this frustration and i hesitate to say what i want to say next because i feel like it is um minimizes the the hurt that we just heard and that's not my intention in any way at all what i want to share is that we are also hearing from community grant recipients that we are causing pain we are also hearing from business owners that we have caused them pain we're hearing from residents that we have caused them pain and it's not anything that any one of us wanted to cause to any of those people and we are in such challenging times that i know that we are all feeling that challenge and these are very difficult decisions that we didn't come too lightly and i want to share that that we all well i don't speak for this for the whole council but i speak for myself and i speak for what i perceived in the discussions that were had that we were all feeling the weight of these choices council member chon has his hand up to you and i believe he has comment council member chon you're still needed sorry um just want to recognize that it is particularly hard for the people that are being laid off i've been in that position multiple talks in 30 to 40 years of my work experience and in many cases under very diverse circumstances where i didn't know we're going to be able to pay for things for our family so i totally understand what you're going through i'd also like to recognize that there's been a lot of sacrifices by the people of work for capitol part of the general employees for the city and so i want to also recognize their sacrifice too and um you know some may be paying more of the sacrifice thank you very much thank you i don't see any additional hands up from council members i believe and perhaps our city clerk can confirm for me that i believe we had a motion and a second already that is correct uh council member bautour was the mover and council member for trans seconded thank you uh there's no further comment from council uh can we get a roll call vote please yes council member for trans i council member bautour all right council member story um i vice mayor brooks hi and mayor peterson hi thank you thank you motion uh carries unanimously that brings us to the end of our meeting tonight um please take care of yourself physically and mentally and emotionally and take care of each other and we will see you again soon


Welcome back to Embedded World and time to focus on asset tracking. Now whether it's moving freight around the world or pallets in an automated warehouse or even livestock on a farm, ST customers want to know where their assets are 24-7. So to discuss ST innovations in this very field I'm joined by Carmen Briganti. Carmen, lovely to meet you and I love the demo but what does it do? Okay this demo represents a complete ship asset tracking management where we are able to locate and monitor the asset outdoor during the outdoor transportation and once the asset arrived to the warehouse we are able to locate with the these technologies the asset inside the different area of the warehouse. When the asset come back outside the journey returns. So what kind of technologies involved? Is this new? Yeah this is the new ST solution, the ST VALA ASTAR 1B providing short and long range connectivities thanks our STM32WB for Bluetooth low energy connectivity, STM32WL for LORA connectivity, the ST25DV for near feed communication. It also embeds a Genesis module for high accuracy location outdoor and it has the high end sensors, temperature pressure and motion sensor and it also has a ultra low power management to ensure long battery life. What kind of development environment is involved here? STVALA ASTAR 1B is inside the end-to-end proof of concept. We provide firmware and hardware packages, a steam mobile apps and a web-based dashboard to start to develop your own asset-taking application. So we provide all the bricks from the scratch, the hardware and the firmware and the ecosystem to reproduce a complete asset-making solution. You know it's just like a child's toy but it's an incredibly complex process with enormous potential, enormous potential. Carmen, lovely to meet you and thanks for talking to us. Thank you.