OK, zelo smo začniti. So, pilično je hrabe, da povedemo Prof. Frans Fračnerič iz Univeristija Fjubljana, pilično je se odnočil te nekaj kompleks analitici v rlješku poznatnih sorceri, če pletima, vsak z nekaj terbe. Če neznam, da so imamo, Klaudija, Aretzov, za invitacij, je to pilič, da sem pripravil, da sem regična. had one very pleasant experience here three years ago for two weeks we had a program a conference in a PhD school. Okay so today this is on, right? It's okay. So today I'll give a rather easy and non-technical talk about a subject that was new to me in 2011 when I first met these two people from Granada, Spain, Antonio Lercón and Francisco López and then we became collaborators and whatever I know about Riemann surfaces, I mean about minimal surfaces is basically they taught me and maybe I taught them some complex analysis we discovered that putting these two things together was extremely fruitful. So the talk is, as I say, it's going to be very non-technical kind of survey. The first 10 or 12 slides are kind of promotional. If you've never seen the stuff before then it's okay. If you have seen it then please just skip this part and then later I'll tell you about some new things which we did together using complex analytic methods. So first this subject of minimal surfaces is quite old. So in fact the first time when the notion of minimal surface was mentioned was by Euler in this year 1744 and this was on a kind of intuitive level.  And so Euler in fact gave an argument that the only area minimizing surfaces of rotation meaning you take a curve and you rotate it and you get surface then these are planes and catanoids and here is a picture of a catanoid. Then this topic was picked up by Lagrange who did first analytical work on the subject and he actually came up with this equation. So this is an equation for a function. You look at a graph of a function over a domain and you'd like to know when is it critical point of the area functional and this precisely means that if you take one per meter family of perturbations and you keep the boundary fixed, you look at the area each time so you find area as a function of t then you differentiate the time 0 and you ask that it be 0. This was the first time when variational approach was used so Lagrange came up with this equation which has to be satisfied which characterizes the stationary points of the area functional. This equation if you look at it, it looks like Laplace, but in fact it is the metric Laplacian, in fact we'll come to it later. And then we see a few years later showed that he was the first one who made a connection with this main point namely that the mean curvature function has to vanish identically. So we say today that a smoothly immersed surface in R3 is a minimal surface, if it's mean curvature function is identically 0. So what is the mean curvature? It's simply at each point of a surface you have two principal curvatures which means you look at the curvature of the curves going through that point in different tangent directions and there are two special directions, one is along which the minimum will happen and one is along which the maximum will happen, they are orthogonal to each other and those are principal curvatures then you take their average and that's called the principal curvature. The product is of course called the Gaussian curvature. So the connection is that the surface is a minimal surface even if and only if it's mean curvature function is 0. If you look at surfaces in higher dimension it's still a little bit like this except that then you have more normal vectors, not just one. If the surface is in R3 you have only one, you need normal vector or actually two, you can take depending on the orientation but if you're in dimension four or higher then you have N minus two independent normal vectors fields and you have to actually choose one. One of them you fix it and then you can calculate again this quantity is the principal curvature, principal curvature is the mean and the Gaussian curvature with respect to that normal direction but the theorem is still the same that all these curvatures, mean curvatures have to vanish in order that the surface is a minimal surface. Here is another picture of a minimal surface, this is called a helicoid. So you see it's essentially what you do is you take a line in the plane and you rotate it but then at the same time you lift it at a constant speed so it's just a screw, a hem at a screw helicoid. That's another minimal surface and then it was proved by Katalan in 1842 that the helicoid and the plane are the only ruled minimal surfaces. Ruled means they're actually unions of straight lines. Plato introduced a new topic in the theory because he started looking at the question when you take a curve in space, a closed curve, image of the circle, can you find a minimal surface in such a way that this curve is its boundary. This became known as Plato problem and in fact this Plato problem was solved in 1932 independently by Douglas and Rado and they showed that the answer is yes. That every continuous Jordan curve spans a minimal surface. This minimal surface might have some singularities. It's not immersed in general but still. Then Riemann also made a contribution to this topic. He discovered some interesting family of examples. Here is a picture. So this example is supposed to suggest, these are called Riemann's minimal examples. So they suggest properly embedded minimal surfaces which have countably many parallel planar ends. You see them here. End is what goes to infinity. So they are countably many, parallel to each other and every horizontal plane intersects each of them in either a circle or a straight plane. Topologically they are actually just plane domains. Now, let's now do some mathematics. If you assume that M is an open Riemann surface, how do Riemann surfaces come in? Riemann surface is coming as follows. If you have a surface in Euclidean space embedded or immersed, then what you can do is put the Euclidean metric, you pull it back, you restrict it to the surface. So you have a metric on the surface. Now let's say that the surface is also orientable. We know that metric, it determines conformal structure and that also determines the complex structure together with orientation. It determines the, because you have a unique, for each vector you have a unique orthogonal vector of the same length and so you determine the J operator and that gives you complex structure. So it's very natural to look at these things, at least if they are orientable as Riemann surfaces. And if you do this, if you take this metric, which I told you that you just pull back the Euclidean metric to the surface, then actually by the very definition your embedding or immersion becomes conformal from this structure which you defined on the surface by this metric to the Euclidean space. It's conformal. So it's natural to look at conformal maps. And then it turns out that if you look at the smooth immersion from a Riemann surface to Rn, we define it to be conformal if it preserves angles, the usual definition. Now, if we denote by H its mean curvature vector, I told you before what is the mean curvature if you have a surface in R3. In that case, this is a scalar function but you multiply it with a normal direction so you get a vector field. In higher dimensions you have also a vector field which is called mean curvature vector field. And in fact, there is a crucial relationship that the Laplacian of your map is simply twice the mean curvature. Here this Laplacian has to be the metric Laplacian. So it's this metric.  But you see, this simplifies actually this formula because usually you like to work in what's called an isothermal coordinate. There is a classical theorem going back to Gauss where the metric on the surface the simplest coordinates you can choose so the metric becomes as simple as possible are called isothermal coordinates and they are coordinates in which the metric is simply a multiple this is a function of the Euclidean metric. And if this is the case then it turns out that the metric Laplacian is related just by this factor. This is a positive factor of course with the usual Laplacian. So it doesn't matter really which Laplacian you use either you use the factor or you don't but here basically you have this formula. So the main conclusion that we now have is that if you look at a conformal immersion then it's minimal if and only if it's harmonic because h, minimal means h is zero and harmonic means this function Laplac is zero. And again it doesn't matter Laplacian or the standard complex Laplacian because it's only the scalar factor in between. Good. Now what is the connection with complex analysis? Of course as soon as you talk about harmonic functions or harmonic maps it's somehow clear that complex analysis is involved but it can be made a bit more systematic. So if x is a smooth function of a complex variable now I'll call it zeta which in fact I think of this zeta as some local coordinate on a Riemann surface. Then we have these two derivatives. This delta of x which is written like that this is the one zero part of the external derivative so this is the c linear part and then you have the d bar of x which is the zero one part the anti linear and in fact holomorphic functions are characterized by this equation the d bar so the second derivative is zero this is the standard Cauchy Riemann equations and that's equivalent to saying that the exterior differential is just del. Then x is harmonic if Laplacian of x which is really the same thing as this operator 2i d d bar is zero this is just a little calculation and so this means that another equivalent condition is that del of x is holomorphic because del of x is a one zero form and here you see that del bar of it has to be zero this is just equivalent to saying that this is holomorphic so upshot the two things together is that if you look at an immersion no that's not an upshot this is discussion about holomorphic and harmonic that point namely that an immersion from a Riemann surface to R to Rn is conformal meaning angle preserving if and only if this is true you take partial derivatives of x on u and on v u and v are the real and imaginary part of the local coordinate and conformality means that these two partial derivatives which are vectors are orthogonal to each other and that's clear this is just conformality going from u plus iv to the euclidean space but now the good point is that we can rewrite this condition as follows that we take this delta of the components of x so this derivative the holomorphic derivative and we square them and we sum them up and we get zero and this is very easy to see x is a real map so x is real so this is the real part this is the imaginary part what happens when you square it you get these are vectors so what you get is a dot product of this vector with itself so you get this quantity x on u square and then you get if you look at the real part of these squares you also get because you have this i here so what you get is the real part of this quantity you get precisely the difference of this length length of the x on u square minus x on v square this becomes the real part of this expression and the imaginary part really becomes this expression because the imaginary part is you multiply this vector with that vector and once more the other way around so you can encode these two real equations into this complex equation that is crucial because now so now the upshot is that the smooth immersion from my Riemann surface to Rn is a conformal and minimal both at the same time which means conformal minimal surface if and only if first of all del of x so the one zero derivative this vector this is a vector valued one form is holomorphic and the sum of the squares of the components are zero okay so this is an analytic characterization of what is a minimal surface now I can fix maybe I don't like to work with forums instead of working with one forums it's easier to work with functions so what I do I can fix no over vanishing holomorphic one forum on M and you see such always exists because M has to be an open Riemann surface M cannot be compact by maximum principle you cannot put it a compact thing inside Rn by such a map so if M is an open Riemann surface then it's a very classical theorem going back to 1967 it's the so called Gunning and Erasm can theorem which says that in fact you have such a forum actually Gunning and Erasm can tell you more they tell you that every open Riemann surface admits a holomorphic function which has no critical points so this holomorphic function gives you an immersion of your Riemann surface to see and then you can simply take the differential of this function for this forum so you even get an exact forum that has no zeros and if you use any such forum then you can actually divide you see if you divide del x by such a forum then in fact you get a map into this into this quadratic variety so you define this is called a null quadratic for some reason this is the quadratic of all vectors in Cn where the sum of the squares of zero is following this equation and then the upshot is that how do we represent conformal minimal immersions everyone is of this forum so you start with some map f holomorphic map from your Riemann surface to this null quadratic except I take away zero to take away zero basically means that I will look at immersion solely if you allow value zero then you allow branch points so you go to puncture null quadratic this gadget without zero take a holomorphic map you multiply it with this one forum so we have f theta this is now a candidate for this del x but what is the condition that you can integrate it well the condition is this one that the integral of the real part of this one forum has to be zero over all closed curves otherwise the integral is not well defined and if you indeed know that your thing comes as a differential then of course this is satisfied differential when you integrate it on closed curves it's zero okay so this is called a vajerstels formula and this vajerstels formula which tells you everybody you are interested in all your objects are of this forum so they are essentially integrals of products of a map holomorphic map from m to the null quadratic with your fixed one forum the condition that you can integrate is that that the integrals over closed curves we call these periods real periods they have to be zero then you can integrate and you get a conformal minimally merchant because it's derivative del of this is actually this thing underneath it's f theta okay so this forum is the link so as we see now what is our task if we want to produce minimal surfaces from parameterized by remote surfaces well we have to create maps to this null quadratic without zero such that this condition holds okay and this as soon as I was told this I said well this is something I know about because I had been working the previous decade a lot on something called orca theory and orca theory is a theory about about the problem when you want to know let's say that you want to construct holomorphic maps from open reman surfaces or more complicated objects in higher dimension called stein manifolds stein manifolds into some complex manifold so you want to know when can you construct a lot of maps and the question is having a continuous map when can you change it by homotopy to a holomorphic map and you see this question is rather classical and was studied first by orca already in 39 then by Grauert in 57 and 58 and in fact the main theorem which I need for this minimal surface business but before I get to that let me finish the story anyway let me finish the story so here we were talking of maps to aren we were looking at this condition that the real periods have to vanish but you see you can also look at holomorphic immersions to see n which whose differential satisfy the same vanishing condition and again you conclude that every such is of this form now it's you integrate f theta without taking the real part because now you really want a holomorphic object and what now f should be it should be a map which satisfy now the full period to be zero not only in the previous slide you see we had a real period real part now we have a full period that should be zero so the conclusion is that if we call such curves holomorphic null curves just because they are directed by this null quadric this means the tangent vector at each point is in the null quadric that's why they are called null curves and now we see the connection immediately that the real and imaginary part of a null curve are conformal minimal surfaces in Rn and not only the real and imaginary part but you can rotate these are called associated surfaces and what is the converse well if you have a conformal minimal surface then it is locally in fact on every simply connected domain in M the real part of a null curve what is the problem in general is that as you know with harmonic function you cannot take harmonic conjugate in general you have conjugate periods likewise here this is the same problem you can take a harmonic map so it will have its differential has vanishing periods but it has the conjugate differential has non-zero periods maybe and that's called flux these periods are called flux in this theory so here is a nice example you have seen before the picture of a catanoid and you have seen a picture of a helicoid and it turns out that these two are closely related they are actually called conjugate minimal surfaces because they are the real and imaginary part of the same null curve and here I wrote down the equation this is not the only possible representation of it but it's a simple one cosine z, sine z minus i z you multiply with e to i t take the real part you call it x t so this is a minimal surface and look what it is it's a combination with cosine and sine of these two columns now the first column you see here cosine hyperbolicus this is a catanoidal curve I mean y and cosine hyperbolic y this is the catanoid and this multiplying with cosine and sine means that you rotate it in the space so this is a catanoid and this one is a parameterization of the helicoid so when your time varies at time zero you have this curve at time pi over 2 or minus pi over 2 you have this curve and this is a helicoid it's also easy to see because you have sine and cosine and x so this means you're really rotating a line and lifting it at the same time and now I wish to point out that in fact this connection how to use Weisstrass' formula this has been really known since the early 60s when this particular man Robert Osserman pioneered this approach in modern time so in the early 60s he discovered this and he wrote a very beautiful book which is even today one of the classics in the subject Osserman was actually more like a complex analyst he was student of Ileforce and that's very much prominent in his work and he's a great expositor so this book is really wonderful ok, now I can finally I wish to say that even though this was known in the 60s that complex analysis could be used but for some reason it wasn't used very much so when I first learned this thing this was back in 2012 maybe when I learned this thing from these two people in Spain it was to me incredible how come that these things after all were not so hard to do how come that nobody has done them I was really amazed and also some people from Russia told me that in fact in Moscow they also knew this work of Osserman and they knew that he proposed this approach complex analytic but somehow it wasn't really developed so here is some but let me tell you let me ask you when do I stop so I don't want to one hour I don't want to be too long ok, so here are some topics I'd like to discuss which we found in probably now about ten papers by now together with these people and one two of them also with my colleague Barbara Adrinovac, who is from Ljubljana so one topic is the Runge-Mergelian approximation theorems for conformal minimally merchants so what this means I'm sure you know because Runge and Mergelian these are classical approximation theorems for holomorphic functions but it turns out that the exact essentially the exact analogs don't prove for minimal surfaces next thing we constructed proper conformal minimally merchants so I now abbreviate this as Cmi into Rn and minimally n in some class of domains which we call minimally convex and again this mimics something in complex analysis which is called Levy pseudo convex domains so it's a kind of analog to Levy pseudo convexity this minimal it has the same role for a minimal surface notion plays the same role for minimal surfaces as the Levy pseudo convexity does for complex curves another topic is a very famous one it's called Kalabiou problem but this is not Kalabiou manifolds but it's a it's a Kalabiou problem for minimal surfaces I'll talk about it when I come to it and then we have some new results on the Gauss map and we have other things but I couldn't put everything in one talk so let's start with this theorem here I'm just recalling what is a classical Runge theorem what is the classical approximation theorem in for holomorphic function so we call a set holomorphically convex if in a Riemann surface if the complement has no relatively compact connected components no holes as we say if m is equal to c then of course this is connected in such case and case called polynomially convex so the complement has to be connected and the classical theorem due to Runge is for functions in c so that every so that you can approximate holomorphic functions on compact polynomially convex sets by entire functions and this was generalized to all open Riemann surfaces in this year 49 by Benkenstein in fact they proved the classical result that every open Riemann surface is what we now call a Steinmanifold so it has lot of holomorphic functions and in particular you can approximate holomorphic functions on such sets and here is then our theorem why I give two years it's because Anarkon and Lopez before we actually met they proved this theorem in dimension 3 with some kind of ad hoc methods and after we met we discussed and then we found a much better proof which works in every dimension so the theorem is you take a compact holomorphically convex set in an open Riemann surface and you take a conformal minimal immersion so a minimal surface defined on a neighborhood u of k we can approximate it on k by globally defined minimal immersions here I am outlining a proof I said it will be a non-technical talk I think some ideas are really easy to explain so I am outlining some few things so we start with conformal minimal immersion from some open connected open set u containing k we can reduce to the connected case so now recall the Schwarzschild's formula which says that in fact x is given as an integral with some initial it's an initial value here at some point it's an integral of the real part of f theta where f is a map and this is a mistake it should be u it's a holomorphic map from u into this punctured nullquadric and the real periods have to be zero in fact why I call it d because I have picked now a domain d is now a compact domain smoothly bounded which is between the set k a nice domain then I have a basis of the homology group first homology group is in a finitely generated abelian group so called z to the l I take a curve switch for a basis for this first homology group and I define this so called period map period map takes holomorphic functions on this domain actually maps to cn to this space cn to l where l is the number of these curves and what is the component each component of this period map is simply that I integrate f theta or the jth curve theta is some fixed one forum fixed ones and for all like in the previous slide and then I note that one forum is exact if and only if this map is zero because it just means these integrals are zero and the real part is exact if and only if the real parts of the periods are zero so im gonna use this period map and here is the crucial lemma this is something non technical given a non flat holomorphic map from this domain into the null quadric and non flat just means that it's not contained in a ray you see this thing is really a quadric right so it's a union of rays lines complex lines through zero and those maps which actually are contained in a ray are a little bit a problem in this theory but not really because what they correspond to if you integrate them they correspond to surfaces which are flat which actually lie in a euclidean plane so those are not so interesting so we can avoid those really so if I have such a non flat map then I can find a neighborhood of the origin in this euclidean space and the holomorphic map from the cross this neighborhood for time and this is parameter now I call it t t is a member of this v so for parameter value zero I get my original map and if I look at this family of periods so I vary t I look at the period map and I take the derivative on t at time zero so I take first variation if you like of this period map on this family it's a map from here to there and this is an isomorphism so the thing is that such a thing exists so in other words I'm saying give me any map I can vary as long as this map is not flat then I can vary it through a holomorphic family of maps of appropriate dimension such that my periods vary in all possible directions so periods become period map the derivative of it is an isomorphism and this is not hard to do because how I do such a thing I use vector fields flows of vector fields to vary in different directions so it's a sorry I went too fast okay so this lemma is not hard to prove but now let's go on we can also observe that the null quadratic has convex hull just geometrically convex hull equal to all of cn that's actually a general effect about any algebraic subvariety algebraic subvariety of cn it's convex hull is actually a linear subspace it's not true for entire subvarities but it's true for algebraic and then one can show easily that if you integrate g times theta theta is this one forum over all possible loops c is a circle so we integrate over loops in the null quadratic can you look at what are all possible values of these integrals and actually they are all of cn and this is a simple idea which is the basic idea of Gramov's convex integration theory in cn I can write it as a convex combination of points in my null quadratic and now what I do so I have some weights it's a convex combination some pj times qj times the points then I construct a curve which goes quickly to the first point spends roughly the correct amount I'm there goes to the next point to the next point and so on so that gives me roughly the right integral so anyway it's not hard to prove this so by considering such deformations for all loops in the period basis we can create a smooth period dominating spray so a map like in the previous slide so initially it's only defined over the union of these curves cj is missing here so this is the union of cj which has this map as a core but there is another standard fact that these loops cj these curves can be taken such that the union of them is holomorphically convex so once I created some functions on them I can use Mergelian's theorem to approximate them by functions which are holomorphic on D and that gives me the spray as in the Slema so in other words I get the spray simply by constructing a good thing on the curves themselves and then I approximate ok, how do I use the spray? well, now comes Groward already mentioned him ok, this nullquadrik has a rather important structure namely it's a homogeneous space of this particular group, Lee group it's called a complex orthogonal group so these are complex matrices which satisfy this relation without the conjugate you see, just a at equal to i so identity complex Lee group and this group acts transitively on the nullquadrik, except at zero of course so the punctured nullquadrik is homogeneous space and here Groward's theorem says that anytime you have a Stein manifold Stein manifolds are like open Riemann surfaces or actually any manifold which is embeddable into Euclidean space maps from Stein manifolds to such a homogeneous manifold satisfy all sorts of properties for example the Runge approximation theorem in the absence of topological obstructions so as long as there is no obstruction to approximate map on a subset by map on the whole manifold you can do it so what I can do now is apply this theorem to approximate my map which is this period map which is defined on this domain in m cross v, v is a subset of Euclidean space nullquadrik by a globally defined map and since derivative of x was in fact the real part of my original form we have that the real part of the period of f is zero this is from the initial condition if f is close enough to this approximation map is close to vf then of course when I approximate periods will change but since the initial period was zero and my family of maps was period dominating meaning that in the family I had all nearby periods so once I approximate I can find a nearby parameter value t such that this map f of dot t which I now call f tilde still satisfies the same equation that the real periods are zero and this is crucial because now I can integrate it and after I integrate it I get a map except there is a hitch here namely I can only do this on a bigger domain d prime which is such that it retracts onto d such that d is a retract of d prime in other words if I change my domain in such a way that there is no change of topology then I can approximate map on the small domain by map on the big domain which is a change of topology then there are new curves and then I know nothing about periods on them so I cannot integrate maybe so that's the moment not clear but at least if there is no change of topology I have this approximation theorem so this x tilde already approximates x and here I explain you a little bit how I deal with a change of how I deal with a change of period with a change of topology but maybe I better draw a picture because it's essentially just like this you have a domain and to change of topology is described by something like this maybe this bigger domain so it's described by attaching an arc to the previous domain but that's one particular change but this is the main one that's when you take a more exhaustion function that's the point of index 1 that's exactly what happens that you have to attach an arc so what do I do well, I have my map already defined on this domain now I have to extend it smoothly across this arc in such a way that its integral is correct because finally I'll have to integrate so I need to have that the integral of my forum on this arc is actually equal to the difference of values of the already given map in order that the integrals will come out right and after that I more or less repeat the story which I told you so I put this thing into a spray approximate by mergelian here I have to use not Runge but mergelian on this object and finally I find a map on a bigger domain which satisfies the period condition also on this loop you see there is a new loop here this arc is perhaps a part of a new loop and it satisfies the condition also on this loop ok good so now I am in business and then I just have to do induction I'm not gonna read the rest of the slide ok and then after this theorem we said ok how about proper maps, right that's more interesting because proper means that you put the thing so that the boundary goes to infinity so the image is closed much better object if you look at proper maps and in fact we realized that with this technique which I just explained you plus the usual technique how you construct proper maps we could in fact prove this theorem so there are three parts I take any open Riemann surface first part says that if you want to go to R3 there are a lot of proper conformal minimal immersions in fact proper immersions form a dense subset in the space of all conformal minimal immersions in the compact open topology so you give me any any minimal immersion any conformal minimal immersion from a good subset of M I can approximate it by globally defined one which in addition is proper so the boundary goes to infinity ok I'm talking of immersions here and I will show you later a result that in general this is not possible with embeddings in fact on the later slide there will be a theorem about embeddings but this is just immersions now what if you go to R4 instead in R4 the natural thing you can expect and we also proved it is that there are such things which can only simple double points meaning transverse double points ok if we go to 5 or higher then we can create embeddings always so here is a comment which I mentioned before in fact there has been a lot of works but maybe the crowning work is this one Mix, Paris and Ross from a few years ago they show that in fact planes, flat planes catanoids, helicoids and Riemann's examples which you also saw on one of the slides are the only properly embedded minimal planner domains in R3 as far as embedded you are much more restricted of course now these same people I believe are working on further results because one would like to know how about if it's not a planner domain how about if it has some genius it's a domain in some more complicated Riemann surface well I think they have some results already but it's story is not complete and there is a corresponding there is a problem which we post and we don't know how about it go into R4 question does every open Riemann surface embed into R4 properly as a conformal minimal surface we don't know and there is a connection with complex curves because in fact complex curves are also minimal surfaces special cases of minimal surfaces so there is a corresponding open problem which is well known since long time ago called bell forced and a simpkin conjecture actually conjecture that the answer is yes but I'm not sure that it's true so I rather call it conjecture does every open Riemann surface admit embedding into C2 that's also open of course if you could solve this one yes then this would be also yes or if this is no then this is no because there are more minimal surfaces than complex curves in this context I wish to mention that Erlen Wald from Oslo and myself have two papers in recent period one of them is 2013 one is a bit older which gives in fact the two best known results still at the moment about which Riemann surfaces will embed into C2 for example we prove that every circle domain possibly infinity connected in C this means you take from C countably many disks in such a way that you get an open domain such a thing is embeddable but the question is wide open and while doing this thing this I mentioned just as a curiosity there is a famous example of a mix from some I don't know 15 years ago maybe of a immersed minimal maybe strip immersed into R3 which has final total curvature in fact minimal possible final Gaussian curvature ok we found an example where it's an embedded minimal strip in R4 seems to be the first known example very simple map actually just have to do some checking the main problem was not to check that this map is good it's just how to come up with the formulas I'm talking about minimal strip so here I take rimless surface to be C star and this is map from C star but in fact it's you see it has this property that two points are identified under this map if and only if the point is either Z1, Z2 or Z1 is minus 1 over Z2 bar and this map Z send into minus 1 over Z bar is the reflection around the circle is this reflection around the circle so what happens when you in fact no in fact it's not this reflection this would be it has to be the opposite so it's this one here in fact if you look on the circle itself the antipodal points get identified so that's why you get a maybe strip now I promised something about minimally convex domain so what is this minimal convexity I was asking myself you see since I having background in complex analysis it's natural to ask what is a good class of real functions on which your objects in complex analysis maybe complex curves or complex manifolds on which you have maximum principle so the good class of functions for complex analysis are the supharmonic or the pleurisupharmonic functions right? Supharmonic functions are precisely those whose restriction on curves will be actually pleurisupharmonic are those whose restriction on complex curves will be supharmonic and so we identified such a class of functions on the real domain I call a smooth function minimal pleurisupharmonic if the following is true you take all the eigenvalues of the Hessian of rho so this is a real function you take its Hessian you take the eigenvalues you order them and you take the first two the smallest two and you ask that this be bigger equal to zero so it means only one eigenvalue can be negative or the rest non-negative and then basically the main thing is that we showed that rho is minimal pleurisupharmonic if and only if the composition with every conformal minimal surface is a supharmonic function after we came up with this class we have realized that in fact we were not the first ones we found some papers by Harvey and Lawson where they also treated this class of functions and they did some general things but we at least had a good theorem after that I mean we had a theorem where this was used so what is the theorem I am stating only a special case for R3 just to be simpler so if I have a domain in R3 minimally zero convex in the sense that it admits a minimal pleurisupharmonic exhaustion function so one satisfying this condition then every then if I have a compact border R3 surface so this means R3 surface with boundary with finitely many boundary components then every conformal minimal immersion of the surface into the domain can be approximated by proper conformal minimally mergers from the interior so really this means I can if M is entirely together with boundary lying inside I can now stretch the boundary to the boundary of the domain that's what I'm saying I can create proper ones and in fact simple example show that this is the best class of domains in which I can do it and this theorem is actually which says that if you take a hardtox or levi-zero convex domain then any compact R3 surface which you put inside is a complex curve you can stretch the boundary out to the boundary so you can make it proper but ok so it's similar but in fact to prove it was much harder and there is another idea so I'm left so instead of reading through this slide I'd like to draw another picture we introduced a method which is coming from complex analysis called Riemann-Hilbert boundary value problem it's one of the classical subjects of complex analysis since 100 years ago but we found a way of using it in this theory so what we do with this thing so we do the following let m be such a compact Riemann surface with boundary so it means one of these things m is something like that because of few finitely many boundary components now you map it in some space how do I call this map maybe I call it x I think here I mean R3 but I could be in Rn actually we have versions of this thing for Rn so if you think of this picture let's say that this is my x of m so it's sitting as a curve with boundary in R3 yes now what I do is I also have this map which I call f in the slide and what does this map do map says take a boundary point of this first curve and put a conformal minimal disk attach to it a conformal minimal disk so this is a disk which center point is here this is my f disk f of p c and c is the variable on the unit disk and I attach these disks around so basically here I have also one at each boundary point in fact I do it on an arc and then I say that in fact what I can do with this figure now I can approximate the central curve by a new curve image of m which does something like this it's very close to m most of the place and then when you come to the boundary it will go almost to the boundary here so this yellow thing now is a x tilde of m so this is a new conformal minimal surface such that x tilde approximates x on as big complex subset as you like of this m could then inside but at the boundary you go almost to the stores ok so this in complex analysis has been well known for a long time and it's not very hard to do not to solve the problem exactly but not approximately but it was much harder to adapt it to this minimal surface case but if you do such a thing then you see now you have technique for doing many things because you see the way you attach these disks this means that you are changing the boundary values from these to those so for instance if you manage to pick your disks in such a way that this exhaustion function is bigger on the boundary than on the center then when you do this game you have actually pushed the boundary of the surface to higher level of your exhaustion function so doing this inductively you push it all the way to the boundary that's one strategy but this thing so I will skip all the rest of this explanation but this strategy was also used and this I'll finish with then with this subject I just want to go through the slide and then I finish this technique turned out to be extremely useful in the subject which is called Kalabiyao problem what was Kalabiyao problem so actually let me first read the theorem ok we take a compact border trimmer surface so one of these we take a conformal minimal immersion of the closed surface inside Rn and then we say that it can be approximated uniformly on M by continuous maps to Rn such that in the interior in M interior this is a conformal minimal immersion and in addition it's complete what does this complete now mean this means that it's a complete metric space in the induced metric from the Euclidean metric ok, what does this intuitively mean that it's a complete metric space this means that this reman surface which I created I mean this minimal surface which I created it stays as close as I like to the original one but it is very, very wiggly at the boundary it has lots of waves it's so wiggly that if you take any curve inside the surface which goes out to the boundary point and you measure the length of this curve in the Euclidean metric ok so these surfaces are staying in a bounded set but they are complete and this has been so this question was first asked by Kalabi in 65 so he asked, actually he conjectured that this cannot be done his conjecture was that if you have a minimal surface in a bounded subset of Euclidean space that it cannot be complete ok, so this was Kalabi's conjecture Kalabi's conjecture was disproved in a very crucial way in 96 by Nadirashvili in fact he found a complete bounded minimal disk in R3 there was a later work there were many works in this area why is Yau mentioned because Yau wrote a big survey paper on problems in geometry the so called millennium paper in 2000 and this was one of the things he discussed and he outlined the remaining problems there was a paper by Martin and Nadirashvili who claimed to have constructed complete minimal disk with Jordan boundary but the proof is not complete the way it's written actually we think it cannot be completed so we introduced a new method by which this can be done you see because what we have is not complete but the boundary is Jordan boundary and you have to work very hard to have this Jordan boundary it cannot be better than Jordan because you see by the isoperimetric inequality the boundary will be infinitely long everywhere it's completely non rectifiable curve otherwise interior couldn't be complete yes I also want to mention that in fact Kalabi was partially right because Kalabi it seems that he kind of had in mind embedded surfaces and in fact it was proved by Kolding and Minikotsi in a series of papers that the Kalabi Yau that actually it's true for embedded surfaces not they haven't proved it in complete generality but for instance for surfaces of finite topological type meaning finite genus, finite number of ends they prove that if it's embedded as a minimal surface then it cannot go to infinity it has to be proper they actually have kind of reverse estimate which says that you cannot increase the length very much in a limited space while what we do exactly is to show that if you immerse or if you go to higher dimension then you can wiggle as much as you like in a very limited space as little as a spot fine and then I have something about Gauss map but I'm over my time and you know what Gauss map is it's just the usual Gauss map of a surface right but turns out that for minimal surfaces Gauss map is in fact a conformal map it's in fact a metamorphic function on your Riemann surface you can think of it like that and what we proved is that actually every metamorphic function can be the Gauss map of somebody so for each metamorphic function you take there is a conformal minimal surface which has exactly this Gauss map and this I was also surprised that after so many years of the theory that such a basic thing wasn't known but anyway this was done with these new techniques and okay I stop here thank you very much for your attention


Hello friends, welcome to another problem-solving session on triangles and its angles in the given question PS is the bisector of QPR. So what is PS and QPR so you can see this line PS is a bisector of QPR the whole vertex angle QPR, right and PT is perpendicular to QR We have to show that angle TPS lets Mention the name so TPS is let's say this angle Which angle my friends this angle? Let us call it X Okay is half Q minus R So how to approach this problem? So let's start with writing whatever is given. So given is Angle QPS QPS is equal to angle RPS Is it RPS why because PS is the bisector PS is the bisector Is it Very good. So what next? also PT is perpendicular to QR that means what? Angle PTQ Is equal to angle PTR Correct this is given okay, and we have to prove that to show What is to be shown? Angle TPS is half Angle Q minus angle R So how do we approach such problems? So One angle is given 90 degrees bisector is given One extra information which we know is angle some property. So can we use that? To find the relationship. So TPS Isn't it so you can see there are two right angle triangles And we are going to use both of them, right? So let's say in solution Okay in triangle PQT PQT right angle PQT or I'm simple right angle Q plus angle QPT QPT plus 90 degrees Which is angle T is equal to 180 degrees angle some property angle some property of a Triangle folks, you already know that so what do I get angle Q? Plus angle QPT Is equal to 90 degrees Now if you look at QPT, what is QPT? If you look at very carefully, what is QPT? So and this is angle Q plus QPT if you see QPT This is the angle I'm talking about isn't it this whole angle minus X Is it it whole angle is QPS? So angle QPS minus X Which is our TPS correct This is equal to 90 degrees No problem, right? Now dear friends QPS is equal to Right, so can I not or yeah, so can I not say angle Q plus QPS? Can I not replace it by RPS? Right What is RPS? See QPS is equal to RPS from the very given fact Correct. So this is true. Now. Let me write it here So angle Q plus RPS. Now RPS RPS can be written as see I can write that as angle Q the first one then RPS and And RPS can be written as angle RPT minus X, isn't it? This is RPS RPS is you can check This is RPS this angle. So let me draw with this on this color RPS This is RPS is equal to the full angle which is the full angle this one and minus this X I hope you got it Right So angle Q plus RPT minus X That is RPS and then I had another X. So write another X like that is equal to 90 degrees Isn't it so this implies Q plus angle RPT RPT Right Minus 2x is 90 degrees What is RPT? If you see RPT is nothing but let me write it in a separate note so that it becomes easier So let us this is one now in triangle PTR It's another right angle triangle in this angle RPT RPT plus angle R. I'm simply writing angle R Plus 90 degrees is equal to 180 degrees again ASP Check you'll get that. So angle RPT Right can be written as If you see 180 minus 90 that is 90 degrees minus angle R from this this is 2 now from one and two I Can substitute RPT by to whatever we have got so we can write angle Q Plus RPT I'm removing and replacing it by this 90 minus R So 90 degrees minus angle R Minus 2x is 90 degrees Correct from one and two so this 90 this 90 will go and we'll get 2x is equal to angle Q Minus angle R and hence X is half Angle Q minus angle R and what was X guys TPS? So angle TPS look at the diagram. We'll get it TPS is equal to half Angle Q minus angle R Isn't it? This is what we had to prove and we have proved it you can write hence Proved so what is the learning in this question? Use of angles and properties so whenever some angles are given in the triangle you can always use angles and property to find Other angles of the triangle


traints but insisted that they should be consistent with written laws which they called the fundamentals of the commonwealth so they demanded the liberty to impose these restraints upon themselves in their own way so this is what they meant by public liberty interference by outsiders met fierce resistance public liberty it's not a theoretical idea new englanders were not new englanders were not a warrior people but they showed themselves willing to defend their public liberty even to the death and they also used the word liberty in a second way which is far into our time when it referred to individuals it became a plural now meaning liberties rather than liberty so these plural liberties were understood as specific exemptions from a condition of prior restraint so tenants in a particular places were granted a specific liberty of fishing in the river so the idea of plural liberties as specific exemptions from a condition of prior constraint was carried to massachusetts so the law granted some liberties to all men other liberties to all free men and a few liberties only to gentlemen so these ideas of liberty seem very narrow to modern americans we do not think of liberty as exemption from prior condition restraint but a restraint as an exemption from a prior condition of liberty but the 17th century idea plural liberties became what the founders called the fundamentals of the massachusetts commonwealth the idea of written laws and liberties existed from the beginning of the bay colony then new england puritans also used the word liberty in a third meaning this is the idea of soul liberty or christian liberty so soul liberty was freedom to serve god in the world it was freedom to order what's own acts in a godly way soul liberty and many people moved to the new world primarily in hopes of attaining soul liberty so what they meant was not a world of religious freedom in the modern sense it's not a word of religious toleration but rather of freedom for the true faith so this idea of religious liberty is consistent with the persecution of quakers catholics baptists presbyterians anglicans okay virtually everyone who was not within a very narrow spectrum of calvinist orthodoxy so soul liberty was consistent with compulsory church attendance rigorous sabbath laws even the indians were compelled to keep the puritan sabbath in massachusetts so soul liberty soul freedom meant they were free to persecute others in their own way okay so ordered liberty was uh very big for the puritans so the words liberty and freedom were used in a fourth way by the builders of the bay colony the people of massachusetts employed the word freedom to describe a collective obligation of the body politic to protect individual members from the tyranny of circumstance so this is ordered liberty so there's a difference according to john winthrop between natural liberty to do evil as well as good and civil or federal liberty which is moral liberty it is liberty to that only which is good just and honest this liberty you are to stand for this liberty is maintained and exercised in a way of subjection to authority it is the same kind of liberty wherewith christ has made us free so these four libertarian ideas collective liberty individual liberties soul liberty and freedom from the tyranny of circumstance all had a common denominator there were aspects of a larger conception which might be called ordered liberty now i think uh franklin dorano delano roosevelt during world war two talked about the four liberties so we've got uh tucker carlson coming up in a minute i'll cut to him sizing different parts of their personality and maybe kind of downplaying those parts that they know aren't really acceptable in a specific situation so people who know how to use impression management understand a couple of really key things they have a really amazing self-awareness so they understand themselves they know their behaviors they know their personality they understand how they tend to come across to other people they don't have a unrealistic sense of who they are and what they're like they understand very well what they're like and maybe what parts of their personality they need to upplay or downplay in any given situation they're also really good at reading the room so they know maybe when the feeling in a room has suddenly shifted maybe things have suddenly gotten very awkward or very tense if they're in a situation that they've never been in before they're going to be very good at reading people and figuring out what the proper behaviors are and what behaviors to avoid so they're very perceptive people and they are not trying to be perfect they are still being themselves but they are upplaying parts of their personality that are more appropriate for the situation and downplaying maybe behaviors or tendencies that just aren't very likable or that have no place in that type of situation so for this personality psychologist impression management does get two thumbs up just learn how to use it properly playing sparkle hd light on my ipad all class long and let's see how the numbers came out okay so we should get that didn't show up automatically aha so 56 percent of you uh excellent 56 percent of you have not straight at all but notice that half of you have found yourselves unable to carry through with a commitment that you made and that you provided some enforcement for in the sense that you internalized in your conscience the idea that you had made a commitment it turns out that for many people simply making a commitment in their mind is insufficient for them to stay in the not even an itty bitty bit category for those people who end up here or here or even here it turns out that putting some sort of external constraints in place are useful so the new york times had a piece last week about something called phone condoms whose slogan is zip it lock it keep it in your pocket and the idea is that you take your cell phone and you put it in this little zip lock bag and while you're driving you are unable to gain access to it one of the students in this class and you'll have access to this on the slide emailed me a link to a computer program which is basically computer condom a zip it lock it keep it in your pocket for internet access it turns off access to the internet for a given period of time but for those of you who feel like you want some sort of external reinforcement but you're not prepared to make use of the internet lock up i thought i'd provide something which would help you stick to okay let's see what tucker carlson has to say good evening welcome to tucker carlson tonight ever been to buckhead in atlanta it's a beautiful residential partly commercial neighborhood in the northwest corner of the city it's not a huge place fewer than 80 000 people live there but it's fair to say that without buckhead atlanta at least as it's currently run couldn't exist taxes from buckhead residents account for fully a fifth of atlanta's entire city budget so you'd think the people who run the city would be very polite to buckhead they certainly should be but the opposite is true for decades various mayors of atlanta have attacked buckhead as if there's something offensive or immoral about maintaining a clean and orderly neighborhood for the most part the residents of buckhead have taken this abuse in silence complaining seemed impolite so they've continued to send huge amounts of money to a city government that hates them for politicians in atlanta it has been a very good deal attack buckhead then take the dough but that deal could soon be ending and the reason for that is the current mayor a spectacular mediocrity with unusually high self-esteem called key shallance bottom like so many big city mayors bottoms is an incompetent demagogue she's no idea what she's doing but she's willing to say anything for a while that was tolerable to the people of buckhead like most people who live in cities they know the drill they're used to it but then last summer things changed a man called ray shard brooks was shot to death in a parking lot at wendy's by an atlanta police officer brooks was apparently drunk had fallen asleep in his car in the drive-thru lane when police tried to arrest him brooks went berserk we showed you the video at the time it's painful to watch brooks stole a cop's taser and pointed at him that's when an officer called garret wolf shot ray shard brooks it was a sad story but it was understandable once you watched the tape and you thought about it carefully but key shallance bottoms didn't wait to think about it carefully or at all without pausing for an investigation she immediately fired officer wolf and then she issued a predictably self-righteous press release about it an independent board later found that what bottoms did was illegal but by then it was too late hundreds of atlanta police officers had already left the force and those who stayed understood very well that there was no point in making arrests who'd want to be officer wolf nobody so response times plummeted you call the police they didn't come crime soared criminals found that even when they'd been arrested for violent crimes they were back on the street within days can you guess what happened next here's what happened next eliana kovic and her boyfriend were shocked to learn the man in this video who is accused of attacking them was out on bond the couple was outside of the mercedes-benz dealership in buckhead last june when they say a man started to threaten them he took out his knife that he had in his pocket when he said he had a gun he kept telling us that he was going to kill us kovic says the violent words soon to turn physical came around the other side of the car punched my boyfriend and it's like left temple and then he punched me and then he punched me again and i fell to the ground i like blacked out the suspect fokradov and more was arrested and charged with seven felonies for the attack however he was released on bond five months later and as kovic and her boyfriend did some digging they learned more had been arrested just days before in clayton county for misdemeanor battery charges and then was released and then while he was out on bail from clayton county that's when he attacked us it's pretty unbelievable threatened to kill them punched a woman in the face what they do wrong exactly nothing just on the street in buckhead in buckhead murders year to date are up almost 50 percent that's a lot of new dead people robberies and aggravated assaults are up by nearly 40 car thefts 65 rise lenox square mall in buckhead was one of the first indoor shopping malls in the united states it was a great and famous place now it's too dangerous to visit beginning last year someone was getting shot at the mall virtually every month in december of 2019 a macy's employee was shot during a robbery in january man trying to defend himself around robbery was inadvertently shot by a cop in february of last year a drug deal led to a gun battle outside of bloomingdales in march a dispute over a parking space outside the cheesecake factory led to more gunfire which killed someone and so on so people in buckhead just stopped going to that mall ultimately authorities decided they needed security and metal detectors they put police dogs throughout the buildings they set up a network of security cameras with software to detect firearms it seemed like it would work but it didn't there was just too much violence this past april a gunman beat a 60 year old woman next to her car in the mall's parking lot and then stole her purse a lina police later said that somehow none of their cameras had caught the crime a witness called james glass says he never saw any police officers or security during the assault even though the attacker took his time beating the woman because he was enjoying it so much according to glass quote it seemed like he was celebrating what he had done he started to rejoice and started shouting just like he was happy for what he had done to this lady it's disgusting and it's still going on on monday two 15 year old shot and nearly killed a security guard at the apple store at lennon square mall so if criminals can terrorize the mall in buckhead which is now one of the most surveilled places in the entire city what can they do with neighborhood well they can terrorize it and they are they're attacking people on residential streets in buckhead watch this report from fox five in atlanta neighbors say they're shocked after two crimes hit their buckhead neighborhood atlanta police say on saturday they found a man with a gunshot wound on west westley road investigators say the man and two others were shot at while jogging hours later on saturday atlanta police responded to the collier ridge apartment complex police say they found a man who was severely injured on scene he did in fact strike a uh individual that was taken out of trash atlanta police say they arrested the suspect shortly after that crime now investigators are working to figure out a possible motive the man who was shot jogging on saturday is called andrew whirl you should know there's no backstory here he didn't know the guy who shot him he wasn't doing anything he was just jogging on the weekend like a good citizen and a guy pulled up and opened fire why he's not when one call is difficult to listen to but you should listen to it just so you can understand precisely what keisha bonnips has done to buckhead here's the call hello holla 1211 west westley is this a house apartment or a place to live i'm on the street 1211 west westley i got shot is the assailant still nearby i don't know he drove away okay and it's very serious bleeding yes i'm bleeding i've been shot and he was shot twice something similar happened to a woman called valerie casper on a trail in brookhaven right near buckhead she was stabbed four times in the back by a total stranger in broad daylight as she was walking on a trail casper was pregnant with her second child at the time she was forced to deliver her baby more than three months early because she was stabbed by a stranger why was she stabbed why was in her world shot these weren't robberies the only point was physical injury was terror was brutality the victims appeared to have been chosen purely for how they look but don't call these hate crimes that's not allowed you can't say that privately though many people in buckhead suspect that's exactly what's going on they've been attacked by reckless politicians for years politicians have ginned up hatred of buckhead for political reasons so why wouldn't others take them seriously why wouldn't shootings and stabbings be the end result of that some people in buckhead have had enough of this they don't think it's going to get better two bills currently in the georgia state legislature would allow buckhead to leave the city of atlanta then run its own competent police department and resume being a safe nice place cnn for one hates this idea cnn is headquarters in atlanta and has been for 40 years they know exactly what's going on in the city they don't care you don't like getting shot while jogging well then you're a racist the mostly white neighborhood of buckhead is pushing to separate from the rest of the city cnn's ryan young has more from atlanta the buckhead exploratory committee reports they've raised over a half a million dollars and is now pushing state lawmakers to push through a bill that would allow their cityhood petition to be voted on in the next election the predominantly white neighborhoods movement is gaining traction with republican lawmakers oh you following this did you get the dog whistles mostly white predominantly white got it so if you're opposed to getting murdered outside cheesecake factory you're a white supremacist they always say that here's the funny thing they don't mean it how do we know they don't mean it take a look at how they live how many cnn anchors including the ones you just saw have weekend homes anywhere near section 8 housing let's see we'll check our notes here read around zero in fact not one instead they run to martha's vineyard or they charter helicopters to fight the hamptons on friday afternoon in other words in their spare time they get as far away from diversity as they possibly can not just some of them all of them every one of them that's the truest generalization ever made a recent piece to put a finer point out of the new york post describes how many rich liberals from new york have flooded into the poor unsuspecting state of montana bill gates has been camping out in big sky recently a lot of them have why it's not the fly fishing bill gates couldn't cast a fly rod if his fortune depended on it no it's not that the appeal is the monochromatic neighbors outside of the indian reservations montana has the demographics of 1956 america everyone looks the same it is not a melting pot so no rich liberals don't actually believe that diversity is our strength they hate it so much that the first sign of spray paint they run for the whitest till they can find then they lecture you some more about how you're the racist because you don't like getting shot outside cheesecake factory that's how it works bill white no longer cares what they say bill white lives in bucket he plans to stay there he's not hiding in montana with bill gates instead he's leaving the effort to make his neighborhood independent of a city that's falling apart bill white joins us tonight bill thanks so much for coming on for being brave enough to do that um tell us what buckhead is like now and why you think it should be independent from the city well first of all thank you tucker for having us on and i was listening to your opening monologue and i can tell you we're going to need to have you come down here and take you on the road with us because you've hit all the key points we are living in a war zone that's how we describe living in buckhead and what has happened here in the last several years is an incredibly dangerous spike in crime and a complete vacuum of leadership the police in atlanta are great policemen and women we love them they just want to do their job and they're not being allowed to do that so buckhead and the beautiful families a very diverse community we in fact are the most diverse community in all of atlanta we have decided to file for divorce and the divorce is final and what we are saying to the city of atlanta is we are going to form our own city we have two bills in the state legislature we've raised the requisite amount of money we need right now to move this forward there'll be a ballot initiative on the ballot next november and we are going to take our city back for the great families of buckhead once and for all some of the people ask us what are what are we going to do differently tucker right how is a buckhead city police department going to do anything differently and i think your great team i sent a video in and i don't know if there's time to show it but this video is horrifying it shows a white car driving past a mercedes band truck yep they got on the screen right now okay they're shooting out of the car at these helpless passers-by on far road and they're shooting and they're shooting one of the bullets comes right through the truck and hits one of the people walking and you can see as the car drives by there's two blue lights shimmering off of that reflection on the truck you know what those are those are two atlanta police officers in marked cars with their lights on and my friend who took that video ran down to find out why in god's name they didn't chase that person who attempted to murder all those people and they were told that mere bottoms issued an order not to chase cars around atlanta because she doesn't want the police getting in accidents all those people shot two police right there wanted to go do service and put the smack down on that attempted murderer they did call an ambulance and they rendered help at the scene yeah but i mean so so why should you have to live i mean the same people lecture us day in and day out about democracy which is if i can just remind them self-government the right to control your own government those people are now suggesting you don't have a right or you're immoral not to one on want to live under people like keisha bottoms who should be ashamed of herself yeah how does that work talk to her that all those uh you know uh we're uh racists or this that and the other thing that is all hateful very disturbing unhelpful language that's fear mongering and it certainly will not stop the crime in buckhead what's going to stop the crime is the establishment of the buckhead city police department that's we plan on having 300 police officers for a prolonged period of time there are only 82 cops on the beat in buckhead right now the square footage of san francisco that has 2000 cops so one of the things we're positive of is we're going to put the smackdown on crime once and for all and we're going to make buckhead a great place to live again where you can go to the mall and not get shot taking a jog this poor woman you had on eliana elevich talk or she's a saint she's so courageous she's a pediatric nurse and they punched her in the face and the guy was out on three different batteries so what you're doing is a model i think for the rest of the country people don't have to put up with this this not political it's not racial it's about wanting to not be shot at the mall and every american has a right to want that and work for it i i think bill bill thanks so much for coming on thank you target god bless you sir god bless new footage tonight from inside the wuhan institute of virology proves that a key part of the story that we have been told with a straight face for more than a year is a lie sky news has the world exclusive footage and we will show it to you next okay that's gonna do it for today take care


Hello, and welcome back to the craft box. My name is Brenton today. We're in episode 37 of the privateer walkthrough last time If you haven't checked it out already, we did the hardest dungeon in the game and somehow managed to not die once Yeah, it's actually not the hardest dungeon, but like To me it always feels like that and I was very surprised when I didn't actually die But anyway, we're back. We're in Mushu We entered Mushu last time so if you haven't checked out that video go and do that now. Haha. It was great But yeah, we are making our way back over to the Yakuza cave at the moment as you can see I have all of these sidequests To do so that's gonna be fun. I have how many pages now? 15 pages. Yep. I need to just sit down and do a side quest for about five hours to get down to pages That's actually how long they take I swear side quests in this game are annoying so bad Like I think I posted this on Twitter the other day, but like if you think Zafarium was a 101 is bad. I Swear you're wrong. You have to defeat so many more things and collect so many more things in such a rare A drop rate. It's ridiculous. I was like waiting. Oh level up. Yay Fantastic, we now level 39 as well. So that's fun We're gonna collect this because why not it's a side quest get it done ish good job Oh, see look defeat 12 ships like what that's gonna take forever UG is where's UG? UG question mark, where are you? Kotan I don't even oh, actually, I know where is he's a bit away, but we're gonna talk for a bit So, yeah, as I was saying side quests got to do a whole lot Hopefully I'm kind of hoping that I will be quite high-leveled going into Mali bone. I think that'll be probably a good idea There's always a chest there, so just always come and see behind there There's all there's usually always a chest whether you can open it by yourself That's a different thing because some of them are gloop gloop plunder chests group plunder chest good England. They're good job So, yeah, I think today's episode is basically Gonna be all about crashing the party or the wedding So that's gonna be fun. There'll be another dungeon today We're gonna see how far we can go through again like always because we're not listening to dialogue if you want to watch Or listen to all of the dialogue Go and check out the musketeer walkthrough, which is a complete Completed pirate 101 walkthrough with all the dialogue I mean there's no real point posting to walkthroughs with dialogue. It's just kind of ridiculous in my opinion So yeah, I'm just burning fuel Yes, thanks for telling me So you go check that out. There's a playlist for it. It's great. I spend a lot of time on that.   We're in Chrysalis, which is the fourth last world now. Thanks to Imperia part one But yeah, that's a great Great walkthrough go and I'm check it out. There's hours and hours and hours of their feet Hours and hours and hours there for you to watch if you want to I'm having a real hard difficulty Is it even a proper stream sentences together today? I don't know why It's weird. I need help. I need help so much help. I also need fuel. Where is fuel of vendor? I think he is over here somewhere Um, who cares not me, but yeah, there's a Lot of content there. So if you want to go check that out go do that most videos are like 25 to 40 minutes Ish, there's a couple that are shorter and a couple that are longer I know the one that's coming up on Tuesday my time the next time that's posted is a really long episode It's nearly an hour long Just because we were finishing part one So if you're watching that go do that if you don't care then don't watch it Simple as that, but if you want to that's another option and there's also past series as well Which you can go and watch Most of them are still there. I had to delete one of them. I think because I was getting Way too many copyright claims and I was just getting annoyed with it. I got spammed with emails Ugh Defeating ships Disgusting this quest takes forever. I have a real problem with Mushu and Pirate 101 and how the quests take Absolutely forever like I could be doing this for like an hour and not get Four like it's ridiculous. Honestly. Anyway, I'll be back in probably about an hour once I finish this So I'll be back in just a sec Okay, I am back just finished off defeating those. It wasn't actually as bad as I remember which is good I got it probably I didn't get the first one. I got the second third Fourth didn't get the fifth one. So kind of like I only didn't get it twice So it's pretty good. This house is so much bigger on the inside than it is on the outside I've just noticed it's kind of a shock Um yay another level. Oh look at that. Oh Hello What's this? Oh, that's so annoying one that we have to bend subata for Don't care, but we get another thing. That's good Okay, can we speak to you cool Okay, so We're gonna go and quickly Collect this new power I can't believe we've done two levels and what's been what? Well one level No, we've lived up twice in this video already and it's been like eight minutes. I would they what? Wait, how is that the possible? I do not know Apparently it is though We're gonna quickly hit over and collect a new sort of power or spell or whatever it's called I don't know talent or power. I think they're called in this game probably From mr. Dude over here nearly the end of Christmas a bit of collect those ones Oh another snowflake eye patch. I've Trashed about seven of them because I already had five and then I just trashed seven more so I'm gonna make that eight now Okay, hello Commodore what you got for me today fam We have Vela's fortress present prevents 50% incoming. That's good and slasher weapons three Fantastic look at that and we can also do guns are blazing the big guns And what do we get next 42 45 47 and 50? What do we go? What's elusive three that would be fun and what's that? Oh, that's like better than this and then repel borders Yay, can't wait for repel borders. That's gonna be fun Yeah, we're nearly there two pages left pretty impressed with that one That's pretty good two pages We're gonna go quickly over to the Inmitation and steal that boy Because we can and we will so why not? Hello, let's hit up this way Um Side quest. Hello. Let's collect you Okay Where are we going? That is a great question memory compound. Yes sounds good I always forget. Oh, this is the valley of the Titans. That's cool. That's a fun place It's like the best part of Mushi the valley of Titans. I'm pretty sure that's where it is. Yeah, best part of Mushi I swear, it's great fantastic. Hopefully we'll be there soon. Who knows though Who knows Let us We're how far away is this because I remember it's not too far away depends which way you access it from I Think because you can like there's just so many ways you can go and it's ridiculous honestly in here. Hello Let's go in please please please can we go in please that would be great Okay, here we go. Hello Thanks for telling me gosh. Hello Okay, can we pick it up? Please? Thanks Thank you Ya kooza cave. I'll be back when I'm over there because it is just a bit. Did I not I must have yeah, okay I'll be back in just a sec. Okay. I am back. We made it over to the forger who is in the yakuza cave Who is just up here? Hello? And what are you gonna do for us today? Oh? Gosh, we have to go back to Where are we going? I think we're going to yeah, how about me to garden? So I'll be back when I'm there because it is slightly away away. Okay. I am back once more and We are in Himamatsu again, can you please load sometimes the loading times on this game are really bad like really really really bad But we will power through I suppose Doesn't it turn there nice? I always forget that there's one in this building, but it's like obvious So you should always get that one Okay, town hall is good Town hall should be not too far I don't think I think it was like the first place we went or something along those lines And then I think we just go back and then we go back and then we go over to the wedding. So that'll be fun Yeah, I'm sorry. I just don't like the numbers up there I have a really bad habit of why not say she'd be good habit of just making sure that there's no numbers anywhere like If I have unread mail, nope, that can't be Like on my phone. I have There's probably only like two apps that actually have like three little you have notifications on it thing Thing so yeah, I just really just like seeing numbers everywhere But I'll be back in just a sec once I've took once I've Once I'm at arm the forger again, okay, I am back or once more We have made it over back to the Yakuza cave and over to the forger who is just stuck here. Hello Yes Speak thank you. Give us new quest. Thank you. Give us XP. Thank you. How much do we get from this one? We get 4,000. Oh my gosh. It's that another level. Are we gonna level again today? We're gonna level again today three levels in one episode. I think that's unheard of That's unheard of what are we level one again geez? UG so luckily we left a mark over at suji mora there was a quest there But I'll get that later because I think we're gonna have to come back Yeah, probably So yeah, let us get on to UG who is when this loads Not too far away Over here. Yeah, and here and We should be told to go over to the wedding and Need to crash that wedding. Hopefully if all is correct Which all should be? Yeah, we're right along the shore. We're just these are all side quests. I don't care Fight me Mm-hmm, so yeah, let us hit over to Taru taru taru. I don't know and I'll be back Okay, I am back We are now hitting over here and crashing the wedding. So that wasn't too far away. I thought it was because There's something that's like a long way away, but it wasn't this one apparently so That's good Yeah Let's go on and we will be here for a bit doing a whole bunch of defeating things and stuff I'll probably skip quite a few of the battles just because they're not too important Like the one that we're about to face in a sec But some of that's quite cool, so it'll probably it'll only be the battle so I skip just because why not Why not skip the battles? Why not? I know we haven't actually done a battle yet. I don't think on camera So we'll do a couple definitely Might as well do this one. I suppose Yeah Okay, so two musketeers to swashbucklers. Yep to swashbucklers. We are going to do that We are going to move up. I've kind of changed my Thing in just a bit. We're now using Black Angus and then El Toro and Bonnie Anne instead of Emmett Bonnie Anne and El Toro just because I felt With the amount of Damage that the swashbucklers and puccineers can do I think it's best if we have someone who can block and repost a lot Instead of just taking like Emmett to it and die So Yeah, that's what I'm doing for now. Bonnie will still take damage, but we need her for the crits and double taps and Things like that where she doesn't actually do much of so I'm kind of annoyed at that, but you know, that's fine Those guys have first strike we're gonna be hitting you with that boy Going like that. So basically I will probably focus on the The musketeers with Bonnie just because we don't have like good talents that do that sort of stuff Whereas El Toro can technically block and repost a lot And Black Angus can Block and also vengeance strike as well. Oh gosh. They got some gold here.  We're gonna do that. We're gonna do that and that Done okay That's the best fire we're talking about. Oh gosh This could be bad This could be bad these crits are not cool Thank you for blocking. Don't not thank you for blocking block nice Okay, we need to get some crits on that would be great Vengeance Okay, vengeance and repost is good. Oh no, he only has one repost. That's right Let's forget how many reposts here. I just assume that he has two of two.   We're gonna do that Hopefully we should be good Vengeance thank you. Can you please die? Oh my gosh die I can you're not to just die please This could be really bad I'm really hoping he can do 93 93 isn't a big number, but surely he can At least he's dead, that's fine Fantastic look at that. That's what we want Bonnie more of that please Also a lot more dodge would be great And vengeance again fantastic and And there goes our dodge What we're gonna do is we're going to heal you as well Shoot you because annoyance Okay, who's got decent health now? Okay, vengeance. Thank you. Wow Just block that that's fine. You would wouldn't you yes, hit me, please Please hit me. Thank you I am the one healing them. Please hit me. You know you want to Don't hit Bonnie that is bad Getting Bonnie will cause you to get in trouble with me Okay, they used to should be dead, which is good and then we should have it Whole lot easier hopefully fingers crossed Maybe I should just sit Bonnie on these guys Okay, fantastic Okay, hopefully these musketeers won't take too much to kill No, gosh just saying that is bad because they're just destroying they're absolutely slaying Stop with the first fire not cool How did that that's not fair Okay, money ends a bit here now, which is good The only problem with this is that Al Toro has really endless and that's really handy Whereas black Angus doesn't I'm gonna have to heal again. Good job That one we might die here. That would be bad That would be really bad. Thank you finally some Chris Where have these been for the last forever?  Oh my gosh rip Bonnie. I Wonder how much double-topy. It's not that much brownie. No, okay, that's bad Yeah, I'm gonna be back in a bit. Okay, somehow I managed to finish off there With 96 health on black Angus Yeah, I don't know how but I did I think there's more in here, which is not cool What do we got here who are we gonna be using? Do you have you have Venge and hold the line into the tide, okay? What do you have your command this first right rough bladestorm? I Will see how this goes.  Where has this been geez? This is extremely hard I'm bad So I could just be really bad, but what I'm gonna do is I'm gonna end off the episode here I mean, I can steal a gift whatever and I'm gonna do a whole bunch of side quests because we have how many pages 17 pages nearly 20 So yeah, I'm gonna go and do a whole bunch of side quests off camera save all of these to do Get some more things upgrade some companions and I'll be back next time to do this Yeah, I'm sorry about that, but it's just Yeah, I've just lost all of my companions in two battles So that's obviously a sign that I need to get better. So I'll be back next time Thank you for watching remember to leave a like and subscribe and if a very subscribed person will avail soon I'll see you know exactly when I post another video and remember craft outside the box


Matt here for Leighton Sports Cards, Ripping 2022, Top Steer One Baseball. This is case number 23, pick your team. We have Alicia S, White Sox, Bobby W Giants, Braden A, Royals, Brian S has the Mets and Padres, Devin D, Cardinals, Drew P, Blue Jays, Harper B, Braves, Angels, Rays, Joshua B, Brewers, Joshua R, Nationals, Junyi P, Mariners, Ken I, D-Backs, Kevin H, Cubs, Kevin C, Orioles, Matthew W, Dodgers, Mathias S, Guardians, Rockies, Michael G, Astros, Mike M, Red Sox, Noah H, Yankees, Ryan M, Twins, Ryan D, Pirates, Sean G, Phillies, Suzanne P, Tigers, Tom C, Marlins and A's, Tony K, Reds and Tony K also has the Rangers. All right, here we go. Yesterday when I did my intro to the 11 Boxer, this is already after nine boxes for Steen and a tier one box. I was like, what's going on everyone? Leighton sports cards here. I was like, well, no, that's not right. I think I'm throwing that one all the way down. I know, that is like a, a Hail Mary pass, right there with that. All right, come on big hits. Here we go, box one, first up, Angels Jersey to 399, Tori Hunter and Yankees Otto to 175, Miguel and Dohar. This is Pirates, rookie Otto to 299, O'Neill Cruz, congrats Pirates. Awesome hit, Pittsburgh. That is Ryan D with the Pirates. Great way to start out the case. We'll mag that one for you. It's Astros Jersey to 399, Michael Brantley. Hey Tony, how's it going? D-Bax, rookie Otto to 299, Tyler Gilbert. This is Padres, Jersey Otto to 99, Blake Snell.  The color of two thirds of the vehicles that I drove in my 20s. That is Marlins Jersey to 399, Derrick Lee and Giants Jersey to 399, Brandon Belt. Brewers, rookie Otto to 299, Aaron Ashby. Perez, legend Otto to 299, Tony Perez. What's with his eyes? That is Ray's Bat Relic to 399, Nelson Cruz. What's up, Simvita? That is Royals Otto, rookie Otto to 299, Jackson Carr and Yankees, tier one talent. Otto to 99, Dave Winfield. Nice card, Yankees. I'll just wait until tomorrow, Simvita. You are in for a surprise. NT football tiered random teams is sold out. We only have five spots left for the night. Those are in Flawless Football, one box break number 52, random left side serial number. Holy crap, this is awesome. That is Bat Relic to 99, Josh Gibson. That is a random to all, awesome card. Man, what a great hit. Is it right in there? I've never pulled a Josh Gibson Relic before. That's awesome. That is Cubs Otto to 199, Nick Madrigal and Dodgers Otto to 299, Walker Bueller. Give me one second here, Gibson there. Full season, Simvita, full season. That is Astros Jersey to 249, Jordan Alvarez and Twins Jersey to 349, Byron Buxton. Marlins Otto to 299, Derek Lee and Brewer's Rookie Otto to 299, Jake Cousins. They should wear those like a few times here. Yeah, what's up, Shane? Hey, Shane. All right, I guess if our task is to upstage Forrest and Kenzie from the national, we'll do it. It's Royals Jersey to 249, Salvador Perez, Dodgers, Rookie Otto to 299, Andre Jackson and Giants, tier one talent Otto to 299, Juan Marichal. Nice one, Giants. Pull that Babe Ruth cut. People are gonna be like, oh yeah, where were those other breakers? Wait, there's eight times here? Let's do it. That is Astros Jersey to 399, Kyle Tucker and Rangers Otto to 299, Juan Gonzalez. Ooh, look at this. Cardinals, Bronze Inc. Otto to 25, Nolan Aronado, three of 25. Add those numbers together, you get his Jersey number. One, even one. Nice one, Cardinals. When I break, when I break, we have to mag all Cardinals. All Cardinals, Mace and everything. All in the recount, absolutely. That is Giants, dual Bat Relic to 25, Willie Maze. Couple of nice legend hits for the Giants in this break already. Nice, Willie. Metz Otto to 299, Marcus Strowman. That is Phillies, Bronze Inc. Otto, four of 25, Rhys Hoskins. From the States, Prime, thanks for jumping in from Austria. That's awesome. Oh, that's cool. That's so cool. What service? This is Red Sox, rookie Jersey to 199, Jerinder Rand. Red Sox, rookie Otto to 299, Connor Wong. And White Sox, Otto to 299, Eloy Jimenez. Denver Broncos fan. Nice. That is Giants, Bat Relics to 299, Will Clark, Padres, Bronze Inc. Otto to 25, Joe Musgrove. This is Marlins. That is rookie Glove Relic Otto, six of 10, Brian De La Cruz. Nice hit, Miami. Gripping autographed Relic card. Nice one, Marlins. Yeah, I think that's awesome. What's that? All the things on Thursday. Nice job. Let's go. Awesome. I'm ready. That's Yankees, Jersey, 399, Randy Johnson. Cardinals, Jersey, 249, Nolan Aronado. Dodgers, Otto to 299, Dustin May. And Red Sox, rookie Otto to 299, Jerinder Rand. Right, that'll do it for the case. We are gonna randomize this, we're gonna random. This Josh Gibson, Bat Relic to 99. So we have all 30 teams in. We'll randomize 12 times. Good luck. One, two, three, four, five, six, seven, eight, nine, 10, 11, and 12. Cincinnati Reds, get the Josh Gibson Bat Relic. Congrats, Cincinnati. That'll do it for the break. Thank you, everyone, appreciate it. We have Pristine coming up next.