Det er en plads til at være her i dette event, færdigende Marcel Bâché. Jeg havde nogle meget gode interaktier med Bâché, vigtigere ærlig i min karriere. Jeg remælde rigtig vigtigt, når jeg havde et dækken længe før. I hans plej, der var en eksempel med hende og hende. Det var bare hende, hans wife og mig og Sergio Kleinermann. Det var eksempel vigtigt, og han var altid meget vigtig. Så det er meget vigtigt at være her. Jeg vil tage om nogle resultatere om optimale regularitet for geometriske fløjer. Så dette vil være et samarbejde med Bill Minicotti. Hvis en modell er en fysisk fenomen, så er en ofentlig trakking franser, som vil tilføje en fysisk færdigere spørgsmål. Hvis en spørgsmål er virkelig en fysisk færdigere spørgsmål, det løber til en af den klassiske, forventelige og nødvendigste færdigere spørgsmål. Og så vil man se, hvad der er regularitet af solutier til denne spørgsmål. Og så vil jeg tage om det her, at det er ansatte til dette. Så vi kan faktisk, for denne spørgsmål kan vi faktisk prøve optimale resultatere, så vi kan prøve, hvad det er, hvor regularitet af solutier er. Men efter det, kan man fortælle, at vi initiallyer, at kunne solutierne være mere regular? Kunne de egentlig bevægge det? Og så, at jeg vil tage omkring en minut, at vi ved, at solutierne er altid færdigere færdigere. Den større derivativ er altid begyndt. Man kan helt karaktere, når den større derivativ er C2. I generelt er det færdigere færdigere. Men der er eksempel, hvor det ikke er C2. Hvis det er C2, er det i generelt ikke C3, okay? Så vi kan helt karaktere det. Men selvfølgelig, så fortællet med det her, vi er stadig vondt om, selv hvis de har denne lille regularitet, er det stadig muligt, at solutierne kunne bevægge, hvis de er meget mere regular? Og jeg vil tage til dig, hvorfor vi er interesseret i det. Og så er vi... Så de størrelse er sort af det mest optimistiske. Så solutierne, så jeg bare trækker på franskabel, men vi er stadig vondt om at se, om det er muligt, at solutierne kunne bevægge, hvis de var særlig analytisk, okay? Og jeg vil sige, at jeg håber at sige, at det er en case, hvor vi kan faktisk sige det. Og igen, jeg vil tage til at sige om nogle motivationer. Hvorfor tager vi det? Og der er nogle virkelig reason for det. Så jeg vil sige om, at efter, vi talte om det, så efter, vi talte om denne particular, det generer en ekvation, og om at prøve optimistisk regularitet. again, optimistisk regularitet er... i for at prøve optimistisk regularitet, det købe er faktisk at forstå underliggendegeometry. Hvis du var bare tænkt på denne problem, som det har været før, men hvis du var bare tænkt på denne analytisk, så vil du ikke kunne prøve optimistisk regularitet. Du ville bare, som et klassisk resultat, kunne prøve at prøve, at solutierne er lypset. Så... Men med at forstå underliggendegeometry, vi kan prøve optimistisk regularitet resultat. Men nu, efter at tales om denne equation, vil jeg prøve at tage om... Så jeg vil se, om denne teknik sætter til andre equationer, og nu vil jeg skrive en resultat til denne. Så her er det faktisk ideen. Så det er et klassisk ting, at du har en front. I dette stedet er det perimeter af en fire. Det fire er revolveret over tid. Du er tænkt på at trække perimeteret, som det revolverer. Og du har tænkt på et simpelt eksempel, hvor fire initiallyer har to forskellige brænderser over tid, de brænderser samarbejde. Så du kan tænke på, at det er lidt mere kompliceret behavior. Og så her er et annet klassisk eksempel, hvor du trækker en anden front. Det er et olieplæt på vand. Og så i dette stedet kan dropletes komme sammen, og at dropletes kan tænke på to. Så du har den sort af behavior. Du er tænkt på at trække den. Og så det løber til, jeg tager lidt mere om det, men det løber til, eventually løber til det. Det er ligesom en posterboy, eller en posterchild for denne kind af ekvation. Så det er denne, det er denne nonlinear, det generer eliptisk ekvation. Det generer den, fordi det forbereder dig, at du er tænkt på at tænke på normen af graden af funktionen. Og det er løbt at se, at funktionen har altid kritisk punkt. Der er altid punkt, hvor graden vandrer. Så du er tænkt på at tænke på noget, at tænke på 0. Okay, så det er derfor, at denne ekvation generer den. Så det er den eliptisk version af denne ekvation, og der er et stedet på at generere et parbolligt version. Men du ser på den anden side, det er et relativt simpelt ekvation. Det er en vigtig ekvation, det er en nonlinear ekvation, men det er en ekvation, som er modt af den Laplace ekvation. Men det er en nonlinear, og det er en stedet genereret. Så her er det optimale verden, at solutoren til denne ekvation, til denne eliptisk ekvation, er altid samt befandligt. Og så everywhere, at kritikere points. Så det er der, at denne ekvation er, så jeg skulle også sige, at det er, at hvis du er fuldstændig fra, hvor graden vandrer, så er det klassisk, at en solution her er smut. Så hvad du rigtig kender for, er et kritikere point for dig, hvor den ekvation er genereret. Men så den resultat prøver, at det er, at det er kritikere point, den funksjon er stadig træt befandligt, og der er kritikere points, den hæsken faktisk har en meget simpel form. Jeg taler lidt mere om det senere. Og så den klassiske genereret ekvation, det var først sjovt, at du kunne solge denne ekvation i den viskostighed som 25 år senere. Og så det var en af de, jeg taler lidt mere om det også, men det var en af de, en af de, færdige usager af viskostighed, men det turner ud, at funksjonen er træst befandligt, og det turner ud, at faktisk solutions er klassiske solutioner. Du har ikke nødt til viskostighed, fordi de faktisk solver denne ekvation i en klassisk sens. Og igen, når den seneste drivning er existent, og vi kan sige, at det er altid fundet, i generelt er det ikke fortsættet. Så det er en sådan form af optimale regulæritet, prøve at solutionsen er altid træst befandligt. Så optimale regulæritet er obviously en klassisk analytisk problem, men om at prøve denne optimale regulæritet, det løber på at dække, at forstå den underligende geometrige. De ideer, de ideer i prøve, bruger en mix af geometrige, og også af virkelige geometrige. Og så er det alle sammen med Bill Minkatti. Og så, så når man har denne optimale regulæritet resultatet, så further development on this equation that I'll describe, it connects with, so it's part comes from the techniques involved in proving it, but it connects for further development that I'll describe connects with when the terms work on catasor of theory and singularity theory, and we believe that these results, these further development are the first are the first instances of a general principle, and this general principle is the solutions of many degenerate equations behave as if they are analytic, even though they have much lower regularity. If this general principle is true, then it would actually explain various conjecture phenomena. So I believe that these, I believe if they turn out to be true that this is a general principle, I believe this will have, I mean it will have various applications to kind of well-known phenomena that really current techniques have no way of addressing, okay? And so let me come back to the equations, I'll describe all of these things, but I want to come back to the basics first. So I want to come back to this elliptic degenerate equation, where does it come from? So it comes from that you're looking at a hypersurface, the hypersurface will eventually be thought of as this moving front, but at the moment I'm just looking at a fixed hypersurface, and it's the unit normal of this hypersurface. H is the mean curvature scalar, so this is just the divergence of this unit vector field. And so if you want to write it out, so the divergence is just a sum of the derivatives, and then in a product with the corresponding EI, where EI is this orthonormal basis for the tangent space of the hypersurface at a given point. So this is the mean curvature at that given point. Now, so this is just, this is what the mean curvature is, this is what the unit normal is. Now suppose that the hypersurface is a level set of a function. So suppose that you have some function on your cleaning space, suppose S is a regular value, and the hypersurface is a preimage of S. Well then obviously the unit normal is just a normalized gradient, and therefore the mean curvature is just given like this. So the mean curvature flow is then one parameter family of hypersurface, where the point at each at a given time, the point is evolving by it's moving orthogonal to the hypersurface, and the speed is minus the mean curvature, right? So if you have a one parameter family of hypersurface that is moving by this equation, then it's set to flow by the mean curvature flow. So this is a geometric heat equation. It was first started by Birkhoff, the first reference I know of it, but there might have been earlier, is Birkhoff in 19th teens. Independently it was started in the material science literature already in the 1920s. If you were a physicist you would think of the mean curvature as the surface tension, and the equation has this property that the area decreases the most efficiently. Okay now so here's the simplest examples of mean curvature flow. If we're looking at it in our tree, if we're looking at a one parameter family of round two spheres with radius, so here I'm thinking about time as being negative, and I'm looking at a one parameter family of round two sphere of radius square root of minus 4t all centered at the origin. This one parameter family of two spheres flows by the mean curvature flow. If you're looking at cylinders, so round cylinders where the radius, so this again I'm looking at our tree, I could look at or similarly I could look in higher dimension, but if I'm looking at round cylinders where the cross section, so this is a circle round circle where this have radius square root of minus 2t, t is again negative, then you will see that these here actually flow by the mean curvature flow, and you observe that at time t equal to zero, the surfaces are not any more surfaces, like the first one here have shrunk to a point, and the second family here has shrunk to a line, so this is correspond to that the evolution become extinct at that time. Right after that time it disappeared, it became singular at that time, and right afterwards it's gone. Okay, and here you see these examples, and the even more trivial one, but less interesting is just the plane, where the mean curvature is zero, so nothing happened, it's static under the flow. But these spheres and the cylinders and also the plane, they're self similar under the flow, this means that the shape at a later time is exactly like at an earlier exact it's scaled, in this case it's scaled down. So they're self similar in this way, and so two very simple but key properties are, so one is that the mean curvature, and this is part of why the mean curvature flow is interesting, it's a gradient flow for area, so under the mean curvature flow the volume or area decreases the most efficiently, and the second basic property is what's called the avoidance principle, the avoidance principle is just a geometric formulation of the maximal principle, and so they say that if you have two initially disjoint surfaces, so these surfaces are just illustrated as if they were curved, so the red one is one surface, the blue one is another surface, if you have two surfaces they are initially disjoint under the flow, they will remain disjoint, and again this is just the maximal principle, and the idea is that if they weren't, if they didn't remain disjoint under the flow, then there would be some first time where the red one, the outer one would have just caught up with the blue one, so since it was the first time, this would mean that the blue one would have to lie in, still would have to lie inside the red one, but would touch at just one point, but at that point you see that the mean curvature of the red one is actually smaller than the mean curvature of the blue one, which means that the blue one is moving faster, so it means that if you go back a little bit in time, the blue one must have crossed the red one, so that's the avoidance principle, and this is really another way of thinking about it, like this is just sort of a geometric term, it just, if you write down the equation, it just boils down to the maximal principle, but in one dimension the mean curvature flow is called the curve shortening flow, and the simplest example is that of a round circle, so under the flow it shrinks through round circle, and in finer time become extinct in a point, and then there are a more interesting example that are shown next, which is sometimes referred to as a snake, and so let's just see this, so this first example here, it's illustrate a theorem of Gage and Hamilton from the early 80's, and so this theorem of Gage and Hamilton say that if you take a convex curve, so the outer one here you should think of as the initial curve, so if you take a convex curve in the plane, and you run the curve shortening flow, the convex curve under the flow, so again so the outer one think about that as the initial curve, the first one inside the outer one think about that at some slightly later time, and then all the other one you should think of as later and later times, and so the theorem of Gage and Hamilton say that if you take a convex curve under the flow it will move inside itself, and it will eventually at some time become almost round, and then once it become almost round it will keep shrinking and eventually disappear in a point, and that point if you kind of rescale right before it disappear, it would look like almost perfectly round circle, okay, so this is a theorem of Gage and Hamilton from I believe 84, and so now Gage and Prude it may be 82 or something like that, Gage and Prude in 89 he proved that if you take any, so I should also have said that my surfaces and in this case my curves are always, my surfaces are always embedded, if it is a curve then we usually, we don't talk about a curve as embedded, we usually say that it's simple, and typically my surfaces and the same curves are closed, so there's a theorem of Gration that say that if you take any simple closed curve in the plane, so this here is supposed to be a simple closed curve, that is kind of wrapping around itself a number of times, if you take any simple closed curve in the plane and you run the curve shorting flow, so this here is the initial time, this is at a slightly later time, slightly later again, etc. down here, if you run the curve shorting flow then actually in this case if you take any simple closed curve it will start unwinding, it would unwine, it's going to shrink, and the reason why it will shrink is comes from the avoidance principle, because if you take any simple closed curve you can just encircle it by a large round circle, you know that the round circle in finer time will become extinct, so and whatever is inside it will remain inside it, so that will also disappear, so it's going to shrink, might any simple closed curve is going to shrink, but the theorem of Gration say that it's actually unwinding faster than is shrinking, and so this means that what happens by the theorem of Gration is that it will unwind and eventually at some time it will be quite small, but at that time it will be convex and then the theorem of Gage and Hamilton take over and they say that once it becomes convex then as time goes by it becomes rounder and rounder and it will eventually disappear in a point, so this is the theorem of math Gration from 89, any simple closed curve in the plane shrinks to a round point in finer time, okay so that's kind of the story, that's conclude the story for the curve shorting flow in so for the curve shorting flow, but in higher dimension various interesting phenomena happen, so here's a very simple example of a surface of evolution, this is a torus of evolution, if you're looking at this as the initial surface, you run the flow, the mean curve of the flow will preserve the symmetry, so it will remain a surface of evolution, and it will shrink, the cross section here will shrink and it will disappear in finer time, it will disappear in a circle, okay, there's another example, more interesting example of math Gration, also from 89 and they say that if you take a dumbbell, so take a dumbbell, so this is a dumbbell in R3, so it's rotational symmetric around an axis here, and in this case the two bells here have equal size, there's also actually in this particular example there's also a reflection symmetry around here, so this example of Gration say that if you take a dumbbell like this connected with a sin bar here, this sin neck, then under the mean curve of the flow the neck here will pinch off first, cutting the surface into two pieces and later each of these two bells will shrink to a round point, so this is illustrated, this is actually a computer simulation done by a guy called Uwe Meyer, and so, and he also did, he was also so nice to do the computer simulation for the theorem of math Gration earlier, so here again, so this is the initial time, this is a slightly later time, right here you see the neck have just pinched off, the surface becomes, it's a heat equation, so the surface actually becomes, it doesn't really look like it here, but it becomes instantaneously smooth after the neck have cut off, and it will evolve, keep evolving and it will smoothen out here, and the two bells become actually convex and then eventually disappear in a point, each of them, and so this here, this example here, so Gration had a kind of crude way using the avoidance principle to say that the neck will really break off first and break into, the surface will break into two pieces, slightly later in 91, this example here sets inside a much larger family of examples of surfaces that are rotation symmetric, so all surfaces that are rotation symmetric around an axis like this was completely analysed in around 91 by Altshuler, Ankenen and Giger, and also some of the, they analysed sort of the family in complete generality of surfaces that are rotation symmetric with some further assumption, similar results was proven at the same time by Sonar and Sognidis, so again, so they analyse completely if you have a surface, initial surface, that is rotation or symmetric around an axis here, they understood completely what happened, and the basic method that they were able to apply, which is of course very particular to that example, but still very lovely, is a stormy real comparison theorem for that exists, but it really only exists for a one dimensional heat equation, okay, so there's no other version of that, and so, and that was a theorem, this stormy real comparison theorem was proven earlier, I mean of course there's a very classical one, but that's not for the heat equation, the one they're using is one for the heat equation, that was proved earlier by Ankenen's, okay, so that was the basic technique, and right, and so, but you see also in this sort of picture here, it begs, or already here it begs to make sense of what it means to flow in the case where it breaks into pieces, there is a moment where the surface is not any more smooth, and yet you are continuing the flow through that point, so one of the next slides are going to be talk about how to make sense of the flow past a singularity, and this was this was work of Chen, Giga and Gouda, and Evansen's book, and I'll get to just a single, and so this is really what's called the level set method, the level set methods have been extremely successful numerically, and this was numerically it was brought into play by Osha and Setschen in 89, and they have a method for they're looking at these equations where you're tracking a moving front, and again as I said in the beginning, the prime example of this is where the speed is a curvature, so the one we're looking at, and they develop something called the level set method, it was earlier considered, this idea was earlier considered certainly in the material science literature, was considered but they you know they weren't able to do anything with it, but they in the material science literature people already understood, how to track you know that there's a reasonable notion of tracking as surface past a singularity, but they couldn't prove anything about it, but they just wrote down what is now known as the level set equation, okay, so the basic idea here of Osha and Setschen is that you have this moving front and you want to track what happened to this front as it's moving with this curvature dependent speed, and so you choose a function, so you want to you want to take the geometry away from the equation, right, so instead of looking at the surface as it evolved, you want to reduce it to a question about PD, about some function satisfying some differential equation, right, and so the idea is that you take, you choose an initial function, so you have this initial hypersurface, all our hypersurface are closed and embedded, you take a function that has this hypersurface as a level set, and let's say it's positive in the, so you have this hypersurface here, and let's say that you take a function that is positive in inside, negative in on the outside, and let's say zero on the hypersurface, and now the idea is that you want to flow, you don't just flow this hypersurface that is given to you, you want to flow all the level sets, so not just the zero level set, but all of the level sets, and the simple idea is that if you flow all of them, then you get an equation, differential equation in the function, right, and so now the geometry is gone, it's a kind of, it seems like a simpler question, where you're reducing it more classically to a PD question, okay, and so by the avoidance principle, right, if you're flowing, if you're taking two different level sets, well we know that those level sets will remain disjoint under the flow, okay, so again, so this leads to this degenerate parabolic equation, these have been tremendously successful, numerically I gave a talk about this at Berkeley maybe a year and a half ago, since then there have been more than 2,000 references to this paper of ocean and sejtion, so it's extremely, it's extremely used all over the place, but so ocean sejtion, they, in their celebrated work, they were able to give a numerical scheme to, that would track solutions to this equation, not much later, a couple of years later, in the work of Avensens book, Tjenke i Gangutto, they were able to provide the analytical foundation for this equation, and so they were able to prove that you had continuous solutions, the solution, so it's a differential equation, it's a second order differential equation, and so they were able to prove that actually you had solution in this weak sense, which is the viscustic sense, and that solutions were unique, and that they agree with classical solutions, right, I should say I always say this because of my, somehow because of my childhood, I always feel obligated at this point to pause for a moment and say that there was, so the, so the, this notion of viscosity was in the early 80s, and it was developed by Crandall, Mike Crandall and Pierre-Louis Lyons, this result here again of Avensens book, Tjenke i Gangutto was really one of the major success in the theory of this notion of weak solution, but there was an earlier version of viscustic solution, there, that I'm sure everybody here know, but I still feel obligated to mention it, of Calabi from the 50s, and so their basic idea is exactly the same, but the two notions are a little bit different though, and so they are a little bit different, the basic idea is exactly the same, Crandall and Lyons was unaware of Calabi's work, Calabi had envisioned, actually, there's no doubt if one, it's not long ago I read Calabi's old paper, and there's no doubt that Calabi, he, as I'm sure everybody here knows, his immediate applications was a geometric application, but there's no doubt that he had envisioned that it would be useful in purely analytical problems like this, okay, but it's different, it is different though, it's a little bit different, although the basic idea is the same, so now let's look at where the initial hyperservice is mean convex, so in this case Avensens book, Tjenke i Gangutto by by work of Avensens book Tjenke i Gangutto and Brian White, mean convexity here is preserved under the flow, the hyperservice will move inward and it will keep moving inward, because it keeps being mean convex, it will sweep out the entire domain that the initial hyperservice bound, so you are at a picture that looks, let me just get back to this picture here, so if you imagine that this, you have a similar picture, although you don't have this conclusion in here, but if this initial surface is mean convex, then it will move inwards and it will keep moving inward into itself, it will sweep out the entire domain, but you won't have this conclusion here, it does not hold, if it's just mean convex, in fact all of the examples I looked at så far, sorry, this here is actually a mean convex example, and this here is also a mean convex example, if you do it properly, it's not all dumbbells that are mean convex, but you can, if the neck is thin enough and you construct it right, it will be mean convex, okay, so okay, and so now this leads to what's called the arrival time, so the arrival time is, so this is just that, so if you take a general hyperservice and you use this method of oceanization, so this level set method, then you get, then you, then it leads to a function that's satisfying a degenerate parabolic equation, if the initial hyperservice then is mean convex, then this degenerate parabolic equation is actually equivalent to a degenerate elliptic equation, and that's the one I looked at, and I will focus my attention on that, on the elliptic one, though many of the shots apply more generally, okay, but I'll focus my discussion on the elliptic equation, and so this is what's called the arrival time, so remember that our initial surface here is supposed to be mean convex, so at a slightly later time it will have moved inside itself, it will keep moving inward, and so this picture here, which is supposed to be mean convex, and please don't look at it too closely, this was actually produced by Sethian, but it doesn't look very mean convex up there, but anyway just ignore that little error, then this is of course the dumbbell, and it's like you just slice the dumbbell, and so what you see here are the hyperservice at different times, the arrival time is simply just that you take this domain that the initial hyperservice bound, and you're looking at some point in the domain, and you're looking at at what time does the hyperservice come through that point, that's the arrival time, okay, and the arrival time satisfying this equation that I wrote initially, so I should speed up a little bit to see, and so in the case of for the round spheres, shrinking round spheres in R3, the arrival time would be, this function is quadratic polynomial, this corresponds to that, this is just normalized so that it arrived at the origin at time zero, similarly for the cylinders the arrival time function is this quadratic polynomial again normalized so that they arrive at this singular line at time t equal to zero, and so you see two things here, you see that the singular point, so in this case here the singular point, so the only the singular point is at the origin, but that is exactly where this quadratic polynomial has a critical point, and similarly you see that in this case here the singular for the flow, the singular singularities is just the line, but that's exactly the critical set of this quadratic polynomial, and so you can actually prove this more generally that critical points for the function are exactly singularities for the flow, so if you want to think about it geometrically you think about the singularities for the flow, if you want to think about it analytically you think about the critical point for the function, and the second thing is this, but it's these examples are too difficult to fully appreciate this more general principle that somehow you should think about this function as part of a tail expansion of this, but you don't see any other term so it's maybe a little bit hard to appreciate this point, but this will be an important point later, okay, and so here's this equation, this is the arrival time equation, and these I should also say that, so this is, so here we're looking at where the front is moving monotromely, numerically these have been numerically modeled even more efficiently, this case here has been modeled more efficiently, in particular there's something called the fastmouthing method of session that is dealing with this case where the front keeps moving inside itself, okay, and so this is the equation, and what you do is, so this is what Evanston spoke that and independently at the same time Chen, Giger and Gutter, they revert this equation in this form here, right, and it's when you rewrite it in this form here, then it lends itself to talk about that you have a solution in a viscosity sense, it really comes from dividing it into this form here, okay, okay, Tom Ilmen and theses proved that, published a few years later, but he proved that there are solutions to this equation here that are not C2, so in general solutions are not C2, and Tom's example really came actually from the exact dumbbell that we looked at earlier, it came from a particular dumbbell, sorry, like this that has two belts of evil size, it had a reflection symmetry like this, and you're looking at the rival time there, if you constructed the exactly right, you know, with the exact right symmetry, he was able to prove that the arrival time function is not C2, okay, and so, right, and so it turned out that that's theorem of Tom actually is, so Bill and I proved that, so remember that we proved earlier that this function is always twice differentiable, and at critical point the Hessian is in a particular nice form, the second derivative is always bounded, but we also proved that, so this is this theorem here, uh, and, uh, right, and so let me just go to this, and come back to this thing from a previous slide just because I kind of got a little bit ahead of myself, that there's also a converse to this, and they say that when the function is C2, so that you can completely characterize when the function is C2, right, so in general the function is always twice differentiable, the second derivative always bounded, but when the second derivative is actually continuous you can completely characterize it, and so the second derivative is continuous if and only if the function has only one critical value which corresponds to that the flow only becomes singular at one time, there's only one time where the flow is singular, and moreover the critical set, or if you want to sing it geometrically, the singular set for the flow is either a single point where the Hessian is exactly like the Hessian of the arrival time function in the spherical case, or it's a simple closed curve just like it was in the case of the donut, the rotational symmetric donut, and in that case the critical at the single set was just this round circle, so you can prove that the arrival time function is C2 if and only if you either have one of these two cases, there's a single point where a single critical set, sorry, there's only one critical value and there's only a single point where you have, where that is critical for the function, or there's a simple closed C1 curve that is the critical set for, that is exactly the critical set for this function, and in that case the Hessian is exactly like a cylinder, and in the second case here the kernel of the Hessian is exactly tangent to this singular curve, and so you see that these two here are illustrated in these examples, if you take the sphere then you would be in, then the arrival time function is, well it was actually a quadratic polynomial, so certainly C2, and this exactly falls into the first category, the cylinder of course falls in exactly to this case here, the ring is also C2 and this falls exactly into this second category, the dumbbell, well there are two different singular times, there's one time where the neck broke off, and then there's a second time where each of the two bells disappeared, so it doesn't, there's not just a single critical value, so it doesn't fall into any of these category, so it's actually, it's not C2, but that's in the particular case where it had this reflection symmetry, this recover of course this theorem of Tom, but of course this here I'm nothing to do with any, I mean this is just in complete generality, we're not assuming any kind of symmetry here, I should also say that I'm stating the theorem as if it was an R3, but it actually holds an all dimension with you know the kind of trivial modification, so, but it's just for simplicity, I just stated in R3. Okay, so again we have talked about you know this singular for the flow, why this when the hypersurface are not smooth, singular points for the flow, analytically you would think about them as critical points for the arrival time function, here were these two examples, I'll just go a little bit faster because I also think I won't get to the end, and so now here it's just very briefly, here's very briefly the key points in the proof, so if you want to prove that a function is twice differentiable, well of course you know you want to prove that it's a quadratic polynomial up to higher order term, but if you want to prove that in a point the function is twice differentiable, then you want to prove that there's a kind of tail expansion, so there's a quadratic polynomial plus higher order term that approximates the function in the neighborhood of that point, now of course again as we talked about in the beginning we're looking at this degenerate equation, but it's only degenerate at critical point, so away from critical point by classical result you know that solutions are smooth, so you only need to check this that is twice differentiable at critical point, the second order approximation here, this quadratic polynomial is going to be the arrival time, we want to prove that it's the arrival time of the cylinders, so we have a great candidate for this approximation, so this suggests that the level sets if you're looking at you have this critical point where you want to prove that the function is twice differentiable, so you want to approximate the function in a neighborhood of that point by this quadratic polynomial, which is this arrival time function for the shrinking cylinders, then it suggests that near that point the level set of the original function looks like shrinking cylinders, and that is indeed true, and this was already known before, now what the key point though as you may recall if you are trying to prove that even a function is just one time differentiable, to say that a function is differentiable means that it is up to a higher order term, a linear function on all small scales, on all sufficiently small scales, it looks like a linear function up to higher order terms, on all small scales, and that linear function is the same on all scales, it doesn't do you any good if it looks like one linear function on one scale, then you go to a much smaller scale, it looks like a different linear function, and it's the same sort of thing here that is a key to proving that a function is twice differentiable, and this was really the difficult part here, so this here is illustrating, and this here was proven earlier by Bill and myself, and so this is supposed to illustrate that, so this is, so in the middle of this say yellow cylinder right there inside it, think about that as the critical point, and you want to prove that you would like to prove that a function is twice differentiable at that point, and so imagine that the level set of this function right before, at a time right before, the level set looks like in a neighborhood it looks like the yellow set, and look, but imagine also that at a slightly later time the level set looks like the blue set, and even later time it looks like a red set, if this here was possible, then the function wouldn't be twice differentiable, and so Bill and I proved earlier that this here is not possible, that it actually has, that these cylinders will have a limit as you're going down, right, if you, okay, and so this is uniqueness, and what made this really complicated, this case here and required new techniques is that these level sets here are, it basically comes from that the cylinders are not compact, okay, so right, so that's the key, the key to proving twice differentiable is proving this kind of uniqueness, okay, now, so now I want to get into, I want to get into some more recent stuff, some very recent stuff, and so after we know the optimal regularity of this equation, so we're not done actually, because we wonder our initially, initially, actually initially I woke up, you know, few years ago, Christmas morning, and I was just wondering what if solutions, that was before we've proven all of this, what if solutions were always analytic, okay, why couldn't you conclude, and so actually you can, you can, if it was analytic, you can actually solve a bunch of conjectures that are well known, now of course, now we know that the function is not even C2, okay, but you still wonder, okay, it's not analytic, maybe, but what if it's, but it's not, you know, you don't a priori need the function to be analytic, you just need it to have the property that analytic function has, okay, and so this is where it connects with this work of Renate Thomas, I guess Renate Thomas here many years ago, and his work in singularity theory and care theory, a catastrophic theory, and so, so, but the background is that if you take a analytic elliptic PD, then solutions are analytic, again degenerate equation has much lower regularity, and solutions only exist in viscosity sense, now if we, so now we are interested, and again we think that these are the first interest of this general principle, that for many degenerate equations, solutions, they won't be very regular, but they would behave as if they were even analytic, okay, and so this is, this is the first result that we proved Bill 9, we proved that in the case where the function, remember that if the function is C2, the function won't be C2 in general, but even when it's C2, it's in general not C3, it's, you know, it's like, it's like there's almost no functions where C3, even when it's C2, okay, so okay, which remind me by the way, I should have just, I should have skipped a slide and I forgot to come back to it, that was kind of relevant for this discussion, this was this here, I should have said that in the convex case, in the convex case, Frisken proved in 1990, that the arrival time function is always C2, Bob Cohen and Silver Saffati proved in 2006, that in the plane, if you take a convex curve in the plane, then the arrival time function is C3, and Natasha proved, and Natasha Sesson proved in 2008, that if, so this is in the convex case, these are always solved where the initial surfaces convex, that in general the arrival time function is not C3, in fact her result is a little bit better, I mean it proved that essentially, it's even when it's C2, which it is in this case by Frisken's, it's for even when the initial surfaces convex, if you perturb it in a kind of arbitrary direction, it's not going to be C3, okay, so now okay, so now coming back to this, and so I'm skipping this, I mean the idea of this here is using, you know, it's coming, some basic ideas coming here to prove this uniqueness comes from, we got that break geometry, the key point here was though it required entirely new ideas, and this comes from the non-compactness of the cylinders, okay, and but the first thing Bill and I proved was, you know, to go a little bit further, that when the arrival time function is C2, well then we proved I wear a shape it's inequality, so this here is a wear a shape it's inequality, and we conjecture that you have, even when the function is not C2, you have a similar wear shape it's inequality, it's known that you can't get this power here, this power won't hold, but you only need the wear shape it's only asking for a power greater than one, okay, so now we proved this, this was not too hard actually, you know, once we have all the other stuff with uniqueness and all of the other stuff, so this here was not too hard, this was really not our goal here, but what we were interested in further properties of analytical functions, now I should say though that the wear shape it's, so this is a really basic inequality altogether, but it's certainly a very celebrated inequality in real life break geometry, and so a basic consequence of once you have a function and you have that inequality, then you will immediately get what's now known as wear shape it's theorem, and the wear shape it's theorem say that if you have an analytic function, so wear shape it's proved this inequality, not with this power here, but with a power strictly greater than one, with a constant C, both C and that power allowed to depend on the function, okay, and so he proved this for all analytic function, and he used it to prove, he used this kind of, this is the wear shape it's gradient inequality, there's another wear shape it's, and they are actually closely connected, they are actually kind of equivalent, and there's also another wear shape it's inequality, and he wear shape it's used these, this is in the late fifties, early sixties, he used it first to prove a conjecture on swads, which was called the swads division conjecture, which was about distribution, if you have a distribution can you divide it by an analytic function, so use this inequality to settle that conjecture, Hermander at the same time also proved that conjecture, except Hermander couldn't actually quite prove the conjecture, he could only prove this kind of inequality for polynomiums, and so that's why Hermander's name is not attached to this because he only established independently at the same time, but only a special case of it, and wear shape it's went on, and he used it to a few years later, he used it to prove another kind of well-known conjecture at the time, also in real algebraic geometry, which was called witness conjecture, and it was about deformation re-contract of analytic sets, okay, so this is a kind of famous inequality, and okay, and it leads immediately, and this is kind of the basic point in the, in wear shape it's prove of the witness conjecture, it leads immediately, once you have an inequality like that, it leads immediately to this wear shape it's theorem, and they say that for analytic function, if you take a gradient flow line, and that gradient flow line, so you flow, the gradient flow line has a limit point, then it has a unique limit, in fact body prove this, that this gradient flow line has finite length, and hence it has a unique limit, okay, so now Renner-Thom just to come back to this place, and Renner-Thom, ja sorry, ja 5 minutter, ja, so Renner-Thom was, so in the late 60s, early 70s, Renner-Thom was interested in singularity theory and catastrophe theory, which is all about gradient flow for analytic functions, okay, and so Renner-Thom was interested in a generalization of wear shape it's a theorem, and so what Renner-Thom conjectured was, that if you take a gradient flow line, so imagine that this here is a gradient flow line of analytical function, so if you take a gradient flow line of analytic function, well by the wear shape it's theorem, we know that if it has a limit point, it has a unique limit, so it kind of will go towards that limit, but Renner-Thom went one step further, and he said why not, why shouldn't the secant, so the secant is that you take this point here, you're looking at this point, as you're going in here, you're looking at the line segment that goes from the limit point to the point where you are, so this give you a line segment, and Renner-Thom was saying that if you renormalize that line segment to unicize, that's what's called the secant, and Renner-Thom conjectured that the secant would have a limit, okay, and so this means that this figure here, which is some sort of Archimedean or logarithmic spiral would be impossible, this here is supposed to, the red line is supposed to illustrate the critical set that the curve, it is a three dimensional picture, and so you have this critical set and you have the spiral into that, okay, so that was Renner-Thom's gradient conjecture, there was a lot of work on that, the Renner-Thom's gradient conjecture was finally settled in 2000 in a paper by, in a paper, I don't want to remember, there's three Polish guys in a paper of annals in 2000, Bill and I, but actually there's a stronger conjecture, so Renner-Thom maybe here, but I guess I was too early, but somewhere in Paris Renner-Thom was talking about his idea about singularity theory and catastrophe here, and Arnold was in the audience, and Arnold went at one step further, and that one is still unknown for analytic function, and so the Arnold's conjecture, which is a refinement of Renner-Thom's conjecture, said that the velocity, the unit velocity, as you are going towards this critical point on a gradient flow line, then the velocity of the unit velocity should converge, okay, this immediately implies Renner-Thom's gradient conjecture, that's unknown, that remain unknown for analytical function, but in this case of these degenerate functions, Bill and I can prove this, so we can prove both, I mean even the stronger conjecture of Arnold's, and so in the, and again I want to emphasize that there are, that if this principle is true, that these solutions to these degenerate equations behave like analytical functions, it will solve various conjectures, known conjectures for flow, I just mentioned one immediate one, but there's various other, that it is believed that a flow should only go through finally many singular terms, and this would be, there's no, as far as I know there's no way of approaching this, but this gives you a way of approaching this, so then in the last minute I have, I want to talk about other equations, okay, and so I want to talk about systems of equations, so a particular system is the richer flow, so there the metric is evolving by, so over time the metric is evolving in the direction of the richer curvature like this, shrinking cylinders are the singularity models, by result of these guys just because I'm running out of time, I won't mention them, but they are there in two and three dimensions, shrinking sorotons are spheres and cylinders in higher dimension, there are many more examples, it's expected that in all dimension the prevailing singularities are cylinders, here I would prevailing, I mean in terms of dimension reduction, so these are supposed to be the typical singularities, the second most typical singularities are expected to be also cylinders like that, and so Bill and I can prove that for the richer flow cylinders are isolated as shrinkers, and we believe we also can prove the corresponding uniqueness that if you have a tangent flow that is a cylinder, then all other tangent flows are that cylinders as well, and so this theorem here is related to earlier results that we did, and this is isolated in a very strong sense, which is a sense that is relevant for this problem, and it just says that if you have any other shrinkers that is close to one of these cylinders, it's a shrinker for the richer flow, if it's close to a cylinder like this on a large but compact set, you only need to know a compact piece of it, then it must be that cylinder, so I'll stop here, it just could have all the time. In the dumbbell example, how are you moving past the singularity, are you doing surgery? No, no, so I mean that's where the level set method come in, because the level set methods, it gives you this degenerate equation, and the work of, so that's exactly what the work of Evans and Spook and Chen-Gi-Giangotto is, that you can make sense of solutions even at critical point for this equation, so that's, so that, I mean again, so there are these, you know for, you know, there are these different notions of going through singularities by the sort of class, I mean there's at least three classical, there's one of the Georgi, there's one of Brake, you know, using geometric methods, the Georgi using regularization, and then there's the level set method.


Raise your hand. Luckily, there's some opportunities coming up. There's Mars 1, where you can go to Mars, but without a return ticket. And who applied for this one? Do we have any applicants for Mars 1 in the hall? Whoa, more than we expected. Three, four. So just a few words, brave. But recently also SpaceX announced their detailed technical plans on how to get to Mars and their technical plans for the vehicle. Who has seen the announcement? Raise your hands, yes. Okay, quite some people are informed. So since we already know there's three or four people who would go to Mars without coming back, who would trust his life to Elon Musk and go up there with him? Some more, some more. Yeah, I definitely see a colony up there. Okay, by the way, what's the difference between the Berlin airport and Mars? Humans will land on Mars in the foreseeable future. The next talk is not only about Mars. It's about the 3,518 known exoplanets and how to pick the right one for a civilization attempt. And we'll also learn about how to get there and what technology needs further research. Because as we know, it's not about the target, it's about the journey. Liz George and Peter Buschkamp, who have both provided us with food for thought on priority three events, are now here for us on this stage. They have many things in common. They are both instrument builders for astrophysics. They both work at the Max Planck Institute for Extraterrestrial Physics and they are both working in the same infrared group. And on top of that, they both develop some of the world's most sensitive introvert instruments. Please take us with you on a journey through space. Liz and Peter. So hi everybody. It's great to be here. I think this is by far the biggest audience that either of us has ever spoken in front of us. The biggest one, the biggest one. So thanks for coming out. So today we want to tell you about interplanetary colonization and by extension interstellar colonization, which is much harder. So we'll start out with the question, why leave Earth? Earth is a pretty great place, right? So this is our home planet. It is about 70% surface area covered in waters. You heard in the last talk. It's got a convenient one G of gravity, one bar of atmospheric pressure. The atmosphere is about 20% oxygen. And don't forget the most important part. Oh yeah. It's the only planet we know of that has Wi-Fi. So Earth is pretty great. Why leave it? And we can start with analyzing the past history of Earth, namely that Earth hasn't always been such a great place to live. So this is the fraction of species to go extinct in each time interval as a function of time. So this goes back about 550 million years. And there's a lot of spikes on this plot, five of which are known as the big five extinction events. And you may have heard of some of these, one of which the most latest one here is when all of the dinosaurs were wiped off of the planet. So something else you might not know of is that we're in fact in the middle of the sixth big extension event on our planet where we're losing species at a rate a thousand times above the background. So we can ask ourselves what has caused these extinction events in our Earth's history? There's many things. So the first big extinction event was because we actually had too much oxygen.  Okay, so you can say, you know, we're actually much more advanced than the dinosaurs. We have technology. In principle, we should be able to survive any of these planet altering potential extinction causing events that come our way. But actually in the very long term, we have a bigger problem and that is our sun. So eventually our sun will use up all of its hydrogen fuel and balloon up into a red giant. So first all of the water will boil off of the Earth. That won't be pleasant for anyone. And then eventually the star will grow so large that it actually envelops the Earth inside of the star and completely fries the planet. So given that this is CCC, you can think of it as, you know, making an offsite backup of humanity before your server room burns down. Which in this case- And in the cloud and beyond. Yes. In this case, we know it will. So we actually have a pretty strong motive here to get off of our planet in the foreseeable future. So that brings us to our next question, namely where are we going? So all of you in the audience and our previous speakers have talked about Mars. And we also want to know how we learn about our new home. So this is what Mars looks like from space. And you can think this is a good target for us. It's relatively close in terms of interstellar space. If you want to get an idea of distances, stick around for the last talk in this track about how space really is huge. And you heard from Jan Warner. It'll take about two years round trip there and back. So Mars also has an atmosphere, but it's very thin, mostly CO2. It doesn't have a magnetic field, so it's constantly bombarded by cosmic rays and solar wind. So this essentially sterilizes the surface and means that humans would have to live underground. It's also pretty cold there. So while Mars is close and technically is in the habitable zone of our sun, meaning there's can be liquid water on the surface, it's not actually that comfy of a place to live. Another target that people talk about is Europa. So this is one of the moons of Jupiter. There's several of them. And this has been a very interesting target because it's actually an ocean world, so the entire moon is covered in water with a very hard, thick ice shell on the outside. And conveniently, it also has an atmosphere of almost pure oxygen. So you think, great, it's got an ocean, it's got oxygen, what more could we want? Unfortunately, the average temperature on the surface is so cold that the ice is as hard as granite, and you'll have to drill through 15 kilometers of it before you get to the ocean. It's also sitting in Jupiter's radiation belt, so the radiation intensity on the surface is so high that a human would die in less than 12 hours. So also not so pleasant. And also the monolith tells us not to land there. That's true. Okay, but the really interesting thing is these planets are close, and we actually know an extraordinary amount about them. And the question is, how do we know these things? So in our solar system, we can actually send satellites to orbit the planets and landers that sometimes crash, but sometimes actually land there and give us some information. So this is a picture of a mountain. It's Mount Sharp on Mars, and it's actually taken by the Curiosity rover that is sitting there on the planet right now, taking data. So this is a selfie, Curiosity took of itself. And one of the really cool things about Curiosity is it's got a giant laser on it that it uses to actually vaporize the rocks and analyze them. So we have this probe sitting there, and this is the first spectrum to come out of this rock vaporizing laser measurement device. And what you can see here is a whole bunch of spikes in the spectrum. And each of those, if you have some very good chemist back home, can be matched to different elements. And so you can see our soil on Mars has iron, magnesium, silicon, aluminum, calcium, etc. All things that would be very useful to say potential habitants of Mars, or just in general to know about what kind of life could live on Mars. So Liz, you are essentially saying we are safe in growing potatoes now? Yeah, we could grow potatoes in this, for example. Okay, so again, this brings us back to our previous problem though of the sun is eventually going to envelop the earth and make Mars a hellish desert world. So we got to go a little bit farther than Mars. And that brings us to exoplanets. So this is a graph of the exoplanet detections per year since the 1980s. And you can see, you know, 30 years ago, we didn't even know that there were other planets in the universe other than in our solar system. And in the last several years, we've really learned about an explosive growth of exoplanets. Well, they were there all along, but our detection technologies got better. And they're detected in many different ways. But today, I'm just going to tell you about two different ways they're detected via the radial velocity and the transiting method, since most of our exoplanets were detected that way. Okay, so the first thing you should know is that a planet doesn't orbit a star. So rather a planet and a star both orbit the common center of mass between the system. So in this graph, you see the big thing is the star, the little things, the planet, and the star is wiggling back and forth. And what this means is if you're observing the star as it moves back and forth, the light is Doppler shifted throughout the orbit. And what you get is you get slightly blue-shifted light on one part of the orbit and red-shifted light on the other side. And if you observe it for a very long time, what you see is a periodic signal. And you can analyze this periodic signal to learn about the mass of the planet that's orbiting, how long the planet's year is, and how far it is from the star. So this is a very precise way to detect planets, but you need an extremely precise spectrograph. And this has been done in different ways over the years. Early attempts used vats of boiling hydrofluoric acid to calibrate the spectrograph. It wasn't so great for the instrumentation. These days, people use laser combs. Stick around for the next talk if you want to learn about lasers. So the other way that we detect exoplanets is the transiting method. So this is the method used by the Kepler spacecraft. And basically what you do is you just stare at a whole bunch of stars for years. And you wait for the planet to go in front of the star. And you would see a very slight dip in the brightness that will change over time. And an Earth-like transit of a planet lasts about 13 hours and happens once a year. So on average, you need four of these transits to define an orbit. And so it takes many years of observing before you get the exoplanets. So what you get from this, in contrast to the previous method where you get the mass of the planet, you get the radius of the planet. So you know how big it is, but since you have no idea what it's made of and how dense it is, you don't know how much it weighs. But what you can find from this system, if you know about your star, how much light it outputs, is you can define a habitable zone where the surface of your planet could have liquid water. And what you can do is you can take the orbit and the size, and you can get a good idea of roughly what the planet system looks like and whether the planet is in the habitable zone. So this is an example of the first roughly Earth-sized planet located in the habitable zone of a red dwarf. And the reason the orbit is smaller in comparison to our solar system, so this is the F orbit. And that's roughly comparable to the orbit of Mercury is because the red dwarf star is dimmer. But that brings us to the next question. Okay, so we know it's a roughly Earth-sized planet and it's sitting in the habitable zone, but does it have a breathable atmosphere? Does it have a greenhouse effect that might make it warmer? How do we find this out? And the problem with studying planets around other stars is that they're super faint. So there's actually a planet in this image. And if I hadn't put this label here, you probably wouldn't be able to tell where it is. And in fact, this is a planet that's sitting super far out from its star. It's way past the orbit of even Neptune. And if you wanted to observe an Earth-like orbit, it would be way in here. And it's roughly a factor of a million fainter than the star. So digging out the signal from behind the star is extremely hard. And one thing you should know about telescopes is that the bigger they are, the smaller this diffraction pattern is. So if you make a bigger telescope, you concentrate the starlight closer in. And in principle, you can see these smaller orbits. So this here is the Kepler satellite. That is what has actually detected most of our exoplanets in the universe that we know of so far. And soon NASA will be launching JWST. It's a factor of roughly six bigger. So that's great. And actually, the Europeans are planning to build something called the extremely large telescope. Not a joke. That's actually the name. But it will have a 39-meter dish, and this will enable us to directly study exoplanet atmospheres with the hope being determining whether any of these habitable planets actually have an atmosphere that would be conducive to humans colonizing the planet. So I'll end my section with the Kepler Ori here. And this is all of the known exoplanets from the Kepler mission in their orbits to scale. So the bigger planets are Jupiter-sized, and the small ones are Earth-sized. And if you want to learn more about all of the worlds that we know of outside of our solar system, you can go check it out on NASA's website. They have something called the New World's Atlas where you can learn about each and every one of these exoplanets. So with that, I'm going to hand it over to Peter, who's going to cover how we get there. Yeah, how do we get there? This is actually far from easy. Liz, do you know the first rule of space club? Is it don't talk about space club? No, it's space does not cooperate. We learned this from Mark Watney. Meaning if you do anything with space, better know really what you're doing. As a rule of thumb, you should be treating vacuum like a poisonous gas. That about gets it right, and then you can start building things. Actually, there's a second rule of space club. Don't talk about space club? No, be prepared to quote Mark Watney again. Yeah, exactly. And these are just some of the topics you want to maybe think about. It's not just about should we go at all and if how should we colonize, should we send robots? What propulsion is the right one? Do we need planetary basis on the way? What is actually sociological aspects? Do we get along on those spacecrafts for many, many years in confined space? I don't know, would my husband get on a spaceship with me for four years? So meaning you have to plan your trip carefully. This is nothing new. Even if you do something within our solar system, you have to plan your trip carefully. People have done this before. Well, obviously we have sent probes to the inner planets, to the outer planets before. And the first concepts date back to the beginning of the 20th century. This is actually the way, for example, how you want to get from Earth to Mars fuel efficient. It's a so-called home and transfer orbit. So you start from Earth, which is the blue planet, in the position where it is where the blue field circle is shown on the slide. At the same time when you start, Mars is in the position where the open circle is. But by the time you have then traversed on your home and transfer orbit towards the Mars orbit, Mars will be in the right position that you can actually just land on Mars. If you have more time on your hands, if you want to use even less fuel, there are other options. We've just seen in the previous talk and heard about Rosetta. And Rosetta did several swing-bys and gravity-assisted fly-bys by other planetary bodies in our solar system, using those to then accelerate, get into a different direction without using any propulsion. And actually, since a couple of years, there's another method, which I encourage everybody who is interested in this to just Google and look up. This is the interplanetary superhighway, as it is called. This is a system of how you get from Lagrange points to other Lagrange points in our solar system without any fuel. Essentially, you just have to get onto those tubes, which are in mass speech, I guess, open and close many folds. I'm not a mathematician, but this is what you want to look up. And then you can just coast along these lanes through our solar system. But normally, you want it fast. You mean one of those SpaceX rockets? Well, sort of. SpaceX is fine for Mars, definitely. These are chemical rockets. You can still tweak them a bit, but if you really want to go far out in the solar system or then even talk about interstellar, then you have to use other methods of propulsion. Let's look at these very crazy spacecrafts for interplanetary missions, which have been proposed before. You will see these designs a lot if you start looking into spacecraft design. It almost always looks like this. So there is, at the very front, there is the payload, which could be some crew compartment, if you want to send humans. It could be some cargo, which you want to send to an outer colony, maybe in the asteroid belt or to the moons of Saturn. The big thingy behind it, this is not wings. I mean, we don't need wings in space, but we need to get rid of a lot of excess heat created by our nuclear reactor, which is at the very rear of our spacecraft. So these are radiators. And since space is essentially black and really cold, I mean, it's 2.4 Kelvin. She's the expert. We can efficiently radiate hot stuff away using so-called radiators. At the very end, as said, there is a nuclear reactor. Of course, there are some things like ion engines and whatnot, but I want to highlight one of these crazy ideas. And this is how the reactor actually looks like, a sufficient fragment rocket engine. What you see is, like in a reactor, you might know, it has some nuclear stuff in the middle here shown in green. And of course, there's a moderator around it to get the neutrons to the correct speed. What is the interesting part here is that you actually will use the propellant itself, so the plutonium carbide dust confined with superconducting reflector magnets to get the fission particles out and read some radiation to your back. Of course, the particles have to be charged for that, so gamma radiation will go in the other direction and you better have some shield there, because otherwise you're frying your crew. The fission fragments are pretty fast. They go at almost 2% the speed of light. And we also get a ton of energy out of it to power thermoelectric generators to have excess heat to get electricity on our ship. If you want to go to Interstellar, this is not going to cut it. We have to have high specific impulse and high thrust. So we'll talk briefly about Project Orion. This is a different Orion than the NASA Orion. This is the old Project Orion. Not to be confused with Project Pluto. Don't look at Project Pluto. This is insane stuff. And after 10 clicks on the interwebs, believe me, I know what I'm talking about. I'm from Bielefeld. You end up in a swamp of conspiracy theories. Don't go there. Don't. Okay? So this is Project Orion, which is essentially Nuke us to the stars. We have a payload section again at the very tip of the ship. We have a propellant magazine and then we have a shock absorber at the very rear. And what we do, well, we take some nukes, which we have in our compartment, just throw them out of the rear, have them make boom, and then propel our ship forward. I would like to point out that this is, I think, the most reasonable use of nukes. So it would be perfect reuse of our nukes today. Of course, you maybe do not want to have this on your surface body. Maybe if you want to start from Earth or from Moon or from Mars, because, well, your mates who stay in the lunar base or so will essentially be fried because of your nuclear explosion. So do this in free space where there is nobody else and see that there's nobody tailgating you or something. Watch out. Are there other options? Yes, there are other options. As said, there are these ion engines, which are very nice engines. They are being flown on spacecrafts right now. There you use an electric field to accelerate ions. It's only a tiny bit of thrust, but it's very, very efficient. And if you have a long mission duration and you can just, I mean, a rocket, for example, like the Saturn V or the Space Shuttle, they go for some minutes. But if you have an engine that can just power your craft for years, you will get faster in the end. About laser propulsion, we will talk in the next talk at the very beginning. And all the other things we'll not talk about because, as I said, you get into a swamp of conspiracy really fast. Don't look up EM drives. Just don't do it. The fairy dust would actually be some of the torch ships classified as normally constant acceleration engines. And this could bring you to the stars pretty fast, but we have no idea whatsoever what technology to use in constant acceleration drives.  Believe me, if you are going at 20% the speed of light in the end or something like that, you have to make sure that you understand that space is not empty. Space looks like this. So this means there's a ton of stuff between the stars, the interstellar medium, its neutral atoms, charged atoms, molecules, dust. And if you do a quick calculation, you will see that if you are, for example, at a speed of 20%, the speed of light, you will erode one centimeter of aluminum in front of your ship each year, just from the interstellar dust. So this is something you really should pay attention to. Of course, there are other things also floating around, micrometeoroids, although they're not that frequent in interstellar space. Of course, it's also something we have dealt with before with interplanetary missions with Earth moon missions with things in Earth orbit. And this is, for example, how it looks like. This is a famous so-called Whipple sheet on the left. So you essentially have a sheet of metal in front of it, which absorbs the first impact. And then you just have a spray of debris from your impactor on your actual spacecraft. Or you could do something like metal foams or something. This is actual research being done right now. So yeah, we're preparing for that. But of course, we want to have really nice things maybe. And then, yeah, it's BYOM, bring your own magnetosphere or force field. And yeah, if you, for example, just put a magnetic field around your ship, you could collect plasma on the way, have a nice plasma in front of you. And that would lead to having a nice magnetosphere around your ship, which would then deflect charged particles.  And that's actually why this thing has this wedge shape. This is not because the artist thought this looks nice. No, because this is, here is the deadly radiation. And actually, well, it's a cone around your ship.  The particles are still fine.  So maybe point your ship slightly off of the planet you want to land on? Yeah, maybe. So still, with all of this, as said, the only option of going really fast are so-called taut ships, constant acceleration drives, which could bring us to Proxima Centauri in about, okay, seven to eight years, ship time, Earth time would be about 10 years, because relativistic effects will then come into play. Anyway, it will take long. It will take so long that you better take a nap during the voyage. And this might seem a bit like science fiction, but actually, this research that is being done by medical researchers at NASA, private companies right now, and these artists' impressions, hibernation could be an option. It's now routinely used for people who have had severe accidents. You just cool the people down. And actually, if you, for example, cool down your body temperature from 37 degrees Celsius to 32 degrees Celsius, you're using your metabolic rate by 70%. And this means, well, you don't have to bring a lot of water, food, oxygen on your ship. Last thing. No, this is not space. Sorry. This is not an option. This is no option at all in space. There are so many things which have to work over a so long period of time that we actually, for some of those, have no idea how to accomplish this. We're talking about trip times of around 100 years maybe. So I have not seen a computer that has worked continuously for 100 years, neither a propulsion system, electronic system. So if you have such a ship, you better bring all the spare parts and you have two ships, or you have maybe capabilities on board to do this. Like mining asteroids to get new parts for your shield, which has eroded away. Maybe that. But this is not to scare you away. This is just the challenges. I mean, we have done other things. Mankind has landed people on the moon soon. There will be Mars. And please keep in mind, there's a very famous quote by the father of rocketry, Tsiolkovsky. And this the earth is the cradle of humanity. But no one lives in a cradle forever. So at some point we'll get out there. The question is when, but there's a lot of stuff to do and you're all invited to take part in it. Thank you. We have some time for questions left. So please go to the microphones if you have a question and remember to keep it as brief as possible. Single Angel, please give us the questions from the internet. There's one question from the internet. If you assume EM drives will work and will have usable thrust, how would that change the space travel? Could we shorten the path instead of relying on parabolas? Good question. Next question. Microphone one. If we have magic, then everything is fine. Yes. I have a question that has always bothered me with space travel. That's basically what's the rationale behind aiming for a foreign and far away planet that right now is basically an utter wasteland and hostile wasteland.  Why should we survive on any like utter wasteland if we can't do it here? Well, if you remember earlier, there was the problem that our sun is eventually going to expand and consume the earth. Yeah, but that's like far, far, far away. Yes.  We should definitely do that. But in the very long term, human space exploration is necessity. And so we should always keep our eyes on the future of our species in general. So one last question. Microphone four, please. Can you think the Alcubier drive will change everything? Well, it's a nice mathematical concept, but the stuff you need to make it real is it's not even clear if it's physical reality. Yes, if we have a warp drive, it's fine. Yes. But sorry, it's not it's not going to happen within our lifetimes. I promise you. So if you manage to build a warp drive by the end of Congress, we take Zal 2 with us. Okay, so one more thing is that we've planned for the space track speakers to congregate shortly after midnight at the speaker sofa outside of Sol G. So if you all have more questions for us that you'd like to ask later, shortly after midnight, that's where we'll be. All right. Thank you very much. That was Liz George and Peter Bush camp. Thanks.


BSU Women's Hockey season opener tonight. The Syracuse Orange at the Sanford Center. First period beavers on the power play. Emma Therese centers the puck. Loose out front. Clare de George takes a couple whacks at it, but Abby Miller covers it up and jump ahead to the third. Orange threatening this time. Savannah Rennie a prime scoring chance, but Aaron Dieter says no. She stops 33 shots total on the night. This one goes to OT. BSU's best chance here. Emily Berglund right up front, but Miller makes another big save. Scoreless 0-0, that's your final after an overtime. If you've enjoyed this segment of Lakeland news, please consider making a tax deductible contribution to Lakeland Public Television.


Hey Golden Bears, you're watching CalTV Sports Behind the Hype. Today, we're introducing a new segment called Afternoon Tea with Brandon Lee. And if you haven't guessed it, yeah, that's me. In these segments, I'm going to be sitting down with some of our distinguished Cal athletes. I'm going to be chatting over a nice cup of tea and some pastries baked by you guys truly. Today, I'm going to be interviewing Emily Boyd of the Californians Women's Soccer Team. Stay tuned and peace out. Emily, thank you so much for joining us today. You're my first guest on Afternoon Tea with Brandon Lee. How are you? Great. How are you? I'm caffeinated now. So you just wrapped up your final season here at Cal and you accumulated a lot of hardware and achieved a lot of accolades. You just got named Pac-12 Go Keeper of the Year. You are third all-time in Pac-12 history with 36 career shutouts. You are the all-time career stage leader in California history with 297. And you just got named an All-American. What can you say contributes to your success on the field? I would say I started becoming really successful when I got fitter and I incorporated more on-the-field workouts, off-the-field workouts into my game. So it's not all just about going to practice every day and trying as hard as you can. It's about everything you do after that. So it's okay, I'm going to go to practice. I'm going to beat Brooke. I'm going to breathe, live. I'm going to try and do everything I can every day to beat them. That's my goal. I want to beat them and every day I can't. And so those days are the days that I really want to go out and get after it on my own and do more, whether it be play more, whether it be go run, go do yoga or just do something more so I can feel like I have won that day. So there are so many renowned female soccer players out there from Abby Wombach to Hope Solo to California alum Alex Morgan. Who would you say is your role model and why? I would say on the field my role model has always been my sister. She was in a totally different situation than I was in high school. She tore ACL as a junior and was recovering and her senior year she hadn't committed yet and she committed late to Washington and had this whole freshman year where the first half she didn't play and then the second half she started and from then on she never left the field. So watching that as a high school soccer player was really inspiring to see that she could go on this team, start with little to no scholarship and then become one of their greatest assets like ever. So it was really cool to see that not only being my sister but being my role model because I knew that no matter where I started I could always go up seeing that in my household. So that would be my role model on the field. And I would have to ask was there like a little sibling rivalry between you and your sisters and she is she a goalie as well? No, she's a center back. Okay, but she does motivate you a lot because I know you never skipped a minute in your California career, correct? Yeah, there's been rivalries just because she played at the University of Washington but we're four years apart so we never got to play against each other sadly. That would have been great to see. That would have been so fun. So when we play UW she is a little neutral but other than that she's like just rooting for us the whole rest of the season. So you just wrapped up your final season in a California uniform. What can Cal fans expect next? Because I heard some rumors you're going to the draft. Are you excited for it? What are the ways you're like dream destination? Yeah, I'm going to the draft. It's January 18th in Philly. My sister's birthday. Oh really? Yeah. And I think that we're going to go to the draft, me and a teammate. Yeah, obviously I would love to play in Seattle because that's where I'm from. I love the team but I think that if I get an opportunity to play anywhere I'll be really excited. And I'm all about going to new places, trying new things. So I think that anywhere would be a great place to go. And I'm trying to maybe play for Canada. We'll see. I don't know that's in the works but I'm excited. Alright, we're going to be cheering for you on January 18th. Hopefully you have safe travels over there. Alright Emily, finally we have to talk about your Instagram page on food and fitness because one, I have something for you. I actually followed one of your recipes. Pumpkin, banana, walnut bread. It looks like... It looks nice. Thank you. It can come out so nicely but hopefully I will get some of your reviews. How is it? It's too dense, not great. It's good. I would recommend not full wallets. Not full wallets. Okay. Other than that, A plus. Yes. Success. Success. Let's talk about the page. What motivated you to start this page? Well actually it was one of my roommates told me that she didn't want to see my food on Snapchat every day. And she was like, I don't want to see your food on Snapchat. You need to like start something. And I was like, you know what? That's a good idea. Maybe I'll start an Instagram page. And then I started this page and I really like making food look pretty. So at first I would post pictures and my teammates would be like, that's not very pretty. And so my goal was to start making healthy food look better because I always try and convince my friends that all these weird foods that I make are actually good. So that meant I had to make it look good. And so when I started working with all the healthy food then it went to the fitness and I was like, this is going to be my food and fitness. And then I started sharing stories about my life and my journey through all of this fitness stuff. And people really started like catching on and following. So yeah, shout out to my roommate G for not liking my Snap stories. Well we have to thank G because you have more than 1500 now. You have 2400 followers I believe and counting. Yeah. What can we expect to see next on your Instagram page? Well I'm doing Whole30 which means no dairy, no wheat, no alcohol, no added sugar and no legumes. So all my meals are surrounded by that. And so probably another Whole30 recipe. Okay, I will be sure to follow because this is my first time trying something sugarless. Yeah. And it still tastes very, very good. Yeah, it's sweet. Yeah. Now everything needs added sugar. That is true. Since the holiday season is coming up, do you have any e-boys food friendly dishes that I can make on December 25th? Well I would say for the holidays, good ways to make dishes sweet, like if you're trying to make desserts, would be bananas. Bananas are sweet and you don't have to add sugar since I'm doing no sugar. And for any specific meals, I would just follow e-boys food and you'll get to see lots of hopefully holiday dishes for you to make. All right, you hear that, cow fans? Be sure to go on Instagram and follow at e-boys food. Emily, thank you so much for joining us today and being my first guest on this segment. Cow fans, thank you for tuning in to this episode of Behind the Hype, reporting live at Edward Stadium. I'm Brandon Lee. I'm Emily Boyd. Go Bears!


any type of content that I'm listening to, I'm usually trying to think about what sections are they talking about? So let's say we're learning about a disease and the lecturer is talking about a heart condition because I'm a cardiologist. They may tell me about what that disease is, how it works, how it's treated, how it's worked up, and those are kind of many sections within the lecture. Now with each section, I'm thinking about what questions would I have to be able to answer to thus duplicate the answers or the topics they're talking about. So they're teaching me about high blood pressure and how it happens. I may think of three to four questions that may represent a slide that they're covering in that video. In that case, I may write down a list of those questions and it grows to a list of 20 or 30 depending on how long that lecture is. But if I can answer those questions, I can essentially go ahead and answer what that lecture was talking about in that video. I can't, I can easily say I don't understand these 10 questions that I listed out but I knew were somehow pertinent to whatever they were talking about. Now I can just go back to the video, pause to that portion, write out that answer below my question in my Word document or whatever in my flashcard. And now I basically have a big document of questions that are corresponding to the video.


Hello everybody, Andrea Meiske here. So let's talk about teeth whitening. This is a very popular topic lately. Let's talk about teeth whitening and is it safe and What different types of options do you have? So I'm holding up a couple different things here now I'm going to go through teeth whitening options that you can get Professionally done in a dental office and then take home kits that you can get as well now to get it professionally done That is the best way to do it because a professional is doing it for you. Not all teeth are the same It's like hair color you can get a box dye from the store But it doesn't always turn out how you want it to so it's the same thing with teeth whitening If you go to purchase an over-the-counter teeth whitening product, it might not be right for your teeth It might not be right for your enamel. It might not be right for your shade You have probably heard a lot of horror stories of people being very very sensitive having an over-the-counter teeth whitening product because with a professional we have different types of Proxides mixed in and we mix in different types depending on your current shade and Depending on the health of your enamel some people's teeth. They are very very thin So they have thin enamel they would use a different type of composition Then somebody who has very thick enamel and doesn't need to go too many shades lighter So there is definitely a difference So the best thing to do is to get it done by a dental professional and you can get quite light But if you wanted to say over-the-counter products Believe it or not. There are professional over-the-counter products as well. So I like this one This is the one that I give to my patients and this is the one that I teach to my Students in my teeth whitening course where I teach people how to have their own teeth whitening business I like this one because it has the right mix of carbamide peroxide and Hydrogen peroxide and it will not make the teeth super sensitive if you have sensitive teeth currently I do suggest using a sensitive toothpaste a couple weeks prior. So a couple weeks prior to even using a professional Teeth whitening service or an at-home kit because that will just help to coat the teeth and Protect the teeth from any additional sensitivity because as soon as you put any type of Peroxides on them if the teeth aren't coated with a proper toothpaste if needed it can make things more Sensitive so this is where a dental professional can come in and look at your teeth and tell you if you need a Stronger type of desensitizing toothpaste you can purchase them in store But then there's also some that you can only get through a dental professional if you really need like a high-quality Desensitizing toothpaste so different things to think about too, but I like this one as my Over-the-counter product I mean not over-the-counter product because you can only get it from a dental professional But something to use at home. So this includes a tray It includes a couple syringes not these ones This this is just an example But it does include syringes that you put in the tray put them in your mouth for 15 minutes 30 minutes However, you want to it does include a light with it and the light turns off after 15 minutes So if you just sort of wanted a quick boost then it lasts for 15 minutes or turn it on again for a half an hour I tell my patients that if they're going to get teeth whining professionally done It's still a good idea to upkeep it at home if they want to and use an at-home product that actually works So a patient might have their teeth whined with me Tomorrow I will suggest a take-home product that they can boost their whining in You know two months or whenever they feel they need that boost But the right products make all the difference there are products out there that don't have a desensitizer built in or there's also products out there that Only have one type of peroxide and not two Hydrogen peroxide does help to lighten the teeth But carbamide peroxide helps it to last and helps it to kind of form a thin protective barrier over the teeth So it's a lot harder to get stank So if this sounds overwhelming to you which it can be talk to your dental professional and they can let you know of the Different options other options that I have to just to just sort of upkeep the whining at home There are mouth washes there are toothpastes like there's all kinds of different ones Like you don't have to get a certain kind But this is where your dental professional comes in handy because there are different things We can help you with to help to limit stain as much as possible But also to keep the teeth as healthy as possible If the teeth are thin if you are getting a lot of cavities If you brush too hard if there's Gingival disease in certain areas your teeth aren't as healthy as they could be and that's where a mouthwash Toothpaste even using a different toothbrush can make a big big difference So let me know you guys if you have any questions. Thank you so much, and I will see you very soon


Torus 692 multi-caliber revolver. It's short, it's stocky, it's overbuilt. Seven shot in both 357 38 special and multi-caliber that also means nine millimeter. It is ported. Okay there are four ports right there on each side and that will keep the muzzle rise to a minimum. Grip is kind of a spongy rubber there.  Heavy trigger pull not too bad between six and seven pounds with that single action trigger. Spurred hammer a transfer bar push on that button pull forward and off comes that cylinder that right there push the button and there you go shoot nine millimeter. I think Torus has a winner here. There we go nice oh I love it.


Hello, I'm Dr. Davi Tubshet from the Environment and Climate Research Center at the Policy Studies Institute. We are working collaborative three-year research project with Stockholm Environmental Institute, International Livestock Research Institute, INERA in Burkina Faso, Sokone University of Agriculture in Tanzania, and our institute in Ethiopia. The research project is on research and learning for sustainable intensification of small-holder livestock value chain in Burkina Faso, Ethiopia and Tanzania. It is part of the broader research initiative for on sustainable intensification of agriculture, research and learning in Africa, the SIRLA project. This transdisciplinary inquiry into livestock feature conducted via a participatory process that explored the relationship between social, economic and environment components with a major objective to explore the future of animal production. Our study site is in Azvi, Tagrai region in Ethiopia. The project focus on small ruminant in Ethiopia, dairy cows in Tanzania and cattle in Burkina Faso. The project has involved all actors involved in livestock sector, for example, farmers, trader, input supplier, government technical experts, extension service workers, researcher, decision-maker, community leaders, religious leaders were participated. We also operationalize our projects through an environmental and production trade-off through learning space and using computer-assisted tool. We name it clean tool. The clean tool is comprehensive livestock environmental assessment for improved nutrition, a secured environment and sustainable development along livestock value chain. It provides an assessment of livestock production system change in that poor environment. We found that all actors appreciated being brought together in the participatory workshop to have an opportunity to learn from actors they might not normally interact with. Meeting high-level policy-maker in the transformation game allowed farmers to understand why some policies exist. The national level policy-maker are also well aware of the target-setting policy, which is why they argued for livestock compositions that would, for most, have a significant impact on reducing greenhouse gas emissions. But when they met with farmers in the transformation game, they discovered why some policies do not work on the ground if they do not still maintain livestock production. This sharing of knowledge between the actors was a key to find solution. For example, meat and milk production can be increased without increasing environmental impact from livestock in terms of water use, greenhouse gas emission and soil nitrogen. This can be achieved by moving to improved dairy breeds and substituting cattle fattening with more sheep production while spreading the use of machinery and reducing draft animals. There has been also a high interest in the transformation game and the clean tool and the demand to scale up the tool application in a bigger and wider spatial scale to accommodate different agro-ecological zones because Ethiopia has diverse agricultural zones and it is proposed for further work at this transformation game to be implemented in a bigger scale or at national level to understand the dynamics of livestock production throughout the country. Thanks for listening.