So we had k is equal to 1 on 4 pi t0 minus t to the n plus 1 over 2 times e to the minus x0 minus x squared on 4 t0 minus t. And we had rho was equal to square root of 2t0 minus t times k. And then we had that d dt of the integral over the surface at time t of rho d mu with respect to the measure at time t was equal to minus integral h plus the normal component of the logarithm of rho. Because k is defined on all of rn plus 1, I can write this here. Or you can write k, if you like, because it's the same third rho d mu. And now, the question then is, and it was posed by you, what happens if the right-hand side is 0? Now, so the question is, which surfaces satisfy h equals minus gradient nu in the ambient space of log k, which is, if you take the logarithm here, it's just this term. So taking minus the gradient, the normal gradient, you get just here x, the inner product of x minus x0 times the unit normal divided by 2 times t0 minus t. This is the equation that a surface has to satisfy if the right-hand side is equal to 0. Well, let me show you. Of course, if you just stare at it for a moment, you find out that this is satisfied by the sphere. If you pick the radius right, depending on t0, and see a sphere centered at x0, or also by a cylinder with the right radius, if the point x0 is on the axis of the cylinder. So we have some solutions that satisfy this. There's the shrinking sphere, and there is the shrinking cylinder that satisfies these equations. So what is special about them? And it turns out they are all self-similar shrinking solutions. So let me show you that whenever you have a solution of this equation, you get a self-similar shrinking solution. Suppose the initial surface, mn0, satisfies the equation h equals, and let's take x0 to be the origin, x times nu. This is the position vector divided by 2 on t0. And then let's just shrink it. Then, so f0 is the map from mn into rn plus 1, satisfying this equation. And then I should write here the position vector f0. And then set f of pt to be, well, we should shrink this surface. So with what factor should we shrink it? Well, the right scaling seems to be to take this quantity to the 1 half, and take f0 of p. And then when we compute d dt of f of p of t, and take the normal component of that, well, then we get from here, we just get minus 1 over 2t0 minus t to the 1 half times f0 at p. And the whole thing is multiplied by nu. But we know what f0 p times nu is. This is equal to h times 2t0. This is h times 2t0. OK. We should rescale this in such a way that at time at t0, we get the same surface back. So I have to also divide by 1 on square root of 2t0, so that I get the same surface for t equals 0. And then here, I have to put in this factor times 1 on square root of 2t0. And then this is equal to, because of this equation, this is equal to minus 1 on, I have to be careful, f0 p.  And now I replace f0 nu by h times 2t0. And let's see what we finally got. What did I do here? Does this look right? I'm left with square root of 2t0 divided by square root of 2t0 minus t times a minus sign times h. This is just the mean, but this is the h of the initial surface of f0 at p. Now since I have rescaled this, by this factor, the mean curvature with exactly opposite factor is, of course, equal to the mean curvature at f of p of t. So we find that the normal component of the speed is equal to the mean curvature. So f solves mean curvature flow up to tangential difumorphisms. So in other words, this equation here characterizes self-similar shrinking solutions. So this equation star characterizes self-similar shrinking solutions. And it's clear that we cannot expect that this is true without allowing tangential difumorphisms, because if you're here on the cylinder, and the cylinder is shrinking, and you study this from this point x0 here, then, of course, here the shrinking in this direction is exactly the same as mean curvature flow. But of course, the shrinking from this point here, or from this point, it's only the normal component of the shrinking is giving you mean curvature flow. So that's why it's important that I have to multiply this with nu, only then you see the mean curvature. So this rescaled surface is, of course, not, it has very strong tangential component, as you can see in this picture, where you have the shrinking cylinder. Nevertheless, the flow, this monotonicity formula has the capability of singling out the self-similar solutions. And so the question is, are there more such self-similar solutions? So here are some other examples. It turns out for any equals 1. So just a curve in the plane, there was a whole family discovered. There's a circle, of course, which we just found already. And then there's a family of curves look like that. It's getting more complicated. And there's a two-parameter family of curves that all satisfy this equation. You see, when you write it down for a curve, this is just the curvature of the curve. So if you write it in angular coordinates, this is just a second-order ODE for the position of the curve in the plane. And then you can always solve it for some initial conditions, but then the solution will just chaotically run around. And if you do it just right, you get a discrete family of parameters where the curve nicely closes up. And that gives you this family of curves, which was found by Operation Langer. And then I think it took about six or seven years until people discovered that Malin had studied this about 25 years earlier in the context of crystal growth and found these curves already. In higher dimensions, there is a whole bunch of these self-similar solutions which are axially symmetric. So I just draw the profile of these axially symmetric solutions. So here's the radius of these curves. And then, of course, you have the sphere. You have the cylinder. And then it turns out there is a embedded torus, sort of an oval-shaped cross-section, which was found by Sigurd Angenand. That's why it's called the Angenand Torus. Again, it's a matter of finding the right initial conditions for the resulting ODE in the axially symmetric case and make it close up. And then there are these pictures very similar to these ones. There's a curve like this. And then it turns horrible. You can have, essentially, there are pictures that look like that. So you have all sorts of immersed torus. And then there are some which hit the axis. So there is a huge variety of axially symmetric self-similar solutions. I should point out, however, these are the only embedded ones up to here. And so these guys are embedded. And these ones here are the only ones with positive mean curvature. So all these on this side, all this junk, either has not positive mean curvature or, in fact, it's not even embedded. So you would not expect these guys to appear as singularity models. They would be extremely unstable. So the really interesting ones are those three. And then there are others. I'm unable to draw them. But imagine a cube, punch a hole through in all three axes and make it into a smooth surface, what is left over. And then you get a surface with this cubicle symmetry. And in fact, someone showed numerically that you can solve this elliptic equation on such a surface. So there are embedded, complicated surfaces in R 3 that move under self-similarities and move under these shrinking equations. But again, you would expect that kind of thing to be very unstable. Now, there's another way of thinking about this. Because, see, this weight here is rather precise. But I come back to that later. We can, in fact, interpret these surfaces here as minimal surfaces in a funny metric coming from this Gaussian cone. And then this would be, say, the equator in a sphere. And this would look like the Cliffor torus or so. So there's a way of thinking about these surfaces as minimal surfaces just in a funny metric. But this thing here comes from a conformal factor. But the most important application of the monotonicity formula is that it allows you to understand the shape of singularities. And this only works if you do a rescaling procedure. So I have to tell you about the rescaling procedure for the mean curvature flow. And let's see. There's two ways to do rescaling. The first attempt. And you see, here, for these self-similar solutions, we have rescaled with this factor here. And we found this, we scared the initial surface, we found the solution. So now suppose we have a singularity. Assume that mean curvature flow forms a singularity that is similar, I will tell you in a moment, to spheres, cylinders, and all these pictures here. We should rescale, if this is the case, the right magnification rate that we should put on our microscope should be exactly this factor. Just we do it now the other way around. We want to rescale with this factor to the power minus one-half, s, t approaches t0. And t0 is a singular time. This is our old Tmax, a singular time. We have a singular time. And by the way, singularities can happen. As Carlo told you yesterday, recall, there are these singularities that come from convex surfaces. And we expect that if we train our microscope in the right way, that we see the shrinking spheres. And he also showed to you this argument with a barrier argument that you get a shrinking neck in here. And we expect that if we train our microscope on this region, that if we do this, we somehow should find the shrinking cylinder. This is the picture. And what we want to find out is how generally it's there. But if the picture is right, then we should rescale with this quantity. So the answer is to set given f from mn from 0 to t0 to rn plus 1, set f tilde of p and t to be this factor, 2t 0 minus 1 to the minus 1 half. We are blowing up. If t is very close to t0, this will be a huge factor times the f of p of t. If we expect the singularity at the origin, expect singularity at x0 equals 0, otherwise I would take here minus x0. But let's make things simple, rescale around the origin. Then it turns out best to introduce a new time parameter. And to get the constant right, s is equal to minus logarithm of 2t0 minus t to the 1 half. So as t approaches t0, this goes to 0 and minus the logarithm goes to plus infinity. You see, it's much better. We want to study now the asymptotics after we rescale. And to understand the asymptotics, it's better to make the time interval infinitely long. And it turns out this constant is just chosen such that if you now take the derivative of f, add the rescale thing with respect to the s variable, just do the computation. It turns out you get, from here it just scales right, the mean curvature vector of the rescaled surface. But then you get an extra term from this front thing. And this extra term is simply f hat at p s. If this is not there, this is the pure rescaling. If the mean curvature wasn't there, this is pure rescaling pushing the things up. So this is one way to rescale the flow. And notice that, of course, the fixed points of this new form flow, fixed points satisfy, in particular, that if you multiply with the unit normal, that h. So if you have fixed points in normal direction, so dds f hat normal component, if you want that to be 0, we always have to work modulate the tangential motion. So we only want the normal component to be 0. And then this just means that you get minus h, because mean curvature vector is minus h times nu. So you get minus h plus f hat dot nu equal to 0. And this is exactly the equation star as before. So we have indeed created, now, by rescaling a flow such that the condition that gives us the self-similar shrinking solutions are exactly the fixed points of this new flow. And then when you compute what is now, now let's see what the monotonicity formula is telling us in this setting. Well, if you said, well, let's just compute it, d dt rho d mu over mn. This is equal to 1 on 4 pi t0 minus t to the n over 2, right, times e to the minus f squared divided by 4 t0 minus t times d mu. Now, the d mu multiplied with this vector is exactly the rescaled d mu. Up to a 1 over 2 pi to the n over 2. This is just the rescaled d mu hat. And this thing here, over 4t0 minus 2, is just the rescaled e to the minus f hat squared over 2. d mu hat over the surface mn hat. So with this rescaling procedure, our weighted integral has just become an integral of the Gaussian, a fixed function which does no longer even depend on the parameter s or t. We've really gotten the Gaussian integral up to a constant. And then when you compute now under this new evolution equation, what is dds of integral mns of e to the minus x squared over 2 d mu, it turns out you get minus integral mns of mean curvature hat here minus f hat nu squared e to the minus x squared over 2 d mu. Mn hat of s. Well, mn of s, yeah, you can put a hat if you like. These are the rescaled surfaces. And the time parameter is now s. But now the time runs, note that s runs between the time when you had t equals 0. So this is minus log of 2t0 to the 1 half, which is some number, but it runs all the way to plus infinity. So if our rescaling produces a smooth limit, then this smooth limit must satisfy our crucial equation. So if we get a smooth limiting surface, then it must satisfy equals f dot nu. In other words, any such limiting surface, it must give rise a self-similar shrinking solution. So the question is, under what condition can we produce that smooth limit? Yes, this is the rescaled measure. Yes, everything is. No, no, no, this is just the, this is just, now I'm back on Euclidean space. This is just dx, if you like. This is just the ordinary, the induced metric on this surface. Yeah, I should write d mu s on this surface. Because this together times d mu gives this measure. Because the measure scales like distance to the n. And here I have rescaled distance with this factor. So the measure rescales with this factor to the n. And this is exactly this term here. OK, so the issue is, when do we get this limit? Well, when do we get a limit? When do we, if we can bound the curvature, right? So we need the curvature to be bounded. So what happens to the curvature under this rescaling? Well, the rescaling, the curvature scales exactly the other way around. So what we need is that the curvature, as we approach the singularity, behaves like the curvature of a sphere or cylinder. So need for this, that the soap of the second fundamental form, as M and T approaches the final time, that this is less than some constant divided by t0 minus t. As t approaches t0. So it's allowed to go to infinity, but only at this rate. Or another way of putting it, of course, is that the soap of a squared times t0 minus t. M and T is less than c less than infinity. If we have this assumption, if this is true, this condition here is called it's a type I singularity. If this condition is satisfied, we call the singularity a type I singularity. So the shrinking spheres, all these things that have been there on the board, all these pictures are type I singularities, because they are self-similar shrinking solutions. And of course, for the shrinking solutions, if they are self-similar, the second fundamental form clearly satisfies this. In fact, this is constant, because it's just shrinking. Now, if this is true, then we know that from such a curvature estimate, we get higher derivative estimates. So this means, of course, that the a hat squared is bounded by c, because the a hat squared is exactly this quantity. So in fact, this is the same. And then we know that if this curvature is bounded, we know how to do this. All the higher derivatives will also be bounded. So we have curvature bounds. We have gradient of curvature bounds. And we have, very important, we have some volume bound. Here, this is sort of the, it's a weighted volume. But still, it's a positive density everywhere on Rn plus 1. So this means if our initial surface is compact, this was finite, it is decreasing. So these surfaces cannot pile up. They cannot get high densities. So now we have no compilation, since integral rho d mu s over mns is less than this integral over the initial surface. The only thing that could still go wrong is that the surface disappears.  And all these curvature estimates are useless, because I get nice surfaces, but they all disappear. But this is easily fixed. Either I recognize my mistake and I just translate it back to go through the origin. Or in fact, you can see that if you really approach this point with the singularity, then because you have a type 1 singularity, the mean curvature, which is the speed goes like 1 over the square root of this. And 1 over the square root of this is integrable, and therefore it cannot move arbitrarily fast, even if you push out with this rescaling, so you get a limit. So let me just write, it does not disappear by choosing the scaling point. And then you get, of course, whether you have a unique limit or no claims, but now with this control, you can show that you get a subsequence which converges smoothly to one of these solutions of the self-similarity equation. So let's collect that in a theorem. Any singularity of mean curvature flow can be rescaled, a self-similar shrinking solution of mean curvature flow. That's the first classification of singularities that you get out of the monotonicity formula. Those of you who have studied minimal surfaces, the comparison, you can view this as a parabolic cone. In minimal surface theory, you have a monotonicity formula. It's essentially the same one, except that H is 0, the formula, which tells you if you rescale a singularity, you see a cone, a minimal cone. Here, we see a self-similar shrinking solution of mean curvature flow. It's the parabolic analog of the minimal cone theorem for singularities of minimal surfaces. Can one classify these self-similar solutions? I have drawn this picture telling you that, in general, probably it's rather hopeless. What I can tell you is there's a theorem that I proved a long time ago that if the mean curvature is positive and if it's self-similar shrinking, then it is, well, first of all, you have Sn, the shrinking sphere. Secondly, you can have Sn minus k cross Rk, the shrinking cylinder. And the only other possibility is the very special case coming from the Arbor-Schlanger-Mullins solution. You can have one of these curves of Mullins and Arbor-Schlanger gamma mal, they're bad curves, cross Rn minus 1. And of course, these are ruled out. This case does not occur if you're in the embedded setting. There's no such thing in the embedded setting. So this I proved first around, I think it was around 1990, for bounded curvature. And I think it was proven by calling minicossi in the most general setting around 2016 or so, without assuming bounded curvature. Right, so this is the, all right, I think I should tell you one more. There's one more, very recent result, which is also very nice. So there was a result theorem by Brandle in 2017, I think, where he showed that if M2 is sitting in R3 closed, embedded and self-similar, and to diffiomorphic to the sphere, if it's simply connected, then it is the sphere of some radius. It's a remarkable result because he does not assume any curvature assumption. So he replaces positive mean curvature by embeddedness. And the fact, and the topological condition. He uses a topological condition and the sphere, and the embeddedness. I told you about this cube with the three holes punched through. That's an embedded surface, and it's a self-shrinker. At least it exists numerically. So this topology condition is absolutely necessary. And the embeddedness condition is also necessary, because we've seen all these other. But the engine in torus also shows you that the topological condition is necessary. And this gives some hope. If you cannot show that for these types of surfaces in R3, there's no type 2 singularity, no other singularity than the ones we've discussed so far. And we know the surface we start with a sphere, then the solution must exist until it has shrunk down to a nice round sphere. And that means you can deform with mean curvature flow any embedded two-sphere in R3 into a round sphere with this PDE. This would be a complete reproof of PDE methods of Hatcher's theorem, a famous theorem in topology. So this is an open problem. Can we use mean curvature flow to give a completely new proof of this famous theorem in topology that embedded two spheres? You know, the Jordan Curve theorem, one dimension higher. At the moment, it still seems out of reach. But this is a hint that we may have a chance. The way Brentley proved this is he saw that the self-similar solutions, this equation here. So the main idea, there were many ideas, but the main idea here was that the equation equals f dot nu is the Euler Lagrange equation for the weighted area, for integral e to the minus x squared by 2. If you compute the Euler Lagrange equation for a hypersurface and you want to locally minimize this with this weight, you end up exactly with this equation. But this here is just the area with respect to a, and I shouldn't use hat, maybe I use bubble tilde, of a, this is just the ordinary area with respect to a conformal metric on Rn plus 1. You take the standard metric times e to the minus x squared over 2n. You take this conformal factor, then this area becomes this weighted ordinary area. And therefore, this here is the minimal surface operator in this metric. So self-similar solutions, minimal surfaces with respect to this metric g double tilde. And then you have the whole power of geometric measure theory, for example, to attack this and check whether you can really have a minimal surface of type S2 in R3 with this funny metric. Yes, I come to that in a second. Next thing is translating solutions, so yeah. I just have to remember your question. OK, so this is rescaling first attempt. So now comes rescaling second attempt. If there is a type 1 solution, there has to be a type 2 solution. So we are not that lucky as in that setup over there each time. So in case we don't know what the blow up rate is, we can still do something. Namely, we can do the following. We can just ask very generally, how can we rescale our flow? So given m and t, mn cross time interval into, yeah, let's do it in Rn plus 1. Don't want to rescale the ambient manifold, which we could also do, but let's do it in Rn plus 1. Then, and given time t0, where we want to rescale, and maybe a sequence of points or events, ti in mn cross 0 t0, can define some parameters lambda i bigger than 0, f f f hat, p, and tau. Introduce a new time parameter tau again. And here we take lambda i, multiply now with lambda i. And we take f at p. And here we take tau lambda i to the minus 2 plus ti. And we subtract f at pi ti. This is what we do. So where does this tau live? Well, the t's are these guys. So this guy is t, the old t. So the t was between 0 and capital T. Then this new guy tau must be between minus ti lambda i squared. And it must be from above less than capital T minus ti times lambda i squared. Time interval. And the lambda i here and the lambda i to the minus 2 here are just chosen. So I've just chosen in such a way that here that if you compute d d tau of f hat i of p tau, then the lambda i here times the inner derivative there just fits together with the df dt to spare you the calculation. You just get the mean curvature vector back again at p and tau. In other words, this is a rescaling. There I really rescale now in some sense more carefully. Here I've rescaled sort of just the space. And then I introduced a slightly stretched time variable. Here I don't try to do it all at once. I just take it each time at factor lambda i and rescale the space and a rescale time. And at the same time, I'm shifting the time a little bit such that my new tau equals 0 is at the selected times ti. And that my new value at tau equals 0 is exactly the origin. So I've shifted space and time to make it convenient to make the 0.00 in space time special. But I've got, again, a solution of mean curvature flow. I don't have an extra term. So in some sense, this is the cleaner, the fairer way of rescaling the thing. You have mean curvature flow again. In other words, I'm really just multiplied everything by factor lambda i, space and time. And now the question is, how do I choose? How do I choose the lambda i's? How do I choose my microscope? Well, I want to see something which is not trivial, but smooth. So I have to rescale sufficiently strong. But if I rescale too strong, I just see a plane. I just see a flat surface. So I have to choose the size just right. And the right thing to rescale seems to be just opposite to the curvature. So I can choose lambda i to be the soup of the curvature on the surface cross the time interval from 0 to, well, I want to go a little bit beyond ti. So let's go to t plus t minus ti over 2. So this is somewhere between halfway between ti and the end of my time interval t0. Sorry, I should take here t0. This is the point around which I want to rescale, right? Here this is the end of my time interval where I have the singularity. If I do that, then I get, first of all, that the rescale surfaces have bounded curvature. And then it is bounded by 1 on mn cross the time interval up to a little bit beyond this ti. So I go to tau is equal to t minus ti times lambda i to the 2. So if I put 2, if t is less than ti plus this, this is less than lambda i squared up to t, if I subtract ti from this, I get t0 minus ti over 2, t0 minus ti over 2. I get the bound on this set, which in turn is, oops, I forgot lambda times lambda i squared, of course. Sorry, i squared. And I get that f of pi ti, if I put ti in here, f of tau equals 0, f of pi ti just goes out to 0. And I also get that the second fundamental form, so I've shifted everything to 0. And if I want to, I could have rotated the thing also. So you could think of the surface somehow sitting around the origin in our n plus 1. And at time, somehow we have rescaled this thing, it looks like that. And since these guys are bounded, I can also get, after rescaling, I get all the derivatives of the AIs bounded by some other constant depending on n. And therefore, I can use, since I've translated everything, and again, since I have area estimates, it is not so hard. You have to work a little bit, but it's not so hard that, again, you get a subsequence which converges, as i tends to infinity, get a limiting mean curvature flow. Surfaces m infinity, n of t, or tau, I should call them. And they solve mean curvature flow again. And on what time interval? Well, what happens to this time interval? Here's this time interval. Now, since we run into a singularity, we know at the singularity, we can pick the events such that the second fundamental form goes to infinity. That was our theorem in the first lecture. If we have the singularity, then the curvature has to come unbounded. So the lambda i's tend to infinity. The t i's go to capital T. So this thing here certainly goes to minus infinity. So we certainly get a solution that has existed for all negative times. About here, now, all of a sudden, it makes a difference whether we have a type I singularity or not. You see, this lambda i, we choose the maximum of the curvature on intervals that get closer and closer to the singular time t naught. So if we have a type I singularity, then we know that these guys remain bounded. So we get that this is less than some t max, less than infinity, if the singularity is of type I. But we get a solution on minus infinity up to plus infinity if the solution is not type I. Well, if it's not type I, people call it type II. Why is the curvature, well, the way I've chosen here, this interval here, see, this interval also, if it's type II singularity, this, the 1 half doesn't destroy this. This has to go to infinity. So I get my curvature bound on arbitrarily large time intervals. So I really get a completely smooth solution of mean curvature flow, which exists all the way from minus infinity to plus infinity. So that's the advantage of the second approach. We can still do type I, then we get a solution that exists from minus infinity to some finite time. Well, this is the convex case, type I. You have this convex surface, you rescale, and you get a sphere that comes from huge radius, time minus infinity, and it contracts to a point at some finite time. Or if you have a cylinder or a neck pinch, you have this neck, and you rescale here, and you see the cylinder coming from huge radius at time minus infinity, and it converges to this line at a finite time. So this is this picture, finite time. Turns out there are some other singularities which are of type II and which can be rescaled to have a solution on infinity time. What could that be? Examples. So example of type II singularity. First example is if you have a double loop, and if you let that shrink, what happens is that the small loop shrinks faster than the big loop. And you develop a cusp rather than a neck. This is in the case n equals 1, a curve in the plane. And when you put the microscope on this thing here, what you see is a solution is y, and here is x, and here is pi over 2, and here is minus pi over 2. And you get a solution that is asymptotic to these two lines, and looks like that. And it just moves by translation. Infinity 1 moves by translation.  You get that x is equal to x at t is equal to x at t and y is equal to minus log cosine y plus t. So it translates with speed 1 from right to left. And therefore, of course, it exists on minus infinity, less than t, less than infinity. You get a solution for all time because you get a translating solution. The second example is something like this happens is if you try to do this neck pinch picture that we had before, but now you do it in such a way that the bubble on the right is not so big, but also not so small.  If the bubble on the right hand side is much bigger, then we get a nice neck pinch in the cylinder like that. But if you choose the size of the second bubble just right, then what happens is it gets stuck. It gets stuck in here and forms again a cusp. And when you point the microscope, this rescaling procedure that I explained at the tip of the cusp, then you see again something like this looks like a parabola, grows like a parameter. It looks like a parabola, grows like a parabola, but is not a parabola. But it moves by translation again. So this is a bowl moving by translation. There's one thing I didn't stress. It's inherent in what I wrote down. You have to pick a sequence of points. I've shown you so far in all the, is assuming I have a sequence of points which actually goes to the point where the curvature is maximum, sort of a sequence of points sitting at the tip, sitting at the tip here. There's more subtlety, which I didn't have time to explain. But if you don't choose your points right, of course, you could see something else. If you choose your points where you rescale and do this procedure somewhat differently, then you might rescale it a little bit next to the tip. And if you do that, then you see the cylinder again, which you see in the ordinary neck pinch. And of course, this is the cylinder that is attached to this parabola. If you go on a parabola far out to infinity and rescale, you get a cylinder. So this rescaling thing here is not a, it is extremely helpful, and it gives you a lot of information. In particular, you get these self-similar solutions and you get these translating solutions. But it still depends a lot on how you do it. And you may see there's no uniqueness whatsoever. So you have to be extremely careful how you pick your sequence, what kind of limit you get. But there are some theorems. For example, there is a theorem by Richard Hamilton, which says that an ancient solution, which is convex, and where the maximum of the mean curvature is attained, or the maximum of the second fundamental form, is a translating solution. Sorry, not an ancient, an eternal. An eternal. These are all ancient solutions because they started ancient times, but only the second one here is eternal. These names have all been made up by Richard Hamilton. He loves giving these names, and I think they make sense. And so in order to speak, this theorem applies in the two pictures I've shown you. You get a convex, of course, you have to prove that you get this convex solution here, or you get this convex solution there. There's a lot of work to do to prove that. But you can do that, and then you get this solution. And of course, if you pick the points right, we can make sure that the maximum is attained here or there. And this theorem then says, yes, we got a translation solution. And now you have a complete picture. Not a complete picture, I said there are subtleties. But now we have sort of a chance to understand singularities or rule out singularities. Because if we want to show, for example, that the flow has beautiful properties and doesn't have singularities, now instead of analyzing mean curvature flow step by step and controlling every single little piece of the complicated surface, this theory says all you have to do is concentrate on the singular times. Just assume you have some singular time, and only then you look at the microscope, and then you just have these two possibilities.  This reduces your problem, for example, in the positive mean curvature flow, to just study these spheres and cylinders. You have a type II solution, and then you just have to analyze the translating solutions. And how that works, I'm going to show you in the next lecture improving Grayson's theorem. So we will combine these techniques and a sketch of proof of Grayson's theorem. So next time I'm going to prove this theorem by Matthew Grayson. If gamma zero is an embedded curve in R2, is embedded, then mean curvature flow, which is also called, in this case, curve shortening, deforms gamma zero to a tiny circle. So you've got a proof, an analytic proof of the Jordan-Koff's theorem. Any complicated, no matter how complicated, curve in euclidean two-space can be deformed with this flow until it's nice, small, and convex. And shrinks to a point exactly. In other words, the way we're going to prove it is I show you that the only singularity that can ever happen is this one. I'm not going to analyze what this curve is doing, how it is winding in R2. Just saying, if something goes wrong, something will go wrong because it's enclosed in some circle, which dies eventually. If something goes wrong, this is the only way it can go wrong. And therefore, just before it went wrong, it must have been a tiny circle. And I'm done. So the whole thing of Grayson had a completely different proof where he really followed the curve and he counted how many turning points there are and so on that had a beautiful way of analyzing that. It was a fantastic paper. But you can't do it this way. You don't have to do all that. You want to be lazy. You just look at the singularity and say, this is the only singularity that's possible. Therefore, it must have unwinded. OK, that's for tomorrow.


It's up! Welcome back! Do you like to do a builder make it? So do we! And we have new videos each week, every Tuesday and Friday! Thank you for joining our session today! We thought we'd share a little more about us before we get started! We are small business owners, full-time crafters! We started with a YouTube channel about five years ago, we grew out of our home, and we grew into some warehouse space where we have production in the bag, and we offer DIY workshops in the front. You can also find us online at KNGmakeit.com where we offer laser cut craft kits, paints, and accessories. Today we're going to show you how to elevate your laser crafting game with some 3D door hangers. Step one, we're going to gather all of our supplies. We needed some quarter-inch MDF for this project. This is where we're going to cut all of our little pieces out of. If you're doing it on a desktop laser, you're probably going to need three or four boards. We're also going to use some of our Foxy Hughes outdoor craft paints. We're going to use some Varathane stain, Varathane's briar smoke stain. Then we're going to use some Starbond thick glue. This will get in all the little cracks, hold everything together. Step two, we're going to make all of our cuts.  We use late burn, but our files work in any of the major laser applications. Inside late burn, we're just going to import our image. We're going to use an SVG for this one. We do include an SVG, an AI version, a DXF version, a desktop version, or is in most of them. We also include a picture so that you know what you're putting together in the end. We're going to import our SVG. Let's arrange it on our board. Now I'm using a 100 watt laser, so for this one, well for all of our files, we use red for cut, blue for score, green for a deep score, and black is usually fill. So let's go ahead and set up parameters. I want 10 millimeters per second. This is a score. I want 200 millimeters per second and 15 percent power, and then for my cuts, I'm going to want 10 millimeters per second at 50 percent power. Like we said, these work in almost any of the applications. Here's Glowforge. We'll go ahead and import into Glowforge. Now for this one, I'm going to use the desktop version. That way, if you don't have a passthrough, you can still make this at 16 inches, except this one includes a puzzle piece. So here's the bottom puzzle piece of your backer, and here's the top puzzle piece of your backer, and then the banner will sit over top and cover that puzzle piece and give it some more strength. And again, here is X-Tool Creative Space. Let's import that same desktop one. This is if we're using our X-Tool P2. We don't want to scale. So again, here is our image. Here is the bottom. Here's the top. Inside X-Tool, I use 6 millimeter MDF. That's the settings I use for the cutting.  Thick draft board will cut through at a quarter inch. No problem. All right. Back to X-Tool. Let's go make our cut. Step three. Now we stain and paint. Like I said, we're going to use this Faritheen Briar Smoke. Did you even know you could stain MDF? You can, and it looks great. It gives it an extra whoa. Yeah, I don't know how I did that. I'm just using a simple paper towel, and I'm going to dip it in my stain, wipe it on, stain on, stain on. Yeah, that's exactly it. Wipe it on, wipe it off. We just don't want to leave it on there. It dries. If you leave it too thick, it dries kind of sticky. So if it doesn't look even, don't worry. It'll even out when you wipe it off. We don't have to do the middle here. If you see that I'm not staining the middle, that's because I'm going to put the center banner over this area, and you won't see it. Won't see it. And this way the glue will stick a lot better. So I'm just making it so I can kind of see the grain a little bit, getting all that excess stain on. Perfect. Put this out of the way. Now it's time to paint all of our top pieces. You'll see that we have score marks in some of our pieces. This is so that you can place the pieces that go on top of them with perfect placement. We're going to paint right over them. Don't worry about it. You'll barely be able to see them in the end, and that's really all you need, and that's really all you want. We're going to be using again our Foxy Hughes Outdoor Craft Paint. So this paint has UV properties, weather resistant properties. It's perfect for these outdoor signs. I'm just going to use this little FIFO bottle, scored a little at the top of our plate. You can see here it's just at the top of the plate. And that's so when I use our little 4-inch foam roller here, you see we reuse our rollers. It's fall. We're going to use this orange a thousand times this fall. Easy. So we're going to put our little handle in here, push it on, and then I can grab the paint from the side of my little, what would you call that? Paint blob. My little paint blob right at the top. Now you don't want too much paint on this handle. I'm going to show it a close-up on the camera. This is too much paint. It's going to end up oozing over the sides of your pieces, and you don't want that. These look the best when you can kind of keep the sides clean and dark. Ooze free.  I've painted hundreds of these sides. I still can't paint it perfect. So we have a little trick for you at the end. I'll show you how to clean them up. All right, like Garrett said, we're just going to paint right over our little piece here, right over our score marks. You can't see it for a moment there, but as it dries you'll be able to see the image so you'll know exactly where to put the word fall. Now we'll set that aside and we're going to give that a second coat in just a minute. Now we're going to do the bottom half here. We're doing this in a light brown. Thank you. Yes, thanks. I had a second roller, but I think it handled it. I'll be your lovely assistant. Thanks. Again, I'm just going to put the paint on the edge of my plate. Roll it. Just show them what it looks like when you have enough paint in there. Yeah, you can still see the spunky. So here you can still see the foam sponge. You can still see it's porous and doesn't have too much paint on it. And I'm just going to roll right over the top, right over all these little whole pieces. It doesn't ooze down in there because I only have a little bit of paint on my roller. And a little bit of pressure. You're not pushing too hard. True. Very true. Just trying to paint the top. Alright, so we got that one done. Now for our smaller pieces here you can see we have some tape. When these come off of the laser we add this little Peters tape and we make ourselves a little tape pallet. The painters tape getting them off the lasers keeps all the pieces in the frame. We package these in these little frames so that they're easy to ship. And they're easier to get off the laser that way. And it kind of keeps everything together and protected during its travels either to their paint room or to someone's house. Even to the front to the workshop. Yeah, or to the workshop. Alright, so I'm just going to lift up our frame. All our little pieces come out. I'll make sure you see this little triangle right here. Alright, so let's do the word fall. Put that here. We're going to do this in antique white. I could feel this already has. Because I'm reusing the rollers, they already have paint in them. So sometimes you don't even have to add any more paint under your roller. Again, you're trying to keep those clean edges. So the less paint you have on your roller, you want just enough paint. It's hard to describe. I don't want too much. I don't want to ooze it out. I don't want too little so that I have to push down really hard. Last thing I wanted to show you is that you can mix paints. So for these other two leaves, so I don't have to add or find another fall color. I'm just going to mix a little bit of the red and a little bit of the orange. And I'm just going to dab on here and hopefully give it like a little fall leaf color. See, pretty cool, right? I'm going to dab the sponge in the yellow. And I'm going to give this one kind of a yellowy orange backer. All right, now I have four different fall leaves. And then we could just kind of put them together. You're ready for our super secret tip to how to keep these things looking sharp. Keep these edges looking black. It's a black, sharpie paint pen.  You know, you might not be as skilled with the roller as Kim. So you just take it, run it down the side. And it's not about getting up to the edge. It's about cleaning up any drips. It gives that visual appearance of a straight edge. Yeah, all you want to do is make it look like a nice straight line. And then your eyes going to glance over it. And it's not going to look like it's over the edge at all. Yeah. But I'm so good, you're having a hard time finding edges to clean. Now, usually you don't have to clean the bottom edge because it'll be hidden by the bottom layer. But I'm just gonna go ahead and just demonstrate. That's it. That's it. I think everything else is clean, right? Yeah. You did a great job. Thanks. Very little cleanup. I feel like you're looking. You're trying to find something to clean it up. What is that? What is it? Oh, that's gonna cost you, babe. Good to see you. Not all heroes wear capes. Moving on with that star bond thick. And we're gonna work our way out of the sign and then up from the backer. And we're just gonna use a little bit of glue. Kim likes to use the dot method.   The thing with the star bond is it adheres so well, the two pieces of MDF, like they are not coming apart. But it does dry with a little bit of haze around it. So if you've got a dark backer like this, you do not want to put too much glue and you don't want to ooze it out from underneath or around the sides. So you want just kind of a medium amount of glue, medium amount of paint and a medium amount of glue. And I can still see my score marks. So lining this up is super easy. There's also two ways that we do our glue in. You can glue the piece or you can glue the backer and then put the piece in. If it's got score marks, it makes it a lot easier to know where to put the glue. This is Garrett and I, we go round and round. I'm like, oh it's easier to glue the piece because then I'll know, I won't, I'll know exactly where it goes because it's only a piece itself. And Garrett said you can glue inside the score marks. Yeah, glue inside the score marks. That's how I like to do it. And here you can see we're gluing these bottom pieces or this bottom layer again from the bottom to the top. And that way everything stacks one on top of the other and you don't, less than learned, you don't want to put your pieces at the top, put your pieces at the bottom, and then try and squeeze something in the middle because inevitably it starts drying and it's too close together and you can't get it in there. There's no moving it. You got like 30 seconds to move it around and get it in place. And then after about 30 seconds it is there for life. That is where it lives now. Yeah here you can see Garrett's just doing it right on the backer. And again these score marks allow that perfect placement so nothing ends up wonky. Your L's aren't laying backwards or forwards. Things are trying to come with me in pieces. That's right. So we're not going to put fall on yet because we don't have this back layer of acorns on. Working from the leaves and acorns on. Working from the backer up.  Yeah. To me. Lessons learned for sure. It's definitely an embarrassing amount of times. Do you want to do these? Like then you do like to do special wires. Like cock-eyed off-center. Well no you see line the bottom of the leaf up with the bottom of the leaf there and line those up. Like that. Really just gonna add ribbon slots that we have and then a bow on top. We have tried every method for attaching a hanger to the door signs. We glued them. We glued it. We nailed it. Screwed it. Screwed it. We could not get anything that would look nice that would hold. So we found these ribbon slots. Now this ribbon slot you just poke it through. Tie a knot on the back side. You're gonna let me get closer to it. No. Forget it. I don't usually do it standing up but for the video I'm gonna do it standing up. Now I'm just gonna tie a little knot in the back and this is what's gonna actually hold it. It's never coming off. It's the best method and it's the simplest method. No extra gluing, screwing, nailing, nothing. And if you want to swap it or steal that ribbon for something else you just untie the little knot in the back or clip it. There we go. Now we got ourselves a little hanger. And then this I pre-made our bow. This is the world's easiest bow.  We have a video tutorial on our YouTube channel. Show you exactly how we make this. The easiest bow. And I, which side is it going? This side. Left or right? Well right because of those leaves. Oh yeah. Alright I guess from that view it's left right. Left right. Alright I'm using these cute little zip ties. They also make it super easy. And my little pair of scissors right here. I'm gonna snip off the excess zip tie. And there you have it. So we're all in at about $16 for this thing and they go for about $55 in our neck of the woods. That's right. If you're using your laser as a side hustle this is a great item to make with your laser and then be able to sell at craft shows or social media online. Let's say on Facebook to your family and friends. And it's a nice little profit in each one of them. Yeah and it actually gets easier if you start to batch them out. If you can do multiples at a time. That way you're doing a multiples of backers and then painting all the little top pieces and gluing everything together at one time do two to six. It's even more efficient. More efficient. You'll just pump right through them. Well thank you again for joining our session today. We do appreciate you joining us. If you're looking for us you can always find us over at K&GMaking.com. You can find it at Kim and Garrett making it on our YouTube channel. Or we have a huge Patreon following with a great group of folks. So if you're learning and new to laser crafting or you're interested in doing more door hangers we have a nice community and a Facebook page where folks go on and share what they're making, share their tips and tricks. It's a great community.


This top-deckers is striking a moving ball. Start with the ball at your feet. Take a touch towards goal and shoot before the ball stops. Approach the ball slightly side-on. Place your standing foot next to the ball with your toes pointing towards the goal. Strike the centre of the ball with the laces of your kicking foot. Keep your head over the ball and follow through with your striking leg. Have you got top-deckers? Go and practice and try the task for yourself.


Mujlai opens up about her rap career Mujlai has opened up on her rap career and how she has been an inspiration for other upcoming female rappers. The star has been able to dominate her talent in the entertainment industry from being a TV personality to a female rapper whose career is on the rise. In 2018, Mujlai won an award for Best Female in Rap at the SA Hip Hop Awards and has since consistently been nominated in that category. Many of her fans are proud of how far she has come. I haven't released a project that hasn't got me nominated in the Best Female category at the SA Hip Hop Awards, so the fact that I was able to keep the consistency despite what was a hectic year as a musician was very exciting, she told a local publication.  It was the first project that I worked on in my house as an independent artist, so the fact that it was well received means that we're growing and the numbers are getting higher.


So currently our equipment is tracked manually, the only way that we can find where a piece of equipment is is by actually going out and finding the piece of equipment. We scour the flight line and random hangers looking for our equipment that could have been taken by other units. Currently our flight is spending 60 to 100 man hours a week tracking down equipment, whether it be for inspection wash or maintenance. While manual tracking may work in smaller units, here at Anderson we have the largest WRM fleet in pack half and have flown more Solaris last year than Red Flag and Red Flag Alaska combined. GPS locator, we would just be able to ping and then instantly know where this piece of equipment is. So not only would GPS locators trackers help with finding the equipment for the aerospace ground equipment flight. Here at Anderson we have a wash contract where it's contracted out to complete all the equipment washes because of the corrosive environment. Now in that contract it's broke down where we have to spend extra money for them to pay a driver and track down this equipment so that they can bring in for these washes every 90 days. Now if this wash crew were able to use this locator they'll be able to instantly find each piece of equipment that's due and there would be no delay bringing these pieces in for the wash that's required. The GPS trackers that we're looking to install are non-invasive installation. They have an 8 to 10 year lifespan and they have been tested for the corrosive environment here at Anderson. Once we've had time to utilize and test this GPS equipment we will be able to share what we've learned across the base and other units would be able to utilize these same locators whether that be vehicle maintenance, tracking vehicles, or CRG tracking assets and getting real-time locations.


Hello and welcome to the session. Today I'll help you with the following question. The question says, draw a map of your school compound using proper scales and symbols for various features like playground, main building, garden, etc. As you can see we have drawn the map of the school compound. This is the main gate of the school area. This is the cycle stand. This is the main building. These are the swings. This is the basket ball court. And this is the playground. So hope you enjoyed the session. Have a good day.


Everybody forced it ripping it's two boxes of the 2022 leaf multi-sport jerseys here for Henry KC. Thanks, Henry K and good luck This is a jackham on the first one here Hall of Fame 88 inscribed jackham. There's the inscription right there Nice No Steelers piece there JSA authenticated Yeah, one more As Walt Fraser Nick's legend right there Walt Fraser, it's got a Beckett authentication there Sticker as well You go nice Had hams you're hoping for eggs on the next one because you had ham on the first one Well Fraser jackham Walt Fraser. There you go Enrique. Thank you again. Get those right out to you


So, I'd like to go ahead and get started, just to be mindful of everybody's time. So welcome everyone to 100 years of exile of Romanov's search for her father's Russia. This is a virtual talk with Tanya Romanov, and my name is Taryn Edwards, and I am one of the librarians here at the Mechanics Institute of San Francisco. And I have to say, one of the things that we love to do is highlight our mechanics members, creative endeavors, and this is certainly one of them. For those of you who are unfamiliar with the Mechanics Institute, we are an independent membership organization that houses a wonderful library, the oldest in fact designed to serve the general public in California. You don't have to be a mechanic in order to get something out of your membership here. We are also a cultural event center and a world-renowned chess club that is the oldest in the nation. Right now, due to the shelter in place, almost all of our activities are virtual, but I encourage you to consider becoming a member with us. It's only $120 a year, and with that you help support our contribution to the literary and cultural world of the San Francisco Bay Area. Tanya Romanov is an award-winning travel photographer and the author of three books, Mother Tongue, A Saga of Three Generations of Balkan Women, Never a Stranger, which is a travel story collection and the book she will talk about today, which is One Hundred Years of Exile. Tanya's work has also been featured in multiple travel anthologies and translated into Serbo-Croatian and Russian, and when she is not on the road, she splits her time between San Francisco and Sonoma County. Tanya is an active member of the Mechanics Institute's Writers' Community, and her books are available via Alexander Book Company and at your local bookstore. Now, the way the event tonight is going to work is Tanya's going to speak for a while, and then later we will take questions. So I encourage you to put your questions in the chat space, and when she is done presenting her book, we will go through and attempt to answer all of your questions. All right, so why don't we go ahead and talk to Tanya? Thank you. Can you all hear me? Yes. Well, thank you, Taryn, and thank you to the Mechanics Institute, one of my favorite organizations in San Francisco, with my favorite stairwell. An incredibly beautiful place and wonderful events. So I'm very thrilled and honored to be here. It's really strange to launch a book in the middle of COVID, but every once in a while, you get a benefit you don't expect. And about five minutes ago, a face appeared on the screen in front of me. My best friend from when I was six years old. I did get her to laugh, and she lives in Washington, DC, and her name is Tanya, too. And I haven't seen her face in probably 50 years. So COVID has brought us together. And so thank you to all of you. I will talk for a while, and then you can post questions that Taryn will coordinate at the end. 100 Years of Exile is a book that I put out. And I think it was a book that had to be written. I don't think I knew that until I finished writing it. And I'm really nervous as heck because nobody's ever asked me to talk about my writing process. And so I worked hard on trying to share some of this with you. The book came about, I had published the book about my mother. Three generations of Balkan women. I grew up in a refugee camp. My mother was Croatian. My father was Russian. We came to America when I was quite young. And so I had written this book. And I grew up in a community that wanted me to be Russian because we were a bunch of Russian refugees. Everybody spoke Russian. We went to Russian school. I was supposed to marry a Russian. I was supposed to be in Russian. And in fact, my father was a refugee twice in his life. That's what this book is about. And by the time he got to America, he had been in a refugee camp for five years as an adult, not knowing if he could go anywhere. And that was the second time after he'd been very successful in Yugoslavia. And so I think he just never believed that this was for good, that we wouldn't be evicted again at some point in time. And so I was supposed to fit into their concept of the world. And I resisted that very much. And so I released this book about my mother. I am doing a lot of launches in person. I married an American. I have American stepchildren. And I was doing a launch for my grandkids in Colorado. And my stepdaughter was introducing me to everybody that she invited. And I suddenly figured out that this woman that she just told me I could speak Russian with was Marina Romanova, Marina Romanoff, who was and is the closest living relative of Tsar Nicholas II.  You know, it was like, really, this person is real. But yes, she was. The amazing thing, though, was not that I met Marina, but I didn't meet her in San Francisco with a bunch of Russians. I met her in Colorado in a small town in the mountains because her daughter and my daughter were very good friends. So it was sort of like the launch of a very strange set of circumstances. And they did continue because this was in May of 2018. Within a few weeks, I received an erroneous email, aimed at someone called Atsom Amacheyev. Now I write as Tanya Romanoff because I was told that if you're going to publish a book and you have a name that people can't spell, pronounce or remember, forget it. So Amacheyev didn't fly. But my grandmother's maiden name was Romanoff. So don't mind the way it was. But I got this email that was, can you hear me? Now you're muted, Taryn. OK, so I don't know where I got muted, but I'm just going to continue with the story. So I got this erroneous email to a man called Atsom Amacheyev. Amacheyev is such a rare name that there are no Amacheyevs outside of Russia, who are not my immediate family. There were three brothers who left. That's it. There are no others. So I ended up connecting with Atsom Amacheyev. So now we're talking about June of 2018. And within 10 days of my hearing for the first time from Atsom Amacheyev, my entire Amacheyev family in San Francisco had agreed to go to Russia and meet with his family. So this book was starting to take shape. And I started writing about my father's family. I just had to do it. And what I decided to do was write the history part first. And then when I went to Russia, see what would have happened, in a sense, to us if we had stayed. That was really the concept that I had. And so the other amazing coincidence that happened, so there were two already, the final one was that today is November 10th, 2020. On November 13th, 1920, my father's family, my grandparents and their sons and daughter, fled Russia for the final time. So we're talking about 100 years to the day. And so if you look at my book, it says publication date November 13th, 2020. And so all these things conspired to make sure I was writing this book. So we agreed we would go to Russia and I started writing. And what I did was the deep dive into the history of my grandmother's family. I started in 1910. And I just started writing. Now, people say, how do you write? You know, do you do it at a certain time? Do you do it at a certain place? I am not a very disciplined person. And I actually write by talking to my telephone. And I'm a crazy walker. I walk for hours every day and I walk and talk. And that is my primary writing vehicle. The drafts of everything I do happen while I'm walking and talking. So that's what happens. And what's interesting is there's a wonderful writing community in the Bay Area of travel writers. And a man called Don George organizes it. There's an annual conference, a book passage. And one of the things that Don George insists on, if you work with him and you're doing travel writing, is travel writing has to be honest. It has to be true. You have to verify all the facts. And so if you're not writing while you're there, you won't remember the facts. So it turns out that that lesson was really important to me in terms of how I write and when I write. So I was writing this book. I started with my grandmother. And I wrote about her and about her youth. Now, I grew up on stories with my grandmother. My grandmother was really mean. But I spent a lot of time with her. Because when I came to America, I was young enough that I was in kindergarten. And my afternoons were with my grandmother in the park. And she told me stories. And so she told me stories about how she didn't want to marry the guy she was supposed to marry. She picked someone else.  We're talking about Russia in 1910. And she decides who she's going to marry. It wasn't exactly normal behavior. So I started writing on that. And I got to the point of understanding it with my first book and with this one that what I write is called creative nonfiction. Or at least that's what I call it. So I check all my facts very carefully. I check my memory very carefully for all the facts. But I also want to write about the people I'm writing about as human beings. So they have conversations. Clearly, I was not part of the conversation of my grandmother and her father when she told him she wasn't marrying Mikhail. But it's much more fun if you have a story with human beings in it. So that's my creative nonfiction part. And it's interesting. I've been working with a group of writers. And my friend, Mike Bernhardt, came to me and said, you know, I want to write like that. I have this history stuff and it's getting boring. And so he went away and he tried it and he wrote me shortly. He said, writing history by imagining scenes is freeing. And I thought, yeah, that's absolutely what it is. It's just freeing. You get to do, you get to write fiction almost in the middle of a nonfiction story. So the other thing is I also do a huge amount of fact checking and research. And the thing that is interesting about that and that I would recommend very highly to anybody who's trying to do something like this is do not assume you know what you're going to find. I know that sounds confusing. But studying history can teach you some of the most amazing things. So I'm writing this. I started writing this book and I started doing the research. I obviously had to research Marina's parents because I was going to weave my family's story, The Russians Who Left Russia, with Marina's story, The Royal Romanovs, and the Emma Chayevs who stayed in Russia. And so I was doing research on Marina's story. And Marina's grandmother was Princess Xenia, Tsar Nicholas's sister. What do I learn about her? She wouldn't marry the guy she was supposed to marry. It was like, oh my god, really? How could history fit in with my story so well? So we had these two women in 1910 who follow the same path. It was such a beautiful thing to learn. And I find things like that a lot. I'll give you another example. My family ended up. So my Russian family, Daria, that's my grandmother, and Ivan, my grandfather, were Don Cossacks. And they lived in an area of Russia that's in the southwest of Russia near Ukraine. And now I lost my train of thought of what I was going to. Oh, yeah. And during the Revolution, they were the so-called white army, the Cossacks were. And the communists were the Red Army. And that's what the war, the ultimate civil war, was about. And so they had to flee. And they ended up in Crimea. And in Crimea, they spent a long time, months, waiting to find out what would happen. And during that time, the civil war went on. And when it was over, they had to leave immediately. So the evacuation was a three-day process. They were in a small city called Yefvetoria. Most of the people were in Sevastopol. And the flight was really intense because the land that links Crimea to the mainland froze suddenly, typical Russian story, a quick early freeze. And suddenly the Red military could come in. And so they had to flee. And so I grew up on stories of this. And my grandmother is telling me how the ship was late. And then my cousin, Helen, and I were really interested in all of this. When Helen's father, so that's my uncle, my father's older brother, was in his 70s, she sat down with him with one of those old movie cameras. And she created a movie of his memories of the flight. He was a child. He was about 10 years old at the time. And so he's telling this dramatic story of this flight and they have to leave. Their father is not there because he's out trading for food with the Taktars. And it's the ships have all left and he's not back. And finally, all of a sudden, this last ship shows up. Well, you hear a story like that. And you say, yeah, right, BS. I mean, it's dramatic.  This couldn't possibly be true. So again, I go do the research. And this time I'm on the internet because it's in the last year or two. And I find out that there is a ship. It's called the Trashikty 412. It was a ship that got lost in the storm, came in at the last minute. And my family is on the ship's manifest. So it's like the research matched the reality. And it just convinces me that real life is actually just as interesting as fiction, if not more or so. And it sort of got me committed. So my book is meticulously researched. I've also learned that no matter how meticulously researched, and my mother's book was the perfect one to learn this with, when you think about the Balkans, Serbia, Croatia, do you imagine anybody agrees on that history? No, so you think you're writing the truth, but you haven't got a clue. But it's still worth pursuing it and getting as close as you can to your perspective of the truth. So that's what my book was about. So the concept of the story was I wrote from 1910 until the present time and finished all of that before I went to Russia. Because I wanted to document it so that I could understand what I was looking for in Russia. And then our whole family went to Russia. And this is what's so interesting. So, the young man whose email was misdirected to me, because he misspelled the last name. Of course, you know, Russians use the Cyrillic alphabet. So you can spell a Machaev, lots of different ways. He put a Y in the middle of his, and the economist left out the Y. So this is a complete stranger. He's 30 years old. I'm 70, my brother's my age. My cousin's considerably younger, but she's got two children who are 30 and 20. And that's the group that's going to Russia. So we have two 30-year-olds. We have people in the middle, and we have us retirees, and we're all going together. And at some's family whom we've never met, we don't know from Adam, Igor and Ira, invite us to stay at their dacha, their summer house when we get to Russia.  We head off to Russia. Now, I don't know how many, well, a friend of mine told me, don't publish your book right now because relationships between Russia and America are not that great. You might want to wait. And I said, I could wait till I die. Relationships between Russia and America have not been good in my lifetime. I don't know that they're heading for any better. So, you know, I think I should publish it. But now we're visiting this Russian family and we want to go to Crimea. The State Department in 2019 says, do not go to Crimea, you will become a, you will get abducted. It's very dangerous. They went on and on. We were still determined to go. Now, Atsom's father, Igor, is an engineer and he works for the Russian government. So now here's somebody showing up at his house who's an author who documents every single thing that happens. And now I'm going to write about Russia. And what you have to think about is, how am I going to do that in a way that doesn't put him in a bad position? Most of my friends in America write about Russia in a way that would make you think it was hell. You know, from the leadership on down.  And that's the challenge. So that was one of my challenges. I'm going to read a little bit about what we did because what I realized was that if I just wrote about what we really did, it would work because we didn't discuss politics. So I'm going to read to you a paragraph about that. Our countries spend a lot of energy trying to make us distrust each other. Much of it is deeply embedded in all of us. And even our mutual skepticism about the media or governments in general could not make us oblivious to it. But it didn't seem to matter. We do not spend much time debating the cosmic divide. We don't discuss Putin or Trump. When we talk about Ukraine, it is in the context of Igor's mother and sister who live there. When we discuss Crimea, it is about the place where I've learned to swim as a child. The final place where my parents sheltered when fleeing Russia and where my friend Marina Romanova's grandparents had their summer palaces and walked between them. We just talk. As you might with a neighbor who stops by for coffee or as I do when I walk with my friends in the morning or meet them in the neighborhood bar for happy hour, we just talk. And so that's what we did. We just talked and whatever came came. The political discussions were very minimal. And I do mention one of them. And that is my brother and I were in a taxi in Moscow. And I say, I wondered for much of the trip how people in Russia view the United States. My Amachaya hosts are so polite and friendly that I hear almost no criticisms. We also hear little of local politics. One Moscow taxi driver casually says, in your country, if very few voted for a president, could he still win? Sasha and I just look at each other. For sure our current president got elected without a majority of the population voting for him, but we're not going there. Someone else tells us Putin probably won't run again when his term expires. That was wrong, but that's what they thought then. Who might succeed him? I ask, surprised. Ani, my father's famous they haven't decided yet, is the reply. But most people avoid politics as much as they had in my previous visit 40 years earlier. So we find ourselves in Russia talking to human beings about normal non-political things. In fact, it was quite relaxing not to discuss Putin for a month. I'm sorry, not to discuss Trump for a month. So it was really wonderful. And the only time I got into a truly awkward conversation about politics was a dinner I went to visiting my cousin's friends from San Francisco who had moved to Russia. And what was interesting was it was the first time somebody started throwing out sort of really nasty comments about America. And it felt so awkward at first. And in fact, I got into pretty much of a debate. And when I first wrote it up, it sounded dreadful. It sounded like I was just dumping on this person. And then I realized, okay, if you wanna publish something it has to be something that can be read by the person you're writing about without him wanting to come shoot you. So let's think about this again. There are two perspectives to everything. So let me try to understand his perspective. And when I rewrote it, I realized that the reason he was willing to share with me a point of view that the Russians really wouldn't was because he was an American. He wasn't a Russian dumping on America. He was an American with a different point of view than I had. So we could have been debating Trump versus Obama but we were just debating America versus Russia. And I rewrote it and it's completely acceptable. So I find that you should write everything that you feel like documenting. And then you can decide whether to take it out, keep it in, modify it, but you should write it first because controversial issues are the ones that in the end make the story. So that's a rule that I've now learned and I really stick to it. That's gonna be a little bit more of what I'm gonna talk about here. Another thing that happens to me anyway is I learned secrets. And then you say, oh my God, what am I gonna do with these secrets? What if they could hurt someone? I'll give you one from my previous book first because this one is an incredible story. So my mother had six sisters and my grandmother during World War I was sent to a camp in Hungary because her husband worked right next to the Italian border in Croatia in a shipyard. And so that was Austro-Hungary at the time. Italy comes in on the other side of World War I. They're like 10 miles apart. So they send the women away. My grandmother suffered during that time and on her way back, she was abducted. She comes back and she has a child in January of 1919. And when I'm writing the book, and this is one of my favorite ants, when I'm writing the book, a cousin of mine tells me, well, you know, she couldn't have been legitimate. This is another cousin.  How could she have a child in 19 that was legitimate? And I thought, oh my God, what do I do with this? So I sort of danced around it. Those of you who have read that book can read it again and see if you see an infant or not. But I sort of did a delicate dance around the whole thing. And then much later, after the book was out, I talked to one of my cousins, a daughter of my aunt, and we did a DNA test and we actually found out she is my cousin. Her mother was completely illegitimate. And then I went to this small town medallion in Issa where my mother was born and found out from a historian who was detailing the whole thing that the men went on strike unless the Austro-Hungarians brought back their wives. And so their wives were returned in the first quarter of the year. So it's like, if you just keep pushing on it, you learn incredible things. Well, I learned all kinds of things and I'm gonna read you an example of a little bit. But what's interesting is I'm going to read something and did I see that there was somebody called Faina? She's gone. There was somebody called Faina on this call. And that was the name of Igor's aunt who stayed in the village where he was born. And I couldn't believe it because it's such an unusual name that somebody was on this call for a while. Anyway, I was, the story about Igor's family and Uhtam's family, is it turned out that his grandfather was born three miles from where my grandparents were born. So we figured we have to be related. And while we visited the village where my father was born, which is again, as I say, three miles from the collective farm where his grandfather was born, Igor found the grave of his mother and grandmother, of his grandmother, sorry, like a hundred yards from the house where my father was born. So, we now had even more reason to believe that we were probably related. And so we had this amazing experience. We got very close, we traveled together, we learned some amazing things. And now it's the next day and I'm walking in a small town where we stayed as close to the village as we could get. And I meet a woman just out walking in the streets one morning. And she's the one who asks me what I'm doing there. And I tell her that I'm visiting a family. She actually asks me the name. And I said, I'm a Chayev. And she said, oh, I know the Amachayevs, they lived in Sifima lot. She said, that's where I came from. She said, I knew Faina Amachayev. And I said, nobody comes up with a name like that by accident, so she really did. So she keeps talking and she starts telling me this incredible story. And I actually have my phone with me as always and I turn on the record button. And so I record the whole conversation that we have. So then I come back and I'm now talking to Egood, okay? The Egood is finest nephew. And so when I get back, Egood is packing up the car. I regale him with my story and he looks at me a bit skeptical.  He laughs, humoring me. I mean, we'll talk about a village of, you know, less than a thousand people 50 miles away. Yes, I did. She was a good friend of your aunt Faina. Now Egood is alert. This is not a common name unlike Eda or Anna. I tell him about our conversation. And then I realize I have to let him listen to the recording. When he hears the part about the tragedy, he just nods. He listens to the end, then stares at me and shakes his head. Another Tonya adventure. I can imagine him telling his wife. She mentioned something about a tragedy. I say to him a bit later about your grandmother. A distant look comes into Egood's eyes. Yes, he says. She died when I was young. I think about her gravestone and realize Egood had been around seven years old when she passed. I remember the rustic wooden cross in the ground nearby which he simply said was also a family members. I think about the fact that he had never before been to the cemetery that he hadn't known his great-grandmother was also buried there until he heard this recording. And that Artsum hadn't even known that his father had been to Serpimola as a child. But I also know that if there is a mystery here, it will remain one. No Tolstoyan epic will be shared. There would be no Maudland storytelling followed by a drunken brawl. But there is something almost surreal about my meeting his Aunt Fina's friend on a back street of Uruppinsk, the official nowhere of Russia. Our experience at the cemetery takes on a new perspective as Egood and I look at each other and we both understand that we are on an unusual journey, one in which the past is reaching out and communicating with us in some unexpected ways. So I talk about the fact that there's a mystery and I can't share what it is. So sometimes you just go with part of the story because it was incredible to me that I met someone who knew him but I could not share the story because Egood is a very restrained person who doesn't share privacy. And the reference I made was to someone quite the opposite who shares everything. And so it's what I'm trying to point out is sometimes you share things and sometimes you don't. When I was walking one morning, I met a drunk man and I'm gonna read you about him because this was the opposite experience of everybody else in Russia. So his name is Zhenya. We're walking along a street in a remote part of Russia near Moscow. We move quickly beyond religion and he asks my age. I round up to 70 and a startled look greets my answer followed by a smile. Again, any expectation I might have of a response is confounded. You look so good I could jump in bed with you. It takes me a bit to comprehend what he means as I have not heard flirtation in Russian since I was a young girl but it was not said in a lewd or unkind way and I understand it as a compliment. He apologizes not for any comments but for using the familiar you with me. He realizes I'm old enough to be his mother and he switches to the formal you and tells me again it is all meant as a deep compliment. He moves quickly beyond the seduction that has briefly distracted him continuing a wide raging conversation nearly solo and requiring little encouragement which satisfies this writer's wildest dreams. He tells me about how his 23 year old grandmother had fought and died in the battle of Minsk one of Russia's largest victories in World War II. His other grandfather who had two daughters suffered a radically different fate. He earns some money and bought a bicycle you can he says then points to his own bicycle. They found it accused him of earning money illegally and he was taken away by the NKVD. I remember this with Stalin's secret police. Yevgeny then tells me his grandfather wrote his two daughters once from the Gulag saying I have lost my teeth, I cannot eat, I will soon die. As he talks I realized that his grandparents are the same generation as my father. He could have been describing the fate of my family had they not fled the country in 1920. These were the stories I had come here searching for. Not understanding yet that I would hear none of them from people who not affected as Yevgeny was by too much alcohol always retaining the privacy they sow treasure. Tearing up he tells me no one ever heard from his grandfather again. I touch his shoulder and tell him he is a good man. But as he looks at me I see the responsibilities of being the bad guy in this world fall back on his shoulders and now we are both near tears. So I meet Yevgeny out in the middle of nowhere. He's drunk as a skunk. He's fallen off his bicycle. He's flirting with me, he's telling me stories. And I also learned later that he is a respected lawyer in Moscow in another life. And now I'm thinking what do I do with this story? And I finally decided that first of all, I loved Yevgeny. I mean, he was wonderful. And secondly, I had to share the story because he was the only one who told me some of these things that have disappeared in Russia because you couldn't talk about him. So I wrote it and then I changed his name. So that was what I finally decided was what I would do. And someday I hope to be in Moscow and talk to him about it. But I feel good about what I did. So that's kind of what I have to say. I will say one other thing that I think is important in writing, and that is you have to leave some mysteries to the very end of the book. You want people to have something to keep reading the book for. And so, for example, we obviously thought we were related to the Amachayevs. I actually snuck a 23andMe DNA kit into the country and snuck their spit out of the country. But you don't find out until the very end what we learned about that family. So that's what I'm gonna close this on and give you guys a chance to ask questions and steer me whichever way you wanna go. Well, thank you. That was quite a journey. So if you have questions for Tanya, let's put them in the chat space. To start things off, we have a question from Matthew who wonders if you could say more about how the book is about the search for your father's Russia, his experiences and how the writing of it, the researching of it, how that really affected your relationship and your understanding of your dad. Well, it's really interesting. I don't think I knew until I finished the book what the book was about. Now, all of my friends in my writing groups knew what the book was about, but I didn't. I think I had to write it. And the book is really a work of atonement. It's my apology for not understanding my father.  And it was really in the writing of the book that I understood that. And it is interesting because one of the things that I forgot to mention and I will now is that I think one of the most important things a writer has to have is a community. And my writing groups are just so wonderful and so supportive. I'm looking at some of them right now. It's an incredible gift. And it's also very hard.  And then you take it to your writing group and they rip it to shreds and you say, oh, I guess I have to keep working on it. But they kept pushing me on what was I writing about? And it really took till I was finished almost writing it. And there's a section in the book where I go into a church in Russia. And I am not a church person, but I love churches. I'm not a religion person. I take it back. I love churches. I think they're beautiful. And I go into this church and a woman, you know, one of those women who clean the churches tells me I can't take photographs. And another one, a wonderful one says to me, you can, if the Batsushka Bluggislavit, so if the priest gives you his blessing and I go in and I get the priest's blessing. And it's like all of a sudden when I finish writing about getting his blessing, I realized that this priest will always be with me. And the blessing he gave me was my forgiveness and I carry it with me. So it was like, I don't think I knew what the trip to Russia was for. I don't think I knew why I was writing the book. I just had to write the book and now I have such deep peace and joy about having done it. So it was well worth it. Thank you, Matthew for asking. And I think you've made it clear how important it is to have a community of writers to help you kind of see through what did it, see the point, what it is, are you really writing about? You need help doing that because it's easy to get into a tunnel and not be able to see the big picture. It is extremely easy and it's also extremely easy to just sort of write the surface and to keep poking at it is hard. And it's really wonderful to do it once you do it. It's not wonderful to do it, I take it back. It's wonderful to have done it. It's the experience. All right, George has a question. He's wondering, will there be an audio version? Oh my God, I can't believe you just asked that. Oh my God. I went for a walk because I was so stressed out about this call. I went for a walk with my friend, Ann and we just had to choose one of three segments to approve for the marketing of the audiobook. It's just being finished. Well, good way to go, George. I can't believe you asked that question. Sorry, I'm a bit giddy.  That's when you know you've, you're a success when you have the audio version coming out. Right, I did the audio book of my first book because my grandson wanted to listen to it instead of reading it. I thought, all right, I don't care how you get it. I'm going for it. So that's a wonderful question. The book, by the way, is also, I mean, this is again, now I'm starting to get to the part I'm excited about. The book is gonna be published in Russia. I can hardly believe that. And again, this morning I was approving the final edits of the Russian version of the book. So it's being published in Russian, in Russia. And I'm really excited about that. And again, you know, Crimea is right now celebrating the 100 year anniversary of the flight of the white Russians. It was hundreds of thousands of people left. It was the end of the civil war. And it's a momentous occasion. And in fact, there's an organization that's making a film about it, about the people who were there and my cousin Helen, who I now just saw. And I are gonna be in that video. And we were supposed to be there for it, but needless to say, COVID has us here. But we'll be doing some Zoom events in Russia and in Russian. There's a Russian bookstore that's gonna do one in San Francisco even. To me, that's incredible. That my father's book is gonna be in Russia 100 years after he left. That is wonderful. And you know, as a San Francisco has such a rich Russian heritage that's gotta be really validating for you. There is a question from a listener who grew up in Cold War, during the Cold War in Alaska. And of course, Russia at that time was considered the enemy. And she's just curious how your thoughts are on how that mood might have shifted these days. Well, you know, I think in those days, it was a lot harder to be a Russian. You know, when I was a kid, I've said to people, you know, I would go try to raise money for my girl scout troop. And people would say, a Russian girl scout troop? Why would we wanna give money to the communists? You know, it was just different. But on the other hand, it passed pretty quickly. I mean, that was in the 50s by the 60s. A lot of that had settled down. San Francisco always had, since I've been here, always had a lot of Russians. And so I think that that started shifting pretty quickly. I've had that question asked by other people. And in fact, there's a woman who's doing her PhD paper on it right now. And she's reading my book, and we're gonna be talking about that subject because she said her father had to change his name. I can't remember. I think it was from Boris to an American sounding name because he was having trouble finding work. But, you know, he's considerably older than I am. He would be a hundred right now. So I think that that Cold War period, it was really more the McCarthy period that made it very hard. I hated being a foreign, but that was different from, it was, they were choosing to be Russian and I didn't wanna do that. But it wasn't because, you know, I thought it was sort of bad to be Russian per se. I just didn't wanna be foreign. So you're fluent in Russian then? I am, I am, yep. I discovered in Russia that I'm still fluent in Russian. You grew up speaking Russian in the household? Yeah, so yeah, I had a very unusual household. So my mother was Croatian, okay. My father was Russian. He grew up mostly in Serbia. She grew up in Serbia and Croatia. So she marries this Russian, moves from Zagreb to Belgrade.  So that's three years after the wedding. My mother didn't speak Russian when she married my father. But she learned it very quickly. Well, they both were committed to making sure the kids knew their language. And in fact, Mother Tong, the book about my mother is about that issue of speaking with her. And that book's been published in Serbo-Croatian and it's called Ponoššamo, which means in our way, in our language. So when I learned how to speak, first of all, I didn't speak for the first two years of my life and everybody in the refugee camp thought it was all my mother's fault because they were all Russian and she insisted I speak Serbian with her. So when I finally started talking, I spoke Serbian with my mother and Russian with my father. And if in the middle of a sentence, I switched my eyes from one face to the other, I changed languages. And that was our lifetime process. I could not speak to her in any other language. I could not speak to him in anything but Russian. So it was a very strange, but I'm fluent in both languages. It was a real gift, a real gift. And you can tell I'm fluent in English reasonably. And I don't have an accent. And what we learned, my friend Tanya came to San Francisco probably about the same time I did. If you were under 10, you did not have an accent. And if you were over 10, you could work for the rest of your life, but you'd never get rid of it. I think it just kind of works that way. I don't know if other people have these experiences. Well, that's an interesting observation. Let's see, Barbara has a question. You discovered many things about your family as you did the research for both mother tongue and 100 years of exile. What's the next mystery that you wanna uncover? I don't know, I don't know. I am working on another book, but I'm not sure I'm gonna publish it. It's a book about walking across England shortly after my husband died. And it's actually why I started writing. So my husband died almost 10 years ago and we walked across England along the Thames Trail with two friends. And I started writing then and I wrote a book about, I thought I was walking across England to heal from my husband's death. And what I learned was when I wrote about walking across England, I started healing from my husband's death. But then what happened was he invaded the book. It started being about walking across England and it's now called Crossing Bridges because it ended up being about crossing bridges in my life and so I got stuck. But I decided I had to learn how to write. That's how I became a writer. So now I may finish that book. Who knows? Maybe I'll travel again someday and I'll write another travel story. I've still got lots of projects and I may just become a photographer again. No pressure, but it's a great premise. All right, Colleen has a question. She says, my Aunt Marina Casey and her son George Casey lived in San Francisco and Aunt Marina's father was also a white Cossack. And she and her family left Russia when Aunt Marina was two. She spoke both French and Russian fluently as does George, who's about your age. Was it common in the Romanov dynasty for people to speak both French and Russian? All right, now we get to what kind of Romanov I am. So I will tell you that when I met Marina, I said to her, I apologize for taking your name as my pen name. It was felt really strange. And she said, oh, Romanov. She said, in Russia, that's like Smith. It's the most common name in Russia. And so I said, oh, okay, I got it. Well, the difference between me and Marina was that my grandmother, Romanov, was a migrant worker. I mean, they were obviously serfs until not very long before she was born. And the royal Romannums, and there are many, many of them, were a different class of people. In Russia, that was what divided the classes. If you were upper class, you spoke French. There was no question. If you were a peasant, you spoke Russian. And so no, we didn't have any friends. I did speak French. I lived in France for two years, but that was not a family trait. Well, if we keep talking, we'll find out how many other languages you speak. Yeah, we will. A few more. Let's see, Elizabeth has a question. Listening as someone whose third or so generation in the United States, she feels on the cusp of that transition and she feels that it is a beautiful homage to how welcoming the country is, the United States. And she's curious about what your thoughts are on, and impressions are on, I guess, the welcoming nature of Americans. You know, obviously, I'll tell you what, I've had more time to think about that in the last four years. I'm sure we all have, you know? And to me, America is a country beyond, you know, any expectations anyone might have. We can say all the nasty things we want, we've done some horrible things, our leadership has done some horrible things, but this is still a country of opportunity. And I mean, I was clearly fortunate in when I came, but it's, I think it's really important to study your roots and in this country, more than any other, you can hold onto them as long as you want. You know, San Francisco, half the people speak Chinese now, they can keep their Chinese.  So I think we're a country that allows you to be who you are. And I'll tell you one thing, over the last months, people started saying, oh my God, I might just have to leave this country.  You know, I mean, there were so many things. And I always said, I'm never leaving this country. And I'm sorry, but we elected him and he's my president because I'm an American. And nothing will make me not an American. I mean, it's like, it's a gift. I talk about this somewhere, I think it's in the book that, so I'm a world traveler. I'm constantly somewhere. I've been to India every year for the last 10 years. I've been to Bangladesh, Myanmar, all over Europe, all over Africa, I love to travel. And here's a really funny thing. I meet Russian people all over, including on the streets of San Francisco, constantly.  You must have just left. I said, yeah, a hundred years ago, we left Russia. And every single time I say that, they say, umna zeleli, they did a smart thing. So these are people who live in Russia. Moscow is as fancy as Paris right now. You would not expect to hear somebody say, oh, you're lucky your parents came to the America today, but they still do. We are blessed to be living in this country and I'm gonna start crying if I keep talking about it. So I have had every opportunity in the world. It was worth living in a refugee camp. Glad I finally got my father's forgiveness. I will thank him forever for what he did. Well, that sounds like a, that's a great place to end, I think, I mean, that's amazing. Yeah, I got a bit. No, every now and then you just have to take stock and be thankful about whatever. I mean, I'm constantly thankful to be living in such a cosmopolitan place as the San Francisco Bay Area and cheers. Yeah, cheers, toast to all of you. Well, I wanna thank all of our guests today and I wanna thank you, Tanya, for a wonderful presentation and I'm so glad that I was able to host you. I know we talked about this. Maybe it was your first book or a mother talk. Yes. And then I don't know what happened, but here we are. So I'm really glad that things have come together and that, yeah, and just wanna thank you for sharing your experience. Well, thank you all of you. You know, I can see everybody. The first time I did one of these zooms, they had me stuck in just looking at myself. So thank you all. It's so wonderful to see your faces. Yes, thank you, Colleen, Casey's Clapings Bank. Hi, I don't even know you, Colleen, right? But thank you and Simana Beach. What a wonderful name. See, I wanna talk to you all, Lucy. I can't believe it. Lucy Han from Minnesota now, right? Are you in Colorado? Anyway. People from all over the country and the world. Hey, Rosenberg. I'm gonna go ahead and open it up so people can actually talk to you if you want that. People can turn your mics on if you want and say hi. Helen. Lena, Tanya, where's Tanya? If you want to, you can unmute yourself. Tanya, you can unmute yourself. Tanya and Lena say hello to each other. Hello. Hi. Hi. Oh my God, that's great. Hello, wonderful to see you. Thank you, everyone. Lena, can you say hello? Hello, hello, wonderful presentation. I enjoyed it. Thank you. That's my cousin. And she was in Russia with me and wonderful. Lucy, hi. And Ann, where are you from? San Francisco? San Francisco, yes. But I'm a second generation born here. My mother's family was Ukrainian. My father's family was Russian, which has created a certain amount of friction in the family, but anyway, here I am. There you go, you're gonna have to write your story. I couldn't write a story. It would be full of drama and intended brides who were spurned. And anyway, a lot of that story intended was reflected. Yeah, that's great. Hi, Lorraine. Hi, Tanya. And Kenny, hi. This is great. Thank you, Tanya. Thank you. Thank you. Thanks. That's great. Okay. Bye. What should I say to you? It's like being at a party. I'll say the Svidanya. Bye, the Svidanya. All right. Thank you, everyone. All right, and I'll see you around at our next virtual meeting. Yes. It's up there for real. Yes, someday. Bye, Frank. Bye. All right. Thank you, guys.