Okay, thank you for being here. Sorry about that, let's translate in the title. Anyway, this is about the Gardening Team. What's the Gardening Team? So the Gardening Team is an effort we started last year. We had a lighting thought in the academy and I said we really needed the team to focus on pinpointing the important bugs and finding people to fix them, taking a bit of care of review boards, because it's silly that we asked people to commit patches and then we never have a look at them, do some bugzilla work and also find a notification in the screen. Finding projects that needed some love and giving them some love. So what we did through the year, we did some work on K-recipes. K-recipes was a bit stuck forever in 2.0 Beta 3, so we just gave it a bit push and got 2.0 out and then we also got 2.1 out, picked some bugs, migrated some infrastructure from surfboards to KD, which is great because surfboards kind of died that week. And we also did basically the same for KGV. KGV was on 2.0 Beta X for a long time with lots of bugs fixed, but nobody was using them because it wasn't released. So we did the release fixing some extra bugs. We also, after that, we chose a project which was working on the documentation for the web that got a bit stuck and it's been in that stuck mode for like 3 months. So yeah, that's all good. We also did the bug of the month that was mostly done by Crystal Tech, which was basically going through all the pilot bugs we have and fixing the ones that are really important and affected lots of people. We did that for 4 months in a row. Two of them were fixed. One of them was, I think it is fixed, but I cannot be sure of it. And then again, we had another bug which we didn't fix and again, we got stuck on it, which I am so great. So summary, we fixed some nice things. We need to convince more people, like we got stuck because basically the people working on there was me, Jeremy Wiping, Scarlett, and then some other random people. We convinced some people, we did a Saturday back day and we got some people there, but we need more people for the effort and yeah, we did decide what to do with those projects that are stuck. We'll probably just skip them and find new ones because yeah, it's not good if we are stuck. And I have 30 seconds of questions. If not, you can find me. Maybe we have a second question. Do you have, like, a need when I see a beginning of two months to decide what's going to be done? Because Crystal, I want to join. I can't shout, I guess. So, no, we are not doing that at the moment. We were discussing those things in the mailing list, but we can as well do an IRC meeting. Yeah, whatever, just to get a kickstart.


Welcome to Toffy TV, Carl Angelotti has held his press conference ahead of tomorrow evening's game against Manchester United at Old Trafford. He revealed that Jordan Pickford will still be out with the injury that kept them sideline for the win against lead to Ellen Rhodes and Robin Olson will continue as the number one. He also revealed that when Jordan Pickford will be fit he will be back to being the number one but obviously you know if Robin Olson has another great game like he did at Leeds that will make it tougher for him and that's exactly what we want, we want Robin Olson putting pressure on Jordan Pickford and having to make Carl Angelotti make tough decisions so Robin Olson in goal for this one. He also revealed that Alan started training today which means that hopefully he'll be available for either the game against Spares in the week in the FA Cup or against Fulham next Sunday so that's really good news and the squad is getting back into some kind of shape now we had a decent squad against Leeds we should have a decent squad against Manchester United and that leaves lots of options available for the manager and the manager was talking about that he was asked about dropping Michael Keane and James Rodriguez for the gaming interleeds United and he said you know there will be rotation you want to stop players from getting injured with a lot of gaming because we've got a lot of gaming next few weeks you know got Manchester City as well just being added to the mix so for the next couple of weeks we've got a lot of big football matches which we want our players to be fresh for so he said there would be more rotation in the game tomorrow night so it's going to be interesting to see who he brings back with obviously Josh King available as well and he'll have had a couple of days training with the first team as well after making a cameo performance against Leeds United Fabian Delphys back Allen coming back soon he did say Gabamon was back in individual training didn't put a time on it this time i think he should just stop putting times on when Gabamon will be available but um yeah he said he did so it's it's good we've got a strong look strong looking squad just at the right time as well with players in form as well especially that back that back line you know the other night Ben Godfrey was superb Mina was superb Dean with another assist um you know i'm sure he'll find a way to bring someone like Keane back whether he moves Ben Godfrey back to left back or right back i don't know but Ben Godfrey was fantastic he's undropable really um and we'll have to we'll have to wait and see we'll have to wait and see what happens but really positive to have so many options now available um and once Allen gets back and is available to play that's going to bolster that midfield as well so really good news on the injury front um yeah i mean that's really all he had to say on on that matter uh one of the pleasure is on the bench at the moment as well as Bernard he missed out on a deadline day move to Al Nassar that apparently was due to the fact that he couldn't agree terms with the club so he stayed at Everton and will most likely be on the bench for the rest of the season as well and again he's he's a decent option coming off the bench so um yeah well we'll we'll see how much how much party has to play in the games now to the end of the season he is an option obviously Anthony Gordon went out on loan as well so he is an option to play the games to come on um and maybe try and get himself a move to a bigger club than Al Nassar in the summer after waiting to see the day you go this has been the Everton news daily uh tomorrows game obviously eight o'clock old trafford it's going to be tough but Evan have won their last four premier league away games so that's uh room there's room for optimism there even though Manulator 190 in their last game so there you go make sure you give this video a like subscribe if you haven't already if you want more great videos daily live videos join us over on page and i'll be live straight after the Manulator game my instant match reaction so make sure you check that out as well see you later you


So, I'm very pleased to have this opportunity to speak at this meeting, celebrating the life and work of Marcel Berger, Marcel had a big influence on me. I won't say more about it now because we're going to talk about it later in the afternoon. So, I want to talk about some work, its joint work with Aaron Naver and Wenshui Zhang, and it concerns the structure of non-collapsed Gromov House-Dorff Limit Spaces with Richie Bounded below, so too as the non-collapsing condition. So I want to discuss some results on the structure of such spaces. Various things were known and I want to talk about the new stuff. So HK denotes House-Dorff Measure, K-Dimensional House-Dorff Measure, and then the basic result that I'll talk about today is that given such a limit space, you can decompose it into two pieces, one of which is Ars of Epsilon, the Epsilon regular part. So this is the part that's not exactly smooth but where the tangent cones are very close to Euclidean space in particular. Ars of Epsilon is a manifold. It's bi-herder to a smooth Riemannian manifold and the other part is a piece of the singular set and with just a lower bound on Richie the singular set could actually be dense. Singular means those points where the tangent cone is not Euclidean space, not Rn. It could be dense. You could make up starting with a polygon and put more faces on it and so on. But almost all of the singular set in a certain quantitative sense, the sense of the theorem, the singularity is very weak so the Epsilon regular part also contains points where the tangent cone is not Euclidean space but very close to it in the Gromov-Hausdorff sense and then the piece that's left over is, well it has a certain structure, it was known previously from our work with colding that it has Hausdorff codimension 2 but now this is a much stronger statement that first of all this set is rectifiable, second of all, which is 4, it has a definite bound on the measure, let's say if you take the part lying in a ball of radius 1 and minus two-dimensional Hausdorff measure has a definite bound and more than that, a tube of radius r about this S epsilon has a bound with the exponent r squared that you would expect if it was just a sub manifold, a nice sub manifold of codimension 2 and a Riemannian manifold. Later in the talk I'll mention how what was just said compares to what was previously known, what was previously known was seemingly quite close but the difference between this result and what was known previously requires new techniques and so on. So just to be clear about the meaning of rectifiability in this context, it means for us that say a metric measure space is rectifiable, the measure is Hausdorff measure in this case, say K rectifiable if you can decompose it as a countable union of measurable subsets that is decomposed up to a set of measures zero and these subsets are by Lipschitz to positive measure subsets, these are the A.I. of R.K. So it's sort of something like a manifold but only in a measure theoretic sense and in fact it wouldn't be true in the context of the previous slide that this S. Epsilon is a manifold rectifiable is the best you can say. So the background what was known, so in work with Tobi-Colding we showed that you had such a decomposition where the Hausdorff codimension was indeed N minus 2. Dimension always means Hausdorff dimension. I should have said the Hausdorff dimension not codimension. And the methodology we used, we didn't really understand this I think when we started but it was the classical methodology that was introduced in the context of minimal varieties by the Georgie Federer Fleming and later maybe brought further by Olmgren. But to implement this idea in Riemannian geometry required new techniques which was what Tobi and I developed in a word because you don't have a background metric. So in particular what's constructed is a filtration of the singular set. So the singular set that's just the set of points where the tangent cone when you keep magnifying it and pass to a weak geometric limit also known as a Gromov Hausdorff limit you don't get Euclidean space. So that's the singular set and by definition. So to repeat what I just said, the tangent cone is a pointed Gromov Hausdorff limit of some sub-sequence that you get by fixing a point in your limit space and blowing it up that is rescaling the metric by a sequence RA inverse which goes to infinity. So one of the basic results from the work with Tobi is if you do that on a non-collapse space then what you'll get is a metric cone. So it need not be unique in general although of course most of the time it's just going to be RN but in any case whatever you get is a metric cone. I'll remind you of some cross-section. So here's a picture of a metric cone in two dimensions. Here's the idea that this one is a limit that if you could have convex surfaces that are closer and closer to it and then some obvious weak geometric limit sense because the cone is no longer smooth even though the manifolds in this picture had non-negative curvature two-dimensional surfaces is what's indicated in the picture. So now the filtration in this case SK is the set of points such that no tangent cones wits off our K plus 1 isometrically. This is something if you want a picture in your head you can think of the closed K skeleton of a simplicial complex and then as would be the case in that picture that is in a simplicial complex if you thought of SK as the closed K skeleton. We would have that the dimension of SK is less than or equal to K and this was what was shown in the earlier context by Alngarn in the sense of Hausdorff measure and it continues to be true here and the method classically was blow-up arguments. I mean the cones are already defined by blow-up in the sense of this rescaling and passing to Gromov Hausdorff limit in this case and to obtain this dimension inequality, the dimension of the K skeleton and the dimension bound SK as dimension of most K you need to do iterated blow-ups. So what's the point of doing a blow-up? So the point of these arguments they're arguments by contradiction so there's a part that comes in from geometric measure theory and it will be said in a little more detail on the next slide and the point is to show it's an argument by contradiction you want to show this dimension bound and what you try to show is if it were not true it would not be true for some cone. A cone is a more special object than a general limit space so that's helpful and the cone is always in this form of the K skeleton context including the original context minimal services harmonic maps or here they exhibit radial invariance so we'll kind of say in a while where this comes from but a cone obviously so I mean a cone with an arbitrary cross section which means geometrically I take these scaled copies and I stack them on top of each other they exhibit radial invariance and in our context more specifically what it means a cone is the formula for the distance in polar coordinates so it's topologically our plus cross something the cross section which could be any metric space in principle and the distance squared is given by the law of cosines that's what I call a metric cone some people call it a Euclidean cone and in this context of limits with Ricci bounded below the diameter is at most pi and in fact you can think since the cone is a blown up limit space and blowing up rescaling by a large constant makes the lower bound smaller so in some general sense the cones have non-negative curvature in some limiting generalized sense and therefore the cross section of the cone should have curvature Ricci curvature I'm talking about bounded by n minus one times the metric just as it would be when the cone is Euclidean space and the cross section is the unit sphere and then that's consistent with this result that I always get into trouble with this I think I'll just use these arrows okay so the bound on the diameter by Meyer's theorem is consistent with the idea that the cone the cross section should have diameter should have curvature bounded below by n minus one in fact this also strictly speaking comes from the splitting theorem with without which it wouldn't even be clear the cross section was connected okay so why is this helpful in proving the inequality is because any point on the cone except the vertex lies on a ray which is in the sense of Riemannian geometry I have infinite geodesic going out to infinity each segment of which is minimal so if I take a point other than the vertex lies on a ray now suppose I blow up again now the ray becomes a line and if you like by the splitting theorem the line splits off isometrically so we can keep doing this iterated blow up always avoiding the set of vertices now we did it once the original we got something across a line so whatever vertex was in the non-line factor now everything on here is a vertex but if we avoid it we can keep blowing up again now the geometric measure theory it's just a density argument said if we had if the inequality failed to begin with it would fail for a generic point on one of these blown up cones and then by iteration again it would fail but eventually we split off R K plus one and then every tangent cone on such a thing that splits off R K plus one of course would also split off R K plus one and then the fact that this S K was not empty but that consisted of things that don't split off R K plus one and so we would have reached a contradiction so that's how the iterated blow ups go now the next thing I want to say because it's important to say it is why are tangent cones metric cones so we can look at the volume ratio up here where this point is in this simply connected space curvature identically equal to H some hyperbolic space for example and this is the volume ratio and this ratio as a function of R if we fix H right is going to be monotone decreasing a priority bounded that is it starts out in a limit as R goes to zero in the smooth case it just becomes one because manifolds are infinitesimally Euclidean but if we go out to let's say R equals one the non-collapsing assumption just says that it's bounded below so it's bounded above by one and bounded below by something involving this V that we assumed in our non-collapsing and then there's a third thing which is crucial which I call coercivity and will be explained on the slide after this one so the monotonicity that's just Bishop Gromov inequality the boundedness as I mentioned is just the non-collapsing assumption and the last point is something we proved with Toby the thing I'm calling coercivity this refers to the almost equality case of Bishop Gromov so first the equality case which is much easier and was probably understood that this ratio which is monotone the script V sub H suppose first that actually it didn't change at all as a function of R well then as in earlier results in Riemannian geometry the proof can be by a string of inequalities each of which would have to be an equality if it didn't change in the end and then you see that it must be a cone actually strictly so I'm going to go ahead and I'm going to go ahead and it must be a cone actually strictly speaking in the Riemannian case it would have to be smooth but if instead of the volume of the ball used the corresponding thing the volume of the end minus one dimensional volume of the boundary of the ball and you assume that that ratio which is also monotone didn't change between two fixed values of R then the conclusion would be that it has to look like an annulus so here I mean strictly speaking an annulus in a cone now when I say that I'm assuming the lower bound is zero but since we're talking about rescaling small balls in the limiting case it's just as if the lower bound is really zero in our context and otherwise it would be whatever it was in the warp product with warping function like in the space of constant curvature H so the coercivity however says that something much stronger is true namely what happens if it's almost constant on some interval that is this thing which is monotone then in the Gromov Hausdorff sense we're almost in the situation of an annulus or depending on what we're saying if we're the volume and we're in a limit space where it actually could be a cone then we almost have a cone now this turned out not just to be a matter of following the string of equalities and imagine what would happen if each one were almost an equality it really needed new ideas which will be mentioned presently but this is true and this kind of stability if you have the almost the hypothesis you have almost the conclusion is much much stronger as we'll see so now let's make normalization may as well after an initial scaling say H is say minus n minus one that of n dimensional hyperbolic space I think I meant here H is probably minus one is what I meant which means the lower bound on Ricci is minus n minus one so that's a slight misprint and let's call something of the form two to the minus j a scale so now if we think about it we have this quantity it's a function of r and let's evaluate it on each scale well it's pinched between one and this constant that comes from the non collapsing but there are infinitely many scales in between two to the minus n equals zero and two to the zero which is one but because of the monotonicity on almost all of them it hardly changes at all right because it's like a series where all the terms are positive and it's bounded and there are infinitely many terms so all but a definite amount are extremely small and therefore we're in the situation when it's small of it's almost looking like a cone so strangely in my opinion this idea probably the quality case was understood and the almost equality case which is non tribletal to prove at least could have been guessed earlier in the game and then if you just thought about it you would see this picture the implication being that on almost every scale all apart from a definite number in a quantitative sense the ball must look like to the naked eye like a cone after magnification so this really could have been done had anyone thought to do it I mean not prove but at least conjecture but I don't think that actually anyone did that okay so now I want to emphasize a certain point that is going to come important in the rest of the talk and that is that in order that we needed this effective estimate which is what I was calling coercivity and one place where that's used is in passing the estimates on smooth manifolds under this weak convergence to tangent cones so you need that for example to conclude that tangent cones of limit spaces are metric cones you need the qualitative version so you have an effective estimate and I gave one consequence of it that on most scales it looks very close to a cone can't tell which ones are bad but you can tell there are only so many bad ones and yet here when you say the conclusion is that all cones are tangent cones that's an infinitesimal conclusion from an effective estimate so this suggests that the effective part isn't being completely used so then here at the bottom is just the remark that there's also an effective version of the splitting theorem which in the non-effective case for non-negative curvature we did with Gromal and it does apply to even possibly collapse tangent cones because they have non-negative reach in the generalized sense okay so now let's I said earlier that in fact in the non-collapse case the singular set had co-dimension to which was part of our work with Toby and how would you prove something like that well it's really part of this general methodology that went back to the minimal surface or minimal varieties story and in fact the most famous version of that I would say was what was eventually concluded by Jim Simons that's minimizing hypersurfaces the singularities are co-dimension seven so the point is that once you know that if you have any singularities then the tangent cone and a singularity must be an actual cone then you can ask yourself well what kind of cones might I have so they have to split off a certain Euclidean factor the ones you're interested in so what could they be and so in our case if you want to show the singular set has co-dimension two then you say well if I had a cone which split off Rn minus one what could it be well the cross section has to be either two points or one point it can't be three points it would contradict the splitting theorem so it's really Rn minus one cross a line in which case it isn't singular at all so we don't have to worry about that case or it's R plus cross Rn minus one that is to say a half space now the point is if you think about it so the half space is a perfectly good cone right well I know this is a cone on a point whose cross section of the point cross this it's a perfectly good cone but our question is can it arise as a limit space and the answer is no and the reason why if you think about it so the point when I say a pointed limit the point is over here on the boundary right so a ball then looks like this half of a ball right it has a boundary in here but on the other hand the balls of which it's a limit don't have any boundary in there the boundary is out there so there's something wrong with this picture and that you can make into a proof so there's a perfectly good cone but it doesn't arise as a limit space that was the and that's always there was the same thing with Simons you know he showed that the geometric measure theory technology said you would the cone you would get if you had singularities had to be minimized Simons with the Simons equation showed how to deform it to something smaller with smaller area and so it couldn't have arisen in that way was a perfectly it would be a perfectly good cone over a compact minimizing smooth thing in the sphere by induction but he showed up to the appropriate dimension where it turned out not to be true anymore that you could deform it to something smaller and that was the proof is the same idea here okay so some of the take so I said that in the Riemannian case it wasn't just a matter of following a string of ODE inequalities you really needed new techniques in the work with Toby and some of them were involved a particular kind of regularization the idea being that distance functions are the natural functions in Riemannian geometry and of course in Euclidean space they might be smooth in fact a coordinate function in Euclidean space or maybe in a Riemannian product would be a harmonic function even with constant norm of the gradient or distance squared from a point Laplacian of that would be constant and equal to 2n so but on a Riemannian manifold of course distance functions need not be smooth on the cut locus so the idea was to approximate them by the solutions of these equations with on the region you're looking at the same boundary values and then measure how close the solution of the elliptic equation was to the corresponding distance function on your manifold the point of this approximation is that these guys actually are smooth and you have control of things like the point-wise norm of the gradient by the Cheng-Yao estimate and Bachner's formula gives you L2 control of the Hessian and then you can pass those results to control of the first derivative part that you wouldn't have otherwise in the form of integral estimates initially so you can use the regularized thing to get some kind of integral estimates on the actual distance function that are the relevant ones for the geometry and then there was basic technique for turning the integral estimates into the estimates on the actual distance estimates and that involves something for instance called the segment inequality so that was some of what was in the paper was Tobi a considerable amount of the important part and now you know there was also something about two-sided balance so suppose you now in addition to the lower bound you assume you have an upper bound on the Ricci curvature and upper bound is not significant in and of itself but if you it adds something basically because you have more regularity when you assume it in addition to a lower bound so there had been a co-conjecture going back to the 90s in particular due to Mike Anderson and probably some of the earlier pioneers like Gong Tian maybe Nakajima that the singular set in the case of a two-sided bound that actually occurs in Mike Anderson's ICM talk should have co-dimension four rather than co-dimension two and eventually we were able to prove this with Aaron neighbor it stayed open for a long time so it was understood here's what was understood here maybe I just take a minute so using the same sort of idea as I just explained over there now suppose with the two-sided bound you wanted to show the singular set had co-dimension four well it was actually understood if you could show it had co-dimension three there was a topological argument you could show then that it had co-dimension four so what it then came down to was ruling out a tangent cone like this so here's the paper cup cone cross Rn minus two and you had to show this couldn't arise as a limit space this perfectly good cone was not a limit space with the two-sided bound on Ricci now in dimension two of course it's kind of obvious that it can't but with the right from this picture but to show that this can't arise as a limit space with Ricci zero because of the blowing up part turned out to be quite hard but there was a rather technically difficult argument we were eventually able to do that so this was the easy part in going from three there was the hard part in going from three to four was known to be easy if you could get co-dimension three so now also in 2012 some new ideas came in in the work with Aaron and that returns to the point I mentioned earlier that if you have effective results like this what I was calling coercivity this function monotone almost constant means you're close to a cone in the Gromov house or sense you would like to make more use of that and you would like to make more use of that and you would like to make use of that and that's what we pointed out in this 2012 paper or realized maybe I should say that you are in fact able to do so in the co-dimension four with the two-sided bound what we showed was that the singular set you actually had an estimate on the tubular neighborhood which is stronger than a bound on the house dwarf dimension and so it had co-dimension four but what we showed there is that so where are we here? Okay probably going to continue to happen because it happens to me in every talk so notice if this were a smoothing co-dimension four manifold then the volume of a tube around it would look like this but without the eta and without the eta over here it would just be four and a constant so we almost showed that in the sense that we showed for any constant the tube would look like this and this was the first result of this type which just wasn't something about the house dwarf dimension but there was this annoying point that the constant in front of the estimate blew up as eta went to zero so it seems like a small thing but it actually turned out to require totally new ideas to get rid of it which I'll come to at the end so then the corresponding thing is in the co-dimension four case that we showed that you get a bound on see so suppose you're outside maybe let me just say it in words I forget what might be on the next slide so here's the bound on this tube around the singular set but what this says in effect is if you're not in the tube namely your distance r away then you have a definite bound on the Riemannian curvature. Not just at the point that you're looking at but on a ball of comparable radius around that point so this is the soup on a ball outside the tube again with this annoying eta but nonetheless so this is much stronger than just having a bound on the center for example so this was a kind of new take on this subject not just how stored measure bounds and the idea being that if you have this soup bound and you have some equation and these are all equations right the same kind of story held in all of these cases partly joint then when you have an elliptic estimate and you have this sort of soup bound you rescale it to unit size and then you get bounds on all the higher derivatives automatically the appropriate bounds so for example in the harmonic map case where the a priori bound from the definition was an L2 bound on the norm of the gradient which you assumed then this led to an L3 bound sorry an LP bound for every P less than 3 and LP is actually wrong so the idea both of the tube controlling the size of the tube and a definite amount of regularity in this sense over here outside of the tube so this we call this the regularity scale sometimes called the curvature scale that's the thing where you have such a bound with the two there is a crucial part of the discussion and has applications and the way that certainly I came to this was in work with Bruce and the sophomore in a very different context and there was earlier work using this the ideas of something like this by Dorn Soro, Jones, Sims and Rebay each in a very different context from what I'm talking about here and the point about this in my opinion is that this work that I'm talking about here stems from is from the 80s but the most elementary case of it just applies to a function of one variable with a bound on its derivative and then you look at all dyadic intervals and you ask yourself for a given epsilon what's the sum of the lengths on which this function after magnification to unit size like if you looked at it under a microscope is not epsilon close to being linear the linear function is not its derivative is just the best linear approximation so one way of saying that a function is is differentiable is right like the earth is flat one way of saying it is if you look at it under a microscope it starts looking like it's tangent line but if all you know is a bound on the first derivative you might have a function say with a high frequency but small amplitude and if the frequency is high enough and the amplitude is small enough it still has derivative bounded by one and this guy although it doesn't look like it's tangent line until you go to an extremely small scale it does look like a linear function most of the time and that linear function is zero because the amplitude is you know that is it's a constant linear function so that's the the idea and this this simple idea was somehow not taught to undergraduates which I think it should be and therefore in these more advanced contexts stemming from the 60s the kind of results I'm talking about here were missed it wasn't understood because most people have made a survey of various very distinguished analysts and they've never heard of what I just told you over here so it's an interesting sociological point so now we started a little late I'm not sure how much time I have how much time do I have 15 great okay so here are the new text techniques that were used in the work with Aaron that I've described up to this point but this this technique is what you need to get the result but still with this minus eta and the coefficient depending on eta so there's a quantitative version of the stratification or what I call the filtration which I'll come to a quantitative version of what I call colon splitting I'll explain what that is something called the energy decomposition a certain covering argument which is not so difficult and novel epsilon regularity theorems so now I want to explain what these terms refer to so this was in the work with Aaron from 2012 so we want to take this filtration and just make a more detailed analysis of a more effective definition some people believed that that's all we did was make a definition but I beg to differ with that okay so let's introduce some parameters into the SK we call SK epsilon and that means the point is in SK epsilon are if we look at the we look at the interval say between R and 1 and we say no ball on its own scale is epsilon close so that's what this means on its own scales epsilon close to splitting off yeah is K symmetric but really what I want is it's not K plus 1 symmetric up to an error of epsilon on its own scale the R means on its own scale so then that's a quantitative version so all balls down to radius R have to not be epsilon close on their own scale to splitting off an R K plus 1 factor isometrically and then it's easy to see that if this happens down to the intersection of all of these is just the ones where tangent cones are epsilon away let's say on a ball of size 1 from splitting off R K plus 1 in fact this is the epsilon K where K is N minus 2 that appeared on the very first transparency and then well or maybe this one I think there shouldn't be yeah this is a misprint there shouldn't be an epsilon there okay it's the union of these okay so the whole singular set the same thing it's Kth element to the filtration okay so these things are very easy to check but nonetheless fundamental they allow you to introduce these quantitative ideas so now another ingredient was cone splitting and by that I mean the following so first in a version of this actually occurred in this earlier work say on harmonic maps you can find that there are some notes for example by we on Simon from these whatever they're called at Utah where they have these summer schools whatever there are some nice notes and a very nice one line explanation in the context of functions of the point I'm going to just explain geometrically so in our context if you take a cone and cross it with let's say a Euclidean space it's still a cone and moreover if this was your original cone let's say two dimensional any point of course since it's an isometric product is now a vertex right so you can regard it as a cone it has a whole N minus 2 vertices once you cross it the converse is also true that's that's the point here that if you have something a space and you can regard it as a cone in two different ways that is where the vertices are distinct then it actually must split off a line by cone I mean a metric cone must actually split off a line isometrically in the two ways you read the two cross sections of your cones are are isometric so I'll call that cone splitting right if I have a space I can regard it as a cone in two distinct ways then it must split off a line isometrically must go through the two vertices and this is the iterated version of it so that's cone splitting so then as you might expect since we're emphasizing the effective version another ingredient is an effective version of cone splitting so it means if the cones are almost cones and we almost have these these vertices where the points are separated then it almost splits off so you could you know just fill in what it would have to mean it's an effective version of the previous statement and this is now this is the most effective version of the previous statement maybe it's worth saying so you could check this where what am I talking about here so I have I have these points which look like their vertices up to a small error and now I'm saying I can actually find this regularized coordinate function by taking the difference of so if it's going to split off a line for example so just think of the following illustration of what's supposed to be on this and then just look in the plane so that splits off the x-axis never mind about the y-axis for the moment take these two points say one zero minus one zero now look at the squared distance from each of these so this one is maybe the r0 this one is r1 over there so so what we're looking at is now x squared minus one minus x squared plus one and if I divide it by four I should just get x which is the coordinate that's what this refers to up here so that's an analytic version again of what I was talking about over there so the difference of squared distance functions is illustrated over there and that's that's what's occurring over here so so this is now actually a generalization of what was on the initial transparency namely the initial transparency was the case of this but for the whole singular set because the whole singular set was really sn minus two so something similar actually this is saying something similar is true for all of these quantitative strata of the singular set so that's true more generally the thing that's special say to the n minus two or n minus four if you have the two sided bound is the regularity outside of it so and then and also the bound on the measure because of the regularity outside so this is just really a generalization basically of what was on the first transparency but without the rectifiability statement from the first transparency and with the eta in there okay so I just I won't be able to say what's involved in getting rid of the eta except maybe a word or two it's too technical but I will say a little bit more about how we arrived at this with the eta using the ideas which have been explained so far so in particular what I said early on was that there was a bound on the number of scales where it wouldn't look like a cone on everything else it looks as close to a cone as you want and we and that wasn't made full use of in the earlier work so so now here's the thing so there there's something which in the 2012 work with Aaron which we call the energy decomposition and it simply was so we fix an epsilon and then we fix a point and then we make a record of the scales such that it doesn't look on that scale within an epsilon error on its own scale like a like a cone so at every point there are some bad scales but only a definite number but the the bad scales which ones they are depends on the point right so I can't say from point to point which are the bad scales only that there are a definite number having fixed at most the definite number having fixed epsilon so what we did was we can if we wish group together if we wish group together all of the points where they have the same good and bad scales right now if you think about it no matter how far down I go so I'm I'm taking some finite r from the previous transparency but it could be extremely small but yet the number of bad scales if I fix epsilon is bounded so the fact that it's bounded means if I think of these collections of points which have the same good and bad scales it's not as bad as you would think because it can only grow so to say polynomial the collections it would be quite different if I were grouping them together and I didn't have an a priori bound on the number of bad scales then there would be many more groups is what I'm saying but here they're only growing polynomially that's this end to the queue the number of groups because I have n scales but only a definite number independent of n are bad so they're only roughly end to the queue possibilities for the number of groups so that's a crucial point ok now why do I want to group them together because of the following two things when I'm on a good scale and I look at all those points if they're close together and they look very conical then I will have this idea that it almost splits off a line ok they're separated you know fairly close together they look very conical then that will mean the geometry splits off a line see if not it will be considering both points where it looked good and maybe other points where it looked bad so that's why I want to at least on the good scales which is almost every scale then I get this kind of line splitting or Euclidean splitting so I want to use that in a covering argument and there will be finitely many bad scales and then I'll be able to do a very cheap kind of covering on that scale so to get the estimate on the volume of the tube so here it kind of says on the good scales the set we're trying to cover will look like if we're talking about sk a k dimensional Euclidean factor namely this where it's k dimensional and the set I'm trying to cover will look very near to this so I roughly need only half to two to the k balls to cover it and go down to the next scale on a bad scale it's not close to a k dimensional thing but I can cover it with two to the n balls and as they dimension and they're only finitely many so that will just put a constant in front of the whole estimate also because there are so few of the groups right because of the a priori bound to deal with the whole thing I can fix my intention on one group and then just add the estimates because it's true there are infinitely many but much smaller it's it will be absorbed into the the eta so that's kind of the idea and so this is just saying the same thing again in words that the covering argument to estimate this size of the quantitative stratification or filtration I cover and then I recover with balls of half the size and so on and if you think of it I'm trying I'm sort of covering the thing by a generalized canter set you can think of it in that way where when I go from one scale to the next which is almost every time having fixed a particular collection of good and bad scales then I only need when I go down a scale if it's l less than k the worst case would be k I only need two to the l balls of half the radius whereas on the bad scales I just recover the whole ball that's a factor of two to the n but I only have k of those so it just put something in the constant and that's how the proof goes and then there's a kind of soft epsilon regularity theorem which tells you that outside of the tube when you're at the top stratum you really it's really regular so in what what this would say in words would be that let's say we have the two-sided Ricci bound case where the singular set has co-dimension four so singular point it's allowed to split off r n minus four but if we're at a point where it's efficiently close to splitting off r n minus three on a ball then it must be smooth that's the relevant epsilon regularity theorem let's let's imagine it's Einstein so it but it's the same it's really the same just with C1 alpha C1 alpha regularity if Ricci is bounded so it's a kind of actually relatively soft epsilon regularity theorem if it has enough symmetry the tangent cone then it must absolute then it must actually be smooth this is where you have an equation which corresponds in in words to a two-sided bound and this is a contradiction compactness where the compactness is in a weak topology and the equation meaning the Ricci formula and harmonic coordinates in this case allows you to get back over the weak convergence as in the blow-up arguments that were introduced by Mike Anderson so now I don't know what do I have a minute or two or I'm done yeah yeah what I'm over time I know but I know I started late this I kept careful track of so in okay in the last minute or two I just want to say what could be the source what where could this Ada come from what was lost in this argument well mainly the idea is when you do the recovering maybe the ball stick out a little bit and something is lost there you can't quite cover a ball by balls of half the size but you could probably get away with that by being more careful about the domains not balls but what are sometimes called Chris cubes or something like this but the energy decomposition which was so pleasing when it was discovered it was discovered is actually a source of the error at worst you would still have a logarithmic error say log R instead of in front of the R to the fourth log R to some power from this energy decomposition because the idea was you did have quite a few of these groups that I was denoting script C just they weren't growing fast enough so they could be absorbed by the Ada so you really need a different idea to get rid of them get rid of the Ada and in a way you could think at a completely formal level that as I said before you can imagine you have a series of positive terms and the sum is some definite C well the first thing which was the non effective estimates about tangent cones is the first observation is this implies a to n goes to zero the second observation is well actually by Markov's inequality there can only be a definite number of terms since you know C which this probably should have a C in front of the X inverse okay but a definite number of terms that can be only so big if you have a bound on the total but still that statement is weaker than simply what you started with was that the sum of all the terms was C so to get rid of the Ada you have to somehow take advantage of in our context that the sum of the terms is C and not just this which leaves you with the Ada and this is possible but certainly not easy so there is a paper by neighbor and Danieli Veltor to that introduce some new tools one to deal with the rectifiability which we also of course didn't know was a Reifenberg type rectifiability theorem which much weaker hypotheses than previous ones and a very non trivial covering argument as well and also the context the concept of what's called a neck decomposition which I won't have time to say anything about initially I thought I might but then I decided it was hopeless and then they proved this in the context of harmonic maps they got rid of the Ada and they proved minimal varieties harmonic maps and they proved rectifiability and got rid of the Ada which was still left from our previous work and then a really remarkable paper in the Riemannian geometry context but assuming a two-sided bound by neighbor and when she was young where they proved again the rectifiability this time in the context of the two-sided bound so and more over this tube type bound which is much stronger are to the fourth no Ada and completely remarkably an a priori L2 bound on the curvature so from we we had done this in the work with Aaron in dimension four where the singular set consisted of points and we had done it in general but with the Ada and one reason why this is particularly remarkable is in the other context of harmonic maps and so on this is not true what you get is it's in weak L4 as with the earlier result was in LP for every P less than four that was from the work of Aaron and myself the harmonic maps and so on it's in weak L3 but here it's actually in curvature in L2 so that's special to the Riemannian case so this is a very fundamental remarkable result in my opinion and the last thing I want to say is that so now what I was talking about today was the case where you just have a lower bound and this actually requires completely new estimates from the two-sided bound one because they use an essential way something they called a super convexity estimate which doesn't hold with just a lower bound so the neck decompositions certainly play a crucial role I didn't get to say what they were but it requires new estimates and something in place of the Reifenberg argument which used a more canonical version having to do with harmonic splitting functions and their behavior on neck regions and in particular a sharpening of something that played a key role in the 2015 paper with Aaron something called the transformation theorem so okay I'll stop here. Any questions? Is there any chance that your main theorem characterizes non-collapsing from cost of limits among method measuring bases with lower rigid curvature bound? Try to make sense out of what you're asking maybe you could ask it a little more precisely that would help me maybe I'm just not thinking clearly so it characterizes them among what? So if it's every metric measure space? No I don't think that that's right I think that you know there is a synthetic theory of Ricci bounded below and I think it definitely I mean which includes as an assumption the fact that it's infinitesimally Euclidean as opposed to Finsler but I think it's I think it's a stronger condition actually there may be another thing that one could say in this context that's relevant so there well alright so first of all there is a synthetic theory which is kind of closer to what I think you're asking and also after almost 20 years 17 or something of the synthetic theory it finally starting to prove structural results as opposed to just more and more inequalities which was something I advocated for quite a while so without answering your question precisely I suspect if I understood that the answer would probably be no but anyway something is coming closer to what you asked I feel. Any other questions? Yes, can you improve your results if you replace the Ricci bounded by bounds and the sectional curvature? Yeah I think then well let's see I mean well the lower bounds I mean the two-sided bound of course then there's the whole theory of bounded curvature and then you would have compactness where the limits would be smooth if they were non-collapsed and collapsing theory. But, well I mean certain things are true like tangent cones are unique but I think there so you can in certain ways but I think in certain ways it kind of resembles the Ricci case also. I mean definitely there are some stronger statements and they're not my results. I mean there's the theory of Alexander of Spaces maybe Karsten as a person to ask. But it's rather similar in a way I mean one crucial point of course is tangent cones are unique and you have strainers and there's this whole discussion there. Any other questions? Thanks for speaking again.


The point of care technologies are important in our particular area of interest, which is low resource settings, because many times people come with symptoms of a disease, travel many miles, kilometers down the road, often down a path, unpaved roads, perhaps carrying their children, missing a day of work to get a diagnostic and treatment. And point of care technologies are very important because without them that sample that's taken that day may have to travel days to go to a laboratory for results. That person goes back to their village with their child and perhaps is never seen again for treatment once those results come in. So our objective with some of the technologies we're working on is to really bring that much closer so the diagnosis and treatment happens at the point of care so that those people can go back to their communities treated and get back on with their lives quickly. We've been working on a platform called NIA, stands for non-instrumented nucleic acid amplification and the whole reason for this technology coming to being is because a lot of the worst diseases that plague say sub-Saharan Africa, for example HIV and malaria are two really good examples, occur superimposed with a map of the least electricity available, density, you know, there's just not a lot of electricity available in these areas. And so the traditional means of high accuracy diagnostics just aren't available to the majority of people in some of the areas that we're most concerned about at PATH. The core technology is called chemical temperature control. So what's happening with our NINA core technology is that we're using a chemical reaction to exothermically create heat. We create a nice stable temperature profile for the chemical reaction by thermally coupling that with what's called an engineered phase change material and as long as we have two phases we can achieve isothermal temperatures. What happens in the process of a person coming into a clinic is they give a very small sample through a finger prick so 50 to 100 microliters of blood will go into a test room that has what we call the master mix of all of the reagents necessary to amplify RNA and DNA. That requires heat so that master mix will then go into our chemical heater for 30 to 60 minutes and then the results are another technology that requires electricity free detection and we're pairing the NINA technology with some of the most innovative detection technologies that are out there.


As I clicked through the pictures, I could hardly believe what I was seeing. Here were horses flying with invisible wings. This man, Alexander Nivzarov, a former war correspondent wounded in the line of duty, and a current governmental advisor, researcher and historian, had found a way to be with horses who were free to express themselves in natural collection. When I read that he worked with the horses without using any punishment, force, or restrictions on the horses' heads, and complete liberty, something in me was shaken, how is that possible? That was when I knew Russia would be my next destination. I used to be a horse trainer. I used to be a riding instructor. Until the day I realized that somewhere I had gone off course. My days were filled with training horses, teaching lessons, and judging competitions. I was good at what I did. I could teach my students how to control their horses. But there was something lost in that process. I looked back at pictures and drawings from my childhood. In the beginning, there was a passion that drew me to horses. A closeness I felt with these gracious beings. Where had the beauty, majesty, and freedom of horses gone? I started to dream. If I could go and meet anyone in the world, who would it be? I looked at the books that inspired me and made a wish list of the authors I wanted to meet. These people broke with tradition and focused more on what we can learn from horses than what we can teach them. My first stop was Colorado. To meet Mark Rashid, a tough-looking cowboy with a black belt Nikito, who talked about passive leadership, softness, and the value of a still mind. A lot of people refer to that as the one we want to emulate. And if you've ever spent any time around any large herds of horses, it's really easy to find the alpha horse, the boss horse, because they're generally off by themselves. Because the other horses don't want to be around them. So for me, I don't know why I would want to emulate that behavior in relationship to training and working with horses. But in a herd of 30 horses, there will be 29 followers and one leader. And the leader is generally an older mare. You say in a wild herd. The stallions come and go. But mayors stay for life. And so it's not necessarily going to be the bossy mare. It's going to be the one with all the life experience. It's the leader that can be trusted, whether it's with people or with horses. The way to develop trust is through consistency. If you're consistent, then you're dependable. If you're dependable, you become trustworthy. If you're trustworthy, the horse will be at peace with you. And if the horse is at peace with you, they can become soft. Softness comes from the inside of the horse or the person. Lightness is just on the outside. You can achieve lightness through training. Meaning you can get a lot of things done. And they look really good. Right up until they get into a new situation, we see horses all the time that people will refer to as soft or light. And they'll bring them into a new clinic. Everything's going fine at home. And they bring them into a new situation and everything unravels. With lightness, the things that are trained into the horse are available when things are going relatively well. With softness, everything is available all the time. Through training, we take all the softness out of the horse and we spend the rest of our lives trying to put it back in. And it's there. It's still there. We just have to look for it and develop it and nurture it and bring it out. Horses who are in captivity are missing out on their activities of being with the herd. And it's my desire that we fill those needs for our horses. If we're looking back to our forefathers, they say that we have something like a realistic world. Things you can see, things you can touch. Klaus taught us exercises to bring us into the present moment, which is where we needed to be in order to connect with a horse. He compared a horse to the ocean, something that we had to come into harmony with, unlike a guitar, an instrument created by man. Until I became aware of how horses mirrored the things in me that I needed to look at, I would continue to be frustrated by my expectations. The last person I went to visit, I didn't find by reading his book. I saw his website. As I clicked through the pictures, I could hardly believe what I was seeing. Here were horses flying with invisible wings. This man, Alexander Nivesarov, a former war correspondent wounded in the line of duty, and a current governmental advisor, researcher and historian, had found a way to be with horses who were free to express themselves in natural collection. When I read that he worked with the horses without using any punishment, force, or restrictions on the horse's heads, and complete liberty, something in me was shaken, how is that possible? I don't have a preconceived notion of where I want to go. Then I would be so focused on that that I would miss the flowers on the sides. I think it used to be at one time about what I can do with the horse, but it's not that way for me anymore. There's a certain way that I'm trying to go through my life, and over the years there's been a shift for me in that it isn't about me and the horse. It's about what the horse is bringing to the table that can help me get to where I need to go. You don't need to put horses in different categories. They're not aliens, they're exactly as we are. They're the same kind of mammal. We are similar in almost every physiological parameter. And also it's quite dangerous to put a horse into some cosmic range of incredible and unseen alien as to put her down to a mere piece of sports equipment. But a close, noble, and concerned relationship with any living creature always gives you amazing results. Everything in life, everything that lives in this world has the same rights for life, for freedom, love, happiness. And a horse gives these possibilities to the human.  You get into a completely different position. In all my travels, I saw that the most important journey we can take is to go within and see ourselves for who we really are. The more freedom within ourselves, the more freedom the horses have, and the more we are awake to the source of life. The horses are our reflections.


This is Stadium in 60, a quick update on the Titan's new stadium. The new Titan Stadium is being designed with sustainability in mind. Most stadiums built in the 1990s were designed to last 30 years. This stadium is being planned with a much longer lifespan in mind. Adam Neuse is a Titan's senior vice president and chief revenue officer. This is a legacy project that is going to be something that is going to impact generations to come. And when you think about that, you think about functionality, long term, sustainability, long term. Some of that comes from the fan experience, not just for our kids, but our kids' kids. Kellyn DeCorsi is the Titan's new stadium project executive. For the Titan's organization to have a home that the Titan's fans can just embody and own and have that home field advantage every given Sunday, I think is amazing. But then just what it means for the events on the next level to have these events that we don't get March Madness College football playoffs, a Super Bowl hopefully, those are things that I'm just so excited about. Nashville is rising, but this will take us to the top. For the latest news, visit titansnewstadium.com.


Andrew has a cool stuff found sort of encased in a quick tip here. He says, I recently picked up a Samsung smart monitor and have been using my M1 MacBook Pro in Clam, M1 Pro MacBook Pro in clamshell mode. But I found that I couldn't adjust the volume using the keyboard controls on the Apple external keyboard. Then I was poking around in Rogamiba's sound source and found an option called super volume keys. I turned it on, boom, that's all it took. Now my volume controls work. Yes, this is one of these magic things. And I had forgotten about talking about quick tips that we forgot about. I had forgotten about this option in sound source, but I use it. And as soon as he was talking about it, I'm like, wait, why am I able to control my volume with my volume keys? And it was like, oh, right, sound source. Yes, I also use this. So yeah, sound source is great. In addition to that one little option that is truly kind of buried, what sound source really lets you do is choose which, choose a different audio output device and volume level per app. So you can say, oh, I want alerts to go, well, you can do that in system preferences, but you can say I want my music, Apple music to go through my big speakers, but I want Safari sounds to go through the speakers in my Mac or whatever. Or I always want this to go to my headphones. Yeah, it's a great little app. And it's just one of those things that is so fundamentally necessary that I just can't even think about it. And then you can set effects on each output, right? So you can say, well, this app, I want to have some EQ on it and this app, no. And so yeah, it's one of those things that is, I can't even, I couldn't live without it. And that one feature that Andrew, what's it called, smart volume keys, that is huge. Like, if you're using a third party monitor, that's the way to go. So yeah, yeah, yeah, yeah, yeah.


Hi there Psych2Go fans, welcome to another feast for your inquisitive minds made possible through your continuous support. Do you ever wonder why each person reacts differently to a similar situation or why each person's approach to problem solving is unique? People operate on emotions by varying degrees. Some are slaves to their spontaneous emotions while others have mastered their control. Being strong emotional maturity is an asset to navigating the ship of life. What exactly is emotional maturity you ask? Emotional maturity is the ability to clearly understand and manage your emotions. Emotions play a huge part in our everyday interactions, so it follows to say that exercising control over them becomes all the more important for meaningful interactions. There are seven signs of emotional maturity. 1. You can put a name to the face of your emotions. If at any given time you can pinpoint the exact emotion you're feeling, congratulations you have officially entered the realm of emotional maturity. Knowing exactly what it is that you're feeling is the first step to trying to manage your emotions. When you can name them, your emotions get validated. This makes it easier for you to understand your emotional state and you can then formulate a plan to deal with it. A study also found that those who have a strong capacity to regulate their emotions should be better able to maintain good health, psychological well-being, and a healthier outlook on life. 2. You can draw healthy boundaries. Are you a highly sensitive person or an empath? You find yourself easily influenced by the demands and emotions of others. If so, then learning how to manage your own emotions and saying no to things could be a sign of emotional maturity. There's a fine line between being selfish or emotionally mature. If you can put yourself before others without putting them at a disadvantage, you are counted as being emotionally mature. Your boundaries are for your protection rather than nuisance to others. 3. You aren't afraid to own up to your mistakes. Everyone makes mistakes. It's how you behave afterwards that makes all the difference. According to a New York Times piece by Paul Krugman, we as a nation are suffering from an epidemic of infallibility. No one, president or not, wants to admit to wrongdoing, at least not publicly. A big sign of emotional maturity is being aware of your own actions. So when you make a mistake, you're able to admit that you were wrong. Recognizing when you're wrong and outwardly acknowledging it shows you're mature enough to try and mend a situation or relationship. Valuing your relationships more than your ego is a telltale sign of an emotionally mature person. According to Lewicki and his collaborators, when you admit to wrongdoing, you show that you value the other more than your own pride. And being able to look beyond self-serving motives is something emotional maturity is all about. 4. You think before you leap. Emotional maturity is at its full bloom when you overcome the habit of automatically acting on spontaneous reactions. It's when you take time to understand and analyze the situation before reacting to it. Emotionally mature people are able to step back and take a bird's eye view before responding to difficult situations. According to psychology today, emotionally mature people are able to control their impulses and are less prone to emotional outbursts and aren't quick to anger. They do not lose their cool in the face of adversity. Rather, they determine how to react through thoughtful deliberation. This is an added feather in their cap of maturity as they're less likely to do or say something they'll regret later on. 5. You have more success in getting your point across. According to a study, emotionally mature people have an effective communication style. You are aware of your needs and wants and can convey them to others in a healthy way. You know what you want and are able to express it. You can get your point across as there is no conflict between your thoughts and actions. Emotionally mature people have a good feel on the how, when, and where the optimal time is to express their emotions. You feel your emotions instead of becoming your emotions. This is what brings stability and objectivity in your communication with others. 6. You are self-aware. Emotional maturity is not intellectual. Rather, it's a higher state of self-awareness. It goes beyond intelligence and includes our senses, intuition, and heart. Self-awareness refers to the capacity of becoming the object of one's own attention. This self is defined as an integrative structure comprised of cognitive and emotional elements. When you are self-aware, you can focus on your emotions more clearly. A clear understanding of your own emotions leads you to acknowledge your needs. You have confidence and don't necessarily need validation from other people in making your decisions. 7. You know how to listen. Since emotional maturity comes with empathy, it automatically makes a person a pretty good listener. It enables you to lend a sympathetic ear to others. You know how to avoid being triggered by situations and getting self-absorbed. This lets you approach the situation objectively, being able to focus on the person talking instead of getting wrapped up in your own needs. So you can listen actively rather than being judgmental or condescending. Emotional maturity is a key ingredient to adding meaning to the recipe of life. It's not a skill acquired overnight, but rather requires a prolonged simmer on the gentle heat of experience. However, that's not to say it requires years and years. With the right kind of positive approach to life, emotional maturity is just around the corner. Which in turn can be effective in keeping emotional burnout at bay? You can watch our video on 7 signs of emotional burnout linked in the description box below to help you understand that better. Were you able to relate to these signs? Have you noticed similar traits in those around you? What are some ways you can incorporate these signs into your life? Do let us know in the comments below. Also, remember to like and share this video with those who might benefit from it. As always, thanks for watching. Until next time.


I'm ready for the Super Bowl class today at 12 o'clock. I'm gonna talk about how to actually trade the Super Bowl You probably like you could trade the Super Bowl. Yeah, we're gonna have a I'm gonna do an event on 12 o'clock Free event so everyone's more than invited. We'll let we'll post it in the room Everybody here on YouTube and old social media network. So I'm more than happy to join us. Please go out there and share it Listen, you want to know how to trade COVID you want to know how to trade earnings You want to know how to trade Fed rate? Well, you know what? Yes, you can trade the Super Bowl and we've done very well So don't get to miss that. Anyway, look in what's going on right now. My god. I feel so bad for Lyft I love that company. I always use it all the time Uber's been on a really good move, but Lyft got Crushed on earnings. My god. This thing got talking about a catastrophe one thing I could tell you about Lyft It's not a pharmaceutical stock. So we know they don't come back. So lift could come back We'll keep an eye on that. Anyway, this stock was a you know One thing I do like about Lyft It's at a major support level and just let everybody know for everybody who's here.  So look in we'll check We'll talk a little bit more about that on later on the show But anyway, let's talk about what's moving and what happened yesterday because Yes, we had a couple of good moving stocks, you know We had a couple of nice little movers and I want to cover them first before we go through the watch list and everything else I like to start off with the the KPRX So KPRX, you know had a really nice move Not only from a little bit of a yesterday But the day before and you could see it right here that the stock had a really really good move right here From 450 went all the way up to a price around eight dollars about a hundred percent had a couple of little slingshots going back and forth and You know not as easy as the day before but you know We I know some of you here were trading it in the cyber group room That was we had a nice little move on that one the real big winner was this okay? Was this HPCO I mean This thing had writing all over it on a slingshot It also had a good hold had everything this stock started this morning at a dollar 20 It ran all the way up to about four dollars and fifty cents now for some of you in the cyber group room in my trading room Remember I was talking about right around here around ten o'clock, and I'm like It's just holding here too Often and it started making a little bit of a foul-stow flag and it had this slingshot that came right back down But when it got back past this resistance levels around 350 look what happened it ran from 320 All the way to 440 great great runner up there, and you could tell I'm in new lower lows and it finally made a new high We had a consolidation going on here right around that one two o'clock, and then we had that big push So there was so many opportunities out there, and if you've been following us on Twitter If you're in the cyber group room, hopefully you got the mobile app where we send you the alerts But we were talking about it, and we were on the stock all day yesterday now listen Granted down to a dollar 50 who cares you know who literally cares I don't give a crap about this thought because I don't own it We were trading yesterday, and we did well with it then and no matter what time you showed up If you're an early bird gets gets the money if you're You know if you're a lunchtime trader or if you're a West Coast trader You know you have plenty of time to get the opportunity to go in it, and it was fun It was moving and you know what? Listen these are things we're getting every day, and you don't want don't worry about it Don't worry about it. There's going to be a number one today There's gonna be another one on Monday. There's been one that they're every day.  I don't know. Let's see on social media. Anybody like Snoop Dog.  He did that and he changed his image He's always feeling like he's always smoking you know weed every time the weed king. Yeah, Mark. There you go I'm surprised people on social media have no idea because they're not no one's actually even saying anything in there But he's a great rapper too, right? So that that was the news and this is the reason why I bring this up is this, okay? Could you imagine just that one little news made the stock go from a dollar to four dollars? You know what I mean?   What was the trump stock everyone? Who could tell me what was this trump stock that we traded? What was the trump stock? President trump stock. There you go D. W. A. C. Right. Remember D. W. A. C. Oh, it's at 15 dollars But you know what we never problem when when the stock when the news came out because it was the state the same Exact news with the Snoop Doggy dog trade right here. Look at this Company came out when public that one day ran from 12 to 52 and the next day it ran from 67 to 175 Now if you're a trump hater or you're a maga, whatever you want to call these people Um, you know what if you were a big trump fan and you jumped into that stock You got you know, you could you could have did well or you could have got crushed if you are if you were You're an anti-trump you short the stock you would have got crushed, you know at the end of the day I don't care what you what your political beliefs, but the same thing happened with W. D. W D. W. A. C. Is the same exact thing that happened with Um, the h poc you would never even I spelled it wrong H. Um the h pco is the same exact thing that happened with this Okay, these little these little things are what make these stocks run and that's what it's all about So anyway, let's go find some other stocks out there to move this morning because we got about five minutes And then we're going to be back here and we're going to start trading in the morning And then don't forget 12 o'clock today. I'm going to do an event How to trade the super bowl? Okay, so you don't want to miss that So anyway a couple of stocks on this watch list I want to start with the biggest percentage gainer and the most active on the list So you have the stock right here that's making a nice little move sql Now I brought up level four and it was happened to check it out There's not really that many sellers out there. I mean these guys started showing up around 8 a.m You could see that these 34 000 share sellers and 23 000 have started trending down right here So, you know, that's why we're making lower lows. So the stock has gapped up pretty big. It's up 127 Um, it does have major support levels around here at a buck 80 The only issue is I do not see anybody at a dollar 80. So To have a potential support You got to have those buyers out there to support it and as of right now, I don't see it But it's got traded 15 million shares and I'm going to I'm going to keep an eye on that so sql We're gonna I'm gonna I have it on my watch list because it is a friday now hills is going up pretty decently Now hills had a couple of iceberg orders out there that just got executed They were out there around 955. I was watching it earlier And you could see them they were right here and they started getting executed So we're starting to make a higher high got a little buyer here for 13 000 right around here Uh, you could see them right here around 17 and whatever. So got a couple of buyers out there Got those supports right there. We got a nice little push. So we'll see how What we got going on on this stock right here um Not bad. Not bad. I I'll keep an eye on this one too. Um got good volume 15 million shares up 82 percent Hills, why not, you know So, oh, yeah, by the way, that's a good one too. Um, Mike Mike Allen on on On youtube says the stock also the ph un Yeah, that was the other one that if you missed If you missed the dwac you could have got also that one back there when that w the dwac went And that thing took off the next and that was least a lot cheaper a lot more A lot easier to trade same thing brian. I saw in the cyber group room in the trading room the same thing Anyway, let's go back to our stocks that we're watching This stock has gapped up about 44 percent. They're all really nice tradeable stocks nice spreads You know nice little moves, but um, not really You know, it's kind of flat so I don't know It had a good little push for the last couple of weeks a couple of days Was that 210 went to almost four It's it's trying to go up a little bit. Um, if I bring up the day before you could see right here It was a great stock that ran yesterday going into the close Which by the way, I forgot to mention that one too But um, you could see it right here great great run up here And it was after hours earnings that made the stock run, but she's not really going anywhere She probably tested support levels right around here 280, but i'm not seeing anyone out there. That's my issue You know, I'm looking out there. There's just nobody out there I see sellers I see sellers. I don't see any buyers. Okay, so no buyers No, uh, no long That's the way I look at it edibles This stock is actually moving we we trade the stock just the other day And it looks like she's moving again So this stock took a very very big hit. It was our $15 stock got destroyed Just got hammered. It could be a little dead cat bounce.  Anybody know what dead cat bounce means Actually, I I talk about just give me a yes or no anyone here on on social media cyber group members Anybody know what a dead cat bounce means? Dead cat bounce. What does that mean? See if you guys could figure that out daniel, do you know what it means? Anyone else? Uh, brian, you must be a really good typer. There you go. No one on social media here knows look at all the traders though That's why you guys on social media. You got to know how to trade. Okay. This is the problem You got to you got to know the lingo of trading That's right. So basically it's when it falls and gets back up and runs. So basically what they say is Because I have cats. That's what I love cats and I'm uh, I like dogs too, but I don't know I just think cats are easier. So anyway, if you hold the cat upside down and you hold its legs a cat spine is um Flexible like it could take it could put his feet down and it's and it's legs up I wish we could all do that right could you imagine we'll probably have a broken back Actually, we would have that means we do have a broken back So anyway, if you hold it upside down you drop the cat it actually if you hold it'll act dead And if you drop it it could actually land on its feet and it could run away So that's the that that was just a whole thing about Market makers say, hey, it looks like a dead cat bounce. Okay. So, um, basically that's basically what that means um So anyway, that's what I kind of looking for stocks that have dead cat bounces. I think Edible could be a little dead cat bounce because it was that was up in that $15 price range at $4 I want to keep an eye on it. What else do I have here? So sad stories about lift, right? I can't take a personal let me tell you something I I had lift I know we a lot of you here remember we traded lift a long time ago And I was trading the stock around the $40 price range and I was such a fan on it and believe it or not I lost money on it.  I like the stock I I use lift all the time not always big uber fan sometimes, but I use both But mainly lift I guess I get the points on delta because I've always used it to go to the airport But um, if but if I let my emotions get involved And say, you know what there's a real company. I like it. I love the people that work for them You know, and I know some people here That are students of mine, you know, once again, some of you are lifting uber drivers And I always try to I I know how hard you guys work and I know how they make money But um At the end of the day, I can't be I can't go down with the sink and ship, you know So I think I was trading around like 47 I ended up getting out of like 42 and I was making these lower lows right here I'm like, you know, I like the stock it's got to come back because I remember when public You know, and you could see it here on the long-term chart I mean the stock started uh when all the way up to 75 So we were trading it in this range right around here because we had resistance levels back here We had supports, but you could see it started making lower lows.  The buyers are gone. Let me just get out of it I'd sure enough. Yeah, there it is 42. That was a price and then I once I got around 35. I said, you know what Party's over. I said next support level look what ended up happening when it'll be here, but I might be a buyer around 10 I might be look right support levels here Uh, it could get a little bit of a bounce here, but we all know when you break all-time lows That's not a good sign. Anyway, so lift is on our watch list L y f t and with lift it's part of the industry just likely do with ai Uber's going to be on that watch list too So uber obviously See what's nice about uber is uber had a nice rally Okay, uber had a really really good rally and you could see it right here Right around four o'clock. It came down when those earnings announcements came out So uber could still probably listen they they trade side by side I mean, I know uber has uber eats and all that good stuff, but um uber could still be a short Yo could be a potential short if this lift keeps going down Because you're talking about $37 stock versus $15 was it 15? I said it's 13 11, I'm sorry $11 stock. So we're gonna keep an eye on those two stocks. All right Anything I'm missing. I did not call out Uh grand is said hills making h o d h i l s. Yes hills. Like I said, that was the second stock on the watch list Thank you very much grant. Yep. It's holding there Let me go check over here on level four and see what we got going on Yep, I told you like listen looks like we took we took those sellers out right here and looks like we're breaking out here So looks like 240 to 250. We got some decent some sellers not anything super You know crazy, but you do got this big uh 67 000 share buyer at a buck 50. All right, so Let's get ready for the market open everybody. We got less than 15 minutes We will be back here live again if you're watching on social media We are going to broadcast live again at 10 a.m 30 minutes after the open if you want to join our trade room Just like us and friend this we're live here every morning at 9 a.m And uh 10 o'clock and at 2 30. So please don't miss out and at 12 o'clock I'm gonna teach you how to trade the super bowl. So you don't want to miss that too So make sure you are at you have the access and and they'll get our alerts cyber group members Listen, it's nothing to convince you make your days pay make your money once you're done move on We'll see you back here on monday. Enjoy your weekend because that's what it's all about It's not a full-time job. Just make your days pay and uh move on. All right guys. Good luck everyone. Happy trading I'll see you back in a little bit. All right


I appreciate your concern and interest in the budget and what we're doing about the federal budget deficit. I understand you've been briefed in some detail on our fiscal year 1987 budget proposal, and the reasons why we believe this proposal is the best way to reduce the deficit. I know they had just a little bit of a discussion here. I think that 86 is a year of decision. The year to decide whether we have the will to last bring federal spending under control, remove the last remaining obstacle to a future of growth and prosperity. It's the year to decide whether we can make the tough decisions that we agreed to last year when the Grand Roman College legislation became law. And it's the year to make clear to the world that our commitment to peace through strength will not be compromised. It's the year to maintain our continued opposition to raising taxes, which I believe could harm the economy and take money from Americans in order to protect government programs that we don't need and can't afford. Today, deficit reduction is the law of the land. Grand Roman College mandates the declining deficit path lead to a balanced budget in fiscal year 1991. So the deficit will be cut. The big question is how? Grand Roman College's automatic spending cuts are only if Congress abdicates its constitutional responsibility and fails to pass a budget that meets the deficit targets. In other words, Congress still has a choice. It can be responsible by making those careful and sometimes difficult spending decisions or it can simply drop the ball to not be across the board cuts take over. The budget that I will soon be submitting to Congress was carefully prepared by taking into account the defense needs, the proper role of government and the Grand Roman College's deficit targets. It will meet the 87 deficit target of 144 billion without harming the economy through a tax increase or compromising our national security. It will preserve the social safety net for those in need it will not penalize older Americans by reducing social security benefits and as I'm sure Don, Cap and Jim had mentioned our budget will include a modest increase in defense to sustain recent improvements in capability. Now some will say it has too much for defense. The proposed funding levels for national security in our budget are consistent with the real growth rate, the growth path that's agreed to by myself in the Congress last year. And Cap has already told you, in spite of all the drum beat of propaganda the defense budget is a little more than a fourth of our budget and that is counter to a history, well historically back through the years defense was considered to be basically about a half of federal spending because it is the number one task of the federal government. Higher taxes are no cure for the deficit. Reducing the deficit through tax hikes I think would impose substantial new tax burdens on American households and as experience has proved, reducing incentives for Americans to work and save and invest and thus choke off the creation of new jobs. And the experience has also shown that higher taxes do not necessarily go to reduce the deficit. More often are used to justify increased federal spending. With one line that was spoken in a debate in the floor of the Senate to the back in 1913 when they were passing the income tax amendment. And I think it's something we should always remember that one senator stood up and deliberately pronounced that we had to have the tax not for government's needs but for government's wants. He thought that was a good idea but I think we've had too much of a government's wants over recent years. The current economic expansion is whether it was third years you know over nine million jobs have been created since November of 1982. Inflation, I hesitate to use the term under control. It has been brought down to a third or less of what it was when it came here but I don't think inflation is under control until it's zero. And we've been having some meetings around this table as to just how we're going to accomplish that even though we've been downing four percent now for quite some time. Nominal interest rates are coming down and are well below the levels that were reached in the 70s and early 80s. Excessive federal spending and a large federal deficit for the last major obstacle is I think blocking our path to lasting prosperity. The budget that we're sending up on February 4th will put us in the right track to a balanced budget as I said by 1991. Then I feel that we should go all out if not before then for a constitutional amendment that bans forever federal deficit spending. So we think that we'll reduce the deficit while promoting economic growth and meeting our natural security requirements. We've made the tough decisions and now it's time to see if Congress is up to the challenge of passing a responsible budget. I need your support and those of all of your colleagues and I encourage you to make your views known. We're in for a tough battle but together we can reduce the deficit and set America on a path to permanent prosperity. Now that's enough for me. I'd like to hear what's on your mind and what questions you might have. Good afternoon Mr. President. I have a very special quote to introduce you to. This is Dennis Fox. First place, best in the world on advertising design Mr. President. Congratulations. This is Kenneth Anderson who is third place in significant investments in radio and TV player. Congratulations. This is Anthony DeCellis fourth place certificate of excellence in milling. Joe House fourth place in industrial electronics. Ricky Kelly fourth place in all reveal repair. This is Aaron McKenzie fourth place in turning. This is Oscar Mercado who won a certificate of excellence in bricklaying. This is Larry Johnson an old friend chief executive also. This is Harold Lewis who is the chief director of the U.S. Geo Olympics with Vicka. And this is James Voorhees vice president of General Motors who represents 500 corporate sponsors of this program. Now I think I'm supposed to be at the 100th or something. Yes, I had. Someone who did. That was in 1983. We often play that speech. We met with two of your secretary Bennett and secretary Brock. So the touch faces and two of those important agencies involved in training. So that's good we can do quite a job on education. I just asked. Right. Great pleasure to meet you. Thank you very much. Thank you Mr. Behind you. President Alonson. Thank you. Thank you. Thank you. Thank you. How did you get in? Thank you. You read about it? I read about it. How did you set it? Good. Good. How do you do? This is an Alger here. Send me an Alger here. I don't know. I don't know. You want to show that gentleman there. That's the big Kyle. That's all. How did you do? Roger. Roger, on the screen. Congratulations, Mr. Alger. Thank you. How are you? It was great. Yes. I'm sorry. Sure, I'm happy to have you here. I'm very happy and proud. Great pleasure. I'm close to being there, and I remember that I used to have a test with my son growing up. I was just one month old. I used to do two laps in the pool, waiting for the day when he was good. I didn't know he was a boy. I didn't know he was a good player. I was ready. I didn't have another answer to it, though. I switched from two laps to one. I was not worthy of dignity on my turn. I would like to present to you this helmet that I wore last year of winning the race. So we're in winning the championship. I'd like to present it to you, and then we can wear it whenever we need it. Which is probably all the time.  I think a good place to get that over is if you're not wearing it. That's great. You're very proud to have it in here, I'll tell you. There's something that we mentioned. All the answers in the court of criminal are in all of us happy. Not to everyone. Yeah, it's surprising. We used to wear a lot more. And here's a jacket for you. I'm not sure if you can do this. That's a thousand of that here. Thank you very much. We'd like to have everybody group around the president board.  We have a book about the whole family. Thank you very much. Read that web ads. I met you in Seattle. We were waiting for a free lecture. You would call us? Yes. No. You're going to have to come to some of the races now. Between your times, right? Since I've been here. How could you get a son and a president? Super.  No. Watch this way, please. We would have you be with me. I'm still traveling by horse. And a sculptor has done those. And a dozen of them are. The first company that has history is the development of the western center. It starts with a 1540 spending. They're kind of interesting to begin with. In the west, in the early west, there were no fences. That's right. So the lasso came into being. And then the first time it came away. And they used to use the English war saddle that the Spaniards had left. There was a big kind of a dome on the saddle. And the first time, if you want to put a lasso on a spear, and then left the saddle head first, they began carving a notch around that big thing so that they could doll it up real quick because they now do it with a horn. And that's what led to the horn. It wasn't the same. Well, I think they won us over here. Yes, they did. This is my bottom. It won't be there if you don't open it all the time. And we used to go over every year on vacation there. And we used to go on now. You're mounting my picture. This was a card showing. Thank you. Thank you. Thank you. Thank you for your time. Thank you. That's the one I've never seen. We need to have newcomers. Probably the most fantastic sporting event you've ever seen with a crowd. I know you started the race at Daytona last year for a feature from your aircraft. Yeah. It's a real thing. You'll have to be after them to come only if you have a promise to answer the question. Well, yes. Let's see. Well, baby. Well, that's great. That's terrific. For any president.  But the first time in the history of the race, it's always been in the right. It's great honor for us to be here today. I'm very proud and happy to be here today as a race driver that picked you up and sort of invoke his name and when I was a sportsman, I was in Des Moines and Arne Orphi came there. He worked for Chrysler at the time and he was sport convertible and what he would do, it was a state fair. So he'd come there to appear and have to lecture on safety and driving. So I interviewed him on our station and then I had to go out to the fair and he was to us. So go out with him and you won't believe it, just go to him.