Ahora les quiero contar un poco sobre la trayectoria del profesor Silvestre Eifinger, profesor de economía financiera en la Universidad de Tilburg, Holanda y profesor visitante de economía en la Universidad de Harvard. Durante su trayectoria investigativa ha enfocado sus análisis en la política fiscal y monetaria y a la integración económica y financiera en Europa, siendo profesor visitante en los bancos centrales de Alemania, Japón, Francia e Inglaterra. Termina siendo asesor especial del Fondo Monetario Internacional y de la Comisión Europea. Sus publicaciones se encuentran publicadas en las principales revistas científicas como el Journal of Money, Credit and Banking, el Journal of Banking and Finance, el Journal of Public Economics, Oxford Economic Papers, el Open Economics Review y el European Journal of Political Economy. Bienvenidos, profesor Silvestre. Gracias por tu buena introducción. Gracias por estar aquí. Quiero agradecer al doctor Santiago Castro por su invitación. Hace un año aquí en la Convención, en junio, el S.C. Es un placer volver, no solo a Colombia, sino también, especialmente a Catarina. Estamos muy, mi hija y yo, muy contentos de estar aquí. Gracias mucho. La investigación que haré, o que presentaré, es que veis mi nombre solo ahí, y no es apropiado en el sentido de que el papel fue hecho juntos con Jonathan Malagon. Y la historia de ese papel es la siguiente. En la Convención de la Convención de la Comisión de la Colombia, el año pasado, presenté un anuncio de riesgo y laquisidad de los mercados globales. Y recibió muchas preguntas de las personas en la audiencia. ¿Cuáles serán los puestos, o que serán los puestos para países latinoamericanos? Así que tenía una gran discusión también con Jonathan. Como dije, Jonathan, podemos hacer un buen papel sobre la política de la Convención de la Convención de la Convención de la Convención, por el sistema de reservas federales, por el ECB, por el Bank de Japón, y vamos a probar eso. Vamos a probar los puestos de los grandes bancos centrales, el sistema de reservas federales, el ECB y el Bank de Japón. Y ¿qué significa para los países latinoamericanos? Creo que estas son preguntas muy importantes. Y la buena cosa es, Jonathan es muy entusiasmante, es que fue una buena cooperación, y Jonathan ha hecho muchas de las estimaciones, el profesor no tiene mucho tiempo para eso. Y lo sé, y estoy muy agradecido a Jonathan que podamos trabajar juntos en este buen papel. Ok, entonces, aquí vamos. Y tengo un montón de slides, así que tengo que ser muy, tengo que acelerar ahora. Aquí tenemos, ok, si no, voy a tomar este mic. Aquí tenemos la introducción. Voy a seguir aquí, eso es más fácil para vosotros. La introducción es muy breve aquí, vamos a la crisis, 2008-2009. Es financiero, originó en las economías desarrolladas, especialmente en los Estados Unidos, y, por supuesto, esta fue la crisis, que fue, yo diría, lo más fuerte desde la Great Depression. Si veamos a la posibilidad de monitorar la política aquí, veas los grandes bancos centrales, así que la eurozone, Estados Unidos y también Japón, y veas también que este intercambio fue al domenio negativo. Y lo que podemos ver ahí es que las opciones para los bancos centrales fueron limitadas. Y que tenían que ir de la política de monitoría convencional a la política de monitoría inconvención y no convencional, porque de la cero cero. Ok, entonces, ¿qué pueden hacer? Y aquí veas también otro problema. Y, por supuesto, la política de monitoría no era posible, porque el hecho de que había un defensor fiscal en Estados Unidos de más de 12%, un enorme ratio de ADP-GDP en Japón y más de 200% aquí, y, por supuesto, el defensor fiscal en la eurozone, que tenía que hacer, por supuesto, con el hecho de que la eurozone también fue confrontada con los constracciones de la estabilidad en el riesgo, que no siempre ha sido tomado en serio, yo debo decir, pero, entonces, sólo había una opción que el Bank de Centro se desplazara. Y lo que sucedió, el Bank de Centro se desplazó por la política de monitoría en Japón y se desplazó, y lo que sucedió, el Bank de Centro se desplazó por la política de monitoría. Aquí veas, las economías desarrolladas, como las de los Estados Unidos, la eurozone, la economía y Japón, han ido a una política de monitoría inconvencional, una política de monitoría inconvencional que no es solo inusual, pero también, yo diría, más allá de las mandaciones, en el caso de la ECB, es dentro de la mandación del sistema federal, porque tienen los objetivos de la inflación baja y de incrementar la output y la empleada. Aquí vemos los assetos totales de los Estados Unidos, la eurozone y el Bank de Centro de Japón. Ahora veas que empezaron con 100 y ahora se desplazó, se desplazó. ¿Puedes imaginar? Así que significa que 40% de la GDP está en términos de las secuencias de los bancos centrales de las balas de balance. No es solo inconvencional, es incluso bizarro. Aquí vemos la división, el U.S. fue, por supuesto, el primero para hacer eso, luego el Bank de Japón, pero cuando los japoneses hacen algo así, lo hacen muy bien. Y luego, aquí tenemos la Unión Europea y la Unión Europea empezó tarde, entonces es detrás de la curva. ¡Bienvenido, Jonathan! No me han perdido las buenas palabras que le he votado a usted, pero de todos modos. Así que aquí vemos la división y el tamaño de esta policía desconvencional. Ahora, dos resultados de QE en el U.S. y eurozone en Japón, así que la policía desconvencional ha sido maybe successful, especially the first round, the first round of QE was more successful than QA2 and QA3, and why? Because it's announcement effect, which matters. If the central bank enters the capital market buying bonds and giving liquidity, the money market is small, but the capital market is enormous. So the announcement effect, the expectation channel, the signaling channel, how you want to call it, is very important. That is the reason why the first round of QE was in the U.S., but also in the eurozone, more successful than the later rounds of QE. And of course the larger liquidity had significant effects on emerging financial markets. And Latin America was not an exception to that. Okay, there we go. Now, the question we wanted to investigate is, is there a differential incidence between conventional and unconventional monetary policy of developed economies on Latin American financial markets? That's the big question. Do we see the difference? Now, here we have the incidence on Latin American financial markets measured in four dimensions, so two are related to asset prices, and two other are related to interest rates. Let me take you through this. So here you have this monitor. You have a conventional part, you have some unconventional parts, here you have the financial market. There is one caveat Brazil, Mexico, Colombia, Chile, and Peru. You know that this group of seven larger countries, there were two countries which had not reliable statistical information. Venezuela, you can understand that maybe, and Argentina. It's not very shopping in that sense. And here you see the asset prices, fixed income market and equity market. And here you see the interest rates, monetary policy rates and active interest rates in terms of lending rates, so both aspects. The hypothesis, what were the hypothesis which were interesting? First hypothesis, unconventional monetary policy developed economies have a similar statistically significant incidence on Latin American fixed income markets, bond markets. Second, conventional unconventional monetary policy in developed economies have a similar statistically significant incidence on Latin American equity markets. Third, in Latin American countries where the exchange rate pass through is low, but both conventional and unconventional monetary policy have null or low incidence on monetary policy and credit market rates, lending rates. In Latin American countries where the exchange rate pass through, that is the fourth hypothesis, high transmission mechanism of monetary unblocked conventional, unconventional monetary policy. In developed economies have statistically significant effect on interest rates. Both monetary policy rates and lending rates, in contrast unconventional monetary policy, is to be expected to have no effects on the interest rates. And then fifth, since the US is the main trading and commercial partner of Latin America, the financial spillo's monetary policy of the Fed are expected to be greater than the ECB and Bank of Japan. And then last, free trade agreements between developed and Latin American economies are likely to exhumate the financial spillo's of international monetary policy. Those are the six hypothesis. Now, literally I will skip that a little bit because of the sake of time, because I always arrive in negative time. We have negative interest rate and I always arrive in negative time and I try to minimize it. So here we have the literature, commercial channel, trade channel, exchange rate channel, credit channel, portfolio rebalancing channel, signaling channel, financial marks channel and liquidity channel. You see that these channels are mostly important for the interest rate of the interest rates and these three channels are mostly important for the asset prices here and then there is one, the liquidity channel, which is mostly important for both of them. Well, I will skip this, otherwise you can read it, otherwise I don't want... Of course I'm a professor, but I don't want to have a whole exposit of the literature. Ok, the methodology. Variables, this is very... So we have of course exogenous variables, so explanatory variables, you could call that conventional monetary policy, which is in this case defined as a one-year government bond yield, unconventional monetary policy, monthly asset purchase by central banks. Then we have endogenous variables, so the depending variables, if you like to... So the bond market, fixed income market, equity markets, monetary policy rate, so... And of course the active interest rate, credit market interest rate or lending rate. You see that those are defined by the ten-year government bond yield, stock market index, monetary policy rate and nominal active interest rates. And here are the sources, you can see that. And of course we have tried to have comparable proxies for that. Oh, that's too fast, control variables. Control variables are necessary to make a good estimation of that. Of course, which you always find in those controls are the CDS prices as normal. CDS is more or less a kind of measure of systemic risk. However, be aware that CDS prices nowadays are less informative than in the past because CDS prices are sometimes used by central bank as financial supervisors information about systemic risk. And then you get the lucas critique, or good or slow, that if you use that information, the value of that information, CDS prices, as a measure of systemic risk becomes less. And so be aware that you often have to be cautious. Then this market volatility indicator fix, this is from the Chicago board options that measures the volatility in the market. Of course, the output gap, difference between actual potential GDP, inflation gap, difference between actual inflation and target, inflation rate, oil prices, commercial integration, you could say trade integration if you prefer that. So it is for free trade agreements, dummy. I will come to that, it's not so successful and I will explain you why. But it's necessary as a control variable. And then the effect of the pass-through, also measured by a dummy, all calculations. So here we have also some new item, I had, sorry, that's the shadow rates. And let me explain a little bit about the shadow rates. I don't know if everybody is aware about the shadow rates, but it's a very important issue by Wu and Xia. It goes back to 1995, Fisher Black. He was the first who spoke about shadow rates. The point is that you cannot, you have to zero lower bound. You cannot go in the negative domain with your normal interest rates. But if you have, for example, quantitative easing, the effective, effective interest rate goes in the negative domain. And how do you measure that? You can measure that with all kinds of forward rates. Wu and Xia, I can really recommend, did that, for example, for the Federal Reserve System. So when the effective Federal Reserve Rate was, you know, between zero and 25 basis points, you cannot go below zero. Then how do you measure then the effective interest rate in the negative domain with the QE exercise? So, and here you see this, this is very nice.  Here you see the aeonia, that's the money market interest rate in the Eurozone, which is comparable to the federal funds rate. And you see here that, of course, the aeonia went to zero, but the shadow rates really tumbled in the negative domain. So that's also a kind of benchmark for the effect of conventional and unconventional monetary policy. Question is, but that's a very difficult question, are both policies independent, orthogonal, as econometricians will say. That's a very difficult question to answer, like symmetry and asymmetry in the expansionary and restrictive phase. But that's maybe something for the future. Okay, then we go here to the approaches. I will do that also very briefly. Here you have the panel date approach. What is panel date? Very simple. Cross section time series. You maximize the number of observations. It's, with five countries, a limited number of observation in time. It's the best you can do. And by the way, I think it's one of my favorite approaches. Why? Because you know what you're doing. That's something different than if you go to a factor autoregressive model, or what now people like to do at central banks, DGSE models, makes no sense, because you don't think anymore. It's measurement without a theory. Like Comans, Charlie Comans once said. Measurement without theory, because you have to have a kind of a hypothesis. You have to have something you want to test. Hypothesis, in our case, six hypotheses. And the moment you put everything in the computer, and then especially when you measure only the first and second moment exposed with value at risk of a structural var models or DGSE models, then you lose a lot of information. So here, the differential effect of developed economies spillovers over Latin American countries, eight specifications. What you see here, so the gorgeous variables. So here are the variables and linked to exogenous variables. Now let me explain here bonds, equity prices, monetary policy rate, active rate, landing rates, again. And here you find the conventional monetary policy, C. And here you find the unconventional monetary policy. U.S., E.U., Eurozone, and Japan. So that's the structure and here you find all the control variables. So that's the structure of how it's cooked. You always have to explain the recipe book. Okay, the robustness check is also possible because I spoke already about the shadow rates. Very interesting to check the shadow rates, but the problem is there are no shadow rates for Japan. They are not available. They are only available for U.S. E.U. So here the bond rate, shadow rates, equity rates, monetary policy rates, and active rates. So and here the control variables. So that's the structure. That's just a robustness check. It's a robustness check of the test of the asset purchases. Now, this is already a nice picture where you summarize everything with the dots. Here you have direct, significant, and strong effect. That's the real green dot. Direct, significant, but weak effect in terms of coefficient, indirect, and insignificant. Now, here fix income market, equity market, monetary policy rate, lending, interest rate, conventional monetary policy, so the interest rate policies, unconventional monetary policy part one, that's monthly purchase of bonds, government bonds, and sometimes corporate bonds in the case of TCB. And here the testing of unconventional monetary with shadow rates. Now, what is the outcome? You see that conventional has strong effects, of course, on fixed income markets and monetary policy. That's not surprising, but what you see is unconventional monetary policy, both by the monthly asset purchases and measured by the shadow rates, have, in most cases, direct, significant, strong influence on the bond market and the equity market, and in this case, direct, significant, but sometimes a weaker effect on the victim. But they all have a direct and significant effect. It's very important to see that results. Now, first, this is the question, what is the differential incident, conventional? That's the main question. There we go. So first this, what is the outcome of this? Now, forget the equation. There is a statistically significant relation between the conventional monetary policy in the US eurozone on Latin America, long term interest rate, the bond market. The effect of conventional monetary policy in Japan is statistically not significant. And then here you see the coefficients, which are explained, but I won't elaborate because of the sake of time. Otherwise I go in the negative domain and we won't, don't want it. So conventional monetary on equity markets, so relationship between conventional monetary policy in the US, in eurozone, and the Latin American equity price is indirect. This is observed through long term rates of sovereign bonds, government bonds of the region. The effect of conventional monetary policy in Japan is statistically significant. And then here you see the effects if you're interested in that. I suppose that this presentation also available for people. If they want to, if you want the presentation, I can ask you on the top for that, or the Salma. Okay, so then we have here conventional monetary policy on monetary policy rates. There we go. There's significant relationship between conventional monetary policy, US eurozone, and Latin American monetary policy rates, but the effect is low. Incident of conventional monetary in Japan is not significant. That's also understandable because of the trade and financial relations with Japan are limited. Yes, I know, I see it, yeah. That means that we see that there's effect from US, Fed, the Federal Reserve System. We see that there's effect from the eurozone, but Japan is negligible, that you can forget. Okay, then conventional monetary policy to the lending rates, very important. Here is the result. Relationship between conventional monetary policy in US, eurozone, and Latin American. Lending rates is indirect, so it serves through monetary policy rates. Effect of conventional monetary policy again for Japan is not significant. Now we go to unconventional monetary policy. For the fixed income market, here we have it. There is a significant relationship between unconventional monetary policy, US, eurozone, and the long-term rates in Latin America, but the effect is very low. The incidence is not significant for Japan. Now we go to the equity markets. Relationship between unconventional monetary policy, US, eurozone, and Latin American. Equity price, direct, significant. It means, it's very important, that there is asset price inflation. So the asset prices in Latin America are blown up by QE, for example, of the very reserve, and later on by ECB policies. So that's important to make proper interpretation of these asset prices. Here the effects on monetary policy rates. So there you see that unconventional monetary policy and the policy rates not significant, and lending rates is also not significant, which you already would expect. Okay, results, robustness check, exercises, there we go. Now using the shadow rates, and you know the shadow rates are meant to measure the effect, not only of conventional monetary policy, so at the zero lower bound you cannot go in the negative domain. I can go in the negative domain with my time, but the central banks cannot do that. That means that you have to measure the effects of asset market purchases, quantitative easing for the interest rate, what they would have been. Now, you see using shadow rates as a measure of unconventional monetary policy, you find the same results as before. So it's a robust. That's important to understand, a robust. So there's a clear relationship, not only in asset market purchases, but also in shadow rates. Here we find the effect with equity prices, which is clear that the shadow rates in the US are significant, however in the eurozone, not significant. We have to think about it, but maybe the reason is that in the US, the trade and financial relations, that could be the one explanatory factor in that, and the second one, the US Federal Reserve started earlier with the whole process of QE of asset market purchases. That means, of course, that maybe the measurement of the effect for the equity price in Latin America, you should have longer series beyond 2016. Previous estimates now, the relation between unconventional monetary policy US, et cetera, and interest rate is not relevant. And then we come to the conclusions, and I see that I'm still in positive time. So that's remarkable. But then I use my time and go a little bit in the negative domain, if I may. So here we have the conclusions. Conclusions about the first hypothesis. So you see that this fixed income markets are really influenced by unconventional monetary. That means that the globalization in terms of monetary policy, especially of unconventional monetary policy, is really a serious affair. However, fixed income effect is low, but it's significant. Same way, conventional, unconventional monetary policy have, and that's important, strong effect and significant effect on the equity markets. That's important to know how you make the interpretation of these asset prices. There is asset price inflation. So be very careful with that, with the interpretation of these prices. And then, unlike the third hypothesis, who you know that was about exchange rate, pass through that's not significant, that has not an important meaning, you see also that both, in both cases, exchange rate pass through, was not relevant for the endogenous variables. So let's go to the conclusion, the last set of conclusions. Conventional and unconventional monetary policy, and you have a greater effect than the eurozone. Yes, I'm almost ready. And that is explainable by the trade and financial flows, which are, with the US, much more important. However, it's changing. Trade and financial flows with the eurozone are increasing. That means that the effect in the future, in terms of globalization of monetary policy, both conventional and unconventional, will increase for Latin America. And then we have the, sorry. Then we have the hypothesis six about the free trade agreements, not as controls, they were meant as controls, but not so important. And why not so important? The reason is the following. It's like capital market restrictions. People always look to capital market restrictions. I did a lot of work on capital market restrictions. That's, those are dummies, zero, one. That doesn't matter anything, because capital market restrictions may be there, but not effective. It's much better to look to covered and uncovered intersparity. Covered intersparity is a much better measure for capital market restrictions than the restrictions as such, because they don't necessarily have to be effective. The same applies to these free trade agreements. They may be there, but not effective. And what you want to measure is the effective commercial trade integration. That's what you want to measure. So be very aware about interpretation of that. And then the last, and then I'm finished. That's, of course, a very interesting conclusion. Of course, the financial markets in Latin America are influenced by QE in the US and QE in the Eurozone. But there's good news, eh, there's good news. The good news is that the shocks, the vulnerability to external shocks in Latin America has become less, has become less, has diminished. And then the question is how do we explain that? Now, one is maybe the macro protection policies there, but also the monetary policy rules. And I think that has also to do with the fact that Latin American countries have learned that they need to reduce their vulnerability to, I would say, the shocks coming from the US and shocks coming from the Eurozone. And I think that's good news. And I think Latin American countries, central banks, should continue on that path. Thank you very much.


Hey everybody, we are back at a very hot and steamy Murph 2021 and I stopped by the tripod's garage booth here. I'm with John. How are you doing, John? Okay, how are you doing, Jim? Oh man, I'm doing great. It is very toasty in here. It is. Well, my glasses are fogging up and as you can see in the videos today, I'm going to be moving my hair a lot because it's very warm. The floor is actually wet in here. Oh yeah. You know why you see these here? Yeah. I already fell. Oh no. I'm durable. Well, we were talking and we saw the drones you guys have been streaming building the droids, I mean. So tell me a little bit about the droids. Okay. Well, this is a Michael Badley design. You actually had a creator on your live stream that uses some of these designs are great. So we were doing a stream every week, every two weeks about how to build these and we've done four drawings so far. Four. Okay. Two small ones and now that was the medium and now we're, I'm building a drinkbot droid. Oh wow. It's called BR2U, Brew2U. So it's pretty cool. Nice. So that's in parts right now but yeah, this is, we could go ahead and take the dome off if you'd like and take a look at the inside. Sure. Let's check it out. So while you're taking the dome off, how long does it take to build one of these? Oh, I have about running on three printers, about 100 hours into it. Oh wow. Yeah. All right. So between painting, electronics, soldering, testing, the tracks are all TPU. Okay. 95, the drivetrain is carbon fiber, priline was a polycarbonate and the rest of it is all just regular PLA. PLA? Okay. Yeah. So the dome took some time. I wanted the dome to actually look decent. It's not my best paint job but I think it came out pretty good. Yeah, that's not bad. Yeah. So that was lots of priming and sanding and body fill. I'm going to have to say that's better than I would do so. Okay. Well, that's appreciated. That is awesome. No, but I mean you have a bunch of really cool stuff that you do on your channel. Where can everybody find you if they wanted to find you? Okay. Well, I have a tripod garage on YouTube, tripod garage, I'll spell out one word or you can just paste it out. I'm on Instagram, Twitter and Facebook. So and you're one of my first people I ever watched on YouTube. So thank you. That's awesome. Well, I tell you, thanks to Thangs for bringing us out and Greengate. Have you printed with Greengate? Actually, we got a roll. Nice. It's a little wet here. Yeah, it's a little wet here. So we're good. You know what? I got printed at home because actually the wife went and bought two rolls yesterday. That's awesome. Well, again, thanks to Thangs for bringing us out to Merf 2021. Right here, they actually got a couple rolls down here. I can see them from here of the Greengate 3D PETG in their booth and we're going to be talking to them shortly. But again, thank you guys for having us at the Tripods Garage booth right here at Merf 2021. How's it going guys? Good. You like that? That's awesome. Take a sticker. It moves. It actually moves and drives. It's pretty awesome. Well, John, thank you so much. I appreciate it. Good to see you.


Hi, I'm Kunde Boer, this talk is about random self-rediscibility of ideal SVP via Arrakelof random walks. This is joint work with Leo Duca, Alice Palémarie and Benjamin Wieselowski. This talk will be about a worst case-to-average case reduction for this shortest factor problem in ideal lettuces. We do this by considering the space of all ideal lettuces which is called the Arrakelof class group. In our main result we consider the Hermit shortest factor problem instead of the ordinary shortest factor problem. In the Hermit shortest factor problem we ask for a vector that is short relatively to the volume of the lattice, which is a more absolute notion and to avoid having the volume of the lattice in the equation we just assume that a lattice has volume 1 which is also particularly nice in the context of Arrakelof theory. So in our context the Hermit SVP problem reads as follows. Given a lattice of volume 1 find a vector shorter than gamma where gamma is a positive real number. Our main result specialized to a cyclotomic range of rank n reads as follows. If you can solve gamma-hermit SVP over random ideal lettuces you can also solve gamma-prime hermit SVP over any ideal lattice. But with the cost of a small blow-up factor. Under the extended Riemann hypothesis and mild assumptions on the class number of cyclotomic fields this blow-up factor can be shown to be within a big O of a square root of n. This theorem essentially states that worst case and average case ideal lettuces in cyclotomic rings are approximately equally hard to solve hermit SVP on. So there's not a big variation in hardness of solving hermit SVP in ideal lettuces. How does our result relate to the literature? Well in 2010 Gantry also derived the worst case to average case reduction for ideal lettuces. We improve on this result in three ways. 1. Our reduction is tighter. The blow-up factor is smaller. 2. Our description of the average case distribution is mathematically more pleasant. We use the uniform distribution over the Arkeloff class group. And 3. We do not require a factoring oracle. So our reduction can also be done in the classical world. The machinery of Arkeloff class groups is actually already used in recent cryptanalysis, though not explicitly stated so. In those works it is viewed more in an algorithmic sense, using a matrix of relations in a combined class group unit tourist object. I would like to stress that the strategy of the proof of this self reduction in this work is exactly the same as those for the discrete logarithm problem in elliptic curves. In those works they show that the discrete logarithm problem is essentially equally hard for all Isorchini classes. This is proven by a random walk over the Isorchini graph. To define ideal lettuces we need orders. Those are sort of discrete rings. And to get started we are going to consider real quadratic orders first with the principle ideal property. Real quadratic essentially means that the order consists of just the integers with some extra number, a square root of a positive integer. Think of the square root of 2. The principle ideal property means that all ideals are principle. That means that they can be degenerated by one element. Coincidentally this property means that the class number is 1. Orders being real quadratic means that they can be embedded into R2. And by R2 we mean the ring R2 with component wise addition and component wise multiplication. As the order must contain z, the ordinary integers, we have to put that first. And the only possible choice is along the diagonal. 1 sends to the pair 1 1. It acts as the unit on both axes. There's also the square root in this order, which is sent somewhere not on the diagonal, in a very specific way. And this fixes the entire embedding. Note that this embedding makes the order into a lattice and a ring. Having the order properly embedded, how do we construct an ideal lattice? To obtain an ideal lattice we just multiply each point in the order by an alpha in this embedded space, the ring R2. Because we have the principle ideal property, we can reach any ideal of the order in this way. By varying alpha, we obtain many different ideal lattices, like this. Though in the beginning of this talk we mentioned that we only wanted to consider ideal lattices with a fixed volume. But as this lattice is more sparse, the volume, or rather co-volume, is increased. And to resolve this, we scale the lattice back. So if we want only to consider lattices with a fixed volume, we must constrain ourselves to alphas on the hyperbola, where the negative branches are omitted, because we can just multiply by minus 1. So up to scaling we can get to any ideal lattice in this way. But we do not only want to consider lattices up to scaling, we also want to omit lattices that just look like, that are isometric lattices. They have the same lengths, the same angles, so they are not substantially different in a geometric sense. So we can forget about the left part of this hyperbola as well. So to define an ideal lattice up to isometry, we just have to choose an alpha on the right part of the hyperbola. Using a parametrization of the hyperbola, we can actually parametrize the space of ideal lattices with a single real variable. But there's more. Not all of these t define different ideal lattices. In fact, after a while, two different t's define the same ideal lattice. It's a periodic thing. So to parametrize the space of ideal lattices, we don't even need the full real line. We just need a segment with endpoint joint. And that's a circle. So the space of all ideal lattices up to isometries is just a circle, for real quadratic orders with the principal ideal property. The space of ideal lattices up to isometry is called the Arakailov class group, denoted here with Picard zero. For these specific real quadratic orders with the principal ideal property, the Arakailov class group is isomorphic with the one-dimensional torus, the circle. If we remove the quadratic assumption, that means if we allow orders of higher degree, the embedding space will have a higher dimension as well, say r. Then the Arakailov class group will be a torus of dimension r minus one. We lose one dimension due to the rescaling, the renormalization of the ideal lattices we consider. If we remove the real assumption, the order needs complex components to be embedded in. The c here denotes the number of conjugate pairs of embeddings. The Arakailov class group is still a torus, but slightly bigger in dimension. By the way, note the loss in dimension again. The ambient space has dimension r plus 2c over the reals as a vector space, but the torus only has dimension r plus c minus one. This loss is because of modding out the isometries. On each complex dimension there is a unit circle that doesn't really change the length of the respective components of the lattice. If we remove the principal ideal property, that means if we do have a non-trivial ideal class group, things get slightly more involved. In this case we get many copies of the torus, as many as there are class group elements. I want to stress here that this direct product of the torus and the class group is only meant in a topological way. The underlying group structure is not a direct product. Why Arakailov class groups? Well, we want to think of ideal lattices as points in this group that are related by the group structure. So it will be of importance how to move or how to walk from one lattice to another one. And this random walking is how we get from the worst case lattice to an average case lattice. So our reduction algorithm randomly walks over the Arakailov group, like this. When it reaches the average case it asks for a shortest vector, v. Then it moves back along the path, but the vector v will become bigger each step back. So it is of importance that a walk isn't too long. The longer the walk, the larger the increase of the length of the vector v. But how do we move or walk in the Arakailov class group? There are two ways of walking. The first way is very continuous. We already have sort of seen it. Parametrizing the set of ideal lattices as a circle, moving along the circle continuously, transforms one ideal lattice to the other. If we try to do a worst case to average reduction only with this continuous walking procedure, it wouldn't work. This is because this way of walking distorts the SVP problem very much. The torus is just too big to reach with only continuous walks. So we need another way of walking. This one is more discrete. The walking happens by means of sparsification. This consists of removing points from the ideal lattice in such a way that the remaining points also form an ideal lattice. In this way we get a different but related ideal lattice. Because of the sparsification though, we increased the volume. So we need to scale it back to get back again in the Arakailov class group, which only consists of ideal lattices with volume 1. Calling the SVP oracle gives a solution in the sparsified lattice. To get a solution in the original lattice, we need to undo the scaling and place the points of the original lattice back to undo the sparsification. Note that again we increased the size of the vector slightly when going back to the original lattice. But you can see that the defect or deformation of the shortest vector problem is way less severe here than in the continuous step. So to summarize, there is a continuous way of walking and a discrete way of walking by sparsification. By combining these two ways of walking, we can sort of cover the entire Arakailov class group without deforming the SVP problem too much. The intuition behind this is as follows. The sparsification walk looks like jumping to random points on the torus. So after a few sparsification steps, we have a number of points quasi-randomly distributed on the Arakailov class group. The continuous walk sort of blurs out those points. And the hope is that having enough dots and by blurring enough, one finally reaches the uniform distribution. In the remaining part of the talk I would like to explain the following. How to prove that the combination of discrete and continuous random walks on the Arakailov class group indeed tends to the uniform distribution. Let's consider the one-dimensional torus again. Now seeing it as a segment. We will now look at distributions on this torus. We want to show that the distribution that comes from the random walk tends to the uniform distribution. In the current image, we have a distribution where all of the weight is at a single point, a single ideal lattice. This is also known as a Dirac distribution. After applying a continuous random walk in the form of a Gaussian noise, the distribution changes into this Gaussian distribution. By the way, it does not really matter whether to start with the continuous walk or with the discrete walk. Remember that sparsification consists of taking an ideal sub lattice. On the Arakailov class group, the torus here, sparsification has the effect of shifting the distribution by a specific shift. In our random walk, we will sparsify with primes and each prime has a specific shift. The primes 2, 3 and 5 have, for example, these shifts. Sparsifying by a randomly chosen prime among those three results in a sort of average over all shifts, like this. And if you apply and apply this again, this yields the uniform distribution or something close. Taking a distribution and averaging over different shifts can be considered as an operator on the space of distribution. This operator is what we call here the averaging operator. It acts like this. Note here that the uniform distribution or any constant function is fixed by this operator. In other words, the uniform distribution is an eigenfunction of this operator with eigenvalue 1. The constant function is not the only eigenfunction of this operator. All imaginary exponential functions with integer frequencies, essentially cosine and sine functions, are also eigenfunctions of the averaging operator. By these manipulations here, you can see that the eigenvalue of these imaginary exponential functions is the average of points on the unit circle. And it has to be a complex number with norm less than 1. We call these eigenvalues lobster n. But imaginary exponential functions are very special. They are the basis functions in Fourier analysis. We can decompose any function on the circle into an infinite sum of those imaginary exponential functions with specific coefficients. This is called the Fourier decomposition. So we are going to do that with our initial distribution, the Gaussian distribution. We write this Gaussian as a specific sum of imaginary exponential functions with integer frequencies. Because the Gaussian distribution is so smooth, only the low-frequency functions really matter. The high-frequency functions only have a negligible part, as you see I stop here at frequency 10. The averaging operator has the effect of multiplying all the imaginary exponential functions with their respective eigenvalue. Indeed, because these exponential functions are eigenfunctions. Note that the constant function is fixed, and all other eigenvalues are in absolute value strictly smaller than 1. This means that if we apply the averaging operator many times, the eigenvalues get raised to the power k and tend to 0. So eventually we can remove all non-constant terms, and we are left with only the uniform distribution. So essentially after enough applications of the averaging operator, the resulting distribution is very close to uniform. But we know that every specification step, every application of the averaging operator, disturbs and deforms the shortest factor problem in some sense. So we would like to have as least as possible applications of this averaging operator. And for that we need the eigenvalues to be sufficiently bounded away from 1 in absolute value. Let's take a closer look at those eigenvalues. Essentially those are averages of numbers on the complex circle, a sort of center of gravity. The specific places of those numbers depend on the frequency n, but also on the specific shifts that are associated with the prime numbers involved. To have this eigenvalue lambda n reasonably small, we need a sort of equidistribution of those prime shifts, a sort of regularity on the prime numbers. And that is why we need the extended Riemann hypothesis, to show that those prime shifts are equidistributed on the torus. So assuming the extended Riemann hypothesis, we know that the absolute value of the eigenfunctions are bounded above by a half, whenever the frequency is not too large. Therefore we can conclude that we do not need too many repetitions of the averaging operator, and therefore the HVP problem is not disturbed that much, which is exactly what we wanted. Thanks for watching and see you at the next crypto.


It's DDK, and I'm back today with another video. Shootouts you for tuning in. If you want to win, smash that subscribe button, give me a thumbs up, and do not stick those ads. In today's video, I'm going to answer one of the most asked questions to any person out there who does gig apps. There's two questions. Number one question is, how much do you make a week? I already did a video on that. So if you ever watch that, make sure you go tune in and watch me go get them dividend. And the second question is, how much do you spend on expenses a week doing the gig apps?  But first we got to go ahead and get some of this money like it ain't funny. We are going to be using the app called Dispatch and we're going to pick up solar panels from a place called Sun One. You about to have a little fun, son. You about to get them funds and run, baby. Order number one is 29 panels, 15.4 miles for 81, 72. If you hate it, I don't know what you're going to do. Order number two is 18 panels, 22 miles for 70, 15. We're about to pull up and get green and then in between, let's go. We in this, Joe. My God, about to get me right. He are loaded up. Yeah, I'm next. Yes, sir. 299J, G-A-B-E. And then after that, we're going with 138. Why you won? What was the other one? Sorry, I just got to see it. 138. 299G-A-B-E. You going to put the skirts on top or I should grab them? Grab your shoes. Grab them. All right. How many do I need all together? How many I need for both jobs? I got to see how big they are. Okay. Two, okay. I'm taking this cart right here. Skrrt, skrrt. Wham. My guy Steve, about to get us right ski so we can get busy. What we thought he up to play?   Yo, come on. I got Steve, about to whip this joint in here. Go to your right a little bit. Okay. Left a little bit more. A little bit, a little bit more. Right there. You in the game, coach? We in the game, coach. We in the game like EA sports. We in the game like EA sports. He about to slide the stick on there. I'm going to give y'all a different angle this time. Let me come over here. Give y'all a little different angles. I got Steve, about to get us right. Steve, how long you been doing this? This job or forklift? Forklifting. I love the gears now. You a pro. Oh, bro, in my old job, we used to sharpen the forks out of the front. For real? And we'd play a game. We could flip the nickel out.  That's how you be moving so fast like that? We in the game, coach. We're here to drop off order number one. We're about to have a little fun. We're about to get these funds.  Yeah, that's right. We're going to have to break it. Step back and pull it off. Got that. What's your name, boss? Chaz. Chaz. Spell it. CHAZ. CHAZ. And if you could stand right there for me, please. Thank you very much. You guys have a good one. Stay safe. For order number one, guess how much we're received. Thank God, though. We're about to put up the stop number two and get on some more. Let's go. Just like that, we are here to drop off order number two. If you hate it, I don't know what you're going to do. Let's go. How long is it? We'll take out and do this job. I don't know. We're out on a appointment, so we can't finish it. For real? We'll get Eric done, but we can't. Come back another day or someone will try to come. Yeah. Maybe another person. Got to hit him with the photo, yard, noodle, smack. Got that. I need a name, either one. Well, Leslie. What's your name? L.I.S. L.I.S. L.I.E. L.I.E. L.I.S. L.I.E. L.I.S. L.I.E. Yes, sir. Man, a big sign right there. And John H. Harkin? Yes, sir. Right there. I'll see you with the joint. I'll see you with the joint. All right, you guys have a good one. In for the second order, we received 96.72. Getting money, baby. It's just what we do, baby. Now we're going to get into what you guys all been waiting for. How much do you spend weekly on expenses? I get this question. $3,000,000,000,000,000,000,000,000,000 times, baby.  Hey, I hope you know how to add, baby. We probably put it together. Let's get this show. Now, weekly expenses. Number one, we're going to talk about gas. I always get this question. How much do you spend on gas? Our average is $205 a week. $205 a week on gas. The second question, how much do you pay for insurance? Weekly, I pay $21.45. Again, insurance, weekly, $21.45. Now, we're going to get into wear and tear. This includes then expenses, maintenance, upkeep, mileage, and depreciation. How much do I spend a week on all of those things? A total of $432.65 enough money to pay the rent, which equals all together. Weekly, I spend $659.10. Now, in order for you to find out how much profit we make per week, what you're going to do is you're going to go to the other video. You're going to see how much we make per week. Then you are going to subtract the expenses and you're going to find out how much it is. And if you're hating, it ain't your business. I hope you guys enjoyed this video. Again, if you are not subscribed, go and smash that subscribe button. Give me a thumbs up and do not skip those ads. And like it tomorrow, we can message you back. Why we can message you back? The sweet 23, it's a big back season.  I'll see you guys on the next one. We going, we going, we flowing. You already know this, DDK. And I'm on my way.


Okay so okay so let me start. First of all I'd like to thank the organizers for the invitation. So today I'm going to talk about the bubbling of what we call Q curvature equations of all manifolds in a new case. So this is a joint work with the home job from Thao China Normal University in China. Okay let me go straight to the motivation of my work. So I assume that M is a compact remanian for manifold without boundary and suppose that f is a non-trivial smooth function on M. So I will talk about the equation 1f which is the prescribing Q curvature equation. So in this equation on the left hand side we have a differential operator called the Panette operator. This is a force auto operator and on the right hand side we have a function f times exponential of 4u. And the reason why we want to study equation 1f because if you have a solution used to that equation then you have a conformal metric whose Q curvature is f. So I will talk about what is Q curvature but let's start with 1f. So in this case we have two simple but remarkable results. The first one is the non-existent result which says that if f is non-positive then the equation has no solution. But if we increase the function f a little bit so we say f less than or equal to small number epsilon which is small then we have some solution. So we conduct this research project because we realize that if we let epsilon goes to zero then solution to the equation 1f with f less than epsilon will escape to infinity because when epsilon equals zero we have no solution. So we put that observation as follows.  So we conduct f zero plus lambda equals f lambda and then we decrease lambda goes to zero and we want to understand the behavior of solution u lambda to the equation 1f with f replaced by f lambda. Okay so the outline of the talk is as follows. So first of all I talk about the Panette operator is Q curvature and then the PDE. Then I will talk about two equations. One is the Q curvature equation. The other is the Q curvature flow equation and then I will talk about some perturbation or the Q curvature equation. So the perturbation is it comes from the fact that I want to unify both equation and the flow equations and I want to study this equation at the same time. And then for the last session I will talk about the main results. So the main result is to show that there is some bubbling solution to the particular equation and then I give some remarks. Okay so to start with the Panette operators let me remind you the second order case first. So we know that the Laplace-Beltram operator is given by delta zero equals some summation and then if we make you a conformal train in g0 which is 2e2u g0 then the Laplace operator enjoy the conformal train. I show here. Furthermore we also have what we call Gaussian curvature equation which relates the Gaussian curvature of the metric g0 and the metric g0. So for higher order case we have what we call the Panette operator denoted by capital P. So this operator discovering 1983 by Panette on four manifold. So it's given by this formula and the leading term of the Panette operator is Laplace square. So this is by Laplace and we have lower order term here. So in the case of R4 the Panette operator is exactly the by Laplace and on S4 the four sphere it is a product of between Laplace and Laplace minus 2. Okay so the previous two differential operator Laplace Laplace and the Panette operator actually belongs to a wider class of operator. We call this is a conformally covariate operators by degree AB. So these operator enjoy this conformal trade rule. So if you want to compute with respect to the conformal metric g0 then it is just exponential minus du times the operator with respect to the background metric of some some multiplication of of fine here. Okay so in the case of Laplace operator it is of by degree zero two. Okay and in the case of Panette operator it is of by degree zero four. Okay so for convenience I also put a table showing that in fact for other dimensions three four or five and so we can modify the operator the Laplaceian and then the Panette operator to have a set of conformal conformal transformation operators. So in the case of two manifolds we have Laplaceian and then for higher dimension we have conformal Laplaceian. So it is different from Laplaceian by another factor involving the scale coefficient. So you see that for the two dimension this term drops because we have n minus 2 here and for for the case of a Panette operator for dimension three and five and above we have what we usually call the Panette Branson operator. So it is almost the same as a Panette operator but it plus some q curvature there. So in the case of the four dimension that curvature disappear because we have a coefficient n minus four. Okay oops I'm sorry I have trouble with the keyboard. Okay so so now I want to talk about the q curvature so this is due to Branson and his co-author. So like the Gaussian curvatures on the two manifolds we have a dehydrated order underlocked for four manifolds called the q curvature. So the q curvature is given by this formula. Here's the scalar curvature here and then here is this curvature. So if we train the metric g0 by multiplied by is 2u same then we have a conformal transformation rule for q gu. So it is almost the same as the transformation of the Gaussian curvature under this conformal chain. Okay so so now I want to talk about the prescribing q curvature problem. So the problem is stated as follow. So given the smooth function f on m is there a metric in the conformal class of the background matrix g0 such that the q curvature is exactly f. So in fact there's some there is one restriction here a metric must be in the conformal class of the g0. So without looking for a metric in this conformal conformal class of g0 so the problem is determined because we only want equation which is a q curvature equals f but we have times m minus 1 over 2 variables. So this is an entertainment problem. So going back to the conformal chain of the q curvature we arrive at what I just call the q curvature equation.  So geometrically if you can solve this equation to have some solution u then you can construct a conformal metric with the prescribing q curvature f. Okay so why should we interest in the conformal chain? So there's one reason why we are interested in the conformal chain of the metric because on one hand it equalizes the number of equations and the number of variables so that the problem is now determined. The other is we can start with any metric in the conformal class. So I put here a diagram. Okay so for this direction this is the conformal class of g0. So here's the g0 itself and this is another metric in the conformal class. So I denote it by e to v0. So if you start with g0 then you have a q curvature q g0 and you have the q curvature equation computed with respect to this metric and if you can solve for u and then you can construct a metric and then you have the q curvature metric is f. Okay but if you start with the other metric e to v0 then you have a corresponding q curvature equation and if you start you can solve this equation for w and then again you can construct a conformal metric and you have the q curvature which is f and the relation between the solution of these two q curvature equations is that you have w equals u minus v or u equals w plus v. So this diagram shows that well you can start any metric in this conformal class. So usually people start with some metric such that the q curvature is constant. So you turn this equation into a simpler one. So this scalar is now constant, it's not a function anymore. So this leads to the prescribing constant q curvature problem. So this is this study by Alice Giannepolniani in 1990, 1992 and others. Okay so well we talked about the finite operator and q curvature just a high order analog of the Laplacian and the Gaussian curvature. So why why we should study these curvatures? So well in fact I do not know. I try to google this and then there's not much information about why we should study q curvature equation except that the q curvature enjoy the Jones-Gauss Bonnet formula. So on the left hand side you have the total integral of q curvature plus one-quarter of the integral of the y tensor square and on the right hand side you have 8 pi square times Euler characteristic. So so perhaps we can say that the q curvature enjoy this formula so there's some topological information here. And now one key information coming from this formula is that the total integral of the q curvature is a topological invariance. This is because the integral of y tensor square here is also a topological invariance. So so well when you to start with you can see the matrix on the m that you can use the the side of the value of this number. So so for this region when starting q curvature equation we usually split the problem into three cases. The first one is corresponding to the positive invariant. The second is the zero invariant and the third one is the negative one. So so in my talk I'm interested in the second case. So I call this the known case that means the total integral of the q curvature is zero. And in fact because of the the results of Alice Chen and Paul Yan we can choose the background matrix zero such that the q curvature is zero. So so we don't have the scalar q zero in the q curvature equation anymore. Okay so so back to the three cases here. So we have some obvious necessary condition when we started the q curvature equation. So I put here the necessary condition. So for the known case the function f is side-changing. Okay okay so so in order to to study the q curvature equation we need to have some some looks and some we need to realize some the difference between the lower case and the higher case. So at least here are some major differences between the two operators. So first of all the panacea operator is of all the four and the the Laplacian is only two. Okay the second is the panacea operator is not necessarily positive. So here's some some results on the positivity of the panacea operators. Okay so the third difference is the panacea operators need not satisfy the maximum principle. So this is a typical feature of a high order operators. So usually we we should not expect the maximum principle to hold. Okay but remarkably for some very small assumption on the the matrix so the Gershky and McCullis and Hang and Po Yang can prove the maximum principle for dimension three and dimension five level. Okay and then the last difference between the two operators is the kernel of the panacea operator is not necessarily trivial. So you know that the kernel of Laplacian is is what but the kernel of the panacea operator is not necessarily trivial. Okay so now I so for the moments I will talk about two types of equation. So the first one is the Q curvature equation and then this the second one will be the Q curvature flow equation. So let me start with the Q curvature equation in the no case. So there's a very nice result. So the very nice existing result. So assuming that the panacea operator is a positive with a kernel consisting of constant functions then under this two condition the first one is the function f is positive somewhere and the total integral of f is negative. Then we have one solution we have at least one solution used to the equation. Okay and the proof is quite as straightforward. You just construct one energy's functionals and then you look at the minimizer of that energy and they use Lagrangian to prove that the solution to the equation. So okay so now let me let me comment on the the result of the current energy. The the assumption on the the positive of maximum f is necessary as I said before but here we need extra condition on the integral of f. Okay and compared to the case of the prescribing Gaussian curvature equation in the no case the condition on the side of integral f is necessary but in the case of Q curvature equation this extra condition is not necessary. So but we cannot we don't know how to remove this that's fine as I know. Okay so to support these comments there's a paper by Beth and others in 2006. They construct a metric on the four torus and then they compute the the Q curvature and then they prove that in that case the integral of Q curvature is positive. So limiting to the the negative total integral is not necessary. So but if we through or away that's the condition then in fact we can prove the existence of solution up to a sign. So we have a plus or minus here and then we cannot determine which one is the correct one. Okay so now I go to the Q curvature flow. So instead of using the direct method to look for a solution I use the heat flow. So I fix a metric as usual and then I try to I try to form less an evolution of the time dependent metric g lambda t which is conformal to the background metric d0 by the e to u lambda and then at the initial time we can put any initial conformal factor u lambda 0 and then for time the positive time the evolution is is in this screen. So the time derivative of the metric is equal to minus twice of the Q curvature minus some alpha f and the lambda. So if you replace g lambda by e to u lambda g0 then you can transfer this equation for the metric to the equation for the conformal factor.  Okay so the way we construct this evolution is to ensure that this is a negative gradient flow. Okay so using the previous energies the time derivative is not the positive. Okay so so far alpha lambda is just a constant the time dependent constant. So we need to specify what is the alpha lambda. So we choose the alpha lambda in such a way that the integral of alpha lambda is constant. So if we use this then we can prove that in fact along the flow the volume is also constant. So using this evolution we have two constraints. One is the constant of the integral of alpha lambda and the other is the volume. Okay so with Zhang Hong we can prove some existing solution to the flow one. Okay so before I talk about that for example let me let me show why should we study that flow. Okay so suppose that the flow converges namely we have u lambda t goes to u lambda infinity in some sense. Then from the flow equation for u lambda okay so this equation and we pass t to infinity and then we end up at alpha lambda infinity f lambda equals q of g lambda infinity. So that means we have a metric g lambda infinity which has a q curvature proportional to f lambda.  Okay so now let me talk about the these results of conversions. So we prove that the flow has solution u t defy all time and then there's a sequence of times some constant of lambda infinity and we have some conversion up to that time sequence. Okay so initially we expected to have a uniform convergence but we failed to prove this and then we tried several ways but we cannot. Therefore in principle we shouldn't expect to have uniform convergence and perhaps this is the reason why we work on the particular equation that I'm going to talk later. Okay so back to the flow equation if we use the conformal chain of a q curvature apply to the q curvature of a time dependence metric.  So therefore compare the q curvature equation and this new form of q curvature flow equation they are more the same except that there are some extra term here. Okay so putting the q curvature equation and this flow equation together we propose to study a particular q curvature equation. So that particular q curvature equation will help us to handle the q curvature equation and this flow equation at the same time. Okay so here's okay so I try to I try to summarize what I'm trying to to speak. Okay so on the left hand side we have the equation so this is a q curvature equation in our case and on the right hand side we have a flow equation so they are all the same. Okay so so in view of these two equations I try to consider the the red equation here p u equals f e for u plus h e for u and for some reason we expect that there are some blow up a curve. Okay so to to inspire a work of Struve in 2009 for the Gaussian curvature flow in the north case so we propose to study the same problem but for the fourth order case and we try to consider some similar scale like we particularly you know function by a little bit by a little constant lambda and we try to understand what happened if we lose lambda to zero. Okay okay so so the configuration of the problem as I said at the beginning of talk if we fix some some function f zero which is non-positive we perturbed by small lambda and we can see the class of solution u lambda followed by using the direct method so just say just minimize the energy function over some constraint and then we have some solution u lambda so u lambda is solution to equation two lambda corresponding to the f lambda. Okay so obviously uh two zero has no solution of course right because now the function f zero is non-positive and uh and if uh if lambda is sufficiently small but positive then we have some minimal some solution u lambda followed by the Gaussian results I talked earlier so so if we let lambda goes to zero so the solution u lambda will fly away right so we are interested in the behavior of u lambda. Okay so we can prove that the limit of lambda for lambda is bounded formable by 64 pi square and also as a consequence we have a limit of integral of positive part of q curfew of g lambda we also bounded formable by 64 pi square okay so the proof is quite involved so I know okay so so um uh so for for for our case uh we uh we would consider the equation of three k and for some sequence lambda k to go to zero to be determined later and we want to understand the blow up behavior of w k s k goes to infinity so uh okay just we go back to previous slide so so in in view of the the first important fact uh there will be some sequence lambda goes to zero such that lambda of lambda is bounded from both by 64 pi square so I can assume that the limit of lambda k of alpha k is bounded from both by 64 pi square okay and for the term involving the perturbation h k um so uh I make another assumption because that h k we can freely freely uh a piecewise so uh my assumption is uh the L2 norm of h k is little over one okay so uh with this uh behavior this is the behavior of h k with k large uh we can we can apply for the q-coach equation in this case k is zero and for the q-coach flow equation okay uh and in in in the case of flows equation you can we can take x to be the time derivative of u and then I form the analysis of flow equation uh we we know that ut satisfy this condition so we can we can make u this assumption okay so um so uh in in in our work we need to perform uh three uh so the first one is to realize the the blow-off so we need to prove that well the blow-off must occur because solution u lambda goes in phase and somewhere must blow-off and then a second one is to characterize the behavior of uh w k away from any bubble uh any blow-off points and then the the the last one is to guess the right behavior of the w k near the blow-off point okay so in order to realize that the blow-off must occur we need to prove something like we need to realize that the blow-off must occur is in fact equivalent to uh showing that there's some concentration of the curvature okay so uh the concentration of curvature is simply uh you integrate the the positive part of the the q curvature uh some around some more then that one is bounded from below by some constant so in our case the constant is eight pi square so if you can prove this then we can talk about we have some concentration of curvature and and okay then that the blow-off must occur so um so proof is uh quite involved so the idea is to turn uh uh to prove some some some some bounded of the uh the volume and then we use a lot of standard elliptic estimate and then we arise with some contradiction and I say try so so therefore the we have some concentration of curvature okay so I don't want to talk in detail here okay so um so for the first talks the main result is there will be some capital i and and finally many points x infinity i i from one to capital i says that uh the the integral of the positive part of q curvature um around the ball uh over the ball of the center x infinity i is is bounded from below by eight pi square here the radius is is arbitrary okay so uh and uh another the fact is um the x infinity i is the maximum point of the function epsilon okay and uh and because of the um the integral of the positive part of q curvature is bounded from above by 64 pi square from the second uh important uh facts then uh we have an upper bound for the the number of blow point so in this case is eight so okay so so no matter how many uh a maximum point of f zero um the number of blow point in our case is always at most eight okay and uh okay so for the second talks uh we want to understand wk away from all blow up point x infinity i so the theorem is is as follow uh so away from this point um wk the solution of the particular equation goes to minus infinity locally in a uniformly and of course at each x infinity i uh wk is going to plus infinity okay um so so the proof of this theorem uh follows standard uh a blow up analysis away from this point you have a upper bound for uh the deposit part of the two curvature this equation less than eight pi square here of course if we have the reverse inequality then we have another concentration point so uh we must have the less than so for the less than then uh then we have a loss of uh we make you a loss of uh elliptic estimate and some functional inequalities and eventually we can prove that wk must goes to minus infinity so uh so this this condition uh this finding is uh unreasonable because uh at a capital i point x infinity i uh the solution wk is going to infinity so in order to maintain the uh constant volume uh the integral of e4 wk equals one uh the other other point the wk must go to minus infinity otherwise uh we cannot have this uh constraint okay okay so uh so now we we we work on the last part of uh uh the problem so we want to understand the behavior of the wk near blow points so um okay so um so we we describe the behavior of wk near the x infinity i as well that's two scenarios first one is a fast blow up and the second is the slow blow up so uh so at each uh blow point infinity i uh if you pull back um pull back uh the equation uh sorry uh the the the the solution wk to the tangent space at x infinity i and then do some scaling in a dilation then uh the the projected solution uh wk two times will will um con will converge to some limit profile so in the fast blow up case we can write down precisely the uh the form of the blow profile you choose the w infinity hat which is a log of some some quotient and for the slow blow up uh uh we we cannot we just uh we can we just say that we have some blow up for five w infinity hat we show some um force all the equation of exponential non-minority here okay so to prove of this theorem uh we'll uh it also based on the standard analysis of standard blow analysis uh with some some circumstances due to the fact that we are we are handling a higher order case but the idea is to to turn the um concentration of curvature into the concentration of a volume and then if you have a concentration of a volume then you can select a sequence xk i going to x infinity i such that the volume also concentrated at xk i okay so uh once you have the sequence of xk i then you just project wk onto the tangent space at x infinity i and then appropriate uh appropriate scaling and dilation uh you can prove that wk delta uh belongs to some sort of space and then you have some equation of force order and then you use some known classification for a force order equation with the exponential non-linearity and you can write down the blow up for five okay okay so uh before a complete thing in my talk i would like to give some remarks so um so first remark our work is totally inspired by a similar work by strouf recently appeared in his work he considered the bubbling of the gaussian the perturbation of gaussian curvature equation uh of the same form so instead of uh an x operator we have laplacian on the left hand side so the rest is the same okay and uh he he he obtained uh he obtained the the bubbling when uh we take h lambda goes to zero okay of course there are some uh some difference in between some differences between the the two problem uh okay i i don't want to talk in detail here okay um one one one major difference between the finding of these two work is that uh for the for the case of the gaussian curvature equation uh strouver uh can prove that there's no slow blow up so so any block is fast okay but in our case we cannot in fact uh in in principle uh either yes or no can occur okay so uh we we limit ourselves to the case of an odd case but in the net of the case uh the problem was studied by galler betty in 2017 but he purely worked on the the q curvature equation in that case so without uh the extra perturbation here so uh we can we can repeat our argument for this negative case and therefore can give some slight improvement of the galler betty result okay but uh for the positive case uh uh no result is no so far okay so uh thank you so much yeah i finished my talk okay thank you quok for this lovely talk we thank you thanks um very nice talk um well uh now uh we are open to questions if anybody wants to make some questions can unmute yourself or uh write in the chat okay can i ask a question and the browski from sisa yes of course so i have a few questions but let's um first question is uh so you said that these solutions go to infinity but you maybe i mean is understood go to infinity in space or they become singular that's what in some finite concentration points is this what you have in mind okay so the value of the solution goes to infinity at some point okay that's what i mean okay my second question is related to okay so as you know uh the is this finite operator has been discussed only so for scalar functions for functions i understand right has it been discussed for spinor fields also like similar there is a in relation to the dirac operator ford power or something like that or you never know uh i think even for the second on a case uh screw will also rise uh raise the same question for for the question you just talked so no i i say no no result okay okay okay let me just in the moment stop okay thanks okay sure thanks thank you any other questions or comments uh can i ask a question yes of course all right so is there some statement about uniqueness of the solution um what was the solution uh you are talking about the two curvature division yeah for you i mean i mean you have some condition on existence right but what about um oh okay um well uh you mean um this the this results uniqueness yes is it what yeah what does this one is there only one or yeah yeah yeah so uh so this one is i mean at least one not just one yeah but but in general i mean is it known like is there no no no solutions or infinite no unique this result at the moment okay yeah so it's it's not even known if there is finite number of solutions or there can be like family of solutions oh well i haven't seen any result this thing yeah i i haven't seen such a kind of results perhaps there's a lot of uh solution yeah but that's a very good question yeah yeah well i also got kind of a naive question so this uh so you have a you you say that there is a requirement that f should vanish somewhere right on a four manifold necessarily sorry um can can you repeat your question so there was some there was a some as far as the suit there was a necessary condition at some point that f should vanish f should vanish somewhere on your four manifold right um which case because um well um if if f is the mood and yeah in this case then of course f will so my question is i mean suppose there is some nice kind of a nice uh situation this zero locus of f forms some three manifold inside four manifold so does it does this three manifold has some sort of meaning i don't know okay okay thanks okay you're welcome well i have just a question to do but just uh for curiosities maybe it's a silly questions i don't know if you have sense but it is possible to to think this kind of problem to solve this equation in higher dimensions or there is something that maybe um but well um so uh so i was talking about two problems i mean uh in fact one problem two approaches one approaches is a direct method and the other is a variation method so flow method so um for for the the prescribing q curvature problem uh in fact we can we can consider um similar problem for higher dimension manifold or even higher dimension operator so um okay so uh if you continue this this order so we also have a similar operator order six or the eight so and so forth so what will we call these operator the the g uh so the the g okay so gsm p operators okay okay so yeah so for for for other dimension you can start uh talking about uh the prescribing uh with the pannet prance and operator so people also consider this problem uh as well so you can easily find a lot of uh results in the literature okay thank you any other question or comment i i still have dabrowski i still have some small naive questions so i understand this pannet operator is elliptic operator of course right principal symbol starts from the fourth power of right correct so my my question is about the what is the interpretation of the index of this operator well um yeah uh i have no knowledge about uh that part but uh yeah so so yeah so and another related on special manifolds like i don't know complex manifolds or color manifold is something more known about these operators and solutions as i know uh people also consider for cr manifolds but for color manifolds i haven't seen uh yeah i haven't seen thanks for speech to see our manifolds we have yes they call two pram uh uh curvy sure thanks okay well um i think any other question or comment if if not well if not we are going to thanks again quok for this very nice talk and thanks for being here uh today with us thank you so much yeah i want to add a comment yes okay sure yeah sir as i said earlier so this is a lunar new just eve so happy new year oh thank you happy new year all right okay well thank you so much yeah thank you for having me


Logic Bomb was written by Rounder House for Gearsday. If you'd like to read it yourself, you can go to www.sep-wiki.wiki.com forward slash surprise-happy-birthday-9. A link will be in the description. Five senses make up the input that goes into the human mind. Sight, smell, touch, taste, and sound. Everything that exists around them falls into one of these categories, or, as far as they are concerned, it doesn't exist. Of course, senses are the only data that go into formulating the human perception of reality. The mind processes this data to achieve a conclusion and then acts upon it. And this is where things get messy. Brains are, at the end of it, supremely complex input-output machines. Sensory input goes in, conclusions about the world come out. The input is run against a lifetime's worth of rules, determinant statements, a complex algorithm derived from other conclusions, and so on and so forth. It is ever-adapting and all-encompassing. The rare scenarios where it fails at these are, respectively, learning and insanity. But the machine is not immortal. Over time, it deteriorates. Input is wrong, and this leads to exponentially inaccurate conclusions. As mentioned, it's subject to insanity, brought on by the realization that something is utterly beyond the capacity for human conception. It defies understanding by means of its very existence. These are natural faults of the human mind, and they are caused by itself. But artificial faults are far, far more interesting. Viral input. Clarification, they are not just detrimental sensory input. Detrimental sensory input is nails on a chalkboard, odors of decomposition, the acrid taste of cyanide. No, these are much, much worse than that. These camouflage themselves as sensory input, just another of the countless things you could hear, touch, taste, smell, and see. And this is a lie. They are no more true sensory input than a chameleon is the branch it sits on, but the chameleon uses this to hide from predators. Artificial faults are the predator. They are invasive and transformative to the mode of thinking, and for this reason they are called cognitohazards, hazardous to thought. Let us return to the idea of the mind as a machine, a bundle of input like any other that comes in one day. A poster on the wall that is slightly too red, the sound of a plane that is too close, the smell of your wife's perfume, but off. It is processed, conclusions are drawn, but something is wrong. It happens too fast to even notice it in the moment, a simple, utterly banal, but fundamental rule that the rest of our world sits on is changed. The gear has been warped. It is functional, but it throws off the entire apparatus. By itself it is too minor to notice, but there is now a crack in the machine. The mind continues unabated. It gets no breaks, but as it takes on more thoughts, more pressures, the crack widens, expands, the remnants of the thought remain, and they spread to more gears, more rules. They warp, one after the other. The failures build on each other, and eventually something inside breaks. Something goes too far, and then the mind realizes it can no longer trust itself, that an object, some viral factor, something it saw just a second, five months ago, has fundamentally broken its understanding of the universe and its subjects. The human experience is fed by both its senses and the mind's logic for decoding them. When both of these factors fail, there is only one logical conclusion remaining, that the rules it has built over a lifetime no longer matter, but nothing can be trusted, because nothing is real, and this, truly. If you'd like to support the Cancer Research Institute, there will be a link to that in the description as well. Head on over there, and if you might want to consider giving a donation. I know Dr. Gears last year, or was it last year, the year before possibly, it's been a little while now, went through a bit of a problem with cancer, and I think supporting the Cancer Research Institute is a good cause, especially if you're an SCP fan, especially considering that these stories were written for Dr. Gears for his birthday, which is an annual thing that we do for the SCP wiki. Anyway, it's nice to know that I'm not alone out here, and I will see you all again on Thursday.


Wait, rock-paper-scissors, rock-paper-scissors, easy-peasy. There you go. Boom, game! Blouses! Hey, how's it going everybody? Dot-2-2 I hear back with part 2. If you could tell this is rushed, it's because it is. We're fighting the same guy from the last video. Okay, okay, okay. For lore. Why didn't you do it? I was gonna say for lore. Ryuki, Dragon Rush Dot-O's counter. And you know how it feels about that. When you go to Dot-O it's person. it is for I mean you're setting it up like I have a grudge match yeah I wouldn't say that I would say more like I'm very sad because you want to change my I will gladly change during he always does 6m after that but I always think he's not gonna do 6m after that this time and then I get 6m I should treat yourself and just win you know what most of the time you're not spitting but this time you are spitting I literally don't know a single time and I wasn't spitting ah see that right there is where I would have done Krillin's old combo but it doesn't work and now I'm too scared to do it now can he play mix or no mix what is the mix come on down captain Ginyu he spark before a spark again there's a spark you said he watches my content he knows the scouting report the scouting report oh the the the defense yeah maybe stutter yeah I don't know why you want wait did I take her was that a combo I did taking got reset right I am upset Caucasus oh there you go yeah close in doubt when in doubt when in doubt you'll close it out but he dropped it hit you with the homie drop because he got your back as I know what for the content I can't a doubt I get washed here oh hell the coming I'm a hot let's go and you know what I'm gonna give him an opportunity to get me back for that put him in the corner there you go I got you take his guinyu take his guinyu to me you doubt you should take his guinyu and and beat him up I can't take his guinyu he would have to give it to me and at this point he's used you know I gotta hit you with a little some some spicy you know I gotta throw a little mild sauce in there Diablo this counts is also taking his character ah you knew I would ah he knew I wouldn't oh I didn't I've never even seen that the classic the classic you know something special now can he handle mix or no mix on this game show you will decide between mix or no mix what's the move up or down I'm just gonna go to him I was worried he would try something there why am I cursed to never have a good games why did I drop that it's because you said he was gonna try something and you know the juror the juror thing came back to you that's what I'm just gonna 5h I'm just gonna punch me in the face I don't know why that killed but it did but we'll take it okay personally I would just swim and then do a TDF I'm gonna 5h punch benefit he finally did it he finally barred off me I knew this day was coming I just didn't know right here my dad this is so much this is so much Bardock damage this is 100% Rob Bardock enter enter through all the hype music right now watch this come back right now oh come on that block to I felt like a king ah good first game you know a little warm-up match you want to know in a chat they said mix we're switching you know what it's time for a team switch myself I can't always have Jiren you know Jiren is a member of the past okay remember of the past and I think it's time we bring in somebody new somebody bold somebody innovative I'm gonna play Napa all right simply dotto I after analyzing this fight for the past like you know 20 minutes I've decided all you have to do here to you think do you think he's a man of his word will he try to dragon will he try to counter here he definitely will but I think all you know but I want to drag and rush his counter so tell me right now do you think he's a man of his word will he get a man he's a man all right then I'm gonna counter but then he he didn't do it he disobeyed the law how he could disobey the law and get away with it is beyond me dude but you dropped out so this is your second chance no no I really wanted that dragon rush I was gonna find you know what actually this is on me I was trying to find peace through brother's pain right that's that's not a good way of looking at life you know wow that last long when it hits a beam I've always known that but that was crazy okay well good news is still is an armor oh my god my super dash was too late dude where's my armor I need on both combos combos dotto I just realized that it's you who was losing yeah no yeah now he did it dude come on yeah but I mean that's it he should have gave it to me then as an offering of peace you got some big leg you got some big legs do you why is he doing this I got some big legs right he might wake up counter classic Yamcha I was afraid I dropped it to listen I didn't think very highly of myself either it's okay everyone gets one one drop in a video so long as you don't do it again it's fine I didn't tech because I was like why should I but now I realize my mistake okay now with it basically a mirror match right now you cannot lose the program and mirror basically matching because there's no reason not to right right right oh my medium moving fist got stuffed by the other Yamcha I know I don't know how it feels on the back side he knew he knew I would step back okay one now it's time for you don't go to the comeback protagonist as he's been called to him treat oh there you go that's all you need ups and I dropped it another one another one look at that look at that I'm not used to go see the thing is I played Goku back when he didn't have the long reaching to him so it's it's new to me there you go I'm still sad the overhead didn't work I did mash I did I should not have superdash I've learned okay do you think this runs about to start with the same way but the correct way but the counter I really want to play his team do it all right I'm gonna go to ring room I've got to match his energy right now this is right right he's trying to match my stupidity so I've got to go lower right right I've got to go even further beyond yeah yeah yeah yeah what was this team he's using your team I see yeah yeah yeah yeah it does he see what team I'm using if I don't see a level seven in this video I'm going to unsubscribe well that's only part that's part of the plan they know it's gonna hurt I feel like him using my team you would think that I will feel better about myself but it's actually going to hurt me on a very spiritual level what I see him do stuff that I've literally never been able to get my beer is very good I'm worried about if he has any beer is set up other than that you know he does he does I've seen him well that I'm gonna so I'll tell I'll tell people right now my only option against beers because I don't know how beers works is every time beers has me on yes every time beer is on the field I will superdash all right now my main goal is to get away and build up meter right right right and then oh I got I got a guard cancel go on out to that's the thing is once you get the first couple of bars it's time to go oh good D.P. Oh really wasted a bar I wasted I wasted a great use of my bar there all right the question is do I spark because I could save that for Gohan maybe save it maybe save it remember you want to go for that level seven thinking that he was a perfect distance away my D.P. my limited range I do I need to keep go Gita lot oh no I'm dead I needed to keep go Gita live so that I could get more stacks I mean at least you still have your sparking yes but the sparking will do me no good if I don't if I don't build up bar I was making the joke as you lost your spark oh wait did you not spark no I didn't spark I didn't spark okay but you know what you can still do with the best TOD in the game that's fine okay I'm just trying to build up to six bars the goal here has always been go on I'm over here just taking notes try to go for a little too much that's fine we're almost at six bars we're almost at six bars you're gonna go all in on Gohan yeah I mean that's that's been the why I'm is it's too oh it's too is it too to us or just to try I don't know damn it okay it's fine we're at six bars that means I'm at seven bars which means GG game over it's time the conditions have been met I summon fourth exodia I will when I have the time and he knows that I'm trying to rush this game now it's fine it's fine it's fine okay wake up buns would he expect to wake up but no absolutely not and should I wake up buns or wake up deep wake up but wake up but wake up but but he does that be yourself he's gonna drop the yes or yes or I should have woke up buns I should have woke up buns I told you make a button my life 11 there you go all right come back season time editor and hype music right now this is gonna be dottos moment is me oh it's remember for her oh no nope nope nope watch out watch out watch out my gohan strategy almost work it was this close to working editor turn off the music all right let's get two more games and oh well we match fast okay all right no not today this is not under living legend what people always get hit by that oh okay we've got a goal Donald's cracked house cracked house cracked I was gonna go for the kill here yeah all right all right we've accomplished the goal we're playing a lot more solid this time you know we have a gimmick this is your chance your chance do it in between the buttons nope he's going for kill brush oh he was beerus because he knows he knows he's a shot to women that was gonna lose everything in the game wait a minute wait a minute I know what I was doing I know what I was doing what do you think I'm doing I was just trying to get the seven bars as quick as possible but you could go it's not it's not that time the time for Shenron will come the time for seven bars is now he does he does get a punish he does get a punish now people will always say just build it up in threes and that is the more viable way of doing it safer the lot safer yeah but but you got to realize that's not the spirit of the game dude yeah you want to be an idiot we do this cuz why white like what are you doing oh let's go there you go all right show me some go on some little stuff oh stuff right now oh kicks reverse beat look at the combos get put in the blender put in the blender yeah put in the blender put that man to blender oh oh this is about to come back oh oh yikes because you got beer is in that too rock I can rock I okay stupid oh come on I wanted to be stupid I had so much to teach oh you can block this is your game oh there you go when I'm down you see I gotta go lower I gotta be stupider what the hell what is you really trying to M.U. of all people with beer is there you go show that man it's true doesn't mean anything there you go kill that cat in the game go all right what's the final move your daughter come in grab I actually all my plan was that spear bomb so after that my brain was fried I literally have no idea rock how can rock how can easy peasy easy peasy there you go boom get in the why because that was the easiest kill I don't know I wasn't gonna do anything risky oh my god I could get so I got it down alright this is the final game the for real see this is the for real see matches this is the one where we're the losers they can never touch the game ever again I had such a big plan I had so so many big hopes and dreams and they were all dashed and that's just something I had to live with yeah all right now what do you want some in the blend I would personally I would personally use the exit door yeah this team isn't designed to deal with this oh no oh no like go on a time to die go on how to go let's be honest go on how to go on that boy at that you gotta go what do you mean I'm still in the game oh maybe I gotta go bro a combo's never even see jumps yeah no he's he's sick ah the perfect that's insane shut up my little to end dude come on my spirit of my little to have okay okay okay you can still do this level seven with him and yes he does the do-it-mere Christmas so all you have to do is tack both characters in there you go oh no I thought he would run closer than that okay we still spark he chose nothing still have spark I think you're dead mmm maybe but I refuse that no you know you're dead level four they did buff Vigito no wait you're dead no wait hold on hold on hold on hold on hold on hold on no damn it oh the riskiest DP well that was a good way to go out I risked it all I gain nothing remember what I said about this now you have to go and press the home button delete the game that was that was the deal I didn't even I didn't even hear that deal but I was gonna do that after the jeering editor play the jeering getting dragon rush the counter video over some sad music and that'll bring us to a close for today hopefully you enjoyed these matches if you did definitely let me know down below in the comments check out some other videos if you want to watch more right away other than that leave me some team suggestions in the comments and I will see you in the next video


When you look at Saturday night lights, we need to do the prevention because if they can bounce a ball on a basketball court, they can wear black robe on the Supreme Court. Free Wi-Fi and free foam service for those who need it the most. This is a campaign promise made, campaign promise kept. That's what this moment is about. Bring us together as a unit and hear from each other to resolve the issues that we're facing. So I cannot say enough. Thank you for coming out. You are the majority in this city. We're taking the dream of a livable, sustainable and beautiful city and turning it into a reality, block by block. Today we're sending a message. These short-term illegal rentals have a major impact on our hospitality industry and those who are employed in the industry. You're giving folks the counseling they need. You're giving them the resources they need and a long-term medical and mental care that they deserve and that's what we're getting right here in this location. The heart and soul of the city is in our cultural institutions. Denial and cultural groups asked for our support and what was interesting, we heard them. You have been ignored. You have been denied. You have been abandoned. That stopped January 1st, 2022, when I became the mayor of the city of New York. Welcome aboard to this new concept of New York City ferry forward, the Adams Administration vision on how to utilize our waterway. And at the heart of this program is making sure that the system is equitable. This is why we tell folks to join, you know, the FDNY.  And so it just really means a lot when we can pause for a moment and just say thank you for your heroic actions.