I'm happy to introduce our last speaker, Timo Weigand from the University of Hamburg. Hello, thank you very much, good morning everybody and first of all thanks to the organizers for having me for the kind and general communication. I'm very happy to be here, enjoying the interaction and I'm very happy to talk about this project which is kind of a longer program that as far my involvement of concerns started together with Wolfgang LeAsier who is with us now online and Song Julie at CERN, then now in Korea and I'd like to give an update and the more recent developments in this work with my post-work CISAR Quota and Alisson from the University of Hamburg and with Mike Lee, definitely. And since we are all on a busy schedule and may have to leave I'd like to give a punchline first and since we are two audiences, one in physics and one in mathematics and if you get two punchlines. So first to the maybe more mathematically inclined part of the audience, I'd like to argue or give an example of how GV and DT invariants play an important role in physics not only in general but in particular in verifying certain physics conjectures about quantum gravity and especially the tower weak gravity conjecture will be relying on this mathematical structure of DT invariants and their particles. And this is one example of the many profound indications of the Florida program called quantum program which makes predictions about quantum gravity from a physics point of view and to the extent that quantum gravity can be realized in examples by spring theory this makes statements about compact string geometry and compact is important because this is the relation to the gravitational. We're not going to talk about non-compact geometries which are relevant for gauge theories but really the compactness comes from the gravitation. And the punchline for the more physics-oriented part of the audience would be that we prove the asymptotic tower weak gravity conjecture in 5G n equal 1 m theory compactification and this is one example of the many well-known to you applications of tower geometry and also enumerative geometry. So let me introduce the object of interest in this talk. It's the weak gravity conjecture. This is well known to the physicists in the room. I'm not sure how well known it is to the mathematicians who haven't followed this so let me remind us all of the statement. It's one of the general statements about what the quantum gravity theory should have irrespective of whether this theory comes from a spring theory or some other theory. And according to these authors every quantum gravity theory should have the following property. If you have a gauge sector coupled to quantum gravity so for example one gauge theory or young mill gauge theory more generally and I'm for simplicity I'm only focusing on the delian part and you have a certain chart letters associated with this gauge theory so charged object charge state for representations of the gauge proof. Then every ray in the chart letters must support a tower of super extreme of state. That's the claim I'll define in a second what I mean by this and the significance is from the physics point of view this conjecture predicts the existence of certain states with a certain ratio of charge to mass. That's the claim in every quantum gravity theory there should be certain states that infinitely many with a suitable ratio of charge to mass will be more precise in a second and from the mathematics point of view why this is interesting from mathematics because we can use string theory to realize quantum gravity theory if we believe that this is the consistent quantum gravity theory and then the existence of these states can be translated into a very precise statement about the non-vanishing of certain enumerative invariant on compact alabias which are the compactification states of the string or anchor. So this is the logic and this is of course what we'd like to show in the in the first and I need to be more precise for what is meant by super extreme and what is meant by power super extremal. So this is now the physics motivation of course or the physics statements there are first of all two ways of talking about super extremality and so that we don't confuse let me recall them. So first of all super extremal means the particle super extremal if the ratio of its charge i.e the weight of its representation per this week to its mass it's bigger than this ratio of charge to mass of every black hole in the theory.  So there must be particles with this property that's the prediction from quantum gravity and another way of phrasing a related way of phrasing that and this explains the name we gravity conjecture would be giving two particles so two states of a certain representation of certain weights in a gauge group then they must be self repulsive this means the sum of all the repulsive forces or the repulsive force from the gauge group must be bigger than or equal to the attractive forces and there can be two types of attractive forces one from gravity obviously important gravity but if there are scalar fields they can also be in addition attractive forces that go to carbon and this would mean that for these particles gravity or more precisely gravity plus your carbons are everything that is attractive should be weaker than the repulsive force from the gauge sector and this was said to be the slogan gravity should be the weakest force acting on those particles in that sense it's gravity the weakest force acting on those particles and in general we have since we are also talking about modularity with modularity we have mass of scalar fields modularity is one of the topic of the workshop of course in physics this means mass of states mass of states leads to the yukawa interactions and tends to a modification so these are the two versions of super extremality they are not equivalent in general because of these extra yukawa terms but in the limit of weak gauge coupling and certain asymptotic limits of the theory we'll discuss within multiple detail they are equivalent hence we will mostly be talking about this which is also closer to the topic which is the name weak gravity conjecture but we have to be careful that inside the modularity space they are not exactly so this is the super extremality and we said there should not only be some super extreme state but there must be a tower what do we mean by this this means that there must be exist a super extremal particle of charge n times q for every charge q where n should be some infinite so it could be for example for every q there must be infinitely many particles this would then be a lattice version of the weak gravity conjecture or the whole sub a whole sub level should be populated but more generally it would be enough if some multiple nq for n and infinite for every q is populated by so this is a lot of states infinitely many states must be there in such a week in such a credit so i'm not going to talk about the motivation behind this this of course and what led to this also a long time ago to propose this there were considerations of black hole physics and of remnants in cosmology and things like that there's a purely motivated conjecture but the nice thing is that one can translate this into statements about geometry whenever we take examples of the quantum gravities from m theory or for string theory and translate those into statements about the numeracy invariance that we can then prove for this proof so in fact um which types of states could occur first of all one knows that every bps state is automatically super extremal or extremal so if you have a bps state in your theory then these states are automatically of the types of states satisfy this weak gravity conjecture so in particular if you have infinitely many bps states in your theory and along a certain charge direction of charge letters you have infinitely many such states then you're done then for that direction in charge letters the weak gravity conjecture is sadness but what happens if no bps tower exists in a certain direction in charge length this can happen and in fact this is the generic situation the generic situation first of all because in most theories there does not exist such a thing as bps state in non-super symmetric theories there are no bps states and even in theories with minimal supersymmetry there are no bps states in six-dimensional theory for example with minimal supersymmetry there are no bps particles there are bps strings but not bps particles and the same is true in 40 n equal 1 theory with minimal supersymmetry so bps very nice but it's the non-generic situation it's the luxury and in fact even in theories where you would have bps states in principle because you have eight super charges and so well for example 5d n equal 1 theory there you do have bps particles and in some cases this is enough to prove the weak gravity conjecture in certain directions because we can show a tower but not in all and this is what i'm going to discuss about but what happens if the bps states are not there and this is the main result formulated a bit more technically compared to the third slide so let's specify let's take a laboratory we have physicists we make an experiment our laboratory is now m theory compactified on a calabria 3 to be very specific you get a theory in five dimensions i'll discuss this for those who don't work on this in a second what would mean for this gyro mean and the statement is suppose there is a direction in the charge that does not admit a tower of bps particle this can happen then one of the following is either there exists a super extremal tower of non bps states so we can find a tower of states super extremals are charged to mouth based on the condition but the particles are not bps or there is no weak coupling limits for that gate and we argue that this is then a matter of debate we strongly believe that this means that the gravity conjecture is not even necessary to hold but this is a different story but the fact is that whenever there is no bps tower at all either there is non bps tower which we identify explicitly in full generality or one cannot take a recoupling limit meaning that the gauge proof is always strongly coupled the coupling is always at plank scale so it is debatable if the original argument leading to the gravity conjecture holds at all and in this sense we are going to establish the asymptotic tower we gravity conjecture meaning we gravity means particle could charge a mass ratio bigger than that of a blank hole tower mean infinitely many and asymptotic means in theories with the recoupling limit so those theories that we as physicists usually think of as controllable perturbatively controllable gauge proof the asymptotic power with gravity conjecture and this can be shown in in general and to do this we need to address more quantitatively these two specifications here first we need to think about excuse me we need to think about the asymptotic part so theories with the recoupling limit this means we need to go need to consider infinite distance limits and modular spaces in this case modular spaces of kala many folks kala kala we are people so this will be the first part to understand which in which regions of modular space we have to look and then we have to do the state counting them and this will be related to certain gb odc invariant which by means of certain physical realities count also non-vps state in the five-dimensional that we will discuss so that's the part for the talk we first have to understand these recoupling limits and then talk about the state counting and then see if this physics conjecture is actually okay so um i am trying to either avoid jargon or this i will not succeed but i would at least i'd like to explain the jargon to those who do not work on this topic a little bit and if i fail to do so please tell me right away so that i can explain what the jargon was supposed to mean so i start first of all with the physical theory that we want to that we want to conclude so we said it's the compactification of m theory to five dimensions what was meant by this the physicists know all this apologies for those of course but just to set the notation so we start with this with the theory in 11 dimensions that's the super gravity theory in a long way thanks limit so gravity theory in 11 dimensions and the important input is that there is a three-form gauge field in this 11-dimensional theory um so this is like a gauge field but with three-form three-form indices so generalization for the jargon or however you want to do in the balance this is where our gauge degrees of freedom will come from in the seconds and what will be the particles so what will be the charged object there are two types of charged objects in this 11-dimensional theory already first there are the famous membranes these are two plus one-dimensional objects so they have two spatial dimensions one time dimension and they couple to the three-form simply by pulling back the three-form to the world volume of the membrane hence they are electrically charged under the three that's the jargon and that's also a hot stool to the three-form so more precisely the derivative of the three-form has a hot stool in 11 dimensions that's the derivative of the six-form the object that couples to this in the analogous fashion is called the five-spatial and one time direction and both of these will be important so the logic will be this will give rise to the bps sector and this will essentially be giving rise to the non-bps sector in a way that we will do okay so that was the starting point in 11 dimensions and now we so that's the unique gravity theory or effective field theory of what one believes to be a unique gravity theory in 11 dimensions and now the usual gain we compactified it on a compact calaverial three so we think of our base time to be the direct product of one plus four dimensions times the compact calaverial three direction and in this way we arrive at by looking at the low energy at a theory in five dimensions so in this r1 form the effective field theory it's super symmetric so it has eight supercharges let's not worry about this and this is a gravity theory and it's a gauge theory where does the gauge decrease the freedom where does the gauge bundle come from its connection is obtained by simply splitting the three form that we have in 11 dimensions in this fashion so we take a certain basis of the two comma of the one comma one form on the calaverial three and we simply write c3 as the sum of one form in r1 form times two form in x3 and this way we get the one form which is which inherits the gauge degree the freedom of three so this is the gauge potential yes that is well because i'm on the calaverial three that's all i can do so i can i can i can consider the three form in five dimensions but that would be a different object i can't consider it and the second possibility would be i take one leg in five and two legs in x3 because if i want you to think about harmonic form so masters degrees of freedom there are no one i mean one comma zero eight of x3 vanishes because it's calaverial three so that's the only option so i want to so that's also the three form but this place no role for us now i want to think about how do the usual gauge bundles or the connections on the gauge bundle come about in this language and this is how it how it all is okay and let me explain the second parts again apologies to the physicists who know this very well if we have this theory now in five dimensions so we inherit the gravitational part and we also inherit the gauge degree freedom in the sense that you now have gauge gauge groups here and the usual yang-milz theory and or or the one gauge theory and then the coupling and this is important the coupling i mean the gauge coupling this is determined now by dimensional reduction it is essentially given by the structure of two form wedge star of the three form internally and this is a certain combination of certain volumes so the cala geometry of the calaverial three fold controls the coupling what stands here is one over the coupling squared so when this is large then the coupling is small so perturbative and recoupling mean large f alpha beta quote unquote in a certain way and this means large volume and which volumes so these are volumes of certain devices and volumes of certain polymer of certain curves on x so that's the only message here controlling the couplings mean controlling the cala moduli of the calaver of the calaverial okay good and um what are we want let's just remember remember what we want to argue we want to show we want to check are their states are their particles under these you once that i get of certain charges q alpha we don't know yet so you want to we want to check this and we want to check if that charge to mass ratio satisfies a certain inequality which was motivated by physics this is the inequality you don't need to look at the details i just want to tell you that once the charges are known this inequality is again a modular dependent state because the masses of the state whatever they are they will also depend on the moduli we will see in a second why so um that's um repulsive that repulsiveness or super extremality condition other context we can translate into uh what it means in this effective field theory and this simply will give us the certain inequality that this modular dependentism that we have under control if we if this means okay so um yeah good but this was just to show that this can all be done specifically um the details we don't need to know okay so now which so we we talked about the gauge potentials what about now the particle which type of states could that be so the first idea and first source of potential states that would satisfy the constraint would be the bps particle the status already bps particles are known to always be extremely also big and which which are candidates for bps particles in the five-dimensional theory they come from these m2 brains recall two slides ago we had an 11 dimension two-thousand one-dimensional object with membrane which d3 coupled now how we do we get a particle of them in five dimensions again by splitting the world volume in a similar way we'd say the time direction d in r14 sorry r14 not r15 and the uh the remaining spatial two-direction along the internal manifold so the jargon is being wrapped the membrane on the curves you all know this because you keep you all know how to compute g gv invariant and some of it in invariant and this counts at the states and these are the states that you know how to think and so these are particles and they have charges why because the original object in membrane was coupling to the preform the gauge field is just the descendant of this in five-dimension so the charge is just computed by um um um um flagging this into here this decomposition very simple of course the charge is then given in terms of an intersection product between a certain device and the curve so this is a logical number the mass of these particles turns out is given by the volume of the curve because the original object had a certain tension. You put it on a curve, and then the volume gives the mass. And the volume of the curve is something that is controlled by Kehler geometry. And hence, we are again in this business that the Kehler geometry tells us everything about not only the charges, either the intersection numbers, but also the mass. And in particular, if the curve is holomorphic, then the particle is PPS. And then as we said, it's either super-extremal, and we are invisible. Therefore, the first question is, does there, in every direction of charge letters, exist infinitely many a tower of PPS particles? Or put differently, if you take a fixed holomorphic curve class, is it true that there exist infinitely many multiples of this curve class, such that the Gromov-Wittman variant, or rather the Gruppler-Kumar-Baffin variant for this curve class is non-zero? That is a well-defined question to ask. And the answer is no, this is not true, as you know very well. However, what is true is that certain curve classes have this property. Namely, if you take curve classes in the movable cone of your Kehler-V-O-3, then it was conjectured by Murat and friends that for those, the following is true, the genus, excuse me, the genus 0, the genus 0, Gruppler-Kumar-Baffin variant, for any multiple n times this homomorphic curve class in the movable cone is non-vanagic. That is a conjecture. And let me recall for you, or let me recall, you know this better than I do, that the movable curve cone is the cone of curves ruled to the cone of effective devices. So on this Kehler-V-O-3, we have the cone of effective devices, the dual cone, if the movable cone, movable cone, movable because they essentially can move freely in the Kehler-V-O-3 or you can have a family that feels out to the Kehler-V-O-3. So this is not a proven statement, but it's a conjecture. It has been confirmed in an overwhelming number of examples of complex Kehler-V-O-3 fold in future papers. And there's also a good reason why this should hold, because as also the Kehler-V-O-3 friend for those curve classes, the associated particles are such that they are BPS. And the BPS condition is the same as the black hole extremality condition. It turns out this is a non-trivial statement. They show that by BPS flow, they show that for these curves in the movable cone, there always exists a nice endpoint of the flow and that in fact, the BPS condition equals the BPS extremality condition, which is not always true when you have scalar flow. So therefore, it is expected that also since the black hole exists, also the particle should exist and hence the invariant should be non-trivial. So this is an interesting project by itself to try and prove this from a purely mathematical point of view while this is true. We are going to ask a different question. The remaining challenge, therefore, is what if there are no BPS cones? And generically, this is the case, because there are many examples. Of course, the movable cone does not exhaust the class of, does not exhaust the, not every element in H2 is inside the movable cone, obviously. So there are many directions where this simple solution or this solution could be gravity conjecture of no cone. And the simplest example would be the cone default. If you have a cone default, a two-title that you just slowed down in the cone default condition, then clearly the GV invariant for one time to describe as one, but for any positive multiple thereof is zero. So in this direction, the charge letters do not have a BPS power symbol. So the question is, why is this? And does this invalidate the tower with gravity conjecture or what's the situation? That's what we would like. That's the question. And the main result, I repeat it again. Whenever there is no such BPS tower, for example, as for the cone default or many other cases, then either one can instead find a non-BPS tower from a different sort of particle and this is related to a different type of invariant. We'll come to this. Or the U1 is always strongly coupled, meaning that we can never take an asymptotic limit in which this U1 gauge group is a nice weakly coupled theory. And hence, we would claim it is plausible that the original physics arguments don't work. But this is another point. But these two are possible. And again, let me repeat so that you remember what we want to do. First, therefore, need to consider all recoupling limits by looking at the telegeometry. And we are going to show that in all recoupling limits, we will find the super-general tower, either of the BPS or of non-BPS. So we're going to prove this just coming from the other side. Everything that has a coupling limit will be proven to satisfy. OK, question so far? Exactly, exactly. So you pick a certain linear combination of U1 and or put differently, if a curve is charged under, has a charge under certain linear combination of U1, such that a multiple times this charge does not admit these BPS states, then this linear combination can never become recoupled. Or instead, you can find a non-BPS curve that does the job. That's the claim. OK, so now I need to do two things. I really want to get here, but I first have to start there because otherwise, I guess, this is how we have to affect the problem. We affect the problem in the recoupling limit. So we need to understand what the recoupling limit is. But this we can do. Why? Because let me recall, recoupling limit means recoupling means. So we have our Lagrangian. We have a gauge coupling. This is inverse gauge coupling squared, 1 over G squared. So recoupling means G small. This means F large. F is essentially a linear combination of volume. So we need to consider the limits in which volumes under the calorie will become large. However, we have to do so at fixed overall volume. The total volume of the Calabria we must remain constant because, to be more precise, the gauge coupling is a normalized volume. So it's volume of devices divided by the total powers of the volume of the total Calabria. So scaling everything up does not help at all. If I'm just making Calabria larger, this makes these volumes larger, but also this volume larger so this doesn't help. So we are mathematically interested in the infinite distance limits in Calabria-like space at fixed overall Calabria volume. Full stop. That's what we need to do. That's what we need to do. And indeed, that's what I just said. But that's only a necessary condition for the decoupling limit. These entries here have to go to infinity because this is 1 over coupling squared. Because these infinite distance limits in Calabria-like space at fixed overall volume. But this is now something that only probably the fitness will appreciate. This is only necessary, it's not sufficient. Why is it not sufficient? It's a bit harder to explain. The point is just because the coupling goes to zero, this does not yet mean that the theory really becomes weakly coupled relative to gravity. So we need to consider the relation between essentially the coupling strength for the gauge sector and the coupling strength of gravity. And that should be small. That is the precise criterion. And our proposal is that we must consider the following object. This is a physics statement, but it's very important because otherwise we would not be able to control this theorem. The gauge coupling squared times the plan scale, this is what tells us about the gravitational coupling. This has to be small compared to the cutoff scale of the quantum gravity. The cutoff scale of the quantum gravity is the scale where the one loop corrections to the Einstein-Hilbert term in gravity become comparable to the tree level. So where the perturbative quantum gravity, the perturbative effective field theory of gravity breaks down. And this is a quantity that has been discussed in great detail in the physics literature. It goes back to first to Twally and has been a lot of papers studying this because it's so important and it's called decision. So for the mathematician, just think about we have a certain recoupling limit, a certain asymptotic limit in tele-modulized states plus another constraint to check that comes from the physics. And for the physicists, it's the statement that we have to relate the gauge coupling to actually the cutoff of the quantum gravity. And this is very important because this will go to zero as well in the limit. And hence we really need to make sure that the ratio goes to zero and not just the numerator. Okay. So let's characterize first the recoupling limits and then we come to the state count which recoupling is going to occur. So as we said, a necessary condition is that we look at these infinite distance limits in tele-modulized space at fixed volume, fixed overall, overall. And these one can classify, we classified them together with Wolfgang and Soju some time ago and they can be classified in this way. Such limits can be obtained either if your Calabria has a T2 vibration, a genus one vibration, need not be elliptic but at least genus one vibration or it admits a K3 vibration or an abelian surface vibration, a T4 vibration. This should be a T4, not a T3. K3 or T4. And so this is a topological criterion for a Calabria 3-2, it emits an infinite distance limit in tele-modulized space at fixed volume. And when this topological condition is satisfied then the limits always look like this. In the first case, the volume of the base goes to infinity, let's call the scaling lambda and the volume of the fiber goes to zero, just inverse so that the total volume stays finite. And similarly on the K3 or T4 vibration side, the volume of the base goes to infinity and the one fiber goes to infinity. So the only Calabria 3 with an interesting recoupling limit or with an interesting infinite distance limit in tele-modulized space must admit these vibration. And this is very nice because many people here in the room are interested in vibration of either of these two types for completely different reasons. And the statement is, these are the only relevant cases if you ever want to study a weekly couple gauge theory. All the other ones that do not enjoy this are not relevant. Second, these are the ones that have been studied for ages and spring theory because they admit nice duality frame. And this is why these dualities are relevant because they automatically come when we want to think about weekly couple gauge theory. This is a very general statement that I think puts the importance of these vibrations into a very physical context and explains why they are. What the feature, so general fiber in these mathematically or physically? No special feature. An arbitrary gene is one vibration. Need not be elliptic. And an arbitrary K3 or Q4 vibration. Need not have sections or all color be our vibration. Do the job, exactly. All the color be our vibration. The fiber dimension, you mean Ramney's fiber double cover? No, no, no, no. No, no, thank you. No, absolutely not. The dimension must be positive. Yeah, thank you, thank you, absolutely. No, this is just a discrete structure. Here I'm interested in really taking an asymptotic limit in the smooth modulation, in the model. Yeah, thank you, it's only these. Okay, so the claim is these are the only ones that admit a recoupling limit. And now for these, something interesting happens that again it's clear to the physicists. Supposedly we take such a limit. Let's look at this first object. How much time do I need? 15 more minutes, okay, good. I thought I would do much. Anyway, let's discuss this. What happens in this limit? The base becomes large, the fiber becomes small. So what happens now in physics? In physics, we said we have these two brain, two plus one dimension brain. We can wrap an M2 brain on this fiber. This gives a particle. When the fiber becomes small, the particle becomes light. We can wrap it infinitely many times because the gramophitin invariant of any positive copy of the torus fiber is non-zero. This is the general result that you can show with the lecture theory or other means. So this is always gives rise to infinitely many particles, wrapped once, wrapped twice, so forth, resulting with every multiple of the torus while we get the GD invariant. It gives a particle of volume the torus when this goes to zero, lambda goes to infinity and the limit you get infinitely many masses of particles. They are BPS. What is the role of these particles? From a physics point of view, all you see that you get many, many light particles when you switch on your collider, you will find many, many particles. You don't know where they come from, but they come at accurate distance in the mass. The mass of the second particle is twice the mass of the torus. And this means that these particles behave like calypsox line states. This means that effectively a new dimension in physics opens up. This is the decompactification limit from five dimensions to six dimensions. And you end up with a theory in six dimensions in which the torus is gone and this defines your compactification. This is of course the well-known F theory limit. And the quantum gravity scale here for the physicists is of course just the punk scale in one dimension, which goes to zero compared to the original time scale. So this has to be taken into account. More interesting is the second type and the nice thing is that that's the only thing. The second type, what happens now? Now it's surface fiber goes to zero volume. What can happen? In particular, recall, we also have these five frames. These are five plus one dimensional object. So I can now take four of these dimensions along the surface fiber, then I have two dimensions left, space and time. These two dimensions I put into my space time, non-compactified, and I get a string. I get a string which becomes light. The tension of that string is the volume of the fiber, it's small, it goes to zero. And since this is the Calabriault, since the fiber is Calabriault, the string that I get is either a critical heterotic string or type two string. I've also been known for ages in the physics literature. If this were a P2 fiber or a non-Calabriault fiber, it would not be true. But in this way, I get a new string, heterotic or type two string. The tension goes to zero, it becomes smaller than any other scale. So my emergent theory at infinite distance is that of a new type of string theory, heterotic or type two string. So I started with an M theory, the word string has not been used, but in this limit, I end up with a string. So this is the reason why string theories are important from that point of view, because they naturally appear at one of the two possible ways how to go to the coupling and that intimacy. And the quantum gravity scale, in this case, is the species scale of solids with that string, and we have to compare everything to that. And by the way, there's a conjecture. So this is an example where this happens, and the conjecture is this emergent string conjecture, that is always completely general, that every infinite distance limit in every modular state of every Calabriault always either leads to something like this, the decomplication or to an emergent new string theory, new compared to the original theory that we found. Okay, so now we've understood the recoupling limit, the possible recoupling, the possible infinite distance limit. What now remains to understand is which gauge groups in this five-dimensional theory become weakly coupled in these limits, because for these we want to check the regravity condition. And this is what we did more recently, we classified them, and the statement is very simple again. The only you want, which can ever become weakly coupled, are one of the following two types. Namely, when you take a linear combination of you want, you can think of this as being associated with the curve flow. Why? I mean, the easiest way would be, think of a three-form, we had a three-form C3, I can integrate the three-form over a curve, then I have one leg left, left, and that's my one form.  And the curves associated with the you want, which can become weakly coupled in one of these two limits are of the following type. Either it's the torus fiber itself, and only the full torus fiber. So no degenerates components, no exceptional curves inside the torus fiber, only the generic fiber, or the curve is the curve inside a generic K3 or K4 fiber, meaning those fibers that can occur at finite distance in the complex structure of the fiber. So what I do not want is curves that are localized only in say type two or type three hulikop type fibers, K3 fibers that hulikop. These will not lead to the repeat couple of H-theories, but all the other ones, the ones that I can deform everywhere, these will lead to weakly coupled you want. So we have now complete control over all the potentially weakly coupled gatefruits. They are either always associated with the torus fiber or with essentially a generic curve in a K3 or K4 fiber. That's the claim. And to show this one has to look at all possible degeneration and look at the intersection form in these degenerate cases, but that's the other thing. Okay, so now I'm done. Now I want to check the weak gravity connection. Now I want to ask, for these you want, are there BPS towers? And if not, are there at least non-BPS towers? We could start with the first case. The first case, are there BPS towers? And this is very simple. The answer is yes. Of course, there are BPS particles charged under the C1. Namely, the BPS towers are just given by M2 brains wrapping the torus fiber themselves. These are particles. There's infinitely many because we already said that the torus fiber has, any multiple of the torus fiber has non-zero GD invariant. Hence there's a tower. And these particles are precisely the ones that are charged under this corresponding U1. The U1 is the dual device. So it would be the section or the multi-section which you always have. So this is precisely the combination of the two. And everything else, every other curve will be charged under U1 that cannot become weakly coupled. And hence we do not care if there's a BPS or non-BPS. So those that are weakly coupled that can become charged under something weakly coupled. For this, there is BPS towers. So that's the boring case. And I'm skipping the technical for the middle case. The interesting case is now the one of the vibration, of the circle vibration. And I'm only going to discuss the K3 case for again, believe me. So what's the remaining question? The political WR3 admits the K3 vibration. Then we already said the only U1 which can undergo a weak coupling limit is of the type that it corresponds to compactifying to reducing the three forms C3 over a curve which is part of the generic K3 fiber or can be deformed into a generic K3 fiber. And indeed one can check that these curves that the associated U1 satisfy the statement also the more complicated physics statement that the gauge coupling becomes small compared to compactive. So now I need to check finally, what are the, do I have BPS towers? Do I have super extreme? So now we are in the realm that many people here have been working on extensively. Namely, we need to think about the letters, the letters of curves inside here, the generic K2 fiber. And as you all know very well, those letters over the wheels can be decomposed into self-fuel and enter self-fuel parts of positive and negative self-interfection. And this is a letters of rank 1, R, where R is not larger than 90 in the generic K. So I now wonder, are there particles that I can get by wrapping M2 brains on other uncertain curves inside here? Any curve that would satisfy the regretting and then you want it. And the statement is this depends on the self-interfection of the curve that I look at. If the self-interfection is non-negative, then a BPS tower exists. Meaning the GV invariant for any multiple of that curve is non-zero. First of all, as a check, such curves are indeed movable. They are movable inside the K3. If the self-interfection is non-negative, they can move inside the K3. And they can also, since the K3 moves over the P1, they move therefore in the entire X, in the entire Calabria scheme. So they satisfy the conjecture of more random friends. They are in the movable cone, hence it would detect them to admit the power of BPS particles. And this is well known, has been known for a long time because the BPS index is counted by a modular form. More precisely, the number of the GV invariant such in the zero of the curve C is given by this coefficient, the N where N is T squared over two. So as long as this is non-negative, this is the coefficient of non-zero. And not only is it non-zero for one time the curve, but also for any positive multiple of the curve as well. So here we have a BPS tower and the regravity conjecture is done. However, generically, we have these curves C squared of negative self-interfection as well. These curves are rigid in the K3 fiber. They are not in the movable cone. Therefore, we do not expect the BPS tower and there is also no BPS tower. But this is now the claim. In this case, a tower of non-BPS takes over. And these states come not from the M2 range, but from this other source of states that we could potentially look at, namely from the M5 frame. More precisely, we can now consider the following object. Once more, we wrap the M5 frame now on the fiber. This gives us a string in five dimensions, not just any string, but since it's the K3, it's a heterotic string. So it's a very special string, a critical string. This critical string has excitations. The excitations of this string are charged under the gauge group. And the claim is there are always some states that are super extrema that we need to show and that's infinitely many. So this is what we need to show. And in order to show this, we have to translate the counting problem of counting the excitations of this string that you get by wrapping in five-dimensional curve into a counting of BPS invariance of Donald Thomas type of on the conduct of type A string theory on the same color beyond. And this was discussed in detail by people in the room in particular, let me point out the table. So more precisely, I'm translating now this problem of counting string excitations to counting BPS invariance. Those of you who have worked in this program very well, let me just review how this works. We have this M theory on Calabria III. This was the theory in five dimensions. Now we put this theory on another circle. So I go effectively to four dimensions. What I can now consider is the following state. I can consider a bound state of this M5 frame wrapping the fiber, the physical string. The string can wrap many times on the F1 and I can also have what the physical KK momentum. So waves along the string. And I can bound it to the particles that I got by wrapping M2 brains on any curve, any curve in the letters of the case. And this is known to be equivalent under certain string theory duality to the object counted by DT invariance and before DT we do a bound state or object counted by coherent sheets with these charges, R times R for the wrapping number times the class of the K3, Q of the class of the curve and N the class of H0, N copies of H0, so to speak, that would be the zero bound state. This is a well-known reality. And equivalently, you can think of these states being the winding modes, winding and KK modes of the headlocked string that you get by compactifying M5 brain on the string at the KK level N and with charge vectors. And, okay, so far so good. And now the important point is in the special case where R is one, so where you wrap only one, you can identify this KK number for the state in four dimension with the excitation level of the headlocked string. This is because the KK and the excitation level in string theory always appear in the same fashion. So when I know that a certain state in four dimensions like this at certain KK level, then this means that if I go to five dimensions, I don't wrap the string, but the same five dimensions that the same states, same charges exist at a certain excitation level of the string. This is only true for the headlocked string with R equal one because this is a critical string. In general, this would not be true. These states in five dimensions are non-BPS. And hence, they must be non-BPS because we've already shown there are no BPS states that would do the job. So these states have a chance to be two-dimensional and they are counted by BPS states in one-dimensional. So these can be counted by these invariants. This way we get non-BPS string excitation one-dimension higher and we now need to do things. First, we need to show that there is a tower. For these, a tower of a certain type with a relation between the excitation level or KK number depending on how you want to do it and the charge. We call this is two-squared negative, so this is how we do it. And then we need to show that they are super extreme. And then finally, we have. So let's first, quickly, I'm being fast enough, but let me speed up. First, we have to show that these states exist. So the claim is there are special states at KK number N and where the KK number is equal to this charge. Or if you think about the gene variance where the N is minus one-half G squared for this gene variance, these always exist. That's the claim. And they are super extreme. That they always exist. We know by looking at the counting problem. The same statement that I just made more formally just means that the elliptic genus of the z-rotic string is related to can be written as the linear combination of a generating function for the s-number and the z-type forms here. The bbs-numbers, the 40 bbs-numbers are counted by this gene variance. And we now need to show that precisely when N plus Q spread over 2 is zero, that then the corresponding omega gamma is non-zero. Then we have those things. And this follows from the observation that was made very clear in this paper in particular. And I'm looking at a special case. The paper is much more general. That's the generating function for the DT invariance has this nice form. In the specific case, K3 with R equal 1 of this eta minus 24 times vector value model of form that counts essentially all whose coefficients are related to the net elliptic numbers. And you can show that precisely when N plus Q spread over 2 is zero, so that when the corresponding netherlatches number would have to be zero as well, that then the bbs-number is non-zero and the state. So this counting problem has been solved already. We just need to put it together. And now what's left for us to do is to show that the state, this is the tower of states. So tower meaning, if I take the charge Q negative, then I can always find an n such that a state exists. And of course, any multiple of Q also exists because I just multiply then n by the corresponding number. Hence, I have a tower. And what we need now to show that this tower is, in fact, super extremally satisfied with this projection. So we have to look at this form of self-repartitionist condition. We have to look at this specific state. And this we can do with the Hearth of Spring Theory. We know what the masses of states are at a certain excitation level. There are a number of technical complications because we work in five dimensions, so we're taking the consular parameters, et cetera. But when the dust settles, one can show that in the asymptotic limit for recoupling, everything just cancels that was causing trouble and miraculously the state just satisfies the risk-reality projection. So everything falls into places, but only for those states. And these are the only ones generally that do the job. Shall we break up with the final statement? Yes. Here it is. So we've shown the asymptotic version of the gravity conjecture of the tower with gravity conjecture in five dimensions. Similar statements can be made in other theories without non-DQF states, namely in six dimensions and in four dimensions, particularly for the minimally supersymmetric ones here. Quantum corrections, same important role that we also looked at. I discussed the K3 story. The G4 story will go the same way, but it would be very nice to understand better from the countering point of view. That's maybe something you can think about later. And now for the physicists, a nagging conceptual question, what now if there's no tower in a certain direction? Well, what we know now is this is only then the case when the U1 cannot become recoupled. But you could still ask, should the Greek gravity conjecture not be satisfied nonetheless? This is an ongoing debate. We have some opinion on this and hopefully contribute to the discussion. And for now, thank you very much. That's thanks for the question. Is there any quick question for this speaker? Yes. OK, any good team. So for example, heterotic on T4. No, I would assume that the same. I would assume that the heterotic on T4. Yes, I think everything would be OK because this is heterotic on T4, right? No, 5G, 5G, any good team. On heterotic on T4. I'll have to think about it. I'll have to think about it. I would think so. I would guess yet because there would always be interpretable, there would always be VPS tower that could be a KK tower or string excitation tower. So my guess would be that it always works. Yes. Yes, yes, yes. I would think so. I don't want to make a definite statement. My guess would be that it works. Perfect. So let me take the opportunity before thanking the speaker also to thank for this very nice, wonderful conference to Simone, Chuck, and Johannes. Let's thank the organizer. Let's also give a quick thanks to Sebastian. Where is he? He's not the rich person. Yeah, thanks, Peter. That's great. There is a lunch. Yes, it's already cooling.


Chief of Naval Personnel, Vice Admiral Robert Burke and Fleet Master Chief April Beldo flew onto the aircraft carrier USS Dwight D. Eisenhower Sunday afternoon. The two toured the ship and talked one-on-one with members of the ship's crew during their visit. The crew was also able to ask questions concerning Navy personnel policies during three separate all-hands calls. During the calls, Vice Admiral Burke emphasized the sailor's 2025 initiative. But we really kind of divide the effort into three components. The first component is to completely overhaul and modernize our personnel system. The second component is called Ready Relevant Learning and that's about changing the way we do training for our entire enlisted force throughout the continuum of a career. The Eisenhower and its carrier strike group are currently deployed in the Arabian Gulf in support of Operation Inherent Resolve. For USS Dwight D. Eisenhower, I'm Seaman Casey Treach.


So, what we need to explain, we are not kind of conference maniacs, but we are here, but I think this is a very interesting example of how we can combine anthropological and archaeological data, because in my impression when I look at the own fields of publications is that we treat anthropologists like a table provider. And the problem is also that archaeologists believe this data, they really believe these tables, but what Garzina wanted to show is that this data can be easily modelled and actually the final results and also the whole knowledge production process should be discussed. So, we want to show you the case study from the China, which I think is a very interesting example of this cooperation about cooperation of anthropological and archaeological data. So, I'll just make a short introduction and Garzina will continue. So, what we wanted to see was the connection between the deceased population and the living population. And also what we did, our assumption was that graves, even the own field graves, they really differ and we can find these differences of course. And also, can we speak about social stratification only about only basic archaeological data? This is also only on the archaeological data obtained from the graves only. This is also the problem, because this is very general question. Is this real world of the dead people, which is as much stratified as the world of the living population? And also, how can we effectively compare the archaeological and anthropological data? And also, what kind of variables can be taken into consideration when we are looking for some social stratification? And also, is there a link between biological quality and grave apparatus? So, this is just some detailed aims. We took two features of these graves. I will show you why, because they are very not interesting. Because archaeological data is selection of grave goods and we focus only on two things. One was metal objects and mammal number of vessels and also type of grave pits. Because when we look at this cemetery, it looks very much all the graves were the same and the only differences were number of graves and there were some metals, very very few and also some had this chambers and some had not. So, I think there are only variables that can be studied. And also, we have human remains. I use my coffee break to calculate the children graves and we have 137 children graves from this. But this is not total number. The total number is more than 600, but some of the graves which were partly destroyed we are not sure if this bones belong to this grave or that grave were just excluded from the test because the site was excavated 40 years ago or even 50 years ago. It's famous for the fortified site and for the really huge fortifications. And what is very interesting for us that is one of the best dental chronological data at archaeological sites in Poland. So, we have very clear chronology of the settlement and also of the cemetery because the settlement was destroyed and we know exactly when it was abandoned and never settled again and the same was with the cemetery. It was not used again after this date of the mid-6th century BC. So, this is what we have and we look at the chamber graves. So, you can see we don't have the actual remains of the wood, but we can see the very suggestive shape of the outline of the pot and we have some analogies from Brandenburg, from Billentofkultur when these chambers are observed in the excavations. And also, Greggs are mostly pottery. So, from the archaeological point of view, when you want to tell the status, this is the worst situation because the metals we have are very, very poor. This is also interesting because when you look, there are only pins and most of them were fragmented, were bent before the position because even just very fragmented, like the heads only or missing tips or just some parts, they were bent. So, this is also an interesting situation and when we don't have like pins, we have metal lamps which might be pins also cremated together with the body. Just for comparison that these metals, there were so few in the Greggs, not because these people didn't know how to make metal objects, they did. These are some examples from the settlement context and also the area is surrounded with bronze deposits. So, you can see the work on iron and gold and bronze, of course. So, what we have, we wanted to check, can we measure the status, the possible status of this population combining this, comparing this archaeological features like grave chambers, metals and minimum number of vessels and some biological indicators. And this is good enough. So, we use the individual information concerning age of death and this cultural data consists of minimum number of vessels, present or lack of chamber, grave chamber and metal objects. And for this data set, I have selected some statistical methods, cheese square paste, Kaplan-Meier survival function which was discussed earlier during the morning presentation and Cox proportional hazard models in order to capture which of these social and cultural variables we have focused on influence the age of death and which of them had the strongest impact on age of death. Age categories, we have the following age categories of children and adults. As I said earlier, adults who started at age of 15 and social strata. So, we assume that these variables were a proximal social stratification. So, minimum numbers of vessels and for two models, the first three categories and then two categories, 1, 9, then 10, 19 vessels and equal and higher than 20 vessels, 1, 9 and more than equal or higher than 10 vessels. Presence of a chamber or lack of chamber and presence of metal objects and lack of metal objects. So, our proposal of these variables as a proximal social stratification of the population are studied. And this slide presents the distribution of minimal number of vessels according to child and adult age categories. So, as we can see, the first model, three variables of minimal numbers of vessels or three categories of minimal number of vessels and the proportion of the distribution of the minimal numbers of vessels in children and adults. And the second model, two categories of minimal numbers of vessels and children adults and the general conclusion as follows. Greater minimal number of vessels occurred in adults' graves more frequently than in children ones. So, for example, 1, 9 vessels were represented in 72 percent in children's graves but almost 88 in adults' graves. In the second model, 72 percent of one, two minimal numbers of vessels were represented in, or occurred in children's graves but 75.5 percent in adult ones. So, statistical significant distribution in the light of two square tests of minimal numbers of vessels according to children and adults' graves. Here, survival probability, I discussed this slide earlier. It provides information about what the fraction of individuals survived to age X. So, here again, to the onset of reproduction, conventionally, the age of 15 survived 50 percent of people up to the age of 30, for example, 40 percent of people, et cetera. But what is characteristic is the fact that the number of vessels, the minimal number of vessels, distinguished survival probability, estimated survival probability and differences in the estimated survival probability according to the number of vessels contained by two square tests, the statistical significance. So, in general, the greater number of vessels, the higher survival probability. So, for example, in the situation where there were 20 more vessels to the onset of reproduction survived over 90 percent of people, but for example, when the vessels were between one and nine, the first category, 50 percent less or 40 percent of people. So, a kind of stratification in the bracket stratification in survival probability according to minimal number of vessels. Then, according to the presence or lack of chamber, and here again, Gahan generalized Wilcoxon test, confirms significant differences in estimated survival probability. So, the first situation, presence of a chamber to the onset of maturation survived over 70 percent of people, but the first situation without presence of a chamber, 50 percent of them, so significantly less. But lack of differences in the right of statistical test in estimated survival probability according to presence of metal objects or lack of metal objects. So, here, in this case, the fact of occurrence of metal objects in a grave did not influence estimated survival probability. And the second question was, so if there is a group of cultural variables influencing survival probability, which of them had the strongest impact on aged dead? For this, we used cox proportional hazard methods, these three cultural variables. First, I checked the impact of minimal numbers of vessels on aged dead, and then I added next and next cultural variables. So, finally, we have three cultural variables, and let's focus on this red marked line, the strongest impact, statistically significant impact on aged dead was caused by minimal number of vessels. What is characteristic is the p-value here and world statistic p.5.6. So, the strongest impact on aged dead, minimal numbers of vessels was the only factor significantly influencing aged dead. Let's sum up, minimal numbers of vessels occurred more frequently in adult graves than in children ones. Survival analysis, the Kaplan-Meier curve confirmed the impact of minimal numbers of vessels on, and the presence of the chamber on survival probability. There was lack of such impact in the case of metal objects. And finally, cox method, cox proportional hazard models showed that the first variable, minimal numbers of vessels had the strongest impact, was the strongest socio-economic factor or social factor affecting aged dead of individuals. So, that's the group of researchers focusing on this topic. Thank you very much for your attention.


Field life is pretty much the ideal scenario for the best training you can get. It definitely makes days go by long. There's not a really good way to get all the rocks out from underneath you, the kind of everywhere you can't really avoid them. It cools down a little bit at night, but it's hot daily. It sucks about living out here in the field. The worst part is easily the heat. There's no way to escape it.  It's well worth it for it. The benefits of field life for the Marines is that when you put everybody in an austere environment, it forms bonds between the guys. And our unit cohesion depends on their brotherhood, their bond. It's well worth it when you get to get back and finally sleep on a bed. It's something that no one else will understand unless you've been through it. Sleeping out in the field is, you know, as close as you can get is the best training to simulate what we're going to be going through. The worst part about it is probably the lack of being able to change your camis. Going back to the normal life is, it's a lot easier than transitioning into being in the field, but it definitely feels nice to get back into it. Our reserve unit's from Fort Evans, Massachusetts, right outside of Boston. So coming out here is a little bit of a shift. You know, we're not exactly used to 105 degree heat with the sun beating down on you every day, but it's nice to see that all the guys can get acclimated to it and get the job done on you.


Woman removes painting varnish, uncovers husband's 50-year-old secret. Sean stands frozen in her tracks, and then clasps her hands to her mouth. The painting that she thought she knew so well looked completely different. The cleaning had removed varnish and exposed its true content. What she had been told wasn't true. Her husband had misled her. Now, the genie was out of the bottle. And she knew life would never be the same again. Sean Smith and her husband Richard had spent the last 54 years together. The two lovebirds first met when they were just 18, and since then they embarked on a journey full of love, loyalty, and companionship. But Sean had no idea her husband was hiding a secret. A secret that remained hidden away. Until now. Since they first met 54 years ago, Sean and Richard became inseparable, and even now in their 70s the two rarely left each other's sides. It was only when Sean cleaned a painting that Richard had given her 50 years ago, when she realized that the past 50 years wasn't what she thought it was. Sean had spent her life as an avid art collector. Although she refused to pay big bucks for certain pieces, she often saved and treated herself when she saw something she truly loved. So when her then boyfriend and now husband Richard gave her a piece of art, she was thrilled, but she had no idea what was to come. 50 years ago to celebrate their fourth anniversary together, Richard had got Sean a painting that he claimed to have gotten for a good price. Richard was always thoughtful with his gifts and gestures. He knew just how much art meant to Sean. Handing over the painting with his Hollywood smile glistening in the light, Sean felt butterflies in her stomach. But she had no idea there was something behind his Hollywood smile. As soon as Sean saw the painting her mouth dropped wide open. It was gorgeous, but it looked just like a Rembrandt and iconic painter. However, Richard insisted it wasn't, and was merely a painting by an unknown painter. Either way, Sean loved it. But she knew she had to find out the truth. No matter what. Few painters are as revered as Rembrandt. The Dutch genius has been hailed as one of the masters of the art world. Famed during his lifetime, Rembrandt's reputation has only gotten better and better since his death in 1669.  They are rare and highly valued. Sean wanted to know the truth, but she wasn't prepared for it. Although her husband insisted that this painting was not Rembrandt, Sean had a feeling in her stomach it could be. After all, she had been surrounding herself with art for all of her life. Something was telling her to find out more, but her husband thought it was nonsense. The painting, titled Portrait of a Young Lady, looked like a typical Rembrandt work. It had all the hallmarks of his back catalogue. Or so it seemed. Sean's husband told her that he sent it off to Holland for an examination. And when they responded, Sean was completely surprised. Her husband said an official assessment from Rembrandt experts in Holland found that the painting wasn't actually a Rembrandt at all. Sean felt devastated she truly did think it was genuine. This wasn't an original painting by the Dutch master so who was it by? Ultimately Sean never found out the truth until now. Decades had passed since Sean received her Rembrandt painting, which was deemed illegitimate. It was a beautiful painting no matter the hand it was created by. It hung in her living space for years and years. And through all these years Sean did wonder if she would ever know who made this painting? All she had were theories. And decades after, the truth was about to be revealed. It was not uncommon for masters of the great Dutch art era to take on protégés. Many of the best depended on mentors to take them under their wings and show them the ropes. Rembrandt wanted to pass on his knowledge and thus took on assistance. But there was also another theory that Sean had. And it was a theory that left the hairs on the back of her neck stand up straight. When Sean was told by her husband that her painting was a fake she was devastated. But Sean could see the sadness in the painting. She could tell whoever had painted the piece of art had some serious anger and violence pent up inside of them, just waiting to explode. In fact, as soon as she saw it she felt a shiver run up her spine. But then a thought came to her. Did the person who painted this piece of art have a dark history? Michelangelo Caravaggio was one of the great Italian painters of the Baroque movement. But he wasn't just that. He had ended someone's life. Famous for his temper, Caravaggio had to flee Rome in 1606 after committing murder after a tennis match. Naturally, this has complicated his legacy. Could his art still be proudly sold? Sean was conflicted. And she was conflicted for 50 years until one day she took a closer look at this painting. 50 years after receiving the painting, Sean was cleaning her house while her husband was out grocery shopping. As she swept the floor she glanced over at the painting. It sure did fill her with happiness every time she saw it. But then suddenly, a sparkle glistened on the painting and Sean's curiosity shot up. She walked closer to the painting not knowing what she was about to uncover. Straight away Sean started to remove layers of varnish and with each layer removed the painting seemed to reveal its true identity. By the end, all she could do was gasp. The painting looked so different from 50 years ago.  Picking up her phone, she called her husband. I knew it, you didn't believe me. You see, when Sean was removing the varnish, she deduced that somebody had poured it on to create a mirrored surface. However, it obscured the original brushwork and color of this artwork. After undoing that damage, Sean concluded that the last 50 years had been a lie. The true artist was Rembrandt. Her husband had lied to her. But now she had to prove it. When the media were notified about this story, there was only one person they wanted to speak to and that was Sean. It was her keen eye that allowed Sean to spot the mistake. But interviewers kept coming back to the same question over and over again. How did 50 years pass before somebody corrected this error? It's hard to know why it took so long for Sean to spot the error. Her husband insisted he didn't know it was genuine. There's a lesson here, always get a second opinion. One such mistake ended up costing a woman over 160,000 pounds of her own money. Lin Fuss had spotted the William Nicholson painting still life of a glass jug and pears and fell in love with it. However, there was no way she was going to spend hundreds of thousands of pounds unless she was sure it was real. The first clue was to examine a string of mysterious numbers and letters on the back. Using ultraviolet techniques, the authenticator eventually discovered they were trained departures that coordinated with when the British painter was alive and working. The second clue was more important to Fuss. It was the physical paints on the canvas. And for this particular artist, they still had samples from Nicholson's paint box. That was enough for Fuss, not to mention the art community. Since the paints were a match, it had to be real. The art enthusiast shelled out over 160,000 pounds and carefully took it back home to enjoy. Things seemed fine until she made the huge mistake of appearing on television. The British television show, Fake or Fortune was an amazing and free opportunity for Fuss to have her paintings worth properly evaluated. After all, there could be a chance to sell it in the future. The show's main authenticator, Patricia Reed took a close look and frowned. It was a bad sign. She had barely looked at the canvas and already had some very strong opinions that drew tears to the owner's eyes. The first stab to the heart was the words boringly painted. The next was close and disquieting similarities. But the woman was still going to move forward with a thorough examination, including the same tests that had proved it was real. Fuss wasn't the only one waiting in agony. The main format of the show was to first evaluate the piece and then reveal the verdict in the following episode. Everyone was desperate to know the real value of Fuss's investment. As soon as the show aired, you could see something strange in the presenter's eyes. It was normal to draw out details for added suspense and airtime. But at that point, nobody cared that Winston Churchill had been a very apt pupil of Nicholson. The presenter finally pointed to the painting delivered two pieces of devastating news that nearly brought Fuss to her knees. Not only was there not enough evidence to show this was a true Nicholson, but the value also dropped from hundreds of thousands to a mere couple hundred. Everyone watching felt the same stab of disappointment and loss, but none more than the owner. An unexpected twist is that Reed became somewhat of a villain. People vehemently protested that if one person thought there was enough evidence, including paints from the artist's own box, then how was her not enough evidence ruling the Superior One? There were even calls for torches and pitchforks to her front door. Reed fired back with a very common claim. The theory was that Nicholson had many students in his workshop, and one of them probably practiced with his paints. One woman's ruling ended up countering past examinations. Fuss was forced to go home with an old piece of canvas that was worth less than the handbag she was carrying. So much for second opinions.


Or I'm going to show you how to add a drop-down list of other companies that will auto-populate the totals over here. And the way I'm going to do that is come to my tab called Data. There are my companies. I'm going to Control-Shift-Down to select all those companies and then give that a named range. Let's just call it Companies. I'll come back to Sheet 1 and here under the company I'm going to go to Data, Data Validation, Allow List. And here I'll just put Equals Companies. Now that allows me to select the company. The next thing I'm going to want to do is use VLOOKUP to find this company's totals. So equals VLOOKUP. As for the lookup value that's going to be this, the table array I could have named it or I can just come over here and Control-Shift-Down to select that whole table array. The next thing I want is the column index. I want to return the second column, that's the total column, so column 2. And then I'm going to go ahead and make it an exact match by entering false and the parentheses and now you can select the company and it'll change that number. You should go check and make sure it works. I'm going to Control-C this, Control-V here and just change this to the third column. And to finish it all up, let's drag that down and fill without formatting.


Police surrounded the residents of Pakistan's former Prime Minister Imran Khan in the eastern city of Lahore before he was arrested on August 5. Police arrested Khan at his residence after a court sentenced him to three years in prison for illegally selling state gifts, potentially barring the opposition leader from contesting an upcoming election. The arrest is the latest in a series of blows that have weakened Khan's political standing after he fell out with Pakistan's powerful military and his party splintered.


تشرف بالاشتغالي بالعلم ولا تبغي به ما عشت يا دا دا دا لا وياله من شراف عظيمي أنا سيد رضي الله عنه ورقوتي that the messenger of Allah صلى الله عليه وسلم said اe سهور the pre-dawn meal surely there is blessing in suhoor أنا سيد مالك رضي الله عنه قال قال الرسول الله صلى الله عليه وسلم تسحروا فإن السحورة بركة فإن السحورة بركة حديث عن سيد مالك رضي الله عليه وسلم تسحروا صح وين تسحروا من فع الأمر it's a command if the prophet commands something what does it show واجب the prophet commands you something it's obligatory it's صحور واجب based on this hadith no the based on this hadith based on this hadith it's wedging if you forget what did we say for the qaida if you forget something خبنا لا تأخذنا إنسينا او اخطانا if you forget something we're gonna come to it we're gonna come to it صحور is what by the way you have to remember something when it comes to the word صحور and then we have صحور صحور is actually what that timing of the صحور right صحور بتفتح السين if you say صحور that is the action ستفعل صح بتفتح السين is what what's eaten at the timing the timing and the thing that's eaten at the timing whenever you bring back this صحور is nice صحور the dates and the milk if you're Somali you eat sour it helps you a lot by the way صحور I forget it all the time it's that white stuff it's not porridge it's a bit different from porridge yeah it's a bit stronger than a porridge it goes brothers it goes way I don't have it I don't like it but I ask Somali to show you go to the Somali it's if the صحور time they give it some milk to it and sugar whole cornmeal that's the word that's it brothers because it just doesn't digest it's just there so صحور is what's eaten صحور it's a what الفعلو it's what you're doing when you're eating just like تهور and تهور right تهور is the what is the pot and the pan and the water تهور is the action of purifying itself it's all وضوء and وضوء it's the same but then this حديث عارو دو صحور which is a command we have another حديث that came and they diverted it from obligation and he made it recommended this is how you deal with the evidences pay attention if I didn't mention the other evidence to view you this would be obligatory صحور would be obligatory if you don't need your sealant تسع عارو because the prophet commanded it but we realize through other narrations that it's not obligatory so this command shows what and the command of the prophet two times أمر للوجوب كمان which is obligatory أمر للستحباب that is recommended this is recommended with what there's an external other evidence what's the external other evidence that show that it's recommended and it's not obligatory is the fact that the messenger صالح he attached many fastings together didn't he not he didn't want to know this وصال right what's وصال it's the person fast two days or more فلا يفتر and he doesn't make his fasting are you with me brothers the prophet used to do that عبدالله من الزبير ايه that's عبدالله من الزبير and I think it's عبدالله من الزبير or his father عبدالله من العوام رضي الله تعالى عنهما may Allah be pleased with them and I think it's عبدالله من الزبير he fasted continuously for 15 days he did with صال 15 half of Ramadan he did it with what نقل عنه it was transmitted from him من فوائده when they say to you come 15 days because he never lived on chicken and chips and kabab these people سبحان الله they were they were what they were أقوية so that's important for us now now we have to realize that the prophet said فإن السحور بركة السحور is بركة let's mention some of the بركة that are in it number one فمن بركة السحوري from the بركة that are in the السحور is اتقوي علي العبادة you're actually going to become strong for the عبادة and Allah loves that imagine you don't do سحوري and you say I don't want to have سحور and in ظهور you're like I'm tired and asir comes I'm even tired and then if طار comes what did you pray oh there your mind is gone does that make sense is that what the sharia wants you to do the sharia needs you to be strong and سحور brings that what does it bring so it says what it gives you dependence to fast properly praying so your عبادة is not based on قسن and laziness صح the person who tastes the سحور and eats good سحور they actually will enjoy the fasting and will desire to fast more and you don't burn out you'll actually say you know what Allah I actually love fasting it's actually a nice thing and I've realized that the days I have سحور that day I'm ready toe to toe brothers you ready I'm ready that's the days when it's what سحور the day when I don't have سحور I'm like I speak to me I even can't hear properly and that's what happens but the days you fast with the سحور brothers you have in your heart is ديادة من الصيام you have actually passion and love for it you wish you can fast more like this and the sharia and because the issue of when you find something hard the love also decreases and it requires كمالو دولي مع كمالي المحب love with complete humiliation so the love reduces you start disliking it so you don't want سحور helps you brothers are you with me والله الشيخ of my may Allah grant him جنة the first 15 days of Ramadan the first 15 days he gave us benefits pertaining to سحور only every day for one hour and the حقام on your سحور and things pertaining to سحور صحيح brothers I have notes on it I have notes this religion is it comes from many angles are you with me brothers just these two points some of you guys are like wow I didn't realize that and I actually even look at it like that number three the third barakah that he has اتباع السدنة you actually follow the prophet you are actually follow the messenger عليه الصلاة والسلام that's a high station صح you follow him عليه الصلاة والسلام because he is صلى الله عليه وسلم when he mentioned صحوري باركة رسول الله السري I am going to follow him and following the messenger عليه الصلاة والسلام is the evidence that you love الله so he just goes back to the issue of love again قول ان كنتم تحبون الله فاتبيعوني يحببكم الله ويغفل لكم الله غفور رحيم صح so if you love Allah تباركوا تعالى there is a way to measure if a person loves Allah صح what's the way to measure it instead of they said this is the eye of امتحان تستخدموا what your love yeah you love Allah you say you love Allah yeah let's see you follow the prophet you love Allah you look like a you look like a rapper you look like a rapper and you say I love Allah I love Allah you don't look nothing like a Muslim if people saw you on the road they'll say oh you just gave me slams they wouldn't know any sign of Islam on you are you with me brothers that this we're not going to say the love of Allah is not present in you no doubt there is but it's weak the poet said you disobey Allah and you claim his love that's something that the mind can't accept Allah can the mind accept that I love you I love you but I won't listen to you I love you I love you but I won't follow you لو كان حبك صادق اللعطاته your love if it was true you would have obeyed him for very little one who loves obeyed the one he loves number 4 with the سحور give you brothers what did you mention رمضان is a time when the عباد is 30 30 nights صح are you with me brothers if if you come with سحور what do you do before you eat you make du'a right that you wouldn't have got if you were sleeping but du'a that you say بسم الله that come that itself is this is this Bismillah it's not a normal Bismillah in the days of the year this is Ramadan you just said بسم الله الرمضان when you just say that this timing this virtue of this time is amazing right the actions have been multiplied it's not a joke not only that the du'a that you make the gratitude and that which you're showing and the عباد which you're coming with and not only that if a person has what's the chances of you praying فجر it's actually high the fact that you're going to go to the masjid you're actually going to be there not only that you're actually going to pray سنة before فجر all of that are you with me brothers all of that comes through صحور so شريعة is waking you up in advance for the prayer but you're eating for it you're eating for it number five is من بركة الصحور صحور is what أن فيه مخالفة لأهل الكتاب you're opposing أهل الكتاب the people of the scripture and this is something that the شريعة really works towards that the Muslims are distinct and they're not like them are you with me that's why the prophet he said فصلوا ما بين الصيامين the difference between our fasting وصيام أهل الكتاب and the fasting of the people of the scripture is what what's the difference أكلة الصحور أكلة الصحور the eating of the صحور we want to oppose them in everything brothers yeah everything everything we eat صحور just those that we can oppose them they don't eat it we eat it that's how the أكلة الصحور you're all looking to is called اختضاء وصراطة المستقيمة لمقالفة أصحاب الجحيم are you with me brothers ريتم المشيخ وليسلام تهمية وص الشيخ رحمه الله يدزه إصلاة which is مفقود we haven't seen it we've been published yet it's been mentioned so the scholars mentioned it and the references is حكم يوم ريتم باكبن حجر العسكة on you with me he talks about this issue of التشبه بلكفاره الإمام الدهبي الرحيمة الله الإمام الدهبي كان هناك فرصة وفرصة في الثلاثة المسلمين يبدأون أنهم يفكرون بها وإمام الدهبي الرحيمة الله يقوم بعمل في هذه الأشياء تشبه بالكفار ماذا هنا أن هذه الثلاثة بركة أنت تستخدم أنت تقوم بعمل أنت معي أحد الأشياء المسلمين يشبه ومسلمين يشبه هو هذا الأشياء المتبخ هل تكون معي أخي فيه خير وفيه شر there is good and evil in it if the integration means أن لا نحاولهم لا نساعدهم لأسلطهم لا ننقلهم على أي بلد أن نحصل هنا مع المتبخ نحن هنا مع كنجر ونحن في منطقة هذا شيء يمكنني أن أكون في هذه المحاولة هؤلاء الذين هم جيدون يجب أن يكونون سلفتين سريع أقل أقل أقل أقل ما قلت صلاة الجمع في وقتها الفاضل المجمع يأتي إلى صلاة الجمع فاجر يقومون بمعيشة أخي إذا كانت تفعل صحور ما هو أن تجد؟ when the person eats they should eat the minimum don't go too overboard eat what fills you good but your mind can still focus and the prophet صلى الله عليه وسلم يقول نعمة صحور المؤمنة التمروء the good of the person's صحور is what? date some people have صحور they don't have date have dates let it be there the prophet also made what نعمة blessed dates are very good very good brothers so don't go over don't exaggerate and go over and fill yourself and say oh today's a long day after work just eat go somebody just talk still and later when you want to scratch your stomach scratch it from there because it's gone so big that is so basically don't forget the truth are you with me? I remember we were eating with one sheikh one time and we were eating and I reminded the sheikh I said sheikh then the prophet I said I see he was going in he was going in the sheikh so I said sheikh and he said to him everybody knows their tulut the tulut is subjective right my tulut is different from yours I can probably eat all of this is my tulut صح صحة لبن السعد رضي الله عنه who reported that the messenger of Allah صلى الله عليه وسلم said لا يزالوا الناس بخير ما عجلوا الفضرة هذا الحديث is a dileel and it's an etiquette من آداب الإفطار when it comes to breaking the fast and that is what تعجيله والموبادلة to hasten it and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and to do it fast and quickly some people they think it's a sign of righteousness to delay and relax and say I'm fine I'll have it when you're لا لا it's not the sun now the person as soon as sun sets he breaks his fast and there's a lot of خير in that from the خير that's in there no doubt is the following the Prophet صلى الله عليه وسلم وَالعملُ بِسُنَّتِ and you're doing صلى الله عليه وسلم سنة was like he's sunna also there are many narrations that show based on the hadith of أبو درداء رضي الله عنوش which he said ثلاث and three من أخلاق النبوة it is from the etiquette and the manners of the prophets not just prophets but the previous prophets as well تعجيروا الافتار hastening the افتار breaking off the fasting وتأخيروا الصحور and delaying the Suhoor ووضعوا اليمين على الشيبالي في الصلاة and placing the right hand on top of the left حن in the prayer are you with me brothers? but remember placing the hand on the right on the left that's the Ambiya that's the sunna of the prophets so this is not from the sunnas of the prophets to just let your hands placing your hand on but where do you place the hand brothers? there's a book written called it's called فتح الغفور بوضع الايد على الصدور على الصدور فتح الغفور بوضع الايد على الصدور I read that book it's written by Muhammad حيات سندي رحمه الله he's a teacher of Muhammad Abdul Habib he was a teacher of Muhammad حيات سندي when I read that I was focusing on this particular issue very seriously trying to get so much and I was trying to get to the bottom of this issue of placing the hands and took a lot of my time and comparing it to other books that were written and what I came with is the conclusion is there is no authentic hadith whatsoever regarding where you need to put your hand when it comes to there's no authentic hadith at all the one that's so they're all weak but so we're now talking about weakness so amongst all of them like in the one that's weakness is the what the least the least so all weak but the one who has the baby weakness is placing on your chest that's why I've just taken that from the angle because it's the weakest it's the lowest of the weakest does that make sense now the highest of the weakest the least weakest does it sound nice but yeah but there is no authentic hadith pertaining to it from the message are you with me brothers so there is what's what's what's it's a matter which really shouldn't be made into a fuss then where you put it here you put it here I put it here you don't put it here all of that as long as that right is on top of that left that's that's clear from the prophet are you with me so Muhammad Hayat Sindi tries to prove in that book he tries to prove that the hand has to be put on the chest I was looking at his narrations I looked at other people try to bring other narrations for it I looked at the روايات and I tried to cross check them all give it a fair judgment on this issue لا يخلو من مقال every narration from amongst them there could be a question put to it the closest that they have towards the chest and the strongest closest to it is the Rewire of Iqlim which is Mursal the Rewire of Iqlim which is Mursal the closest إن شاء الله و تعالى بإبن الله this is a promise from me I'm going to explain it in the book إن شاء الله و تعالى فتحوا غفور بوضع الإيدي على الصدور على الصدور صدور صدور صدور so this hadith tells us that the three are from the etiquettes and the manners of the prophets from them is تعجيلوا الإفطار hastening the fasting صحبوا this and the Salaf and this ummah for example بخاري brings in the Sahih مثلا أفطل أبو سعيد الخضريح أغاب قرص الشمسي أبو سعيد الخضريح as soon as the sun قرص the sun تهيب رقص no delay didn't really wait for everything to become dark no not at all عمرب الميمون and أودي و ديني ساي كان أصحاب محمد صلى الله عليه وسلم أسلعوا الناس إفطارا و أبطأهم صحورا he said that the companions of the prophet صلى الله عليه وسلم were what the hastiest they were the fastest in breaking their إفطار و أبطأهم صحورا they were the delay they would delay it the most they would take it to its last point when he says كان أصحاب رسول الله كان the word كان فعال that show what it shows مداوما consistency this is how they were second thing that shows us he said أصحاب رسول الله I mean this is all the companions عمرب الميمون and أودي mentioned are you with me what about if a person breaks their fast and when they break their fast they thought when they were breaking their fast that the sunset they thought the sunset و لم تغرب و عليكم سلام so we're talking about a person من أفطر يظنوا أن الشمسة قد غروبت a person who breaks his fast thinking that the fast broke sunset sorry and it becomes clear to him what and it becomes clear to him after that it wasn't the case we'll say فصومه صحيح and he's fasting is صحيح لأنه معضوره شرع in the sharia he's excused he's not excused معضوره شرع but what's upon him is what what's upon him is يمسكوا عن الأكل حتى تغروب الشمس but he has to withhold from eating until the sunset he ate and realized oh no سبحان الله are you with me brothers he has to hold back and carry on his fasting are you with me he takes the ruling of a person who done it from أربنا لا تؤخذنا إن سينة أو أخطأنا هيدانة mistake are you with me brothers هيدانة mistake so we mentioned the hadith صحيح مصر مصر الله say قد فعلت و في رواية it says نعم another last point I'm going to conclude with إن شاء الله you're fasting and you're breaking your fast right now ينبغي للصائم and the person who's fasting is upon him my brothers and sisters take this advice أن يغزتني ملحظاتوا إفطار brothers benefit from this opportunity of the time of iftar when you're breaking your fast why this is أوقاتوا الإجامة this is the time when the dua is going to be accepted so the person should make دعافة يدعو بما أحب supplicate and ask for from Allah that which you love because for verily you have فإن له دعوة المستجابة at this particular moment your dua is accepted and it's the time when it's accepted أبو هرير رضي الله تعالى عنه the hadith is found by Imam Tirmini ابن ماجح والحديث له شواهد he has many other narrations to strengthen it like the prophet صلى الله عليه وسلم he said ثلاثة الله تردوا دعوةهم three types of people their dua is not rejected and it's not turned back to Adam number one number one is الإمام العادل the just leader who is fair to his people and he judges them correctly and by the book of Allah والصائم and the one who is fasting حين يفتر when he is breaking his fast he is fasting when he is breaking his fast also his dua is accepted ودعوة المظلوم and the dua of the one who has been oppressed and that falls in the hadith that we mentioned وتقي دعوة المظلوم فإنه ليس بينها وبين الله حجاب are you with me? are you with me? are you dites yes yes are you with me? yes yes okay okay okay okay okay okay okay okay فهو يفعله و لا يفعله فالبرس عسلم يقول أن الشخص الذي يفعله و في وقته يفعله يفعله و يوجد فرق من دعاءه و إذا يجب أن يفعله فهو ينفعله ابن أبي موليكا و يتابعي من الثقات و روى له أيمة الستة كل سقولات مراته منه يجب أن يقول أن عبد الله بن عمر رضي الله و تعالى عنه إذا أفضل و عندما يفعله و يفعله يقول الله ما إني أسألك برحمتك أو الله أسألك معك أو الله اللتي وسعت كل شيء إن ماسي which has become vast and large of everything أنت غفر لي that you forgive me also what is also recommended to say once breaking your fast is that which عبد الله بن عمر رضي الله عنه also narrated you can find أبو داود and others narrated it to say و بتلت العرق و ثبت الأجر إن شاء الله سم الشيخ they questioned the authenticity of شيخ مغبل ابن هاد الوادع رحمه الله and I think so if I'm not wrong شيخ مغبل weakened it if I'm not wrong شيخ ناصر الدينة الالبانية رحمه الله authenticated it and that which personally I'm inclined to is و بتلت العرق و تبت الأجر إن شاء الله is not the right for two reasons if he narrates with عانا فريواته لا توخذ بس حق السبيع is a model و تدليس is actually one of the harshest تدليس and another غامضه we're not going to go into right now anything which I have said that was wrong any mistake that came out of my mouth is from me and shaytan and Allah and his messenger are free from it سبحانك الله أشهد أن لا إله إلا الله أستغفروك و أتوب إليه tomorrow same time 5 o'clock to 10 o'clock و فنشتبوك إن شاء الله