completely general effectivity result or a method for computing the Brouwer set. So we can do it in practice, but this is a very, very different situation than David Harvey's lecture series. So he's trying to really optimize the running time and the complexity and getting in deep to like how far you can push these algorithms. This is a fundamentally different situation. We're just trying to see, okay, how do we do this? Can we do it in an example? What, yeah, like can I throw all of my knowledge and ideas and and do it in this case? Rather can I like, as the LMFDB people say, can I push a button and run it for like, I don't even know what orders of magnitude examples? So that's a different situation than I'm going to be talking about. So let me just say, so I said here completely general, so that we do have two effectivity results that I describe more in the notes. They're both by Kreschen Schinkel. So in one, they show that a larger subset, the algebraic Brouwermann subset, which I'll define in a minute, is effectively computable. And in the second one, effectivity for surfaces, they show almost that for surfaces, the Brouwermann and set coming from the end torsion in the Brouwer group is effectively computable for any given end. So we do have some results. But again, these are also, these are really effectivity results, not algorithms with complexity analysis. So it's still fairly different flavor than this morning. Okay, so we don't have a completely general method, but we do have sort of, I don't know, an approach slash procedure that people usually do. And as you'll see, there's parts of it that's like, okay, that's all you're telling me. What do I go from here? And the answer is like, yes, exactly. Where do we go from there? That's that is what is making it so that it's that it's not completely effective. Okay, so before we get into computing the Brouwer set, so the Brouwer set was defined as the set of a deli points are orthogonal to every element in the Brouwer group. So first we want to look at just understanding or computing the Brouwer group. Oh, okay, there we go. So first thing just to note, so I might have said this, but it's not apparent from the definition I gave you, but it is true that the Brouwer group of X is a torsion, abelian group. So we're not trying to figure out very complicated group structure. We're not dealing with trying to understand what normal subgroups are or anything. It's very simple group structure. But as you'll see it still can be hard to compute. Okay, so the way we often begin on trying to compute the Brouwer group is by subdividing it into different filtrations. So Brouwer zero of X typically denotes the image of the pullbacks from the Brouwer group of K to the Brouwer group of X. Okay, so these are in the lecture notes. You don't have to like super get them in your head, but let me just remark. So we have X, X is living over K. So we have this structure morphism from X to spec K. And so then we have a map in the other direction. And so these are the ones, these are the ones that you think that they're, they don't come from X. They come from K, right? They're not telling you anything about the geometry of X. They're, they're constant. They're sort of like there and you have to deal with them, but they're not really telling you what's going on with X. So you should think of these, yeah, as not as important. Okay, and then the second step of the filtration is denoted Brouwer one. And these are called the algebraic Brouwer classes. And these are the ones that are annihilated when you pass to the algebraic closure. So the algebraic Brouwer monon set. These are the ones that are slightly easier to compute. Okay, and one thing that you looked at in your exercises yesterday, you might not have gotten to this particular one because there were a lot, maybe you were interested in other ones. But you can prove that these constant classes, they cannot obstruct any idyllic points. And this should fit with the intuition that they're not seeing anything about X. So why should they be able to tell us anything about the points on X? Right, they really just come from K, they're already there. So you know, and then you can also prove that the Brouwer set depends only on this quotient, that actually you only need representatives for the quotient Brouwer mod Brouwer zero. So really when I talk about computing the Brouwer group from the point of view from the Brouwer monon obstruction, and also just the point of view of what is the part that's related to X, we really only care about the quotient Brouwer mod Brouwer zero. Okay, so I will try to say Brouwer quotient over and over again instead of Brouwer group, but I predict that I will fail often. So you can ask if it's unclear, but you should also just think where everything is about the Brouwer quotient. Okay, questions. Yes, that's correct. Okay, so the question was, if X was a field, then passing to the algebraic closure would just kill everything. So Brouwer would just be equal to Brouwer one, and in fact equal to Brouwer zero, but okay, but the important part was equal to Brouwer one. And so I think the question was two parts. One, guessing from what I've defined that that's not true in general, and two, like, is there something that we know always kills the elements of the Brouwer group? And okay, so let me answer the first one. Yeah, so in general, Brouwer is not equal to Brouwer one, but in several types of varieties it is. So if you have something that's geometrically rational, so geometrically birational to projective space, then Brouwer is Brouwer one also by Sen's theorem for any curve. That's not an obvious thing, it's just like a thing that I'm throwing out there, if you want to look at it more, that for curves there is this theorem that implies that Brouwer is equal to Brouwer one. So that helps in those situations, we're in a better situation for competing the Brouwer group. If you had a variety where Brouwer was not equal to Brouwer one, and the Brouwer quotient Brouwer mod Brouwer zero was infinite, then say Brouwer mod Brouwer one was infinite, I don't know if you can, okay, well, if you base change to the algebraic closure of the function field, it will kill everything, but maybe you wanted a smaller extension than that, and I don't know a smaller extension that is guaranteed to always work off the top of my head. More questions. I really want to get the questions going again, so if you ask a question, you are doing me a favor. That is a great question, so, okay, so the question was, if you base change to a field extension, can Brouwer one become like as small as possible? So yeah, so Brouwer one always contains Brouwer zero, so if you base change to a number, like a larger number field, it won't be trivial, but it, so when Brouwer one mod Brouwer zero is finite, there is always a finite extension that trivializes Brouwer, wait, no, am I saying this correctly? Okay, under an assumption which I will mention later, which implies that Brouwer one mod Brouwer zero will be finite, then there is always a finite extension of your base field that trivializes that Brouwer one. Yeah, but lots of times Brouwer one mod Brouwer zero can be enormous and infinite, and so for like, for abelian varieties and for curves, there is no finite extension that will ever make Brouwer one as small as possible. Great, thank you, anyone else? Yes, yes, each piece in Brouwer one still dies in a finite extension, that's right. Okay, the question was does each piece of Brouwer one die in a finite extension? An answer is yes. Okay, I'll pause again shortly so you can keep thinking. Okay, so really we're interested in computing Brouwer mod Brouwer zero, and just by definition, we have this nice exact sequence where we have Brouwer one mod Brouwer zero injecting and then we surject onto Brouwer mod Brouwer x. So normally we think of dividing those two pieces, and then as I explain in slightly more, but not too much detail in the notes, from a spectral sequence, you get some exact sequences or isomorphisms for these quotients that relate it to the Galois cohomology of pick x bar. So, particularly if pick x bar is finitely generated and torsion free, then this Galois cohomology group is fairly computable. Assuming you have generators for pick x bar, then you can compute this h1. Okay, this part is much harder to compute, and that's just the way it is. So you could work on making, yeah, understanding this exact sequence better from a computational point of view. Right now, I think this hasn't been leveraged so much for computational purposes, only for theoretical ones. Okay, so that's the approach, and as I said, you may feel very underwhelmed by this approach, because it's far from an algorithm, you're still left with computing these cohomology groups, you're left with computing pick x bar. Okay, we have, yeah, some effectivity results for different pieces, but that's why you have. Okay, but sometimes you get lucky, and you can sort of guess an element in the Brouwer group and verify that it is actually an element in the Brouwer group. And I outline one of these approaches in the exercises. So sometimes you get lucky, or maybe you're in a situation where someone else has already worked out these cohomology. Now you have the elements of the Brouwer group, how do we go from there and compute the Brouwer set? So assuming we've computed this quotient, how do we compute this Brouwer set? And so let me just put this up here to have it. So this is the set of idyllic points, such that the sum of the invariance of alpha, okay, how did I do note it? I think pv upper star of alpha is zero for all alpha. Okay, and as someone asked yesterday, there were a lot of questions yesterday about this map. So does the pullback depend on alpha? Does it depend on x? How do we compute it? What do we have to know about it? And I said, yes, you have to know all those things. It's hard to compute in general. So this is what we have to figure out. And just, I think I introduced this notation. I often want to think of this as a map from the idyllic points to the Brouwer group that depends on alpha. So I'm going to call this map f alpha from x of kv to q mod z. I guess really this should be like f alpha sub v, but the v will hopefully be clear from context. Okay, so we know some things that are helpful. So we know this evaluation map is always locally constant. So if you want, you can always throw away as the risky closed subset of your x and just compute the evaluation there. And then you know you won't have this really, really bad fractal or ill-described pattern. Everything that the preimage of every value is going to be an open set in the idyllic topology, open and closed set. And we know a little bit more. Also, if you know that your algebra, your Brouwer class is killed by an unrammified extension of your local field, then you know that it's always zero. And that is very, very useful. And particularly, you can see this for elements in Brouwer 1. So you know that they'll be killed. Everything in Brouwer 1 is going to be killed by some finite extension. That extension will be unrammified away from a finite set of primes. And boom, immediately, you know, for all of those places, this class is zero. So that is a huge, a huge bonus that helps us. The downside is that outside of this, so, in quotes, this is, unless there is a reason why not, typically these evaluation maps have the largest order that they possibly can have. So you don't, you don't usually just get lucky and like it's constant everywhere. Okay, so some of you who maybe have read a little bit about Brouwer 1 and obstructions previously, or maybe you're doing research in this area, or you decided that you were going to do the last big problem on the problem set exercises yesterday, you might be thinking like, hey, but we just did this problem and it was constant everywhere. It was always constant. It just like, I worked out and I only had to check at one point and it was, it was super easy. That's, that's because we made it that way. Okay, the examples that you see in the literature are examples that are rigged to be easier to handle. So in the literature, they are often constant all the time. And from looking at that, you could be like, oh yeah, no problem. I'm just going to check this one point and see what's happening. But that is not, not what happens in general, unfortunately. So just don't, don't be led astray. Okay, so what I would like you to take from what I just said, I like threw a lot of these definitions I use, some exact sequences, all these different properties. What I want you to take away from that is that it is doable, but it can be hard. So yes. Yes. Through the local topology, through the viatic topology on XKV. Sorry. Yes. Oh yes, thank you. The question was, what does a locally constant mean for the viatic points and is it the same as it being continuous? And so locally, I mean locally for the viatic topology. So if you haven't seen this before, each X, you can cover by affine covering. So you can think of the different pieces that's sitting in KV to the N, which inherits the product topology and the topology from KV.  And it's like the only thing that you could think of doing if you look at, at some points. And because Q mod Z is discrete and actually for each alpha, it's going to map to a finite group, that's the same as saying this map is continuous. Okay. So what I want to do is, what I want to do for the rest of the talk is talk a little bit about this theme that I mentioned on the first day, like how do these theoretical and computational pieces interplay? And can we, can we try to bypass this somehow? Okay, so what I want to just think about right now is, yeah, so why is this, why is computing the Broward-Mannan obstruction difficult, sort of on a on a heuristic level? Okay, so the way that I think about this is the, the no free lunch principle. So, yeah, I looked this up on Wikipedia. Actually, the history comes from, I guess, like, pubs or something giving drinks for free at lunchtime. No, lunch for free and then people would buy drinks. Yeah, and you just have to pay somewhere. Okay, so if it says it's free, there's something secretly hiding. And what is going on here, like, you cannot dodge, you can't, you can't, you can't bypass the work. When you're trying to compute the Broward-Mannan obstruction, you can't ignore the elements. So that would be getting something for free that you can, like, just throw away some elements to worry about. These Broward classes, sometimes it feels like their, like, purpose in life is to obstruct Adelaide points. They just, like, if they can, that's what they're going to do. So, what I said is obviously not a theorem, but let me show you some theorems that illustrate this. So the, the, the bigger principle is unless there is a reason otherwise, you should expect x of ak, the set of Adelaide points orthogonal to alpha to be strictly smaller than the set of Adelaide points. You should expect it to carve out some subset. So we already saw some reasons otherwise, right? So elements in Broward-0, elements that come from Broward group of k, they don't, they don't obstruct Adelaide points. But that's because they're not about x, right? That's why. Okay, so the theorem, the, like, legend for the theorem is the things in pink are illustrating the reason otherwise. And then the, the theorem is showing that why we should expect this. Okay, so the first theorem dates back to Hari from 94. He says, if you have some family of varieties that has a non-trivial Broward group on the generic fiber. Okay, so it's like for many things in the, in the family, you expect the Broward group to be non-trivial, but the Broward group globally is trivial. Then for infinitely many points on the base, the Broward set is going to be strictly smaller than the set of Adelaide points. There is going to be an obstruction. Okay, and so the, the idea, the reason otherwise would be that the Broward element actually extends to be a, an element of the Broward group of the whole total space, not just your, your single fiber. But if that doesn't happen, then you should expect the Broward classes to carve it out. And this tells you, so this tells you that failures of weak approximation, so that's just when the Broward set is strictly smaller. It tells you that you just, you sort of know for free they exist in families. You often don't have to write them down. If you know that there is a family where you have sort of generically non-trivial Broward group, then you're going to get failures of weak approximation. Okay, yes. Oh, that's V sub eta. Sorry, that's the generic fiber. So this is a1. So that's, there we go. So a1 is speck of the polynomial ring. So I could just view, this is a fiber thing over the function field, over some bigger field. So now, rational function field in a single variable. So this is my eta. And this V sub eta goes there. So you could think of this, say you have, your V could be a family of elliptic curves over the J line. And say you have a Broward class, say I wrote down a quaternion algebra that was like five comma function divided by J minus 1728. Okay, so that would give me a well-defined function, probably a Broward class. But when J is 1728, there's, that function vanishes in the denominator that won't extend. So that would be an example of a Broward class that you have generically but doesn't extend over the whole family. So it's like, you have some, and the reason why you want it to not extend over the whole family, well you need some condition because you need to rule out things that just come from Broward K, right? There's no condition here. I don't have any quotients here. So I have to, I have to somehow throw out these elements because we already know that they don't do anything. But if my algebra just came from the ground field, it would just extend over the whole, the whole total space. The whole family of elliptic curves over the J line. Okay, here, and then there. Go ahead. Yeah, so, okay, so the question was, what do I mean by failure of weak approximation? So the, the short answer is I'm going to say it and then also you can feel free to ignore it because this is just a sort of like bonus thing around. So I said at the very beginning, I'm going to focus on the Broward monon obstruction to the existence of rational points. But you can ask about other properties instead of just asking about the existence of rational points. You could ask about, like, well, how many are there and can I approximate any idyllic point that exists? So that would be weak approximation if the closure of the rational points in the idyllic topology was equal to the set of idyllic points. Okay, so WA is weak approximation. So if this is the closure in the idyllic topology equals the set of idyllic points. But this is one thing you'll do in your exercises today. Well, if you want to do this exercise, the Broward set is closed in the idyllic topology. So in particular, it contains the closure of the k-rational points. So it can not only obstruct the existence of rational points, but it can obstruct the, it can obstruct weak approximation. It can obstruct this density. Yeah, great question. Yes. Yeah, okay, so he writes something, yeah, that's right. So by trivial, I mean that it's equal to the, that it's equal to Broward or not. So what you need, so really the condition is you need an element of the Broward group. And by non-trivial, I mean not just Broward or not. You need an element of the Broward group that is ramified on some fibers. But I didn't want to define ramification. So I tried to translate to the theorem statement. Yeah, but so by trivial and non-trivial, I mean equal to Broward or not. I'm not equal to Broward or not. Sometimes. So this is also a question. So I just started the motivation by asking about, okay, if you want to prove that there's no rational points, you need a way to witness there's no rational points. And if the Broward set is empty, then that proves to you that you have no K rational points on your variety. You can also ask the converse question. Say I compute this Broward set, and I know that it's non-empty. Do I then get for free that there is a K rational point? So that's normally called the Broward-Mannan obstruction is the only obstruction to the Haase principle. So that, that is a different, a different direction. And yeah, in some sense, that's a, that's a harder question because it's giving you the existence of a rational point, which is a fundamentally difficult question. And then you can ask the same thing about density. So it is conjectured that Broward-Mannan is the only obstruction in several classes of varieties. So Kola, Tallinn, and Sonsuk have conjectured for all rationally connected varieties. Broward-Mannan obstruction is the only obstruction. What is known falls far short of that, but Olivier Wittenberg has the best results to date on when, with Yonatan Harpas have the best results on when we know that is the only obstruction. I was gonna say something else. Oh, but we also know that it's not the only obstruction all the time. So there are examples of Skorbagatov and then of Poonin and variations after that that show that it's not the only obstruction. Yeah. And just as you can tell from my very long answer, I find those questions very interesting, but I just have to choose something for three lectures. So that's why I'm not gonna talk so much about that unless you all ask me more questions, which would be great. Okay, more questions. Okay. I think in the interest of time I'm just gonna blow through these next questions, but this is another result of Martin Bright more recently that sort of looks at these things called residues and shows when they give you that the evaluation map has a large image and then Bright and Newton and Rachel Newton is here. She's giving the research program talk later today. They show that you can push this even farther and even show at places of good reduction. We should expect to get these failures and Pagano wrote down an explicit example that demonstrated this. So when you're computing the Brower set, you just you just cannot bypass computing the evaluation map for these alphas except in these other, you know, unless there is a reason otherwise.   Yes, question. That's right. Yeah, so the question was here I'm just saying a single alpha should carve out something smaller but maybe we get lucky and like a bunch of the alphas are redundant. So if we compute one of them, we don't have to do the extra work for the others. And I think that exact question hasn't been looked at but I would guess that if you look at some of these reasons, some of these theorems that I put up by Hari and Bright and Brighton Newton, that if you looked in there, their proof, you could get it to tell you that like, okay, if I took this collection of elements, then the evaluation map for the like going to the product of Q mod Zs from the product of the alphas would have a large image as well. I'm not totally certain but I would guess that there would be some conditions that trigger that. Yeah. Yeah, that's a great question. Okay, so this is why it's hard. It's just like there's all these elements. They're likely to do something. You have to compute them.  But so I think it's not so talked about secret that one of mathematicians greatest strengths is laziness. So if we want to compute the entire Brower set as a subset of the italic points, we have to do all this work. But that's not what we asked for at the very beginning. What we asked for at the very beginning is, can we prove that the set of rational points is empty? So what if instead of asking to describe the whole Brower set, we instead just ask this decision problem, is the Brower set non-empty or not? I don't care what it is. I don't care what points are in there. I just want to know is it empty or not. So why is that a better problem? And probably I can't completely say it's 100% definitively simpler, but why are some reasons why it might be simpler? Well, let's go back to some of the topological facts that we talked about. So if X is projective, then the set of italic points on X. So here I'm taking the italic topology, which is the product topology on the theatic topologies on each factor. So under that topology, this is compact. And then that tells you, so any compact set, if you have an empty intersection, it can be realized as a finite intersection. Finite empty intersection. So that tells us if the Brower set is empty, which is the intersection of all these X of a k of alphas, then there's some finite set of Brower classes, which will exhibit this is empty. Okay, and so there are some cases where this is definitely already better. So if you have an elliptic curve with finitely many points, assuming the chef of each group is finite, then the Brower set is the closure of the set of rational points. And since you have a finite set of points, the closure is just the finite set of points. And to cut down the set of italic points to a finite set, a finite non-empty set, you have to use infinitely many elements. You can't just use finitely many. So already we know that we're in a better position here if we're just interested in this problem. Okay, so you could ask, like, can we hope to understand what this B is, or some properties about this B, ahead of time? And then restrict our search from the beginning to make the computational problem easier. Okay, this isn't going to give us a, well, okay, it's not an idea for a different algorithm. It's just if we understood the answer to this theoretical question, then it would reduce the computational problem that we need to solve. And that would be a win. Okay, so I'm going to introduce two definitions. And so we'll say that some subset, so for the definition, it doesn't have to be a subgroup, but maybe you might prefer that you just take a subgroup. But we'll say that it captures the Brower monon obstruction. If the Brower set is empty, means that the elements in this B already detect it. So I'm not saying anything about the elements outside of B. In this case, I'm just saying it can be, the essence, the obstruction can be captured by the subset. Okay, and then I'll say something stronger, that it, this set completely captures if whenever you have some obstructing set, any subset that obstructs the full existence of rational points, you can detect it from the elements in that subset that already belong to your B. Okay, so completely captured is like the golden standard, but captured already reduces the computational problem. Okay, and as I said, this is really capturing the Brower monon obstruction to the existence of rational points. You can't ask, well it doesn't make as much sense to ask for a capturing set to weak approximation, because that's what I, that's a, you need to understand more things. Okay, so ideally we would be able to find a B that captures or completely captures and that this subset is more easily computable. So, but at the first sense we find a B and maybe if it's, okay, B equaling Brower X is not fun, that definitely captures and completely captures, but if we can find some proper, possibly proper subgroup, then, then we're getting something better. Okay, so this terminology was introduced, I introduced it fairly recently with my collaborator, Brendan Kreutz, but you can look back in the literature and there were results in this direction, so if I rephrase them in terms of this language, also it follows from what, from what Monon did when he originally introduced the Brower Monon obstruction, that if you have a degree D genus 1 curve, so genus 1 curves can be realized or embedded several different ways, but if you have one given to you as a degree D curve, then the D, the D primary torsion completely captures it. You don't need to look at any Brower classes with order co-prime to D and Swinerton-Dyer proved that if you have a cubic surface, then the three torsion of the Brower quotient completely captures. So the, for a cubic surface in this paper, he classified the possibilities, so it can either be two torsion or three torsion and he showed that whenever the Brower group is two torsion, then there's no, actually he showed that it satisfies the local to global principle. And then Kohle-Telenen-Punin proved that if you have a cubic or quartic del peto surface, then there is always a single element that captures the obstruction. So you actually only, you don't know which one, but there's always only one. Okay. Okay, and so let me just say like one thing that's slightly difficult about thinking about this or what you might want to prove is you want something that you can describe ahead of time, that is sort of intrinsically described. But there's usually not that many, because we have a torsion-Abelian group, there's often large automorphisms of these Brower groups, so there's not like really a way to pick out a single element. So for instance, the Brower, in a cubic surface case, the Brower quotient is one of these five possibilities. Okay, and so what we can say is that the three torsion completely captures it and there's a three torsion element that captures it on its own, but you can't hope to say anything more about which element it is because there's sort of like before you start computing, there's no way to distinguish what the different elements are. So we couldn't even like hope to say something stronger, because how do you tell apart the different elements in Zmod 3 squared? Okay, I want to leave more time for questions, so maybe I'll just skip some of the things about like how you might directly compute the n-torsion and we'll just go to some of the more recent results that show what we have. So in this paper with Brendan Kreuz, we prove that if you have a twist of an Abelian variety, so this generalizes the genus one curve statement that I had before, and then instead of the degree of the curve, you put its period that that subgroup completely captures it. So this should just be taken as an illustration of what we know. Why is it? Okay, and then we know for kumar varieties, which are closely related to Abelian varieties, they're desingularizations of a quotient of a twist of an Abelian variety. But anyway, it's something that's related to an Abelian variety and it's related by a two primary cover. So in that case, then Skor Bogatov showed that the two primary subgroup completely captures it. So that's very strong. And then Masahiro Nakahara proved, and all of these, I put publication dates for really like all of these papers came out right around the same time, that if you take a vibration of twists of Abelian varieties, so you could take a family of genus one curves, or you could take a higher dimensional family, then the period of the generic fiber that torsion completely captures the Brouwermann obstruction. So this tells you already from the beginning what primary order you have to look in. And particularly in the case of when you have all these Abelian varieties flying around, Abelian varieties have torsion in the Brouwer group of every order, and the N primary torsion is always infinite. So restricting the torsion there, which orders you're looking at, that's really a strong statement. Okay. Oh, and one more. So this is a subgroup. Something is going on with my remote. But this is a subgroup I didn't mention. It's usually denoted by a surreal like B, it means the locally constant algebra. So these are Brouwer classes that when you base change to KV, they land in Brouwer 0. So they become locally trivial. So for these actually, once you know them, then computing the evaluation maps is easy because you only need to evaluate at a single point. So this is really great. And so Croix showed that if you have a twist of an Abelian variety, then the locally constant subgroup completely captures. And so then actually you can take the intersection of the first result and the second result and it tells you the locally constant algebras of order, a power of the period they completely capture. Okay. So that's the good, the bad. So we would like this. I mean, yeah, we saw all these results were like, wow, maybe this holds like all the time. Like what? And so I feel like there's something missing from my slide. Oh, yeah, here. So in joint work with Croix and Philippi Volak, we've tried to look at this question for curves of hyogenes. And basically, like the paper is just a series of counter examples that show like many things that like any possible statement that you might hope for generalizing from the previous slide doesn't work for curves. So I mean, maybe there's a different way to do it or like that these these subgroups that capture should depend on more than the genus and the degree or in some subtle way, but we could not figure out how to do it. So I think it's it's it's hard. Particularly when you go to general type varieties, it could be hard to get a handle on on these things. Okay. And yeah, I'll just I'll just put up one one slide. So I have a bunch of open questions. I forget from David Harvey's things. Maybe these are like single lightning bolts that are open, but he hasn't thought very hard about them. Am I remembering correctly? So that's what these these are for. So I'll post the slides on the website. We won't get to look at all of them. But I think there's a few other cases that seem tractable that you could ask to what to what extent this behavior extends. Yeah, so I will just go to my summary slide. And end there. Okay. Okay, thank you. Are there more questions for Bianca? Yes, it is Brouwerman. I don't Oh, yeah, sorry. The question was if you have this one of the first examples, counter examples to the local to global principle is this curve three x cubed plus four y cubed plus five z cubed equals zero, do I say correctly? And whether that is a Brouwerman and obstruction. And the answer is that it is a Brouwerman and obstruction. But I do not off the top my head. Remember what the Brouwer class is or even what reference explains it? Oh, Monon's paper. Okay, that was one of the three on the bottom. Great. Yeah. So but does it does Monon's paper write down the Brouwer class or just show that it can be realized as a Brouwer Monon obstruction from theoretically? Okay, right, right. So for as I said for torsors under elliptic curves, we know that Brouwer Monon is the only obstruction assuming finiteness of Tate Chevrolet rich groups. So and then Bjorn is telling me that in Monon's paper, he checks that. Well, okay, this example is a Brouwer Monon obstruction. But I think it's not. I don't know if there's had someone has written down like this collection of Brouwer classes gives you the obstruction. Yeah, you're welcome. Yes, question. So the question was, is there an algorithm that's conjecture, but we don't know that it terminates, but it's conjecture that it terminates for computing Brouwer groups or Brouwer Monon obstructions. I don't think there is. I mean to my it's possible that maybe like something alluding to this is in the effectivity papers that I mentioned by Krush and Schinkel, but I have never actually tried to turn those into an algorithm. They're like very intimidating to me, but I think Jen Berg did you do this for your like sort of made it algorithmic in an example. I think it's pretty far from a general algorithm that is expected to terminate, but we don't know. Yeah, I think it's a different situation. But so someone else can correct me if I'm wrong, but that is if there is an algorithm that is conjecture that conjecture to terminate, it's at least not very easily implementable because I don't know what it is. Is there another hand? Oh, yes, here. So the question was, if I take a Brouwer class that comes from an Azumai algebra, does it help in any way to do any concrete computations? So for me, when I have computed it, the best way, like the way that I typically do is I realize, try to realize the Brouwer class as a central simple algebra over the function field. I guess if you knew that you had it already as something that like, that honestly lives over all of x, so that might be a benefit for an Azumai algebra. Then you can really sort of evaluate, you just take the fiber at any local point to do the evaluation, whereas if you're given a representative over the Brouwer, as a central simple algebra over the function field, there are necessarily going to be, there's necessarily like a Zariski closed subset where you can't evaluate. But since it's locally constant, that doesn't cause a problem, but I think that would be the advantage for using a more global object. Alright, if there are no further questions, let's thank Bianca again.


to avoid any discussion this parliament had become dysfunctional here with a working majority in Parliament then I will get Parliament working again for you people get up at 5 a.m. to get their businesses ready when they risk their own money or mortgage their own homes to develop a new product or a new venture when they have the guts to find a new market at home or abroad we don't sneer at them we cheer for them and do what we can to help because we understand that it's only by having a dynamic free market economy that we can deliver on our program of uniting this country and leveling up with infrastructure education and technology and it's only if you have great public services that you can have a successful market economy so I say come with us that's the choice of this election I hope very much that you will support us let's get Brexit done and unleash the potential of the whole United Kingdom good afternoon I've just been to see her majesty the Queen earlier on and she agreed to dissolve Parliament for an election I want you to know of course that I don't want an early election and no one much wants to have an election in December but we've got to the stage where we have no choice because our Parliament is paralyzed it's been stuck in a rut for three and a half years and I'm afraid our MPs are just refusing time and again to deliver Brexit and honor the mandate of the people I can tell you I've got to the stage where I'd be wanting to chew my own tie in frustration because in a sense we're so nearly there we've got a deal oven ready by which we can leave the EU in just a few weeks it's a great deal for this country it delivers everything that I wanted when I campaigned for Brexit we can not only take back control of our money and yes we'll be able to spend hundreds of millions every week on our priorities such as the NHS we take back control of our borders with an Australian style points-based system so that we can attract the workers from scientists to agricultural workers that our economy actually needs and we can take back control of our laws so that we can do things differently and better if we choose from free ports to free trade deals from banning the cruel live shipment of animals to cutting the AT on sanitary products and we can leave the EU as one UK whole and entire and perfect as we promised and so it's been frankly mind-boggling in the last few weeks to see how Parliament first voted to approve this deal and then voted for delay and I'm afraid that it is clear that if Parliament had its way then this country would not be leaving even on January the 31st and that of course is bad for democracy it's disastrous for trust in politics why should MPs decide that they can cancel the result of a referendum I'm afraid I also think that this delay is now bad for the country and for the economy and with every week that goes by uncertainty is deterring people from hiring new staff from buying new homes from making new investments and if we can get this deal over the line with a sensible majority government we certainly can then we can release that pent up flood of investment hundreds of billions are waiting to pour into the UK and we can inject a surge of confidence into our system and we in this government can get on with delivering on the priorities of the people I'm very proud of what we've done in the last 108 days or whatever it is 108 days or so the biggest programme of NHS investment for a generation lifting up funding of schools across the country at 14 new hospitals by the way lifting up funding of schools across the country putting 20,000 more police on the streets and infrastructure revolution were planning in rail and road from electric buses to new green cycle schemes gigabit broadband in every home and we have the confidence as one nation conservatives to make those investments not despite our belief in a strong private sector but precisely because we champion this enterprise economy in the UK and when people get up at 5am to get their businesses ready when they risk their own money or mortgage their own homes to develop a new product or a new venture when they have the guts to find a new market at home or abroad we don't sneer at them we cheer for them and do what we can to help because we understand that it's only by having a dynamic free market economy that we can deliver on our programme of uniting this country and levelling up with infrastructure education and technology and it's only if you have great public services that you can have a successful market economy so I say come with us that's the choice at this election that's the choice come with us a government that's putting billions into education or go with Jeremy Corbyn and the Labour Party but that's the only alternative we actually want to ban off-stead that protects kids from bullying in the classroom come with us a government that believes in high wages is raising the living wage to £10.50 the biggest ever increase or go with a left-wing Labour Party that believes in high taxes for everyone and that voted under this government for against £7,800 of tax cuts on working people that's what they did come with us put in a points-based immigration system or go with Labour and a totally uncontrolled and unlimited immigration system that would put huge pressure on the NHS and other services come with us a government that believes Britain should stand tall in the world or go with Jeremy Corbyn and the Labour Party who sided with Putin when Russia ordered poisonings on the streets of Salisbury come with us get Brexit done and take this country forward or and this is the alternative next year spend the whole of 2020 in a horror show of yet more dither and delay imagine waking up on Friday the December of the 13th and finding Corbyn at the head of his Technicolor yawn of a coalition they would spend the whole of 2020 having two two referendums one on Scotland because he's done a deal with the Scots Nationalists to assist the breakup of the union if they sustain him in power and another another referendum on Brexit which is meant to happen in nine months time after he's renegotiated supposedly are exit and renegotiated this deal and what is his plan for that renegotiation what question would be put to the public we don't know what are the options we don't know we don't even know what side he would take and we don't know what would happen if the result was either for remain or for leave best of three call it quits we don't know what we do know is that in any scenario the dither and the drift and the delay which is increasingly damaging for our country would just continue and we do know that there's only one way to avoid that nightmare and that is to vote for a moderate and compassionate one nation conservative government and we will make this country the greatest place to live to raise a family to start a business to send your kids to school a country where we lead the world in cutting co2 in tackling climate change in clean technology where we stand up for our values around the world a country where everybody has the opportunity to make the most of their lives and where we work as a government to give them that opportunity from the moment they are born and that is our mission if I come back here with a working majority in parliament then I will get parliament working again for you on day one of the new parliament in December we will start getting our deal through so we can get brexit done in January and unleash this country's potential we'll put uncertainty behind us families and businesses will be able to plan let's make 2020 the year of investment and growth not the year of two referendums I want to thank everyone in the building behind me and across government for all the work the wonderful work they've done over the last three months I'm going out now to campaign across the whole country for those values and for that program I hope very much that you will support us let's get brexit done and unleash the potential of the whole United Kingdom thank you very much


We continue today with Chapter 14, The Reflection of Holiness. The Atonement does not make holy. You were created holy. It merely brings unholiness to holiness, or what you made to what you are. Bringing illusion to truth or the ego to God is the Holy Spirit's only function. Keep not your making from your Father, for hiding it has cost you knowledge of Him and of yourself. The knowledge is safe, but where is your safety apart from it? The making of time to take the place of timelessness lay in the decision to be not as you are. Thus truth was made past, and the present was dedicated to illusion. And the past too was changed and interposed between what always was and now. The past that you remember never was, and represents only the denial of what always was. Bringing the ego to God is but to bring error to truth, where it stands corrected because it is the opposite of what it meets. It is undone because the contradiction can no longer stand. How long can contradictions stand when its impossible nature is clearly revealed? What disappears in light is not attacked. It merely vanishes because it is not true. Different realities are meaningless, for reality must be one. It cannot change with time or mood or chance. Its changelessness is what makes it real. This cannot be undone. Undoing is for unreality, and this reality will do for you. Merely by being what it is, does truth release you from everything that it is not? The Atonement is so gentle you need but whisper to it, and all its power will rush to your assistance and support.  The Atonement offers you God. The gift that you refused is held by Him in you. The Holy Spirit holds it there for you. God has not left His altar, though His worshipers placed other gods upon it. The temple still is holy, for the presence that dwells within it is holiness. In the temple holiness waits quietly for the return of them that love it. The presence knows they will return to purity and to grace. The graciousness of God will take them gently in and cover all their sense of pain and loss with the immortal assurance of their Father's love. Their fear of death will be replaced with joy of life, for God is life and they abide in life.  In this world you can become a spotless mirror in which the holiness of your Creator shines forth from you to all around you. You can reflect heaven here, yet no reflections of the images of other gods must dim the mirror that would hold God's reflection in it. Earth can reflect heaven or hell, God or the ego. You need but leave the mirror clean and clear of all the images of hidden darkness you have drawn upon it. God will shine upon it of himself. Only the clear reflection of himself can be perceived upon it. Reflections are seen in light. In darkness they are obscure and their meaning seems to lie only in shifting interpretations rather than in themselves. The reflection of God needs no interpretation. It is clear. Clean but the mirror and the message that shines forth from what the mirror holds out for everyone to see. No one can fail to understand. It is the message that the Holy Spirit is holding to the mirror that is in him. He recognizes it because he has been taught his need for it but knows not where to look to find it. Let him then see it in you and share it with you. Could you but realize for a single instant the power of healing that the reflection of God shining in you can bring to all the world. You could not wait to make the mirror of your mind clean to receive the image of the holiness that heals the world. The image of holiness that shines in your mind is not obscure and will not change. Its meaning to those who look upon it is not obscure for everyone perceives it as the same. All bring their different problems to its healing light and all their problems find but healing there. The response of holiness to any form of error is always the same. There is no contradiction in what holiness calls forth. Its one response is healing without regard for what is brought to it. Those who have learned to offer only healing because of the reflection of holiness in them are ready at last for heaven. There holiness is not a reflection but rather the actual condition of what was but reflected them here. God is no image and his reactions as part of him hold him in them in truth. They do not merely reflect truth for they are truth. Lesson 113 from the workbook for morning and evening review. I am one self united with my Creator. Serenity and perfect peace are mine because I am one self completely whole at one with all creation and with God. Salvation comes from my one self. From my one self whose knowledge still remains within my mind. I see God's perfect plan for my salvation perfectly fulfilled. On the hour I am one self united with my Creator. On the half hour salvation comes from my one self. Amen.


Hello everybody, welcome to healthy cooking with the sisters. This is Nicole and my name is Sharon and today we are going to make some plant-based milk as an alternative. This is a segment where we shall be showing you a lot of healthy and wholesome meals. So Nicole, would you like to pray for us? Sure. Let's believe and pray. Dear Jesus, thank you so much for this day. Thank you for the gift of life and Lord I pray that even as we do this demonstration on how to make this plant-based milk, dear Jesus, may we do it in the right way to glorify your name and dear Jesus, may we learn from these simple recipes and make our lives better by eating yummy and wholesome and nutritious meals using my prayer, trusting and believing. Amen. Amen. So, before we get started, Nicole, how many different types of plant-based milks do you know? Okay, so I know the soy milk, the famous soy milk, there's almond milk, there's cashew milk, I've seen rice milk as well, I know there's coconut milk, I think that about exhausts what comes stock of mine. I know I've heard of peanut milk as well, which I'd like to taste sometime, yeah, there's hemp milk, most nuts can make milks as well. So today we are going to make something that has three different kinds of ingredients in one milk. So it's basically a nut and grain milk. So we have cashews, so these are already soaked, we're using three and a quarter cups, sorry, three quarter cups of soaked cashews, you can soak them overnight, you can soak them for 30 minutes in hot water or overnight in cold water. Also Sharon, sorry to interrupt you, why do we soak them, why don't we use them directly, why is there need to soak them? Just also to soften them and also just to make sure that they plump up a little bit, yeah. They soak in the water. Yes, they soak in the water. Then we are going to use a quarter cup of oats, I'm sure many people have a lot of oats in their homes because it's usually a go-to breakfast option, so that's a quarter cup and then we are going to use desiccated coconut, that's half a cup of desiccated coconut. So it's a bit hard to make coconut milk just from the main coconut, yeah? Yes. Because you have to, yeah, you have to peel it off from the, they call it coconut nazi, so it's peeling it off from the skin and it's quite an effort, so the desiccated coconut is quite easy to use. And then we're going to use a pinch of salt. Salt is a balancer of flavour, so you just use a pinch just to make sure that everything is well balanced. We may use a sweetener if we desire, but for now I just want to use it plain because this is a milk you can use to cook, you can use as a beverage drink, you can use it as a, just to drink it on its own, with cereal, so we don't want to sweeten it because when we sweeten it we narrow it only to sweet ingredients, yeah, so we put in that on the blender and then we put in three cups of boiled water, okay? So you blend that until smooth and then we shall spit. I know you said blend till smooth, but when do you actually know it's ready? You can see some granules on the inside, if you check on the side, if you do, most blenders are see through, so you're able to see some grains on the inside. You can also use a spoon just to check and see if there's bits and pieces of the cashew. The cashew is what would make it blend for longer because the cashew is a grain, is a nut, sorry, the desiccated coconut and oats would blend really easily, so you can see it's still a bit grainy, so we can blend it a little bit more, maybe for another two, three minutes and then, yeah, then we can sieve it. So I think that's enough. Also the reason we blend it longer, you get more product once you blend it for a while because then it squeezes out and makes everything into a liquid. If you blend it for a short while then the liquid isn't as much. So we are going to use a nut milk bag. This is a nut milk bag. Usually you can buy it from some shops, they have them, Indian shops have them, some shops in some malls have them, especially the ones that sell kitchen equipment. But if you do not have this, you can use gauze. You just fold it into several bits and if you can get like a wide gauze and then just pass the milk through and try and fold it and then just squeeze the milk. It's a bit messy so make sure that you have a lot of working space and also because it's something that you're handling with your hands, make sure that your hands are always very clean. Also, before I point in for you, what's so special about, why do we need to use, where can I use a sieve? Because the holes in the sieve are not as- They're too large, right? Yeah, they're not as small as we would need them to be. So finding even any home, anything at home that is fine enough to get rid of all the small particles, right? Yeah. So you can always makeshift anything. Sometimes when I don't have gauze or the nut milk bag, you know these cloths that you've been getting in the supermarket, you just wash one prior to using it and you can always use that. And you make sure you just dedicate it only for milk making. Yes. So Nicole, you can pour it in for me. So as you can see, there's already some that's coming out.  I mean, this thing is made of cloth. So if you're too tough on it, you may tear it and then that just makes your work useless because then you'd have to start again with a different cloth. So just gently, also Nicole, have you heard of people who use the pulp for other things that remains? Yes. I've heard about it. But I'm not so sure if I remember exactly what they use it for. But I know when people make coconut milk with desiccated coconut, they can use whatever fiber is left on the pulp for baking. So if you have any coconut recipe, you can even put it in a vanilla cake so that it doesn't go to waste. This is always a good source of fiber in our bodies just to get everything in our guts moving well so that stops constipation and all the good things we know about fiber. So you said you didn't remember but you really just explained everything for us. So that's mainly what people use, the pulp that remains for blend. You can use it in, you can even put it in your smoothies as well just to make it more filling. I know people who put them in their smoothies, you can bake it, you can toast it in the oven and then just also just pour it over your food, your soups. I've also remembered you can actually use it for granola right? Yeah. If you're making granola so it doesn't go to waste, you can always put it in the oven with your granola as it's baking and it doesn't go to waste. So I think that is, I think we've gotten quite a bit. Actually this makes about 750 ml to a litre of milk and that's better beating at home. So thank you so much Niko, so you can pour it into the jug so you can see how creamy it is. The reason I like this milk is because soy milk has an acquired taste so not too many people like it. But milk is quite thick so I find it very difficult to make hot beverages with. Then coconut is thin so all of them are just different on their own but when you mix it together it's a very thick milk. You can use it for tea, chocolate, you can use it for cereal, you can drink it just as a glass, you can add in your fruits and then make a smoothie. So it's a really nice thick milk alternative that's why I like it and it's also very wholesome and it has very many nutrients that you can get in it because of the oats, because of the cashews and because of the desiccated coconut. So I'm just going to give Nicole a little taste. Moment of truth. 100%. Yeah. It's actually really nice, I can actually drink it like this. So how you store it, you put it, if you do have a glass jar it would be much better than storing it in plastic, put it in the fridge, probably the coldest part in your fridge so maybe at the back, not on the doors, although most of us would be more inclined to put it on the doors because that's maybe where the space would be. But if you can get smaller just so you can put it at the back so that it lasts longer. And also because remember these items are all different textures, it means that the milk can also separate, that doesn't mean that the milk is spoilt, it just means that the coconut may come to the top, the oats may be at the bottom, so you have to always keep on shaking, just swirl it a little bit every time before you use it and then you can boil it, you can drink it as it is, yeah and enjoy it as your family, okay? So that's all for this session and I pray that you may enjoy it with your homes and that you may be blessed and enjoy this plant-based alternative. Bye. See you.


Hi, my name is Juanita Sanchez and I'm from Pasco, Washington. I was introduced to Hungry Generation on Instagram. I've always gone to church, but I've never really felt that connection to anything. From a very young age, my dad and my mom got a divorce and I feel like that's always been a part of my life and I feel like that led me to want attention from guys. And when I got to high school, my first two years of high school were really bad because my focus was not school, it was boys and get that attention. You need that attention and recently, these last two weeks, I was in a messy situation where it was just hurt. I was hurt and I remember scrolling on Instagram and I saw a Hungry Generation post to join them on Sunday and I decided to come and it felt good to be here but I didn't accept Jesus until Wednesday and I just felt it felt good. I felt like I finally had something. I felt like I had that connection I had not felt in a long time and that's why I want to get baptized. Juanita, do you believe in Jesus Christ as God's Son and your personal Lord and Savior? Yes. Do you promise to follow Him with good and clear conscience for the rest of your life? Yes. Based on your faith, confession, and your commitment, we'll now baptize you in the name of the Father, the Son, and the Holy Spirit.


So, this is now become a question and answer session. So you can ask questions related to the content that was covered, as well as lab, whatever questions you have, I am happy to answer. Yeah. Good morning, ma'am. Actually, a question is related to Wireshark and basically we are running this Wireshark in window-based version. So I want to know that what is the advantage that if you use the TCP dump, because the packet capturing this scan we can do in the window-based GUI and all. Even that ERP request and ERP this also we can do on command prompt window. Yeah. So I mean the same point. But I am saying that whether it is in the main workshop also we have to do in the same way what you have. Actually it is a question of whatever it is you be consistent in what you tell your participants. The reason why I said TCP dump, again I am emphasizing is in Wireshark when you say capture is going to capture a whole bunch of packets, right. And then you have to put a filter where when you sort the packets you have to specify what the filter is and then it will sort it and display the select set of filters. That is perfectly fine. If you are comfortable with it, finally what matters is that they evaluate the packet and they are comfortable with it that is perfectly fine. As I said it is to me to filter I somehow find Wireshark filtering to be generally a bit maybe because I am not that familiar with because I already learned TCP dump I know how to filter. I have not learned Wireshark filtering thereby I don't find it very easy to filter. But if you are starting from scratch and you want to learn how to filter in Wireshark, specify how to specify host, this number, port, this number, ARP, whatever it is. If you learn it and you convey it to them, fine. That I will leave it up to you as a remote center in charge. So I am but which will be the more advantageous that I want. There is a personal preference I am sure there are people here who will say I love Wireshark I will do everything in Wireshark. People who will say I love TCP dump I will do everything. It is good to know both tools. It doesn't hurt what I mean you are learning two tools in one shot. So it's but your personal preference finally what matters is they evaluate the trace. Whatever method you want to follow you follow. For which you become comfortable, you become familiar, you should know how to filter. I think to install this TCP dump and all these on you went to or this Linux platform please provide us some guidelines. Yes, so there will so don't I mean let's not get into questions about that. There is a specific section where I will specify what exactly do you need to do to conduct the labs in the main workshop. There is a specific this is on the final day. Thank you. Madam can we show the DSCP packet sitting at the client side? Like DSCP packet capture can we show? DHCP? Dynamic host configuration packet. Yes. Sitting at client side. Yes there is an exercise for that also. Madam we usually say that IP address is a logical one because we can change it very easily. It is also true with MACID means we can change the MACID. What things happens behind the screen? Okay. As far as IP address is concerned it has been designed as a goal is to change it. The MAC has not been designed if you are changing it it is a hack or it is something you are not supposed to do your doing. So that's the high level thing. When it was designed people came up with IP addresses. They were supposed to be changed based on location. When people came up with MAC addresses they are not supposed to be changed based on. They are supposed to be fixed. I mean it's like a security or not supposed to do something but people still do it. You are not supposed to steal. People still steal but it's one of those things. MAC addresses are not supposed to change but people can hack them for some malicious purposes. Even system administrators may do something like that for valid purposes also. But all I am saying is by default they are not supposed to change. What is the difference between protocol number and port number? Create confusion with that some having the protocol number some not having the... For example TCP having the 617 protocol number I think and UDP having the... Actually it's at what layer you are looking at. So protocol number is something that the network layer uses to determine which transport layer process it has to pass the packet to. That is the protocol number. So TCP has 6 and UDP has whatever. ICMP has 1. So whatever are these this is what the network layer is using to determine which of the other network layer processes which is ICMP in this case or the transport layer processes it has to pass the packet down. Port number is being used by the transport layer to determine which of the application layer. So it's... Port number is working at the transport layer. Protocol number is working at the network layer. My question is related with forwarding. There are various mechanisms or there are various methods of forwarding like host is specific, network is specific, router is specific. I know that each and every method has its own... I don't digest and I just don't digest. But just to confirm here in the lab session, I just want to confirm whether it's like a router-based forwarding mechanisms or network-based forwarding scheme. I just want to confirm like in lab sessions which method it is useful. Useful as in there is... So there is something that is implemented. So when you are talking about forwarding, the one that is visible for you as a user is network layer forwarding. Switch layer forwarding you can't even see because it is transparent. It just goes... So let me just... So let's say this is a switch and this is your host that is connected to the switch. The switch in turn is connected to another switch. This in turn is maybe connected to another switch. And here is maybe... Let me call it router that is there. In turn this is connected to some other. So let's say this is somehow the way things are set up. Now when you are sending a packet, this destination let's say it is somewhere this side. Let's say this router is directly connected to a switch and this is where the host... Let me call this host B and this is host A. Now when host A is sending a packet to host B, what you are going to do is... So the packet that you assemble as I said there are... You have to take care of the MAC addresses as well as the IP addresses. So from the MAC perspective, the source is kind of straight forward. You assign your own source MAC address and the destination MAC address, you are going to assign it to that of this router R. Because as far as you are concerned these switches are transparent. They are not... They will just forward internally but as you see you are not directly connected to the router. It is going through many switches. Now when this... And as far as IP address is concerned with respect to the source you will put your own IP address which is corresponding to host A and the destination will be... Let me call this RM to represent the MAC address of R. This one will be B. Now this packet will come through the switch. If the switch knows where this router MAC is, it will directly send like this. Otherwise it will just broadcast everywhere. Finally when it broadcasts it will come to the router. Router will now look at this packet, figure out that it has to go through this destination B and it knows that this destination B is on this side. So thereby it will again maybe call ARP which is again broadcast. It will get the MAC address of B and it will assemble a packet where on the top is AB, a source and destination at the IP level. At the MAC level it will put router MAC as the source and B MAC as the destination and send it. And B will get it because the switch again if it knows it will forward otherwise it will flood everywhere. So that is how things happen where the network layer is interspersed with link layer in this particular fashion. My question is related to TCP. Like in heterogeneous network suppose multiple senders are communicating with multiple receivers via common bottleneck link. Then I believe that my sender window size has to be equivalent to BDP. Is it correct or it has to be divided by the number of flows? It is deterministic environment for example. So it depends upon how you are defining BDP. BDP is often defined as client specific. It already factors in, so if there is bottleneck bandwidth let's say is 1 Mbps, right? TCP is, there is some fairness to TCP. It's not perfect fairness. Then let's say there are 10 hosts that are sharing this bottleneck bandwidth. Now TCP is, there is some fairness to it. So what it will do is 1 Mbps it is going to divide by 10. So each is going to get 100 Kbps. Now the BDP that you are going to experience through this whatever windowing mechanism will correspond to 100 Kbps. So TCP has it factored into the way it operates. So there isn't anything like the BDP is 1 Mbps. BDP is not 1 Mbps really. The bottleneck bandwidth is 1 Mbps but the bandwidth delay product each client sees will be already that factors in the fairness that will be 100 Kbps which is corresponding to what you should get. When we set this value for example on sender if my BDP is very small and let's say I keep this very small I can exploit the available bandwidth. Right, so what I am saying is I need to set that value minimum equivalent to BDP. You have to set it to a value equal to not don't call it BDP. BDP you call it bottleneck bandwidth along the path. Whatever bottleneck... So for example this tuning of the window size in a typical setting it doesn't occur. The default values are what are going to play but suppose you are using a satellite link where the delays are rather large then it's important to tune because otherwise you are not going to utilize the bandwidth. So the tuning should happen with the knowledge only with the knowledge that you know where the bottleneck is what you are experiencing then you should tune. In general there is no tuning typically for the kind of throughputs that you experience in a typical internet setting they are already tuned to operate they are typically set at a higher value. So that you can utilize the... And how Q-Size will affect the performance in the same scenario like Q-Size on intermediate route or where there is a bottleneck or maybe Q-Size on each sender. Yeah. So what is going to happen is... TCP again if you have gone through the video it's a very dynamically adaptive protocol. So what it does is it... as it gets feedback from the network it is going to adapt its behavior where really it's going to change the window size based on it will increase if there is more bandwidth it will decrease if there is less bandwidth. So it'll adapt to what it is seeing. So currently if it is able to... its fair share is only 100 kbps it'll start sending data corresponding... the window size it'll set such that the delay corresponds roughly to 100 kbps. Now but if other flows have went away the queue has become smaller now we are starting to get 200 kbps. TCP will again adapt it'll increase the window size such that you are able to get 200 kbps. In fact that is the beauty of TCP it's extremely... I mean each time I look at TCP I will... this is... TCP is like the... of all the protocols I have it's a personal favorite. So that's the function of TCP the way it does it. So queuing, so what happens is as condition sets in packets will get dropped because queues have overflowed this is a feedback to the TCP to cut down its window and as condition goes away and you are able to... go through that to increase to see whether it can get more. So it's a dynamic protocol that automatically manages all these things. There are two kind of forwarding or routing that is source based routing and the destination based routing. I think whatever you explained here it is a destination based routing. That is the default. Can you show the mechanism how source based routing is carried out in networks? Well source based routing is not used in the internet that is why we don't cover much about it. So source based routing is used in the majority of the internet we don't use our source based routing because it's not efficient. It's not just that you're putting the burden of figuring out the root on the source and people like you and me who have computers it's not an easy task to figure out what the root is. So source based routing is used in very specific scenarios typically by these the ISPs where they know the path and they kind of know a lot of details where they will install a source based thing to for example ensure that certain traffic goes along a path and not through some other path because of some competition or because of security concerns or whatever it is. So source based routing is not often used just by ISPs in a very specific setting or in the research domain where people like in red-hawk networks that kind of setup you will see internet doesn't use as such source based routing. Ma'am actually I tried number of time to go through this topic but I was not able to understand the concept of actually forwarding mechanism of source based routing I'm clear about this. Source based routing is in fact even simpler than this all you're specifying the entire path you're seeing that for example again let me there are 2, 3 there is lose whatever strict source routing means your packet has to take the entire path that is specified the non strict I don't know what it is called is lose source routing is it has to cover whatever is mentioned but in between it can take other paths. So I'll probably okay so here is the source whatever this is the destination so this let me call it R1, R2, R3, R4 okay so this is so when you are specifying to reach destination you basically specify that this packet has to go through R1, R2, R3 R4. Okay so when it comes to R1 it knows it has to go to R2 and not this let me call it X1, X2, X3 it should not go through X1 so it will forward the packet here. R2 knows that it has to go to R3 so it will forward it may have other interfaces it's not going to send on them. R3 knows it has to go it will come here this knows that this is the destination it will send it here now the strict and lose people have. I think it is some kind of a virtual connection which we use in frame relay or FDDI networks. This is no virtual connect well you are insisting that this is the path it has to take there is no connection set up a priority for this path to take that is not that is virtual circuit switching this is not virtual circuit switching this is regular routing except that you are specifying the entire path so the job of the router in destination base routing is to look at the routing table the job of a router in source base routing is just to look at the header figure out who is the router figure out which among the interfaces corresponding to it and push it out on that. So how this router R1 will come to know that it how to forward only to the R2 and not to the X1 or X2. This source route is carried in the header of the packet that this guy sends yeah so this source sender that is sending this packet there is a header field in the packet right in the network layer header in this network like a header you are specifying the path that it take SR1 R2 R3 R4 and D this entire path is specified as part of the header so a router just needs right now a header has only the destination address which you look on a routing table to forward that is destination base routing here a router will look at the header figure out the entire path it is telling which the next stop is it doesn't have to consult any routing table it will just forward based on that that is source base route. Thank you. We can consider the MAC addresses the bunch of MAC addresses are allocated to different different NIC manufacturers. Suppose if I can consider the Cisco X number of MAC addresses are given to the Cisco but Cisco is manufacturing more number of NICs compared to only the X numbers so is it that the Cisco is supplying the same MAC address NIC1 to the India and same MAC address NIC2 US or UK or Australia. All I know is there are 2 to power of 48 which is a huge number you are not going to exhaust that space anytime soon so there are enough MAC address space. MAC address space is definitely not a problem so often what is done is each company so there is the company portion of it the first 8 bits or whatever is assigned to the different companies within that they can again subdivide it India may have the next 4 bits others may have other bits but what exact mechanisms do these people use to distribute the MAC addresses I don't know but scarcity of MAC addresses is not quite there consider they are manufacturing the NIC 80s onward then again the NICs are life of NIC is only 3 to 4 years normal life of NIC so again they have manufactured even if you use it 2 to the power of 48 is such a I think even if you are doing this for the past 100 years or 200 years it will not be I mean you know the it is something like it is a huge number even the total machine over the globe are including the routers even the 2 to the power of 48 you do the calculation 2 to the power of 48 is a huge number it is like the IP thing when they talk 2 to the power of 128 is I mean some of it is I don't know how much to believe it is like since big bank theory the number of milliseconds that has passed is still less than some such statistic is there with respect to IPv6 but in this 48 because they have to reuse the same thing that is what do they have to reuse life is less if I can consider the IP okay IP usability is possible because it is reconfigurable but if the manufacturer has manufactured one NIC in 80s with the address X1 but that life of that particular NIC is not more than 30 to 4 years that is fine but all I am saying is you worry about the problem when there is address space I don't think the address space is there yet because that number is really huge if it's a smaller number like 16 I would say yes 32 also fine but 48 is still from 30 to 48 I mean each one is a doubling you imagine like it's a huge number I don't think you are going to run out of MAC addresses anytime maybe for the next maybe I have to do the calculation but my gut feeling is for the next few generations you don't have to worry Comparing the IPv4 and IPv6 packet format we are observing that some portions are not present in the IPv6 packet format is it advantage or disadvantage comparing with the IPv4 and IPv6 and there is another small question that is it affect the security of the routing protocol whenever the IPv6 is used can it affect the secure routing protocol or in the routing protocol is it the some fields are not present in the IPv6 and how can you manage the so first of all when people went from IPv4 to IPv6 they know all the troubles of IPv4 so whatever they do I mean if they are intelligent people will be such that it addresses many of the shortcomings of IPv4 and doesn't security by the time is also people are aware of the kind of secure issues people are facing so IPv6 header is a very streamlined header and there is a reason why they have removed some of the fields this is mainly for multiple reasons so for example you do not see fragmentation in IPv6 the reason why they have removed fragmentation is it puts additional load on some of these intermediate routers to fragment the packet and that tends to increase the speed of the router so and then does it mean that there is still a need for fragmentation in other words you cannot really say all packets along a path will support the MTU size that may not be true but the burden of fragmentation has been pushed to the source source has to figure out what is the right fragment to use along a path and then once it figures out all the packets will just pass because you are not going to do fragmentation as far as the routers are concerned so it has streamlined some of these things keeping in mind the capabilities of the routers and so on and so forth in fact I would say it has removed a lot of these options the header also is options are there but they have been cleverly designed such that if there are not options the router is very efficient in reading the header because it's a fixed block but if there are options there then it's been designed cleverly that it's a pointer based system so it again improves the router processing speed the way it had happened in IPv4 is if there are options there you don't know where I mean it's a they're spread about you have to interpret each and everything for you to figure out what it leads to router delays so all these things are nice features that IPv6 has handled now the question is the compatibility now if you're using IPv6 and the fact that you have removed some of the fields what if there is an intermediate router that is IP version 4 and now we have a packet that is IP version 6 now what is to be done about it so that's the migration issues so a lot of the migration issues till the transition has happened by transition I mean everyone shifting to IPv6 from IPv4 till this transition has happened lot of this migration issues are handled a bit manually in other words the ISPs when they change the routers they tunnel they establish a tunnel so such that in case there is so all these are configured specifically it's not automated where things happen no they don't happen so you ISPs have to configure the routers such that the portions that are handled by IPv4 you tend to put an extra header such that they do and they do that kind of thing this is manually configured by ISPs thank you ma'am can we establish TCP between different subnet different subnets you can establish TCP within your own host so there is no question of TCP is just a transport layer mechanism for the endpoints can be the same IP address but different port numbers within the same machine you can establish for example within a lab if you are trying to SSH to your neighboring machine that SSH is operating over TCP which means you are establishing TCP between one host and other host that are neighbors as far as the network you can this is when you do socket programming assignment maybe you will also see you will when you do socket programming for debugging you don't want to involve another machine because that means traffic has to go so often you establish the two processes are running on your own machine and they are using TCP to communicate with each other or sockets the TCP UDP is up to you but you can communicate with different processes within the same machine using TCP TCP is a connection oriented so things and in the network actually IP data packets will be forwarded through different path I think so how this connection oriented we call this connection oriented then so when we say TCP is connection oriented all it is meaning is before you start sending the data the both endpoints have to manage their state or exchange their state so what happens with TCP is again let me so here is a host let me call it A it is going through many routers R1 R2 R3 and here is another host B and now I am establishing a TCP connection from A to B so this is the TCP connection now before sending data on TCP I am going to exchange some information to B so I am going to say for example that I can support SAC or I am using so some of these features you can specify or I am using this sequence number starting from X whatever is this information this is the state that you are exchanging with B only this is connection oriented in other words when you get a packet you just don't send it off to the other guy unless the state it needs to know what sequence number this connection is going to start all this information it needs before it is ready to receive the data connection oriented means this state exchange has to happen before you send the data connection less means you don't maintain any state first packet comes push it now if you see A and B are using this routing infrastructure to exchange this data so A is going to send this information B will send ok fine do whatever you want that kind of exchange happens then data transfer happens now when you look at the router R1 once it receives this packet corresponding to this A to B so the source is A destination is B and this is the packet what it is going to do is it will just send it out it is not establishing a connection with R2 telling expect this packet from it is not it will somehow figure out a path and it is going to send it out to R2 R2 intern also will send it out to R6 it will go to B so connection oriented is in that context the TCP is exchanging information before it can send the data between A and B and it doesn't care whether there is 1 hop 10 hops 100 hops in between because to it that doesn't matter for all practical purposes they are both on the same host also it doesn't matter as far as TCP is concerned but it has to exchange this data that means 2 end process are always ready in router when you do connection less in connection less you are design the protocol such that you don't have to exchange any data and still act on whatever you are getting whereas in TCP the protocol has been designed such that you have to exchange the data before exchange some connection state before you can start acting on the functionality so that is the difference between connection oriented and connectionless ma'am what exactly what is the load balancer and what is the role of the load balancer in transport layer, network layer and the link layer I give an example for my network say I have a lot of lines for the internet I simply connect those lines on the load balancer and I simply pass some lines to various networks for my network architecture so how this load balancer works so that is when you point it out load balancing can be done at different layers of the protocol stack it can be even done at application most of the load balancing that you see at the server levels is at the application level so for example I have a server that is set up and when there are many connections coming I will push some connections on to some other server and some other connections on to another server they are going to handle that is server level load balancer ma'am that is the application level load balancer as a hardware some load balancer as a hardware this work on the network layer also ma'am so the network layer load balancing works in this fashion so for example so there are two ways to reach a particular so for example if you have when you run the routing protocol you figure out that you can reach this destination through P1 or P2 often again what is this router R will maintain only one entry corresponding to D which is suppose this is going via R2 it will say to reach D sent to R2 it typically maintains only one but if you want to do load balancing what you can do is when you do this you may also associate that this path has a cost of let's say 10 and you will say D to reach through R3 this path has a cost of what is it called let's say 20 okay now this is a very crucial functionality of the routing protocol if with this for example if you receive let's say packets you can decide that since this cost is more I will send 80% of the packets on this path I will send 20% of the packets on this path you can do something like that but if your routing protocol is not aware of this for example if you are using distance vector of a routing protocol this can lead to routing loops because it's a dynamic thing each one thinks the path on the other side then you start to looping but if you are using some for example in adhoc routing or whatever where the protocol is not distance vector but some other version of the protocol which is load balancing compatible and you ensure that routing loops do not happen then what you can potentially do is you can send 80% of the traffic and send 20% of the traffic here so that is how load balancing works at the network layer where if there are multiple paths between the same source destination depending upon the capabilities of the path you try to split it ma'am there are one watchguards watchguards simply for example my network is connected with the proxy server and my proxy cannot pass some particular IP but using a watchguard I simply pass on those packets which are having the destination of that particular IP which are blocked by the proxy server so how the watchguard work on the network layer ma'am tightly bombay also we have this proxy server and if you want to access certain websites that are restricted in other words the proxy will not permit the access to those websites and you still want to access those websites so there are some of these other servers you can contact to get that particular content is that what you are trying to circumventing the operation of the proxy ma'am there are two ways either I simply put some filter on proxy server simply filter these packets from the proxy server but for the same subnet the subnet is blocked but for the same subnet only one particular IP can be allowed to pass the packets so for this in our network architecture administrator simply put one IP my IP on the watchguard watchguard is simply a hardware unit there simply put some IP and the packets from my IP pass on through the proxy server okay so I mean I think probably what you are referring to is the firewall rules that so typically whenever there is a firewall which has been configured so for example it may say block all packets coming from let's say 10.105.star.star which means everything is blocked but you can also put an exception saying that the longer prefix but do not block packets from 10.105.10. level which may be your IP address so this is a function of how you set the rules within the firewall to act upon this I mean that's an entire use so most of the the firewall configuration is again a very complicated thing but it provides a rich set of functionality for you to set rules so it depends upon how you set the rules within it to implement the required functionality you want like let this traffic pass don't let that traffic pass allow this user, don't allow that user in fact you can do it at the user level you can do it at the IP address level you can do it at the port level there are many things you can set as part of the firewall but all this are actually going up the protocol stack to the application which is doing this looking at all the things dropping the packets and then again they go down and go ma'am can I identify this with the help of command that particular packet is drop had or drop tell ma'am there are two types of drop had and drop tell in a queue then how can I identify with the help of command that this particular packet is drop had and drop tell as such you cannot so this is a queue so a router typically implement drop tell they don't implement a drop head now if a packet is dropped you have unless the router conveys that information to you in fact you don't even know where it got dropped there is an entire path taking R1, R2, R3, R4 unless R3 tells I dropped your packet through some ICMP message you cannot even know where your packet got dropped so that kind of feedback is expecting a lot from the internet which is best effort which often doesn't convey this kind of information