Okay, welcome back to the lecture series of the Summer School, and the speaker is Luisa Grifo from the University of Nebraska, and she will talk about symbolic powers. Great, well, thank you. Welcome everybody to the very last lecture series of this very long workshop. I am thankful to the organizers for inviting me and thankful to all of you for sticking around until the end. So I am going to be talking about symbolic powers, and in particular, about how we can use characteristic P techniques and tight closure in particular to solve problems on symbolic powers. So we won't really see any characteristic P until Wednesday. So today I'm really gonna be talking about background, like what are symbolic powers? What are questions people are asking? And then we'll use primary prime characteristic tools on Wednesday. Now, this is a very short lecture series, and so there's of course a lot of symbolic powers that I will not be able to cover today or on Wednesday. So if you wanna learn more about the subject, there are many resources you can use. So if you look at my notes, there are many references, first of all, in particular, a couple of surveys. And if you go on my website, even from the main page, you should be able to access long 200 page set of notes that I wrote about symbolic powers. So if you wanna learn more, you can read that. Though I wanna warn you, it is not complete. In particular, it definitely still has typos, and there are several chapters I'm still going to write that are not there yet. So if you look later, you might find a longer, better version of the notes. So anyway, so today I wanna start with background, what are symbolic powers? And the very first few things we need to remember are really things that we already all know. And so I'm gonna go very fast in the beginning, but if you're not, when we get to actually talking about symbolic powers, we'll slow down, okay? So these are just, this is a recall moment. First of all, I need you to recall what our primary idea was. So being primary is sort of a generalization of being prime. An ideal is primary if when you have a product in there. If one of the terms in your product is not in your primary ideal, the other one must at least be in the radical. So we must have some power in that idea. So immediately from this definition, we see that the radical of primary ideal is prime. And so we're gonna wanna say, we wanna record what that radical is. So we'll say something is P primary if its radical is the prime P, and it's a primary ideal.  It's very easy to give examples. So for example, if you take the ideal X square, X, Y, take K to be your favorite fields, then it's easy to see that this is not primary. The problem element of this X, Y over here, but also the radical of I is also easy to compute, it's the prime X. So this is an ideal that's not primary, it's radical is prime, but still. An easier way to see what the problem is with this ideal is that actually being primary is an equivalent to having only one associated prime. And the problem with this guy is it has only one minimal prime, but it has an embedded component. So there's an embedded prime X, Y. So I will say right away, we've seen our first exercise that there are things I'm calling exercises that are suggestive exercises to think about. The ones I intend for you to turn in are actually called problems. So when you look at the notes, you wanna go at the very end and the things that are called problems are the ones that I intend for you to turn in. Things called exercises are just good questions to think about. So anyway, here's another simple exercise that we'll use today. The intersection of primary ideals with the same radical is also primary to that same prime. Okay, so having primary ideals, you can now talk about primary decomposition. So giving a primary decomposition of an ideal is to write your ideal as a finite intersection of primary ideals. Now, if you have, I just told you in this exercise that if you intersect primary ideals with the same radical, you still get something primary. So if you have two components with the same radical here, you might as well intersect them and make just one component with that particular radical. So I wanna think about irredentant decompositions which are ones where I've already done that. I've already collected together all the components from the same prime. And of course, I wanna think about decompositions where I can't delete anything, because if I can delete something, then why even write it? If it's not doing anything in intersection, I don't wanna count them. Now, I'll notice from what I've said so far, it's so easy to see that if I hand you a primary decomposition for something, you can very easily turn it into an irredentant one by deleting things you don't need and then collecting together components with the same radical. Now, one of the, maybe one of the first theorems that we would definitely say this was definitely primitive algebra. If you were definitely doing primitive algebra to do this, it's this beautiful theorem that was proved in full generality by Emmy Noether in the paper where she introduces Nazarian rings, which of course she didn't call Nazarian rings, but this is where they first appear. And it had been proved before by Emmanuel Lasker. And yes, that is the chess champion, Lasker. He had proved that the setting of polynomial rings and power series rings, and then she proved the full generality. And the statement is, of course, give me any ideal in a Nazarian ring, and then it has a primary decomposition. So for my entire lectures, unsurprisingly, all the rings are Nazarian rings. And notes, I could give you examples, but I think you've all seen examples before. I'm just going to remind you that primary decomposition, they're not unique, but there are some parts of it that are unique. So there are two uniqueness theorems. One tells me that whenever I hand you any redemptive composition, the primes that appear as radicals of those components, they are actually the associated primes of your IQ. And so the associated primes of the mongealarmor eye, those are what I call the associated primes of the ideal eye. And those are exactly the primes that appear in any redemptive composition of mine. You might have different decompositions, but what you know for sure is what P's you see a P primary component for. So in particular, the number of components is fixed. It's the number of associated primes. Now, some components are unique. The components that are coming from minimal primes, those are unique, right? Remember, the set of associated primes contains all the minimal primes. And so in particular, the P primary component associated to each minimal primes is actually always the same in any redemptive composition. And it's given by what you get when you look at the ideal eye, you localize at that prime P and then you contrast to active R. So another way to say this is you're asking who are the elements that lived in eye once I've localized P, that locally around P, you want elements that lived in eye. So in other words, the elements up in your ring that are sent into eye by something on P. So these minimal components are important in particular. And that's the beginning of the story of symbolic values. So let's first talk about symbolic powers of prime ideals.  Well, we write it P to the N, but with parentheses around it to distinguish it from the ordinary powers. And a simple way to describe this is to say, this is that unique P primary components in P to the N. So when we write the power P to the N, fun fact, you'll see examples soon, a power of a prime does not have to be a primary ideal. Again, it's radical P, but that doesn't make it a primary. In fact, if the powers of primes were all primary, there would be no such thing as the theory of symbolic powers. We wouldn't be here talking about this. But even though P to the N is not primary, I know for sure there's a unique minimal prime as P. And so I know exactly what that unique minimal component in any redundant decomposition looks like. So the theorem we just wrote tells me that this unique P primary component is what I do when I look at P to the N, I localize it P and I contract to R, right? So writing this more explicitly, it's the elements that live in P to the N up to maybe multiplying by something not in P. Now, it's an easy exercise to check that in fact, this can also be described equivalently as the smallest P primary ideal. Continuing P to the N. So how would women check this? Well, you would take any P primary ideal that contains P to the N, right? And I noticed that you intersect it with P to the N and you still get the same thing because it contains P to the N. But now that's something you can play with your primary decomposition. And if it's something P primary, you can put it together with your P primary component and you'll find that nothing changed. So this guy must have contained that small, sorry, must have contained this symbolic power, which is that minimal component. Anyway, I know I said that fast, but if you've never thought of this before, try to check it a little slower later. Okay, so here's an example that in particular, I will explain what I just said that powers of primes don't have to be pregnant. So take your favorite fields and take any constant you like see. And now let's take, this is actually the same computation if you do the polynomial ring, but let's do the local setting. So let's take the power series ring, k of x, y, z. And then let me take x, y minus z to the c, just some hyper surface. And in this hyper surface, I wanna consider the prime ideal x comma z, okay? Now, I wanna think about the powers of this prime ideal. I claim that this is going to be an example where the powers are not primary. And so we're gonna say something interesting about the symbolic powers of this guy. Maybe let me say immediately that of course, if this is the p primary component and p to the n, or if you like by looking at this line, one can quickly see this is going to be something containing the power, okay? So if I'm gonna show you, I'm going to show you that this symbolic power of this prime is not the power. Maybe let me in fact write that as a remark first, right? So when are the symbolic powers powers? That's exactly when p to the n is primary. So I'm gonna show you that there's symbolic powers of this guy that are not the powers by showing you that the power is not primary. I wanna think about the power p to the c. What is nice about p to the c is that in here, of course, you have z to the c because z is a generator of p. But in our ring, that's the same as x, y, okay? But now notice, of course, that y is not in this prime p. And it sends x into p to the c. So it's kind of hidden, but from this definition over here, we see right away, this tells me that this x is in the cth symbolic power of this prime p. But on the other hand, notice that x is very much not in p to the c. Well, I guess it's important here that I declared that c was at least two, okay? And so in fact, this tells me that we just found an element in the symbolic power that's not in the power. So it tells me the two things are different. And in particular, it tells me that this cth power is not a p primary p. So maybe another comment to make here is that one thing that's interesting, well, okay, let me save that comment for a little bit. Sorry, you can guess what I was going to say. Okay, so I give you a definition for primes. What is the more general definition? Well, I'm going to want to tell you what is the nth symbolic power of an ideal i. All examples we will consider in these lectures is we're actually going to be radical. And that's already extremely interesting. In fact, the symbolic powers of primes alone are already extremely interesting. And if we only talked about primes, there'll be plenty to say. So you don't need to go think about all ideals to have something interesting to say. Now, for technical reasons, I want to have an i with no embedded primes. And the main point is that when the ideal has embedded primes, there's sort of two choices of definitions for symbolic powers that are different. They're both good for different reasons. And we have such little time to talk in these lectures. So I don't want to get into the weeds of all the technical details, it really isn't, I think it's not the point of the lectures. Again, even if you just think about primes, it's exciting. So in fact, most of the time, even you can take this for the entirety of the lectures, I'll take an ideal that it's radical. So what is the symbolic power? Well, again, we write it like this. And what we do is we take our power, i of the n, and now we're going to localize it a bunch of primes and contract back to R. But now we don't have one minimal prime to look at. We're gonna do this by looking at all the minimal primes of i. Now, notice I said my ideal had no embedded prime. So this is in fact the same as going over all the associated primes, right? So I'm collecting what I'm doing here is I'm collecting the minimal components of i to the n. I'm writing the primary decomposition for that power i to the n, and I am collecting the pieces that come from minimal primes, right? The minimal primes of i or i to the n are the same thing. So I'm just collecting these minimal components. So in a way, computing symbolic powers is like computing a primary decomposition but focusing on the minimal components and ignoring all the embedded components. So if you want to write this in a friendly explicit format, you could say that this is all the f's that are sent into the power by now some s that is not in any of the minimal primes. Which is the same as saying it's not in their union. So like before, notice, this is really a question about associated primes. So the symbolic power is the power if and only if the power has no embedded primes. So essentially the thing, the power only has the minimal primes which are actually the same associated primes as the ones that I have. You didn't gain anything by looking at the power. I do note that if I take a radical ideal, so what's a radical ideal? One way of describing being radical, yeah, you can see you're equal to your radical. But another way to say that is your e-code to the intersection of your minimal primes. So these are the minimal primes. When you have a radical ideal like this, what happens when you take I and you localize? I hope you can't hear her but my dog just decided to start barking very loudly. So actually, let me say for simplicity, let me say I take any of these PIs, any of these minimal primes. What happens when I take I or even I to the end? Actually, let me write I first. What happens when I take I and I localize at a minimal prime? Well, all the other components are certainly not contained in that minimal prime, right? So they're all going to explode and become the whole ring. The only component that gets saved is the PI component. So this is the same as taking that prime and localizing it. And now, of course, same thing happens when you take powers. So notice in particular, let me see if I can show you everything at the same time. So now here, when I compute these things, it's the same thing as taking the power of the prime and localizing that and contracting that. So in other words, if you like, we could have defined the symbolic powers of a radical ideal as just taking the intersection of the symbolic powers of the primes, right? So you really only need to know what symbolic powers of primes are in some sense. Now, before we do some more examples, there's a few more comments to make. So so far we've, I think we've said that computing symbolic powers is something about understanding associated primes. So another way to describe this, completely equivalent. So in fact, let me say this like this first. So to compute the Anismbach power of I, I compute a primary decomposition for I to the N and I want to delete all the embedded components. How do you delete certain primary components of an ideal? That's taking a saturation with respect to something. So let me write that down explicitly. So this is, I guess, a separate definition if you'd like, it's kind of a side note. So if you take I and J ideals, the saturation of I with respect to J. So this is kind of like a column but there's an infinity of here. So what you do, you take the columns of I with the powers of J. These make some ascending change. My ideal is an ovarian so that ascending change stops. So whatever is that largest of these columns, that's your saturation. So really you're asking about elements, the sand like J to the N inside I for some N. But eventually you send all the powers of J inside of I. So you are going to show, this is one of the problems you're gonna turn in. And it's not very hard, but it's I think as good as a science to think about. You've never done it before. You're going to show that what such a reading does, wanna write it in words and it's written in math in the notes. What this does is this deletes primary components, certain primary components of I. And which ones is it delete? It deletes the ones, let me say it like this. So with the primes corresponding to them, so associated to primes that contain this J. So to compute a symbolic power, what you wanna do is you wanna delete certain primary components. So you just have to find an ideal J that is sort of contained in the right primes and not contained in the right primes, right? So that it deletes exactly the correct components that you wanna make disappear. So at the end of the day, any symbolic power is just a saturation. So if you wanna make statements about symbolic powers, you can also just study saturations and you're proving statements about symbolic powers as well. Now, we can think about this as sort of a way of computing symbolic powers would be, go find the correct ideal J to saturate with. And then you're in business, saturations are really easy things to compute. The problem of course is finding the correct thing to saturate with is not such an easy thing to do. But in fact, you're going to, one of the problems you're gonna do, no need to copy this, it's in the notes. One of the problems you're going to do is you're gonna show that there exists an element, okay, maybe I should be more precise. So given an ideal I, and maybe for simplicity, we're gonna say it's radical. And you know, whenever I assume our ideal is radical, it's not actually necessary, it just makes the technicalities much easier, okay? You're gonna show that there exists some elements, I'll call T in your ring R, such that saturating the powers with respect to that T give me back to the symbolic values. So saturating with one element is really easy, the hard part is figuring out what elements you wanna take, okay? So I'm really in a way sort of getting ahead of myself because I'm answering a question that I haven't explicitly said out loud, but I should right now. Why are symbolic powers difficult? Because they have something to do with primary decomposition, and primary decomposition is complicated to calculate. One of the difficulties about studying this topic is that you wanna compute examples. Examples can be very difficult to compute. Even if you have a computer, you have Macaulay, if you wanna go through primary decomposition, good luck, right? If your power is low, okay. If your ideal's gonna ring with, not very many variables, let's say, okay, but things get very difficult very fast. So going through this definition is tricky. If you have such a T, oh my God, that's amazing, right? You just have to compute a power and computing saturation is super easy. It's one of the easiest things to do computationally. So for some types of ideals, one can say what this is, but in general, in general, in general ring, the story is much more complicated. But okay, take this as a side note. This is something you're going to do in the exercises. I do wanna point out though, that one thing that helps, okay, maybe I'm gonna point out one thing that's bad and one thing that's good. So I think that's bad. Associated primes of powers are difficult to find. Things that's good, you have a little bit of control because okay, maybe first, let me say problem, the function that computes the associate primes of power is very difficult. So like for example, when you increase the power, the newest associate primes don't even have to contain the old ones.  But one thing that is really nice is that actually when you compute, when you look at all the powers together, even though each power might have an insane, crazy set of associate primes, there's actually only finally many among all the powers, which is a relief, I think, right? It makes things work a little better. So the fact that there's finally many associate primes is a theorem of atlas. Actually it's a line and a proof of something else, people by atlas. But as I know, this is the first time this appears in writing, although if you look at those notes that I was mentioning, I think the proof is much longer than one line. But you know, it's sort of written by Eckhardt-Simmons paper. But then Rodman proved something stronger. So Emmy attributed this to Ratliff in 76 and then Rodman in 79. They show that the set of associate of primes. So this first statement is Israeli Ratliff. This set is finite. And then Rodman showed more. So I was just telling you that when you go from one power to the next, you don't have to contain the old associate of primes, right? I mean, you contain the minimal primes for I, but then crazy things can start happening. But Rodman showed that there eventually the associate of primes stabilize. So if you go look high up and up, the set of associate of primes becomes constant at some point. But up until that point, where they become constant things can be crazy. There's not that much in the literature about what this point, like this stability index is. You'll find some results in a paper by Susan Morey. But in general, it's really hard to say where this stability begins. Okay, so I've been talking about the sort of abstract idea of taking symbolic powers, but I really owe you examples and properties. So let me start by writing properties first because it'll help us with computing examples. So properties. I won't prove these properties. They're all elementary. Anything I don't prove, I should say, you can find them in various places in the literature. One place I can promise you has every proof of every theorem I will say in these lectures is those notes, those long notes that you'll find on my website. So here's some easy properties that you can try to do on your own. Very good. So one, I already said this and this follows immediately from the definition, but think about it for a second if you don't see it right away. Symbolic powers at least contain powers. The first symbolic power is I itself. Well, that's, you know, caveat, right? I'm assuming my ideals have no embedded prime. So in fact, let me just write to make this quick that my ideals are optimal. If you take a higher symbolic power, so if A is bigger equal than B, then your symbolic power gets smaller the same way that the ordinary power gets smaller. And in fact, maybe let me write this number four, I'm not quite following the order, this is in my notes, sorry about that, I just switched to, okay, let me say that R is a domain and I is non-zero. So if you happen to know that you have a power inside some symbolic power, well, we already know this works if you take the same number. So if you take a power that's bigger, this is certainly going to work, right? But this is in fact, and if and only if. So A must be bigger equal than B, for this to make sense. And I'll finally, one that we'll talk a little bit more about today, although I'm going a little slower than I intended, so I should pick up the page. This is the greatest family of ideals. If you take a product of something and symbolic power by something, not a symbolic power, land inside a symbolic power corresponding to the sum of the orders of those two. Now, something that I want to write separately, these are all elementary, the thing I'm going to write next is not elementary, although you can prove it in many different ways. It's sort of a fundamental fact about symbolic powers is that sometimes the symbolic powers are indeed the powers. And one situation when that happens is when your ideal is generated by a regular sequence. So for simplicity, let me say I have a radical ideal, complete intersection generated by a regular sequence, then the symbolic powers are found. Let me say something really fast. Let's maybe explain a little better in the notes. The real statement is that if your ideal is generated by a regular sequence in any room, the associated primes of the powers are the same as these sort of primes of the I. So in particular, if you're ring with Conan McCauley, that says that I has nothing better primes. And so it says that it satisfies, we can use our definition of symbolic powers and symbolic powers of powers. So instead of saying that it's from McCauley, I said that my idea was radical, which easily makes this work anywhere. Okay, now with all this, I'm now ready to give you some examples. So let's say that I take this square free monomial ideal. So I'm in a polynomial being in C variable and I have an ideal generated by monomials where the powers that appear are one and zero. So this is square free monomial ideal. Those are exactly the monomial ideals that are radical. Computing primary decompositions of monomial ideals a little easier than computing primary decompositions of some random thing. One way to see that what I'm writing is correct is you can think about this geometrically, if you like to look at things geometrically. This is the ideal of the three coordinate points in P2 and or if you're in a fine space, you have the three coordinate lines, right? That decomposes, why don't I draw the picture? That decomposes as the union of those three lines and each one of them corresponds to one of these three ideals, right? So for example, this is the axis. Okay, so one of the symbolic powers of this guy, well, I have these three minimal primes and I'm going to compute their symbolic powers. So I'll take x, y to the n, x, z to the n, y, z to the n. Now notice these three ideals are generated by a regular sequence. Their symbolic powers are powers. So this is just the intersection of the powers. Now sadly, or maybe not sadly, it's a good thing it makes symbolic powers interesting. Intersections and powers don't commute, right? I can't fool this power out. So for example, if you compute the second symbolic power, now I should have used the copy face but I'm halfway there. So now it's good to do that. In here, it's easy to see that you have the element x, y, z. Like for example, x, y is in here, like z in here, y, z in here. And I don't know, it's either early or late for all of you. So maybe I won't ask the audience for help. But now you've had a second to see that this element x, y, z has degree three. And that's bad because I is generated in degree two and that means that I square is generated. So I square is generated in degree four. So everything in here is degree four or bigger. So there's no way that this element of degree three is in I square, but it is in the second symbolic power. So this is a common phenomenon that you'll see elements in your symbolic powers that kind of live in the wrong degrees to sort of be in the past, right? You'll see, I wanna see unexpected degrees that you weren't expecting to see in these symbolic powers. I have one more example in the notes that maybe I'll skip for now, but it's a cool example to try to do by hand. It can take the two by two minors of a generic three by three matrix. And that's a prime ideal whose symbolic power is not the powers. So in particular the determinant of your matrix is in the second symbolic power of the month as well. And so that very fast, please check out the example in the notes later. I have 10 more minutes, I think. So I'm gonna use those 10 more minutes to tell you two more things I really wanna cover today. So first, I wanna tell you one reason, okay, there's a thousand reasons to take care of our symbolic powers, right? First of all, primary decomposition is one of the most natural fundamental things that can come in rounds with us. So of course we should care about symbolic powers because they come from primary decomposition in a very natural way. We care about associative primes. We wanna understand associative primes as powers. Those are very natural questions to ask.  I think of the ordinary powers as sort of the natural algebraic motion of power. And somehow some of the powers that appear naturally algebraically actually contain geometric information. So one way to make that precise is by a classical result of Zewiski and Nagata. So this is something that Zewiski proved half of in 49 and Nagata proved the other half in 62 in his local rings book. So let me say that my ring, let me start over C and then I'm gonna delete C and put any field.  Well, actually let me re-organize this. Okay, this is my fed up for now. What does Nussel and Zat say? Well, if there are various ways of phrasing Nussel and Zat, it tells me that this ideal I is the ideal of all polynomials that vanish along the variety that I divide. It tells me that for each point in my affine space it corresponds exactly to a maximal ideal. So to say that I vanished at all the points in this variety, so to vanish at one point is to be in a certain maximal ideal this means this is a maximal ideal. To be a point on that variety means the corresponding maximal ideal contains the ideal I. That means to be a point on the variety that I defined. And this is to be a part of an I to vanish along that variety means that you vanish at every single point in that variety that I defined. But each one of these is like going over every point in that variety. So in fact, the statement I just wrote, you know, if you're now going, don't interpret it. Like, of course I could put any algebraically closed field and give the same interpretation I just gave. But actually it's true more generally that erotical ideal is equal to the intersection of the maximal ideals that contain it. That's just saying that this is a Jacobson thing, this polynomial, but this is through a variety of fields. So the Zarisky and Nagata theorem is a high order version of this. I'm gonna add, I'm gonna decorate this equality with symbolic powers on the side and powers on the side. So you should read as the saying that to be in the end symbolic power is to be a polynomial that vanishes to high order. Let me say along Viva. So think of this as saying that at that point of course point to F, you vanish to order, right? That's sort of the interpretation. You wanna give this in the algebraically closed field saying. Okay, so one can also write this statement in a sort of a projective looking statement. So if you take a homogeneous ideal, you would then instead of maximal ideals here, you could put the ideals of each point which now are not maximal anymore, right? They have like height one less than the dimension of the ring. I could write that explicitly, it's written explicitly in the notes. But since I only have six minutes, I'm gonna use those six minutes to tell you something else. So if you wanna see a projective version of this theorem, please, please check out the notes and you'll find a proof in those long notes in my website. Okay, so on Wednesday, I wanna talk about efficiency P problems. So today to prepare to get there, I want to talk a little bit about what are some of the big open problems related to symbolic powers. And in all fairness, I'm saying something that the problems I'm going to list here are a little more than just open problem. Like if sort of some of them are too open-ended to ever get a complete, you know, nice solution, but sort of the types of questions that we wish we knew more about, right? So here's some problems. Oh, I should write the word open because otherwise there's a danger that it sounds like these are the questions you need to solve for your homework. You should not, do not attempt these at home. Okay, I mean, do but not this week. Okay, so here's one problem. What to me is maybe the first obvious problem is equality. When is a symbolic power just a power? It's just kind of like the best case scenario, right? If the symbolic power is just a power, you know, everything you could want about it. You can say which polynomial is finished or what are in along this variety, right? You can tell me what degrees they have. You can tell me anything I ever wish to learn about them. We have a sufficient condition already, right? We said this always happens when I was generated by write the sequence. You know, module, technical things. And so is that an if and only if? No, not at all. Very, very, very far from being an if and only if. There are many other conditions that might lead to this equality. So you could take this question as how do I characterize which ideals are I that satisfy this equality for all N or you might ask, you might hand me an I and ask for which ends this is true. Is it true for any N? Is it true for all? Is it true for some? You might have equality for some values and not for others. Again, there are all things you can find examples of in my notes, not the notes for these lectures, but the longer ones. And there's many nice theorems to cite here, but I have limited time. So I would at least cite one theorem that is not a statement in character CP, but it's a very recent result. And it was proved via characteristic P like techniques.  So here's an example of that. It's the recent results of Jonathan Montaigneal and Luis Munez Fethicula. This is a 2021 paper that's the publishing date. So you take a field, you take a polynomial ring over that field and you take a square free monomer. You would think that this is easy dealing with structure monomer like deals and polynomial rings. What is there to know about some of these things that we don't know? Well, lots turns out. So maybe take this also as the kind of things we didn't know until 2021. Okay, so this came in as the following. Suppose you've tested the equality of symbolic powers and powers for small values of N. So you went up to the minimal number of generators over two and you took the feeling that's sufficient to show that you have equality for all N. So this answers the question, can I test equality by testing it up to a point? It's given me a finite number of steps to test that will guarantee that I have equality for all values. We don't have such a result even in a polynomial ring. We don't have such a result for all ideas. We have such a thing in dimension three. Wow. There's no theorem correct me if you the affairs that in dimension three, you're either equal, the second symbolic power kind of determines everything. They're either all equal or they're all different. But that's very special, right? If you take more numbers of variables, we don't have a theorem like this in more variables. But the proof of this theorem sort of involves techniques that look like characteristics. You're inspired by Provenius like things. Okay, so my clock says that it's 9.50. I think I've started two minutes late. So I don't know, is it okay if I speak for two more minutes? Or should I stop here? I can also stop here. It's okay if I go two more minutes? Okay. Yes, you can go in for two more minutes. Okay, great. Well, I think I started at 9.02. So I'll go two more minutes. Okay. Oh, no. This one, let me say something very vague. What is the smallest degree of some elements in i-symbolic again? So if i is homogeneous, it's easy to prove the symbolic powers are also homogeneous. So I have a homogeneous ideal and some graded rain. I wanna ask about the degrees of things in the symbolic powers. Notice that since the symbolic powers contain the ordinary powers, giving upper bounds on this degree, on the degrees of things, it's easy, it's giving lower bounds this far. There's so much to say about this question, but my time is limited. So I'll just mention that's an interesting question that people think about. And let me mention very quickly that another big question is to study what is called the symbolic phase in algebra. And here's what might be a surprising difficult question. When is this a finitely generated or algebra? Which when a ring is excellent, is it equivalent to saying that this is in the beginning? And this is also interesting and difficult. This is sort of asking, can you describe all symbolic powers by describing finitely meaning? This is some graded rain, thanks to the fact, the last property on symbolic powers there, but thanks to this fact, that's a nice graded rain. And this being finitely generated is kind of saying finitely many symbolic powers describe all the symbolic powers. And this can fail. So you might have something where this is not an ordinary rain, even for primes in polynomial. And there's a lot to say about that. I may mention this briefly again next time. So what we're going to do on Wednesday is we're going to talk about one problem in particular and some theorems that have improved using character 60 techniques. And so we'll see in particular why character 60 techniques are so useful to prove things about symbolic powers. And now I'm definitely over time. So thank you. I'll stop here. All right. Let's thank Louisa for the nice talk. Are there any questions? All right. So if not, let's thank Louisa again. Thank you. All right, so let's move on again on Wednesday. Yeah. And you are all welcome also to email me questions. Although I am quite busy and this week in particular and I don't promise that he won't take me like a day to reply to you, but I'll turn on this or you can ask him questions on Wednesday during the lecture. Thanks. I'll see you all then. All right. Bye. I think you can stop there with Quentin.


Thanks, Dan. So we are going to talk about service authentication and specifically how to choose whether or not you're going to use tokens or certificates in your authentication. Real quick, who am I? Like Dan said, I'm the CTO of Tremont Security. I've got 20 years of identity management experience, big Fortune 100 companies, governments, various vendors over the years. I've been working with Kubernetes since 2016, wrote a lot of the documentation around OpenID Connect, worked on implementing authentication in the dashboard and in most recently in Keali, and co-author of Kubernetes and Docker and our prize guide. So enough about me. So when we're talking about whether or not we're going to use tokens or certificates to authenticate your services, start by asking yourself a few questions. First off, does your service use end user data? Some services are used by end user applications, and a lot of that will dictate the easiest way to go for authentication. How much is automated? If you're not automating at least something, then you should probably take a step back and think about how you would automate it, because trying to do this stuff manually is going to get you into a lot of trouble. How complex is the network? A lot of times when we start talking about certificate authentication versus token authentication, a big thing to think about is how close is your authentication point for the certificate to the actual system that it's protecting? And then finally, how does a transaction span multiple service calls? Most complex services are more than one call. You're not just going to say, OK, call this one service. Get your answer back. You're done. Often you'll call one service. I'll call two other services to get the job done. Does the transaction span that? Do you have to be able to get user context across those services?  So this slide's really the biggest slide. I'm actually going to see if I can get rid of the. Make it a little easier for me. No, no, maybe not. Am I still sharing? I am not. Try that again. All right, cool. So this particular slide's really the one that you want to pay the most attention to. It brings together a lot of the information on comparing and contrasting certificates and tokens. When we're talking about certificates, the big strength of the certificate is you have no secret going over the wire. Very, very secure when you do it correctly.  You're generating a certificate that's getting trusted. Your service doesn't need to know anything about that private key. There's no shared secret. You look at tokens, the big strength is its ability to be able to move across networks easily. You're able to say, okay, here's my token. Doesn't matter how complex the network is. The other big thing that's great about tokens is that it can generally include identity claim information in it. So you're able to put, for instance, user context information, group information, perhaps transactional information that can all be verified, so you don't have to go back to the source in order to get that data. Now, when you're looking at the drawbacks of these two methods with certificates, the biggest drawback is one that security that I mentioned only comes if you do it the right way. And by right way, I mean, you're generating a private key in a secure store. You're generating a public key and a certificate signing request. That certificate signing request is getting signed properly and then re-imported. Using something like Istio or other service meshes where this is done for you, that makes it a lot easier. If you're generating a P12 file and then downloading it, chances are you're not doing it correctly. So doing that correctly is very, very difficult. Another aspect of certificate authentication that can be much harder is revocation checking. Certificates tend to be very long-lived because you don't have that secret going over the wire. And so with that long-lived credential, there's a chance that that credential's been revoked. That certificate's been revoked. Well, there are only three ways, really two ways honestly, that are standards-based that you can do that. Either you're checking what's called certificate revocation list, might be pulling it from a file system or LDAP, or you're querying OCSP, very old protocol to check to see whether or not the certificate's still valid. Kind of a flaky protocol, truth be told. And it can be very problematic. So if you have the certificate, in order to do it securely, you need to make sure that you're generating it securely. You also need to make sure that you are checking for revocation. That revocation part's very hard to do in most instances. With tokens, the big drawback is you have to send that secret over the wire. So whether that secret is really well-built cryptographic secret or a really poorly-built hand-encrypted secret doesn't really matter, the fact is it's going over the wire. So if somebody gets a hold of that token, they're able to use it without having any other piece of information. There have been attempts to build standards to bind a token to a connection. They had one a couple of years ago that Microsoft was really pushing, didn't really go anywhere. Chrome decided to take it out of its own implementation. So that's always been the biggest issue with tokens is how do you protect that secret over the wire? The biggest way to mitigate that risk is to have very, very short-lived tokens. So if your token's only alive for a minute, chances are by the time the attacker gets a token, figures out what it is, figures out what they can use it for, it's already expired. So that goes back to the automation piece about why automation is so important for your security. And then the other major drawback for tokens is that doing a proper service-to-service exchange can be really painful. There is a specification that lets you do it. It's called the OAuth 2 token exchange. I just realized I'm still wearing my mask, sorry. The OAuth 2 token exchange, it's doable, it works very well, but it's something that you need to know how to do inside of your code, inside of your token. The big risks with certificate authentication boiled down to are you using it properly and then how far away from the application is the certificate. So when you're doing something like Istio where you've got that sidecar that's validating the certificate for you right up against your pod, that's really great because there's very little distance between your certificate validation, your applications. The fact that it's just injecting some identity data as headers is not a big deal. If however your certificate validation point is four proxies up and now you just have some standard contact information going through these proxies, there's no way to validate it, that becomes a real security risk at that point that anybody can inject that information along the way, you can't really track it from level to level. The major risk for tokens is gonna be the fact that it is a bearer token, it's something that goes across the wire, somebody catches that, somebody logs it by accident, that's where you're gonna run into issues. So food for thought while you're looking at which one to choose, don't be afraid to mix a match. There may be scenarios where you're going to start off, let's say you have an application, that application's gonna talk to your services, that part might be token based because it's really easy for a web application to work with tokens, users aren't generally going to have certificates, there are some instances where that's not true. But for the most part, your users will have tokens, they won't have certificates. But then once you get into your infrastructure, maybe you then switch from certificates or from tokens over to certificates, that's perfectly valid to do that transition. Just make sure you're logging it, keeping track of it, things like that. Service meshes and API gateways are really important and the line has gotten fuzzier and fuzzier over the last couple of years as to whether you'd use a service mesh to do authentication or an API gateway, but whichever one you choose, and I'm not gonna say one should be used over the other, it's gonna really depend on your situation. Don't do it alone, don't try and reinvent the wheel because chances are it will go horribly wrong. And I kinda say like inventing your own authentication systems like inventing your own crypto, don't do it. There are a lot of standards out there with thousands of hours of peer review, thousands of implementations, stick with those standards, they might not be perfect, you might not think it suits your need or what you want, but chances are the terrible documentation for those systems is really what the problem is. So yeah, some links and resources. A couple of blog posts that I've written about certificates versus certs for the API server specifically, and then using tokens and pipeline authentication just gets into the real nitty-ditty gritty detail of how service accounts work. And then this blog is written by a friend of mine, Alex Ellis, he's the author of Open Fuzz and Inlet. And this blog's really interesting because it talks about using OIDC with tokens from GitHub Action. So using GitHub Action to securely call Open Fuzz. He based it off of a new release of a tool that lets you use GitHub Action's OIDC to call GKE resource. And this really shows both the benefits and the risks of using tokens because in this one example, you're able to both generate tokens across different mediums, but at the same time, that means secrets are going over the wire. So you're getting better security by not having a secret stored locally, but then you're still getting a security boost because now, okay, I'm running my GitHub Action but I don't have to generate a secret in order to use GKE. And then finally, some shameless self-promotions. Say hi on Twitter. If you are looking for an identity provider, we make Open Unison, open source identity provider to Poise Wright and the Cates. And then our book, Kubernetes and Docker and Enterprise Guide. That's it, thanks. Thanks so much, Mark.


permission to record. So recording has started. Yeah, awesome. Thanks. Hi, everybody. Welcome for this new Jenkins infrastructure meeting. We are the 29th of June. So we have three topics to cover today. So the first one is we have a security release coming tomorrow. So this will be a weekly release and a stable release. So stay tuned. I mean, I hope that we won't have an issue with the release environment. So everything should be fine by tomorrow. Just, yeah, just related to this, please don't update Jenkins instances tomorrow. So we don't introduce potential noise. So the next topic is about analytics. So giving start a discussion on discourse about how we, I mean, if we could use Cook and Analytics or something else for plugins that Jenkins or IO, we explore two options, plausible and immutable. So the first one we just because I'm already using plausible for personal projects, but I have the smallest subscription. We just did an experiment for plugin sites. So yeah, we just have access to basic information. We just stop the experiment because we consume all my credits in just several hours. And so my account is not big enough to cover plugins such as IO. And so I don't want to pay right now. So without to explore, I just want to be sure that we want to use plausible or matemobile for paying or finding sponsors. So if you have an experience here, that would be that, I mean, if you're free to share, I think the discussion can continue on discourse. It's in the infrag category. And what I like, sorry, yeah, Olivier, what we're seeing here is a is a website website statistics report that was generated for plugins that using plausible. So you registered Jenkins.io with plausible. It started collecting data. And this is this is what you got that quickly. Yes. So it's quite simple because it's just a small JavaScript that you have to share in your websites. And so you don't allow them that small JavaScript send information to plausible. So this view is public. So just go to plausible.io slash plug in searching. You see a small peak because if you just look at what happened since seven days, you have a small peak because given enable that service yesterday, this time and I disabled this morning when I realized that I use 2000 22,000 page views and I have a limit on 10,000. So that's why I disabled it for now. But you had a kind of information that we have this page visited sources countries. So it's really basic information. It's not the kind of information that gave in was looking for because giving was more interested about understanding the behavior of users on the website. We could use Google Analytics, but I would like to avoid if possible. So I think that giving as an action attempt to explore Matomoto, which is another option here, but I don't have any experience with this one Matomo sounds more like Google Analytics replacement. So yeah, if you have any insights, feel free to share my only concern. So yeah, in both cases, plausible and Matomo are open source. So we could deploy the services ourselves. But my concern is in first that kind of service, you know, they're to have them useful for the project, you want to create data on a long term. And so that means that we will have to be sure that we we have to be sure that we back up data. And so that would just put extra pressure on the Jenkins software team. And we already have a lot of things to do. So that's why I'm not really confident to host the service ourselves. And also more importantly, as soon as you provide a service with analytics, if the data are incomplete, people take the wrong decision. I mean, people take the wrong conclusion based on the data. So I'm always feeling uncomfortable if I cannot share a service that it's 100% reliable, when you want to collect data and analyze them. So that's why I'm the tree. Yeah, I would like to do it to explore the sponsoring or the beta approach. So that's a small project. Another, the next topic is about migrating archive to Jenkins.io. So we discovered last week that Rackspace stopped sponsoring the project two months ago. And so now we receive invoices. Those invoices were sent to KK's email address. And if I get to share them with us, so we'll have to pay the invoices. So at the moment is around 600 per month. So we have to pay and we have a third coming. So the idea is to migrate as soon as possible. So we are now exploring. So archive is a copy of another machine. So we could deploy that on Amazon, but I would like to have it deployed on another cloud provider. So we have some redundancy there. So we have either the option to deploy in Azure, but we are already trying to decrease the cost of our Azure accounts. So I would like to do that option. And so we have two options here, either deploying that on our Oracle clouds or on Scaleway. I mean, those two accounts offers low cost infrastructure. So that's what we are exploring right now, because Devian is not available at the moment and he has access to the Scaleway because that's a new project that we are exploring there. And we'd be more interested to look at Oracle this week. But yeah, migrating archives at Jenkins.io to a different location is quite simple, because that machine is perpetized. So we just have to be sure that the data is transferred from one location to the other one, which is the one time our sync commands. So that's the current state for archive. Otherwise, I don't really have any other topic. Last week was pretty busy with Silicon and Contributo Summit. So we reduce our time here during that time. So do you want to break any topic? Mark. So Aditya had paired with Damian on some efforts last week. You and I have a plan to do work on archives.jankens.io. Are there sensitive things there that we could we offer to let Aditya pair with us as well on that one? There is nothing critical. I mean, we don't have secrets, I would say. So we won't share a secret in the video. So I mean, everybody is free to join. After the budget. So Aditya, if you're interested, I'll include you in the invitation for our session tomorrow. I think it will be more difficult. Yeah, no, yeah. No, maybe that's possible. Sorry, what do you think may be more difficult? No, I was just thinking in time, but at the time it's fine for Aditya, I think. Oh, well, it's as bad as this one. It's a terrible time for Aditya because it's late in his evening. But it will be right now, Aditya, your time, I assume it's roughly 10 p.m. 10.30 p.m. No, it's roughly 9.30 p.m. Okay, so it would be about 7.30 p.m. tomorrow, your time that we would be. I'll include you on the invitation. You're welcome to attend. If you can't attend, don't worry. That's perfectly okay if you're not available. Just for me, it's great to have other people involved as we're doing these things to help us and assess what we're doing. Just for information, if you want to move it later, feel free to ask and we'll see if we can do that. I think I will be available, so yeah, I would be happy to join. Okay, great. Is it okay if I suggest something? Can we please go back to the docs, actually? Sure. So, when you brought up the topic for plausible and Matomo, you were referring that someone, I forgot the name, actually, they were looking for something like user analytics and user behavior. And we would like to avoid Google analytics. So there's this another tool I'm aware of that's called awstart. If you want to again link, they'll give you the link as well. Yes, I'm putting the link in the chat. Only bad part about it, I see is that UI is very old fashion. So the UI part is not great, but it gives a lot of data about number of visitors, unique visitors, their duration, and rush hours and etc. So that might be something interesting for someone to look into. And I think it was open source, so we can build on it as well. Interesting. Thanks for sharing. I was not aware of that too. Sure. I look at it. I was not aware. Thanks for sharing. There is one last topic that I can bring. I'm not sure if I shared that information during the last infrastructure meeting, but I made some changes to the status page. So I'm going to share them again. Sorry if I shared them two weeks ago. So what I changed in the status page is I removed all the iframe. So it's way faster now to display this page when you go to status. But instead, what I did is I added two different tabs. The first one is services.  But at the moment, we list three services, and the plan is to add more. So we just have to work on the documentation. But if you click on a specific get-a-jink, and let's get-a-jink scenario, we now have access to a description of the service. The iframe is loaded here. Because we only load one iframe, it's just faster. We don't have to wait five minutes to load the full page. And then we put some links to various locations. So in this case, it's get-a-jink-in-zalio. So I'm providing links to the ham charts, to the service, and so on. And so if you're curious about what this look page looks like, you just go to the get-repeatory status page.  Let's show the review. And it's just an ad hoc file with a few parameters. So we just provide a parameter to the file, and that will render. So we have the title of the service. You have description, the monitor views, if you want to let's say include an iframe. So we have a title for the iframe, and so on. So we use the Go templating to generate the page. And so the plan is to add more. So we have right now one for get-a-jink-in-zalio. And the idea is to provide, let's say, for the main website, jink-in-zalio, plugins, and so on. So that's how it changes the status page. I'm currently working on upgrading CS states. So it just changed the way we visualize we have access to the time. And that should clarify. And so if I go back to the root of the service, I also provided another view, which is monitoring information. And so from here, we also have links to other pages. And so, for instance, there is links now to Datadoc directly. So there is one view for every website response time, HTTP response time. And so the plan is to just provide the links to the different public dashboard that we have on Datadoc, PedroDuty, or whatever. I'm just trying to revisit how we deploy and manage the status page. I know that we are putting a lot more information about the different services on the status page, because, yeah, the goal of a status page is just to tell you if a service is done or not. The goal here is, my goal here is to provide enough information for every services so people can look into the different location, understand what's wrong, and maybe provide some, yeah, understand what's wrong, basically. So right now we only have to get the Jenkins that I own, package the Jenkins that I owe, and update the Jenkins that I owe. But the plan is to get, so it has to have more. That was the last point. Any other suggestions? Otherwise, I propose that we close the meeting here. None from me. Thanks for showing status.Jenkins.io and marvelous work. I love it. Yeah, I'm really happy with that. So that's working very well. So then thanks for your time, everybody, and I'll see you tomorrow during the work with archeology.Jenkins.io and otherwise, see you on our scene. Goodbye. Thanks. Bye-bye. Thank you.


Should lekkie a pet tour gate reopen many people have their own opinions on this but personally I believe there's no right or wrong answer however the fact remains lekkie concession company LCC is a private limited entity with investors and loans that needs to be paid back and the only way they can do this is for them to charge the public don't get me wrong we should never forget the shooting that had that took place at the tour gate but we must also have said this issue carefully and logically lekkie concession company LCC as mentioned is a private entity in charge of the day-to-day operation development and maintenance of the road in 2006 LCC won the concession to construct and enlarge the 49 kilometers road and also the construction of the 20-kilometer coastal road the agreement was based on a build operator and transfer agreement on the PPP private public partnership this contract would expand 30 years before LCC had to hand it back to Lego State LCC will recoup their investment back by building three tour gates on the 49 kilometers road but for LCC to achieve this incredible development they had to borrow monies from both local and international banks however in August 2013 after 35 billion Naira was spent LCC lost the concession due to an increase in tour gate fees caused by an increase in exchange rates due to the public outcry ex-governor Babatini fashion I had no choice to cancel the PPP PPP contract which led to Lego State buying back the contract a new agreement now meant that LCC would be owned by Lego State but managed as a private entity with this information I failed to understand or see why people would would think Ashwajibola Ahmed Tanubu owns LCC because Ashwajib was not the governor at the time the LCC contract was signed nor was he the governor when the contract was cancelled why did the protestor decide on the tour gate one would say the protestors wanted to be heard so they decided to to besiege a few public facility that will bring them maximum public awareness a few days into the protest there was a change in their demand from one topic into several others and one of their sole focus was their misinformation about their ownership of LCC and the Oriental Hotel as mentioned above Ashwajib and the owners of these two businesses came out in public to state that these rumors were false one could say that the protest could have been influenced or hijacked by others with their own personal political agenda should LCC start operating again sooner or later people must understand that the that the leaky a pair road and the Iqoi leaky bridge would reopen because directly or indirectly LCC had no control over what happened during the ensar's protest nor did they have control over the police or the military however LCC as a business should have done more to handle the situation better after the event instead of hiding behind Lagos state by them not taking their own stand people see no difference between LCC and Lagos state which is the biggest problem now Nigerians must also understand that LCC cannot ignore its liabilities and debt obligation and neither can we as of today LCC owns local lenders over 11 billion narrow and foreign lenders 31 million dollars because LCC is so business is the maintenance of the road and the redevelopment of other areas if they cannot charge for the use of the roads and bridge how would they pay back the existing loan and continue the much needed redevelopment in the leaky area and we must also understand that if loans like these are not payback this could affect us all in the long run as cost of borrowing would increase and Nigerians businesses as a whole will be seen as high-risk customers well well said I also have share same sentiment the fights or the big disagreement I will know is what fight I was there actually is between we the young persons of Nigeria and the government who are trying to make a statement to them we have made our statement is and the government have had us they are not deft they are not they've had us it's not left for them to do the right thing now about LCC it's true because you know we do something we should not just do something for the sake of our parochial reasons we should also think of our image as a nation let me give you an example when you banned Twitter because you wanted to punish the youth you didn't think that they will see us as a high-risk country in doing business and no freedom of speech you didn't think that you were just thinking that ok is the Nigeria is it's me and the youth we are not thinking about the long-term effect number two when you say don't open LCC don't open the toolkit you didn't think because you think you want to punish the government you didn't think that these are private investors some foreign some domestic you didn't think that some of these people would channel their resources next time to neighboring countries and then we see why going to get countries like Ghana where we are also causing our problem so we should not shoot our safe in the leg indirectly Nigerians were all worse enemies we do things without thinking so let me bring up something so way before this Ansar's stuff right a majority of people in Lagos still have issues with the lucky tour gate or and the lucky express access that lucky access because there was meant to be a full alternative route yes there to date it's still no full alternative route so that is also part of the issue you're forcing people to go through a toll and pay yet you have not provided the alternative route so that's despite of Ansar so people are forgetting people are already upset with LCC and legal state on that point okay the the other thing is also how they did the road there should be some bypass some overhead driving bridges that can accommodate people who are not going to the local areas but maybe going to Haja or to reduce the traffic right now it's still only one road and those are things that were meant to be done so according to their initial mandate they have not fulfilled their initial mandate so that's a beef in itself now when you look at the Ansar's debacle and the loss of life there's an emotive factor it's it's it's people are fired up people are angry people are resentful because generally innocent people died generally the military and police did not do what they were meant to do what they are meant to do serve and protect though there were political biases political things that were going on and people got caught in the middle LCC knowing that their private company whoever's in charge of their marketing and communications honestly terrible terrible terrible terrible they got lazy complacent and they hid what if I was with LCC what we would have done is do a lot of on-ground work away from the state government separate from the state government you have not marked any memorial for the loss of these people who died where your infrastructure is you were not to blame it was in no way was it LCC's fault what happened because they were taken by taken by by surprise exactly but there was nothing putting place to memorialize these people there was nothing put in place to sensitize these people to do some proper you know image work and ground work to get the youth on sign you know to understand and bring them into the conversation I mean so that is why because they didn't do that what to separate themselves from legal state in people's mind that's the problem because if if they were bored enough or so like like you said and I wrote us also is that they should have separated itself by separating themselves then they're basically saying do you know what with you protestors we were not with you we're not with legal state this is our business this is our livelihood what has happened has happened the less less acknowledge it but however we still need to move and they still angered some I want to hear what I as means but they've angered people in lacking lacking residents association because I think they came up with a statement saying they they're opening this is LCC and that they've got the permission of lekkie residents yes lekkie residents to say when did we give you permission when did you we have this meeting so they completely allegedly completely kind of lied and went over the lekkie rest over to you the big contact you guys so I admire thank you guys it's a pure disaster an absolute disaster they don't have any grounds to stand on as it is we need to go back to the table we need to draft this thing and begin to think about complicating the lives that were lost it's not so much about opening that place and trying to be the date alone how much are you going to place on the lives of the people that were lost there there's no amount of money you're going to give the family that's going to go back what was your stand when that happened you did absolutely nothing now it's okay we make mistakes but we must always know when to track our steps and do the right thing as LCC done the right thing so far have they issued a statement to commentate with the family have they done any campaign to identify the issues that were discussed what is their stand on the panels that have been opened in legal state have they been following up with the media they have done absolutely nothing so on what grounds do they want to come back and open that place I'm an accountant so I understand the figures but I also understand that good will is what they need to work that business they don't have the good will how far are you going to go without good will they're going to open that place and something really about the game she's right no you're right because because like is is is the Nigerian disease when something goes wrong we know what to do but we don't do it because there's no there's no accountability and then in this in this situation also LCC hid behind legal states which is a mistake which is which is a big mistake what are they doing now now they should have done I mean whoever is in charge of the PR they're completely useless because they should have taken they should have taken the bull by their horn and everything they should have consulted even if you don't want to talk to the leaky people you can put something in the paper where we are we know that this has happened we are and everything pull it plack them let people know your customer from time to time make people passing that route understand that you know the people that died you're recognizing that we lost some people not a known can turn ties around for LCC you don't have to put a permanent structure you can just say okay every 20th of the year or every 20th of every month we'll just run a 16 seconds campaign of the base of the people that died in the inside people hashtag enter is still alive yeah hashtag we hear you you don't need a scientist a neurologist to tell you that you need to rebuild your brand that means I mean like I said I mean like I said there's no right answer or no wrong answer but the way they should consider look at it from a business point of view the public are your customers the public are the one that we use your facility so consider even if you don't like them consider them up next is Elijah please stay with us


Archmour has told me about this, but thank you all for that wonderful mountaineer welcome. It's great to be in West Virginia, and I'm delighted, I'm delighted to visit your proud town of Parkersburg. Two great American sports are taking place this fall, politics and football. So before I go any further, let me say hello to some great teams, the Williamstown High School Yellow Jackets, the Parkersburg South Patriots, yes, the Parkersburg Big Reds, yes, one more thing, how about them ears? Warm welcome to your Mayor Pat Pappas, greetings to my good friend and your once and your future very great Governor Archmour. I'd like nothing more than to work with Arch so we can see to it that West Virginia shares in the prosperity that is sweeping across America. We were governors together and I know that leadership means more, or he knows that leadership means more than just raising taxes, and special regards to your outstanding candidates for the House of Representatives, Jim Altmyron, Cleave Benedict, and you've got a candidate for Senate who is a member of the Republican Opportunity Team could do a lot for West Virginia. I ask you a favor for myself and for the people of West Virginia on election day, vote against high taxes, vote for creative jobs programs, vote for growth and opportunity by sending John Racy to the United States Senate. The Senate is one key to continuing what we've begun. West Virginia has a fine tradition for rising above partisanship when the national interest calls. A tradition that's well represented by your retiring senator, Jennings Randolph. John Racy has the qualities of leadership necessary to continue that tradition and to do what's best and what's right for America. I need, and I think your state needs, if you'll permit me, John Racy in the United States Senate. Jim, Cleave, and John are great candidates and they'll make even better representatives in Washington, so please send them there. You know, I'm always happy when I visit this part of the country because maybe because you make me feel so much at home. You in West Virginia have always given our country more than your share of greatness and courage. Here is the steadiness of purpose, the fidelity to ideals, the love of country, and I want you to know that George Bush and I not only believe West Virginia is worth visiting and worth listening to, we believe West Virginia is worth fighting for. Abe Lincoln said, we must disenthrall ourselves with the past and then we will save our country. Well, four years ago that's what we did. We made a great turn. We got out from under the thrall of a government which we had hoped would make our lives better, but which wound up trying to live our lives for us. The power of the federal government had over the decades created great chaos, economic, social, and international. Our leaders were adrift, rudderless, without a compass. Four years ago we began to navigate by certain fixed principles. Our North Star was freedom. Our common sense was our constellations. We knew that economic freedom meant paying less of a family's earnings to government and so we cut personal income tax rates by 25 percent and contrary to what my opponent is saying, we cut them from top to bottom evenly across the board. We knew that inflation, the quiet thief, was stealing our savings and the highest interest rates since the civil war were making it impossible for people to own a home or to start an enterprise. We knew that our national military defense had been weakened so we decided to rebuild and be strong again and this we knew would enhance the prospects for peace in the world. It was a second American revolution and it's only just begun but America is back. America is a giant once again powerful in its renewed spirit, its growing economy, powerful in its ability to defend itself and secure the peace and powerful in its ability to build a better future. And you know something? That's not debatable. Yet four years after our efforts began small voices in the night are sounding the call to go back, go back to the days of drift, the days of torpor, timidity, and taxes. My opponent this year is known to you but perhaps we can gain a greater insight into the world he would take us back to if we take a look at his record. His understanding of economics is well demonstrated by his predictions just before we took office. He said our economic program is quote obviously murderously inflationary. Now that was just before we lowered inflation from above 12 percent down to around four and just after just after our tax cuts he said the most that he could see was an anemic recovery. Well that was right before the United States economy created more than six million new jobs in 21 months and and there have been 900,000 new business incorporations in just the last 18 months. My opponent said that our policies would deliver a misery index the likes of which we haven't seen in a long time. Now there was some truth when he said that. You know you get the misery index by adding the rate of unemployment and the rate of inflation. They invented that in the 1976 election and they invented it so that they could say Jerry Ford had no right to run for reelection because he had a misery index of 12.6. Well now they didn't mention the misery index in the election of 1980 probably because it was over 20 and they're not talking about it much in this campaign because it's down around 11. You know my opponent said that decontrol of oil prices would cost American consumers more than 36 billion dollars a year. Well one of the first things we did was decontrol oil prices and the price of gasoline went down eight cents a gallon. I figured out that maybe all we have to do to get the economy in absolutely perfect shape is to get my opponent to predict absolute disaster. He says he cares about the middle class but he boasts quote I have consistently supported legislation time after time which increases taxes on my own constituents. Doesn't that make you want to be one of his constituents? He's no doubt proud of the fact that as a United States senator he voted 16 times to increase your taxes. Now this year he's out done himself. He's already promised of course to raise your taxes but if he is to keep all the promises that he has made during this campaign he will have to raise taxes by the equivalent of $1,890 for every household in the United States. Now that's more than $150 a month. It's like having a second mortgage, a Mondale mortgage. His basic plan has two parts. One raise your taxes and the second raise them again. You know I've got news for him. The American people don't want his tax increases and they're not going to get his tax increases. His tax plan would bring this recovery of ours to a roaring halt because one of the key most significant things in the recovery we're having was the tax cut. I'm now beginning to see more clearly why last week my opponent said let's forget about the past. It's not the past that worries me. It's the future and I'm even more concerned because now last week my opponent used another familiar phrase from the past. Trust me he said. Well the last time we trusted the administration of which he was apart they took five. You can count them. Five different economic plans and nearly tripled inflation and by contrast we trusted the people. You and with just one economic program we've cut inflation by two-thirds. If my opponent's campaign were a television show it would be let's make a deal. You know you would trade your prosperity for his surprise behind the curtain and if his campaign were a Broadway show it would be called promises promises and if his administration were a novel a book you'd have to read it from back to front to get a happy ending. He sees in America in which every day is tax day April 15th we see in America in which every day is independence day the 4th of July but seriously we want to lower your taxes and everybody's taxes so your families will be stronger our economy will be stronger and America will be stronger and I'm proud to say on another subject during these last four years not one square inch of territory in the world has been lost to communist aggression and the United States is more secure than it was four years ago. My opponent sees a different world. Sometime back he said the old days of a soviet strategy of suppression by force are over that was just before the soviets invaded Czechoslovakia and after they invaded Afghanistan he said it just baffles me why the soviets these last few years have behaved as they have but then there's so much that baffles him. Just one year ago we liberated Grenada from communist thugs who had taken over that country but my opponent called what we did a violation of international law that erodes our moral authority to criticize the soviets. Well there's nothing immoral about rescuing American students whose lives are in danger by the time my opponent decided that action was justified the students were home a long time after the Sandinista revolution in Nicaragua he praised it saying winds of democratic progress are stirring where they have long been stifled but we know that the Sandinistas immediately began to persecute the general genuine believers in democracy and to export terror they went on to slaughter the mosquito indians in their country abuse and deport church leaders slander the pope practice anti-semitism and move to kill free speech don't you think it's about time that my opponent stood up and spoke out and condemned the Sandinistas crimes more recently he failed to repudiate the Reverend Jesse Jackson when Jesse Jackson went to avena stood with Fidel Castro and said long live President Fidel Castro and Che Guevara but let me try to put this in perspective the 1984 election isn't just a partisan contest I was a democrat for a good share of my life the majority share but in those days democratic leaders weren't the members of the blame the america first crowd its leaders were men like Harry Truman men who understood the challenges of the times they didn't reserve all their indignation for the united states they knew the difference between freedom and tyranny and they stood up for the one and damned the other to all the good democrats and I hope there are many presidents who respect that tradition I say you are not alone well I know that across this country there are millions of good patriotic democrats who found in their hearts they can no longer follow the leadership of that party today well we're asking you to come we're asking you to come and walk with us down the path of hope of hope and opportunity believe me together and it can be a bipartisan task that we face to make america strong and keep it strong once again we need you and you're certainly welcome you talk me into it you know this this month an american woman walked in space Catherine Sullivan made history and then she returned after that walk to the space shuttle in which some of the great scientific and medical advances of the future will be made cures of for diabetes and heart disease may be possible up there steps have already been taken to to find that out experiments and technology and communication but my opponent led the fight in the united states senate against the entire shuttle program didn't want it to even begin and called it a horrible waste well we support the space shuttle and we've committed america to meet a great challenge to build a permanently manned space station and to do it within a decade now i've been probably going on too long here but that isn't what you meant by four more years well you know the point is we were right when we made our big turn in 1980 we were right to take command of the ship stop its aimless drift and get moving again and and we were right we were right when we stopped sending out sos and started saying usa you know the united states was never meant to be a second best nation like our like our olympic athletes this nation should set its sights on the stars and go for the gold now if america could bring down inflation from 12.4 percent to four then we can bring inflation down from four to zero point zero and we're going to do it if lowering your tax rates led to the best expansion we've had in 30 years and it was the key as i said then we can lower them again and keep america growing into the 21st century if we could create if if we could create six million new jobs in 21 months and then make it possible for every american young old black or white who wants a job to find a job if our states and municipalities can establish enterprise zones to create economic growth then we can elect people to congress who will free our enterprise zones the federal bill that has been buried in a committee in the house of representatives for two years buried there by speaker tip o'neill and bring it out to the floor for a vote so we can provide hope and opportunity for the most distressed areas of america whether they be in the cities or rural if if we can lead a revolution in technology and push back the frontiers of space then we can provide our workers in industries old and new all that they need which is mainly new and proper tools because i believe with all my heart the american worker given the proper tools can outsell anybody can out compete can out produce anybody anywhere in the world anytime if our grassroots drive to restore excellence in education could reverse a 20 year decline in scholastic aptitude test scores and it has then we can keep raising those scores and restore american academic excellence second to none if our crackdown on crime could produce the sharpest drop ever in the crime index and it has then we can keep cracking down until our families and friends can walk our streets again without being afraid if we could reverse the decline in our military defenses and restore respect for america and we have then we can make sure that this nation remains strong enough to protect freedom and peace for us for our children and for our children's children and if we make sure that america remains strong and prepared for peace then we can begin to reduce nuclear weapons and one day banish them from the earth entirely i've been interested in some of the scare tactics that my opponent has been using i've had the privilege in these four years of addressing the parliaments in several countries both in asia and in the united states and in every instance i took the opportunity to tell them that a nuclear war cannot be won and must never be fought and yes a nuclear freeze makes strength if it follows negotiations that reduce the weapons in the world down to a verifiable and equal level on both sides and if we can strengthen our economy strengthen our security and strengthen the values that bind us together then america will become a nation even greater in art and learning and greater in the love and worship of the god who made us and who has blessed us here more than any other people on earth have ever been blessed but now a week ago sunday yesterday a week ago yesterday i didn't get to finish my closing remarks in the debate i ran out of time so i'm going to say it here to you the young people of this country let me say that you you and your future are what this election is all about your generation is something special your idealism and your love of country are unsurpassed i have seen this in meetings of this kind all across the country in this campaign it's our highest duty to make certain that you have an america every bit as full of opportunity hope confidence and dreams as we had when we were your age you know my generation and then there are a few generations between mine and yours but most of us grew up in a land where we took it for granted that we could dream the highest and finest of dreams and if we were willing to try and work we could make them come true that there was no ceiling no limit on what we could accomplish or how high we could fly by our own strength and ability and then we came to a period when some people began telling us that no there was an era of limits that things could never have been again be as good as they once were and that you had to kind of give up some of your dreams for the future well i'll tell you me and those my generation i should say and those other generations i mentioned here not yours we have a sacred trust and i think we're going to fulfill that trust and that is when the time comes to turn the reins over to you young people out there we're going to turn over to you and america that is free in a world that is at peace and your dreams can come true all of us together are part of a great revolution and it's only just begun america will never give up its special mission in the world never they're new worlds on our horizon and we're not going to stop until we all get there together america's best days are yet to come i spoke earlier about sending these wonderful candidates and office holders back to their jobs you know the job that arch belongs in and you know the jobs for these but i have been honored greatly by all of you in your allowing me to serve you for these past four years and i ask for your vote i ask for your support i want to keep on with the job we started four years ago with that new beginning i wanted more than anything i've ever wanted extreme pride that i on behalf of the state of west virginia west virginia university present to you this hat which a bit of memorabilia from west virginia to savor our victory over pin state and we'll savor your victory this election day