Okay. Yeah, thanks. So I will, yeah, so I will get started with the second lectures. Just share this screen. Okay, so, so the goal of. So in today's lecture, I just want to sort of continue with the general algebraic theory of quadratic forms over fields fields of characteristic not to sort of isomorphisms between them and we're, you know, we're going to use some of these tools later in the in the course to study invariance of quadratic forms on housing and Kowski theorem. And so what I want to start off today with is the following is a really fundamental theorem due to bit. And so the theorem is the following. Right, so I guess I should say, well throughout the lecture, we're going to fix a field F where the characteristic is not equal to two. Okay, so the theorem is the following. So, let V, comma W, comma Z be quadratic forms over F. Sorry, so this theorem is due to fit. It's called the cancellation theorem that's cancellation theorem. And the theorem is that, well, let's suppose that V direct some Z is isomorphic to W direct some C. So as quadratic spaces. And then the conclusion is that V is just isomorphic doubly. So it's saying that you can, you can sort of cancel isomorphisms. So if two things become isomorphic after you add the same some and then they're actually isomorphic to begin with. And this is the type of result which is generally not true like, you know, in, there are lots of mathematical objects where you can do this kind of thing you know you want to study whether you want to be or isomorphic. And there's a weaker question which is, well what happens if a plus C is isomorphic to B plus C. So that's some sort of stabilized isomorphism question. And usually the stabilized isomorphism question for various reasons is a lot easier to understand and stabilize morphism process of most types of mathematical objects are much easier to understand than isomorphism classes. So, then this theorem is kind of remarkable which is saying that in the setting of quadratic forms, there's sort of no difference between these two concepts. Right. And so just also a caveat is that when I say isomorphic I mean there exists an isomorphism so that the isomorphisms that you have here are not necessarily going to be related or not going to be related. Right. So, so what's sort of special, you know, so what makes us work for quadratic forms that doesn't work for, you know, for other costs of mathematical objects. It's that quadratic forms are as as mentioned last time they a key feature of quadratic forms is that they, they have lots of symmetries. So with that in mind let's, let's prove the theorem. Okay, so first of all, so Z is diagonalizable. So as we saw last time. That means it's a sum of one dimensional forms. So by induction. Well, by induction induction on the dimension of Z we can assume that the Z itself is one dimensional. So we can assume the Z is the one dimensional form brackets a. So, in other words, spanned by a vector whose product with itself is equal to a. So, suppose, an isomorphism from be direct some brackets a. So it's called as isomorphism T to W direct some brackets a. And what we want to do is we want to cook up an isomorphism from B to W. So to do this let me let me give these vectors name so let's let's say you one is a basis element in brackets over here, and F one is the basis element in brackets a over here. So the basic problem is that. So, in general, T is some isomorphism like this. And in general it's not true that T of E one is equal to F one. So if T of E one were equal to F one, then you could say well you've got an isomorphism that takes E one to F one. And therefore it takes orthogonal complement of E one to the orthogonal complement of F one. And that would just be saying that T carries begin to W. So if T of E one equals F one, or in fact you don't even need TV one to be equal to F one for example it could be minus F one for this argument to work. Then T carries. V which is. So, which is you want sorry. E one perp into W, which is F one perp and induces the isomorphism we want. So the problem is that you have some isomorphism but it doesn't actually direct respect the direct some decomposition so it doesn't have to carry E one into F one. So so that's that's that's well in general in this type of thing that's that's why stable isomorphism is going to be a weaker relation than isomorphism. But so what sort of saves us in this in this quadratic form setting is that you have lots of isomorphisms and even though T of E one is not equal to F one what you can always do is you can pre compose a post compose with some other automorphism to make this to make this happen. So, what you can do is that observe now that T of E one and F one so we can think of those as vectors that they live in the vector space w direct some brackets a. And this is times F one are two vectors of the same length or the same length squared. And now there's this, this wonderful lemma that well I guess it was it was in the exercises from last time, but I want to sort of stated over here. So whenever you have two vectors with this property to that person in quadratic space such that they have the same lens with the same self product, then they're carried. There's an automorphism of that quadratic space that carries one to another. So, so let me state this as a proposition I'll prove the proposition but first I will explain how the proposition applies theorem. So proposition is that if V comma q is any quadratic space, then the orthogonal group of V comma q acts transitively on the set of all vectors V and V such that V dot V is equal to a for any a not zero in the field F. So, so you have lots of you have lots of symmetries of any product space in fact you can go from I mean sorry so I mean a basic example of this is if you have like the Euclidean space and then this is saying that your orthogonal group of the Euclidean space is going to act transitively on the n minus one sphere. So this is a generalization of that to any quadratic space. So I'll prove the proposition a second but now let me explain why the theorem follows. So, apply. So apply the proposition to the vector space V. Sorry, not to W direct some brackets a contain an automorphism you which lives in the orthogonal group of W direct some brackets a such that you have T of E one is equal to F one. And then you observe that you composed with T is an isomorphism from the direct some brackets a to W direct some brackets a carries E one into F one and hence induces an isomorphism from the into W. So that's that person here. Okay. So once you have enough automorphisms to act transitively on these generalized fears then that proves the result. Okay, so now let me explain the proof of this proposition so this again this was on the the exercises for for yesterday. And the idea is that you're going to use a reflection, the idea is that you're going to you're going to use a suitable reflection to go between any two vectors except you might have to multiply by plus or minus one as well. Okay. So for proof, let's suppose I have two vectors so suppose the common W are our vectors in this quadratic space, such that V dot V is equal to W dot W, which is equal to a, a some elements of my, my ground field which is not zero. And so what we want is some, some element T and your talk in all group of this quadratic space such that T of V is equal to W. And so what we're going to do is we're going to consider the reflection reflections through V plus or minus W. So, consider the following two vectors. So consider V dot, sorry. Consider the vectors V plus or minus W. And the, the observation is that because V dot V is equal to W dot W. We have these two vectors are orthogonal so V plus W dot with V minus W is equal to zero so V plus W is orthogonal to V minus W. So as a consequence, either, well if you have two orthogonal vectors, and they're both isotropic, then their sum is also isotropic. So, so as a consequence, either V plus W or V minus W must be an isotropic. So if, if they were both isotropic then V itself would be isotropic, or to V would be isotropic which, and let me suppose for simplicity that V minus W is an isotropic. So I'll explain how to handle the other case. So then, consider the reflection through the vector V minus W so you're allowed to reflect well so you're allowed to reflect through any vector which is an isotropic. It has to be an isotropic because they're dividing by that vector data itself, and observe that this reflection through V minus W will sort of by construction carries V minus W to W minus V. So it's the identity on the line through the minus W, sorry, it's minus one on the line through the minus W, and it's the identity on the orthogonal complement so carries W carries V plus W to V plus W. And hence, it carries V into W, as desired. So, so this work, so this reflection works great, so I could have even just started with this comment. So if, if V minus W is anisotropic, so then you just take the reflection through it and then then you have your final transformation. If, if not you have to work a little bit harder, but if V minus W is an isotropic is isotropic, then V plus W is an isotropic. So if V minus W is isotropic, then V plus W is an isotropic. And you compose the reflection with minus one. So if V minus W is an isotropic, then V plus W, then you have to add an extra sign at the end, but that's fine, because the sign is also the orthogonal group, it's multiplying by a scale. Okay, so that's improved with the proposition and then that, which we're just telling you that you have this transit of action and then that's also the proof of its cancellation. So, so if it's cancellation theorem, and also a sequel which I'll explain a little bit later, is a really powerful tool for trying to understand isomorphism classes of quadratic forms and it helps. And in particular one of the key features that it has is that it lets you write quadratic forms in sort of a normal form. So, right, so if it's theorem is fundamental, because it enables one to write quadratic forms and sort of a normal form. And it also means that, so in general, like, so as I mentioned, so studying stable isomorphism classes of objects when you consider isomorphism systems after you've added or direct something on some other other term is a lot easier. And that's because stable isomorphism classes of objects can often be organized into a billion groups I mean this is sort of the starting point of, of K theory, K zero. And in the case of quadratic forms this leads to a structure called the growing date that ring which is going to come up next time. And so this is sort of like the commutative ring you can write it down and you can study it in various ways. And what that saying is that, well what that cancellation is saying is that you can actually encode the isomorphism class of quadratic forms. Okay, so, so we're going to come back to that. But also sorry first before I want to move by just also want to say just a comment about the exercises so I have to apologize so in. So in the exercises for yesterday.  And I also asked you to prove the cartoon do today theorem, which, which is that any, any element of the orthogonal group is a product of that most and reflections where and is the dimension. And so, well, you should definitely try to that's a theorem and it's, it's, it's completely true. It's still a little bit harder than what I intended to ask which is really just that it's generated by reflections. And so, so the fact that it's generated by reflections is something that you can prove directly from this proposition. There's, there's some more, more elaborate argument to show that you can take out most and reflection so so there's going to be in today's today's exercise set, which is already posted in SoCoCo. It sort of walks you through the argument to prove the stronger form of cartoon to do though in fact that's the precise quantity of form is actually not going to be used later but yeah. Okay. So that's the general theory of quadratic forms. And in particular I want to explain sort of a particularly fundamental example of a quadratic form which we sort of saw yesterday, but which is really central to the theory. And that's a hyperbolic plane. So hyperbolic forms. So the hyperbolic plane is the following quadratic form. So it's a quadratic. So it's going to specify by a symmetric matrix, and it's a symmetric matrix 0, 1, 1, 0. So it's just two dimensional quadratic form. And while you can think of it as follows it has a basis. So it's two dimensional has a basis given by vectors e1 e2 which are isotropic and such that e1 dot e2 is equal to one. So by the way e1 dot e2 equals one, you can replace one by any other scaler any other non zero scale is rescale e1 or e2 appropriately. But the key point is that it's generated by two isotropic vectors that are that have a non trivial inner product with respect to each other. Yeah, so by the way, you can also write this as a quadratic form one zero zero minus one is a quadratic form x squared minus y squared. Okay. So, so this is a hyperbolic plane. And the basic like so the one of the basic reasons this is so fundamental is the following proposition. So, if V is any, any quadratic form over F which is which is isotropic. So V be isotropic quadratic form. Then what you can always do is you can always split off a copy of the hyperbolic plane. So, so then V is, V is isomorphic to a hyperbolic plane direct some v prime for some smaller quadratic form v prime. Well I guess I should give the hyperbolic plan a name I'll call it H2. Whenever you have an isotropic form, you can always split off a copy of the hyperbolic plane and some smaller form. And in fact, so by the bit cancellation theorem, this v prime is actually determined up to isomorphism. So what's the proof. Whenever the top of the company may be assuming to be non is in the nature. Thank you. Yeah, so when I say quadratic forms I'm only going to consider non degenerate quadratic forms. And yeah, also the field is characteristic not not to. Thanks. Okay. So, so how do we prove this. Well, right.  And not zero. And then we're just going to sort of build a hyperbolic plane sitting inside of the so well. So by assumption or quadratic form is known to generate so we can find a vector w in the such that w dotted with fee is not zero.  And well so so w doesn't have to be isotropic, but basically you replace w with w minus lambda v for some lambda in the field f. And if you just make some shearing automorphism then, then, then you can normalize so that you have two isotropic factors such that their inner product is not zero. And then you've got a hyperbolic plane. So this gives a hyperbolic plane sitting inside v. And then you can take the prime to be the orthogonal complement. Right. So, so this is really useful because it's saying right so it's saying that whenever you have an isotropic form so me standard example but isotropic form is just a plane. And saying that whenever you have an isotropic form, you can always sort of peel off a copy of the hyperbolic plane and you're left with something smaller. In fact you can, you can just keep doing this you can keep doing this can just keep peeling off copies of the hyperbolic plane, until you're left with something and isotropic something with no non trivial zeros. And that's actually well by again by that cancellation is going to be determined up to some words. So a corollary is that any quadratic form is going to be isomorphic to a cop some some direct sum of copies of the hyperbolic plane. Plus, some other quadratic form B prime Q prime, where B prime Q prime is an isotropic, and it's determined up to some words. So again by that cancellation is determined up to isomorphism. So if you want to classify quadratic forms up to isomorphism over any fields. So this suffices to classify the anisotropic quadratic forms, because every quadratic form is some some cop some direct sum of copies of the hyperbolic plane plus something and I saw. There's a question in the chat. All right, so the question is what is an isotropic form. An isotropic form means a form that has a vector with zero, zero length. So, yeah, so let me just write that. So recall. If there exists some element of V some vector, some non zero vector with V dot equals zero, and it's an isotropic otherwise. So if your goal is to classify quadratic forms up to isomorphism which is, which is an important goal in the subject, then it really suffices to work with an isotropic forms. And another observation is that basically, like, hyperbolic forms look the same over any field, and for any question involving them is very easy to answer. So it's really the anisotropic forms that have that sort of interesting to understand. Okay, so this is sort of the normal form of quadratic forms that I'm saluting to that that you can always. Yeah, that that anisotropic forms are the building blocks and the ones that you want to sort of understand up to isomorphism. So, yeah, so another sort of observation. That's really a consequence of that's cancellation theorem in this type of machinery is sort of a general observation in the theory of quadratic forms. So, as I mentioned, like if you're studying quadratic forms there's sort of at least three natural questions that you can ask a quadratic forms over a given field. The first question is, you know, so when if I have two quadratic ones when are the isomorphic, what can I do to distinguish them, receive their isomorphic. The second question is when does a quadratic form represent zero so when is a quadratic form anisotropic. And the third question is, when do, when does a quadratic form represent an element of the field so when is there a vector, be with V dot be equal to some elements of the field. And turns out the really the most powerful question and all of this is the question of when is a quadratic form isotropic. So, the really fundamental question that you can ask over a given field is that is to have a tool for determining when a quadratic form is isotropic. And so, for example, this is the form that the, the Hasan Minkowski theorem states. So, fundamental question was a fundamental question I mean fundamental out of the three questions I mentioned. Okay, so. So, I guess what I mean by that is that suppose you have like an Oracle, you have an Oracle that will tell you so you're working over some field and you have you have an Oracle that will tell you when a given quadratic form is isotropic you handed a quadratic oracle tells you whether or not it's isotropic. Then, you can classify quadratic forms over the field sort of inductively, at least. So, in particular, sorry, so when I say the quote, quote, fundamental question I mean that this is a stronger question. Okay, so, so why is this really the fundamental question. So, suppose, become a queue well suppose you have some sort of Oracle or machine that will tell you when quadratic forms are isotropic. And now let's suppose that become a queue, and w comma q prime, or quadratic forms. And you want to know if they're isomorphic. So how do you do this. Well first you have your Oracle so you can ask our V and w isotropic or an isotropic. So if V and w are isotropic, then write them in the normal form. So they're hyperbolic planes plus an isotropic. And now if V and w are isomorphic then the anisotropic parts are isomorphic and the number of hyperbolic summands have to be isomorphic. So V is isomorphic to w if and only if the anisotropic parts are isomorphic. And the number of hyperbolic planes well okay so I guess you know the dimensions so if the anisotropic parts are isomorphic then they have to be isomorphic because hyperbolic parts have the same dimension. So you may as well assume that V and w are anisotropic if you want to have some sort of machine for determining when they're isomorphic. So this is nice sort of trick. Well, if you want V to be isomorphic to w, then any length squared and V so any, any, any scalar that's represented by V has to be represented by w as well. So pick V and V and let a to be little V dot V by construction V direct some minus a is isotropic. So question is w direct some minus a isotropic. So if not, then V is not isomorphic w. But if yes, what you do is you split off hyperbolic planes from the direct some minus a and w direct some minus a and ask if they're isomorphic and ask if the complements are isomorphic. And this is actually a, well, and then you sort of recursively continue this, this process, and this terminates because at each stage you're lowering the dimension I mean this is the complements are going to be dimension less than, less than V. So inductively determines determines when V and w are isomorphic. Yeah, so I guess maybe it won't spend too much time on this but I think that the rough takeaway is that really the most powerful question in this business is, is whether you can tell when a quadratic form is isotropic.  So for example, like, it's often easier to prove the quadratic forms are isomorphic, or to have a tool from showing that they're isomorphic then to prove whether a quadratic form is isotropic. And so often there's sort of two different forms of various results and the version involving when you can solve the equation when something is isotropic is strictly stronger statement. And that's because of this, this type of procedure. So there's a question. What does it say between inductively and when. Right, sorry, so it's saying you want to understand when V is isomorphic to w. And, well, V is isomorphic to double if V is isomorphic to w will then V direct some minus a is isotropic if and only if w direct some minus as isotropic. And then if and only if the anisotropic parts of V direct some minus a and w direct some minus a are isomorphic. So if you have reduced your question of when or to anisotropic forms isomorphic to a question about anisotropic forms in smaller dimension. Using this sort of procedure. What does it say after if and below. Okay, so, maybe instead of, well, maybe I can put this on the problem set or something to think about it more, but I really just want to work through it in. I just want to work through this in an example, and just really see how how this sort of helps us. And the goal, I just want to be really concrete here. The goal is, let's classify quadratic forms over a finite field. So, so maybe rather than trying to, you know, make this very precise and we can ask like what what what do we really mean by an algorithm and so forth. I just want to sort of illustrate this by saying, well, let's let's classify quadratic forms over a field. Application, well, the application of this idea. Let's classify quadratic forms over F over a finite field FQ. So we saw last time that you can classify quadratic forms over C they all the same if they're given dimension over our where we'll basically have the dimension and the signature. And so now we want to do that the other case that's that's simplest which is the case of a finite field. But I think it'll be sort of maybe fun and also relevant to point out that this these methods are actually more general than the case of a finite field. So they will apply in some other examples which maybe will not be used in the course but are quite interesting. And, well, more, so I'm actually going to try to formulate this a little bit more generally. So, what we're going to do is we're going to, well, kind of make the following definition, which is. So, given a field F. The U invariant of F is a dimension. It's a largest dimension of an anisotropic form over F or infinity if if there's no, there's no largest dimension so for example if you're over the real numbers and the U invariant is infinity because x one squared plus data dot plus x a million squared is still an isotropic. So in many fields once you have a quadratic form in enough variables that it automatically has a solution. And that's called that's called the U invariant. And so what we'll be interested in is so so the U invariant is I guess it's some sort of measure of the complexity of some form, at least as far as quadratic forms are concerned of the field. Because if you if your U invariant is small, then basically the classification of quadratic forms is somehow reduced to the classification in small dimensions because because once the dimension gets large enough, then you can sort of keep splitting off on hyperbolic hyperbolic forms. So, roughly, the smaller the U invariant, the easier it is to classify quadratic forms. And I think I put some stuff on about the U invariant on the on the exercises the U invariant is actually is actually connected to quite a bit of recent research in the, you know, in there in in the theory of quadratic forms. There are a lot of sort of open questions involving involving the U invariant. So we're not really going to need most of what we're not going to touch that but what I want to do is is is is just sort of me just uses a specific guide. I mean, if the U invariant is small then it's easier to classify quadratic forms will roughly because if you can show that lots of things are isotropic then it's easier to classify quadratic forms. And in particular, I want to start with the following theorem, which is that any quadratic form of dimension three or dimension at least three over fq is isotropic. So, the U invariant is is equal to two. Okay, so you have to check something which is that there is a binary anisotropic form. But, well, okay, so right, come back to that, but in particular once you have at least three variables there's a root. So once you have at least three variables and a quadratic form over over a finite field. There's always a solution. So this is a basic fact which is kind of why well it's going to tell us that the classification of quadratic forms is also very simple over a finite field. And in fact, this classification is going to work over any field where the U invariant is equal to three. I'm sorry, equal to two. Okay. So let me prove this theorem, let me prove that any quadratic form of dimension three over fq is isotropic. And, well, we can always diagonalize for quadratic form. So we can assume the form is brackets a comma b comma C where we have some non zero elements of the field fq. And so what we want is a non zero solution to the equation. A x squared plus b y squared plus C z squared equals zero. So what we want is an is a root of this quadratic form is an isotropic factor. And that means we want a non trivial solution of the equation x squared plus C y squared plus b y squared plus C z squared is equal to zero. Okay, so in fact what we're going to do is we're just going to take z equals one. So that, that what we want to solve is a x squared equals minus C minus b y squared. And now we're just going to make accounting arguments. So now we're going to make accounting argument. Well, we observe that the left hand side only depends on x and the right hand side only depends on why. So we have a function of x and a function of why. And we just want to show that their ranges overlap. So, left hand side depends only on x. Left hand side depends on why. And we want to show that the two functions have ranges that overlap. So observe that if you look at the collection of all numbers all numbers in fq of the form a x squared as x ranges over fq. This set has cardinality q plus one divided by two. So why is that? Well, you first let x range over all the units in fq and then squaring us two to one on the units of fq. So a x squared is x ranges over the units. It's going to have size q minus one over two. But then you also have zero. So then that adds one. And similarly, the collection of all minus C minus b y squared as y ranges over fq has size q plus one over two. And so that means there has to be an overlap because you have two subsets of fq that each of which is greater than, I mean, so they can't be destroyed because otherwise they we have more than pigeonhole principle. So by the pigeonhole principle, you can solve the equation. And so that tells you that you can you can find you can find an isotropic vector in your three dimensional quadratic space. Yeah, so this is kind of a fun argument. So in fact there's a that's what the univariant is at least two that doesn't prove explicitly that you can't have an anisotropic form of higher degree does it. Um, okay, yeah, thanks. That's, yeah, so I should, you're right, I should say a little bit more so. When we have the highest degree of an anisotropic form is one. Right, so I should, I should, you're right so. That's that you can construct one that is all that is anisotropic for the great to or something. Right. Right, so you can do that because not every element of fq is a square so let you, well, maybe I shouldn't call it you. Let me be an element of fq cross, which is not a square. So then x squared plus V. Sorry, x squared minus the y squared equals zero has no has no solution q equals to I'm always I'm only working in an odd in characteristic greater than two. Yeah. Yeah. So, right, so not every element is a square so you can write down a binary quadratic form that doesn't have any solutions to the un variant is at least two. But if you have a quadratic form in, you know, five variables then it has a subform of three variables. And so, so then, then you get an isotropic factor. So, yeah, yeah, thanks for putting that out. Okay, so there's a question. Right, so why is the size. Okay, so why is the size of ax squared that the collection of all ax squared as x ranges over fq. Why is that, why is that size exactly q plus one over two. Well, let's let's count. So first of all, let's count the cases where x is not zero. So if x is not zero, then the function of squaring is exactly two to one. I mean, the pre image of everything in the image is exactly two points, plus or minus. So so the non zero squares are our carnality q minus one over two. And then you have zero. And so if you add both of those, then you get q plus one or two. So, yeah, so then that also applies to the other case. Okay. Right, so. Yeah, so also, I'll stick around after the, please ask questions now as well but I will just stick around after this. Okay, so there's actually a more general so this is sort of a fun and direct argument but it did use a diagonalization of quadratic forms so it doesn't necessarily apply if you have forms of higher degree, but there's actually a wonderful results that there's something like this in for higher degree polynomial equations as well. And so this is something that you'll explore a little bit on the exercises, which is that. So more generally, finite fields are what's called C one, meaning if you have a homogeneous polynomial. So if you have a homogeneous polynomial p of x one through x n is homogeneous of degree D in n variables. And n is greater than D. So they're, they're more variables than the degree. Then, then there's a non trivial zero. So there's a there's a zero of this polynomial which is not just a zero vector. There's a non trivial zero in the finite fields. So any field that has this property is called called C one. And so actually, I think, so another example, I mean, there are other examples of so in particular C one field is something that has property that any like ternary quadratic form is is a topic. And there are many other important examples of C one field so it's a little bit on. And the problem set so something like the rational function field of the field of rational functions in one variable over the complex numbers. And yeah, so. Okay, so these are sort of particularly simple fields for many of these types of like comological and arithmetic questions if any solutions to equations. Right, sorry, so there's a question of why is it see one and what's the one. Well there's a version of this for this, there's a notion of CR field. Yes. Yes, so it's answered in the chat where you have. I'm just saying that you have more variables and we replace that by d to the k. So, so you need an even larger, they're weaker conditions where you need even more variables to have a root. Okay. Yeah, so, so for example, fq is, is C one but something like a Laurent series field fq Laurent series T is going to be C two. And there is sort of this for the hierarchy of fields that goes up and becomes sort of harder and harder to solve the equations over them. Okay, so I just have a few minutes so what I want to do is I want to explain how you can get the classification of quadratic forms over over a finite field or more generally over any field of where the un variant is is at most two so the binary forms right isotropic. So now, let's classify quadratic forms over fq, or really over any field of you invariant at most two. So, for example, this will apply to any C one field. So if you have a quadratic form and you want to understand what it is up to isomorphism, then well the first invariant is the dimension works over any field. But there's, there's another invariant that you can write down pretty easily which is called the discriminant. So, let's make the following definition. v comma q be a quadratic form. And this is going to work over any field f. So, the discriminant is defined. Well, first you choose a basis. You choose, choose a basis E I of the, and the discriminant of the comma q is exactly the determinant of E I dotted with E J. So, you choose a basis, and then you get a symmetric matrix right so so as last time quadratic form is given by can be specified if you've chosen a basis the inner product is specified by some symmetric. symmetric and by and symmetric non singular matrix. And so you can take the determinant, and that's the discriminant. But the caveat is that well right now so this is not really well defined because, well for example if you scale the basis if you just scale all the basis vectors by, by five, then, then you're going to, you're going to scale this. So, right. Sorry, so I guess what you should do is you should consider this is living in f cross modulo f cross squared. So it's going to be indeterminate, it's going to be independent of the trace of basis.  And that's that's basically because I mean if you think about how the symmetric matrix is determined so so any quadratic form gives you a symmetric matrix. And when you change a basis then you multiply the symmetric matrix by some matrix on the left and by its transpose on the right. And so if you think about what that's doing to the determinant it doesn't fix the determinant, but it multiplies the determined by a square. So this determinant is is not something that's that's independent of the choice of basis. But if you change the choice of basis you always change this determined by a square. So this is a well defined coset in f cross mod f cross square.  So you'll get it if you consider the discriminant if a quadratic form over over a finite field you'll just get a sign plus or minus one. And so, yeah, so let me state the proposition proposition over a field of you invariant, at most two. So I every ternary form is isotropic, then quadratic forms are classified by the dimension, which is, which is a positive integer. And, and the discriminant. So that's the that's theorem. Okay, so what's what's the proof of this. Right. So, I guess basically the proof is that if you want to classify quadratic forms, then you may as well classify quadratic forms of dimension at most two.  And, well, if you do that you'll change the discriminant or the change the dimension by multiplying by subtracting some even integer and you'll change the discriminant by a sign. So without loss of generality, you can assume that you're working in dimension less than or equal to two. So what you can do is well suppose V and w are quadratic spaces of dimension to so dimension one is easy so let's just say it's dimension two with the same discriminant. And then what you need to show is that then V is isomorphic to W. Okay. So, in fact, what you can do is you can show is that if D equals the discriminant. And then V is isomorphic to brackets one comma D. And then so is w. And so why is this will the fact that you're working with you invariant at most to is going to tell you that any two dimensional quadratic form represents one. So, because V has you invariant. So, in dimension two, you can find a vector V and V with V dot V equal to one. So, I'll leave that as a sort of as an exercise. If you have something of you invariant one you can always find a vector of this form. And then let V prime be some vector in your orthogonal complement. And then you can diagonalize. And then what you find is that this two dimensional vector space two dimensional quadratic space is isomorphic well you have a vector of length one. And then you have some orthogonal vector whose length is something, but just by looking at the discriminant that something has to be D times the square. So, okay so this is anyway going to be on the problem sets so. Maybe I should I should stop here, but this is this is how you how you prove this theorem so you can, you'll see you'll explore this more on the on the homework. So, I'll stick around for a little person for questions here. Yeah, any, any, any questions. We can just ask. I have a question. I was wondering why we call the hyperbolic plane and hyperbolic main is there some sort of geometric intuition behind that. Well, I guess it's like, right, I mean I guess if you write it in coordinates it's a quadratic form x, y. Yeah. I mean it's a quadratic form given by x, y, which I guess the level sets are hyperbolic. So I'm not, I'm not, I don't even I guess at least, but right, sorry so there's a question what pairs discriminant and dimension give rise to a quadratic form this and yeah so that's a that's a good that's a good question. Um, but in fact you can get any discriminant and any dimension. Because, sorry, maybe I should share again. So you can get any dimension and discriminant, because what we can always do is you can take one comma one comma dot dot one comma D, and then you can have it any dimension you want and then the discriminant is also what you want. Sorry, sorry, there's a question, why can't we rescale the basis vectors by square. Right sorry so if you rescale the basis vectors if you rescale. Right, sorry so if you have a if you have a if you have a quadratic form in diagonal form. So sorry I think I would say a little bit more about this next time but if you have a quadratic form in diagonal form, then the discriminant is it's a product of all the diagonal terms. And you can change the diagonal terms by squares as you're saying, but the discriminant is really only defined in f cross modulo f cross squared. I hope that helps. Yeah, so I will say more. I think I would say more tomorrow. I will try to get more into detail but when you know how you can, like, how, where do isomorphisms between quadratic forms come from. And really, I mean right so any quadratic form is, you know, you can you can write it down by some, some n tuple dimension and by some sort of n tuple of elements of your field and you can ask when or to such n tuples giving you diagonal forms isomorphic. In fact, you can sort of make that very, you can answer that very precisely if maybe slightly inexplicably. So I guess this chain equivalence theorem of that, which I will explain next time which turns out to be really useful for defining invariants because if you want to check that something is well defined you just have to check that it doesn't change after these sort of very, very particular moves that you're allowed to do. So I think there was a hand raised on this piece. So basically, I mean, if we look at that proposition that we just proved for every field of you invariant less than or equal to two in the case of the the finite fields where this FQ cross modulo FQ cross squared is just that mode to Z. Now that means that well for every dimension we have up to isomorphism preserving the quadratic form. We have basically two quadratic forms right in each dimension one for each well for each element. Okay. Yeah. Yeah, sorry, I should have said that. Yeah, maybe I will pick up on this a little bit more next time, but yeah. So a little bit simpler than over the real numbers for the real numbers you have this.  But here you just have you just have one sign plus or minus one. So so you can also define the discriminant over the real numbers but it's not a strong enough invariant. It's only one to that over a finite field that's everything. Is QP a C1 field for some prime P okay great question. So yeah so it's not it's not going to be a C1 field. But right so okay we haven't defined QP yet I guess that's going to be Thursday's lecture probably. But so you cannot, I think the question to ask so QP is going to be something kind of like a Laurent series it's sort of supposed to be analogous to a Laurent series field over FP. So on the exercises, there'll be some things about Laurent series fields, which are supposed to be a warm up before we talk about QP. And the answer is that, so this was actually an open. So it's not going to be C1 because somehow you have, you have an extra parameter like even extra parameter that comes from. Well if you think about Laurent series you have an extra parameter coming from the order of vanishing. So it's known that these Laurent series fields over a finite field are C2 this this weaker thing that if you if you have a degree D polynomial in it in more than the squared variables and there's the zero. And so it was I guess it was an open question for a while if that's true for QP. And so it is true for quadratic form so so the unvariant of QP is going to be, it's going to be for so this is this is really one of the things that. So, right so this, you know, talked about today is a case of you invariant to essentially that you have you have one invariant given by the discriminant. And you know later in the course we're going to talk about you and essentially the case of you invariant for. And that for example is the case of QP. So so so so degree five quadratic forms have a zero. In general it's not true that it's not QP is not C2 I think this this turned out to be false. So if that's true, there's, there's a famous result called the the axe coconut theorem that if you, if you have, if you fix a degree D, and then you take P sufficiently large for that degree, then. You're going to have to deal with the order of quantifiers. Then any, any form in more than D squared plus one variables over QP has a has a zero has an entrepreneur zero. So, so there is something but yeah it's QP is C3, because as your variant for. Sorry, so it sort of you want it to be C2 I think, which would give you invariant for so so the maximize what is true is that the invariant is for that the. Oh, I understand how the CI naming war. So, right. You're very, and how long to do to my CI numbers. So in general you can't, because the you invariant is a forms of any degree. So, I'm sorry not the C, the C condition is for forms of any degree, whereas the you invariant is only for quadratic forms. And so quadratic forms are special in many ways, for example, diagonalized and so forth. I mean, the, if you, if you have a field that is CN. So I didn't, I didn't define CN, but basically then the you invariant is going to be bounded by two to the end. And they don't know. Yeah, I don't know whether these things are probably not the best possible in general. I'm not sure what this state of the art is here. But I mean, someone here knows it to feel free to. Step in. But yeah, so QP it's it's not it's not a C to field, but it sort of behaves that way for quadratic forms. So quadratic forms over over QP you can still sort of control them pretty well and you can classify them. So so here you're here over over a finite field. I mean you have this one I mean you have the dimension and then you this one invariant the discriminant. So QP you have this you have this additional invariant called I guess house of it invariant, which is in during that, I guess we'll see pretty soon. More questions. Yeah, so I guess I will try this. Oh, sorry, question. Yes. I can't hear you sorry. I can hear you now. Okay, so just to clarify the terminology so when we say quadratic form, we actually means a specific decomposition right or we already have a quadratic space and we there and then there is a decomposition. I guess when I say a quadratic space or quadratic I usually mean the just a vector space with a quadratic form I'm not choosing a basis. But I guess if you if I like if I write out a quadratic form and coordinates for example like a x squared plus b y squared plus c z squared then I guess I'm doing that kind of informally but I'm sort of implicitly referring to, you know the quadratic form, given by this particular basis in this vector space but I think when. So so basically when we say quadratic form that already means that there is a fixed basis and then I guess I wanted to understand the difference between quadratic space and the quadratic form when we say these two terms what's the difference of the these two terms. Well, I, yeah I guess I'm not necessarily going to make a very precise distinction between them I'm only going to. I think everything I'm going to say is going to be about quadratic spaces. And I think maybe I mean maybe sometimes people would use the use the usage as sort of what you said that the quadratic form refers to a specific function. You know I don't I don't I'm not talking about other people I'm just talking about in in our like in in your lecture, because the quadratic form also show up this this term also show up pretty often. And I just wanted to know when you say quadratic form then when you say quadratic form versus quadratic space then how do we distinguish what you mean by the two. I'm only really going to work with quadratic spaces. Yeah. So if I say quadratic for my, I guess I really mean quadratic space. Okay. Yeah, so I don't know if Dustin is going to have. I guess you could say that the quadratic form is the function on the vector space values in the scale. So it's part of the data of the quadratic space, the second part of the data, the first part is the vector space that terminology would make sense to me. Or can I can I think that if there's so quadratic space comes with a quadratic comes with a function the quadratic function right. And so, so if I fix the basis then I will be able to express precisely what their function is. And then also there's a decomposition. So. If you if you choose a basis you get to write down a polynomial, you know a degree to polynomial. Yeah. And if it's diagonal. So if it's just got x squared y squared z squared and so on and I guess it corresponds to a certain decomposition as well. But if it's got cross terms, like x y and such. So when it is over Q yesterday in the yesterday's lecture we know that over R then it's always. There is a one to the R and negative one to the S. So that's, it's always in this phone, but if it is, if it is for a different field, do I possibly have a non diagonal decomposition.  I mean that's, that's saying that you can choose a basis such that the quadratic form looks like that it's, I mean if you start with a quadratic form and if you choose it, like an arbitrary random basis, then you'll probably have lots of cross terms. If you choose a random basis of a quadratic form over our quadratic space over the real numbers, then there's no reason that the vectors have to be orthogonal to each other, for example, but a statement is that you can choose well over any field. You can choose a basis such that there are no cross terms such that the basis vectors are all orthogonal to each other. So in terms of the real numbers, you can even choose them so that the self dot products of the, you know, the EI dot the eyes are either cost or minus one. Okay. Yeah. So I mean if if you want to think about it in terms of symmetric matrices. So like a symmetric matrix really gives you a function on F to the end, for example, then the symmetric matrix when you start out it could have lots of cross terms. But what we're we're studying is a symmetric matrices up to up to a suitable equivalence, and the equivalence is that you, you're allowed to say that a symmetric matrix is equivalent to a symmetric matrix a is equivalent to be transpose times a times B. And that comes from a change of basis of the, in the quadratic space. And then the statement is that you can up to this change of equivalence you can, you can put your quadratic you can put your symmetric matrix in this particular particular. Okay. All right, thanks. So I guess you could phrase everything in terms of symmetric matrices budget geometrically. Okay, thanks. Yeah. Yeah. So I have another question now that I that I was thinking about it. So basically we have for a way we have we want to define a quadratic form. Well, mapping into into some field. And I've seen well two definitions the one that I've seen which is probably closer to the one we're working on. We're working with in this course is that we we start with well an inner product some bilinear form symmetric bilinear form. And then you can just have the associated quadratic as Q of X is X dot X. But there's another definition I've seen which is kind of more pulling which is more polynomial related which says, well, it's just a degree, degree to homogeneous polynomial. And I was looking into well, an argument in a textbook about how to reconcile those two definitions. And in that argument we used again as many times in this whole study, the fact that the field doesn't have characteristic to because we divided by two at some point. How do you, is that is there a way to reconcile. How are things even defined in characteristic to because there's like pretty basic stuff that we're doing where we divide by two. How do you circumvent that when you want to discuss about an algebraic closure of F2 or something. Well, I think there are multiple I mean there are multiple objects you can study in characteristic to so for example, you could study symmetric bilinear forms which are non degenerate. And you could also study quadratic forms in the sense of you have a function Q like as you said a homogeneous degree to polynomial. And that gives rise to a symmetric bilinear form and you could say impose that symmetric bilinear form is non degenerate. And those are those are sort of different theories. And I think one can develop many of these results in characteristic to but I, I would really have to defer to someone else here on the state of that. So even when we talk about how well you have a quadratic form and it defines symmetric bilinear foreign advice versa, even in the polarization identity we divide by, we divide by two. So, well, sorry, but I guess what you can do is you can, you can say that you have a quadratic function. And then you can, well, you can do the polarization without dividing by two so q of x plus y minus q back minus q by, and then that gives you a symmetric bilinear form, and you can ask that that is non degenerate. Oh, I see, I think, yeah. So, the symmetric bilinear form in this case isn't going to determine the quadratic function. So, but so the quadratic function is going to be stronger data yeah so yeah. Right sorry so so I guess the quadratic function that you would consider in characteristic to is roughly it's one half of the construction that you would do in, you know, in characteristic away from to. But so this I mean this is, this is a perfectly sort of, you know, I think having people show you quite a bit that, that you have this function q, which you can think of I think people call as a quadratic refinement so it's something stronger than the symmetric bilinear form. So, I see, thank you. Yeah, but yeah sorry I think for for more about this I really do have to defer to someone with more characteristic to expertise here. In some of my lectures we're going to be doing quadratic forms over the integers, and then you need to be similarly sort of careful, and I suppose I will be when the time comes. I can't say that I'll right now be able to say exactly how it should go. Yeah, hopefully I'll have a straight by the time I give some lectures. Thank you. Well, I guess, for example, I mean you can have quadratic I mean you can have a, you can have like a symmetric bilinear form to actually yeah well. Yeah, so the different things you could. Okay, so you would have a symmetric bilinear form. Non-generous symmetric bilinear form such that x dotted with x is always zero. So, you mean a quadratic form. Yeah, well, or a symmetric bilinear form. Yeah, right. I mean, you can always have that. But you're saying it can have a non zero refinement to a quadratic form. Or what do you say. Yes. Yeah. Well, in particular, but I'm saying I can already have a symmetric bilinear form such that every element is orthogonal to every element is isotropic. I even a non degenerate. Yeah, okay, non-generate symmetric bilinear form. And I guess if you're working over the integers, then the quadratic refinement is sort of a congruence condition, right that x dotted with x is always even. But if I mean if you're over a field of characteristic to then you're you're making some choice. Making. Yeah. You have to choose how to divide by two. Choose. Yes. Any more questions or. Yeah. Okay, so I guess I will try to. Yes, I'll try to drop by the office hours and otherwise, I'll keep an eye on the, I guess, discord. So, otherwise, I will see you. See you tomorrow.


We feel privileged to have you back to this show. I think the Kawaii Human Humane Architecture happening to be our 297th show, and you are the accumulated viewer, which you're seeing down there. And we are back in the fellow windy city of Chicago, Illinois, and us is the Soto Brown, back in his Asupov design home at Diamond Head in Honolulu, Hawaii. You, the Soto, our Bishop Museum Historian and Archivist, Ty. Hello. Good day to you. Thank you, and me, Martin, the Spang, back at the opposite end of the world, near Munich, Germany. And we meet, as I said, somewhere in the middle, almost a stopover in Chicago that we're on to find out why we borrow so much architecture from there. And Chicago is certainly one of the heyday places of architecture, the cradles of, you know, urban architecture in the United States. But that was then and now is now, and we're recently getting lots of high rises from architects from that city. So we thought we sent you back, which we did about a year ago, a little more than a year ago. And if we get the first slide up, we're also going, you know, our minds go back and forth to other places here, cross-references. That's what we want you, the audience, to do as well. One thing leads to another. And this is getting us two capitals, and actually we're missing on the Illinois capitol here, but we have a couple of other capitals. Which are they, the Soto? Well, first we're looking at a large photograph of the dome, the new dome, or the new-ish dome, atop the German legislative and government building, the main building in Berlin, which is called the Reichstag. And that goes back to the late 1800s, and it's been through some major changes. It was badly damaged, twice, been renovated. And when, with the reunification of East Germany and West Germany after the fall of the USSR in the early 1990s, they did this new dome, atop the building, which was to make things, to show that it was a new beginning when the capitol moved back from its temporary location during the splitting of the two Germanys in the city of Bohn. And so this is this famous dome, which I have not personally seen. It's a glass dome, and it's got this mirrored interior section to it. And you're able to, you do have to pay admission, as you explained to me, but you're able to look down on the legislative processes going on below. But we think that the Hawaii State Capitol, which was built in 1969, is better because you can, as a member of the public, simply go up and look in the windows if you want to, or go inside to the galleries to see the legislative processes going on. And so it's much more open to the real people. And we think that's admirable because the people are the ones who the government is running for, and the people are the ones who pay for the government. So the more openness and the more the architecture reflects that openness, the better the process is for everybody involved. Exactly. And the picture at the bottom right is looking into one of the chambers of our state capitol in Honolulu, Hawaii. And we do this because we're charged to write an architectural tour guide about Honolulu for DOM publishers. And so we better get going. And this is what I took when I was just happy to be there. And for this show here ongoing, we use their guide, their Chicago guide, which was freshly out a year ago. And so at the top right is what you mentioned, the BUN one, and this is our friend, Metnoblatt's firm, Banish Architect, and in fact, the founder of the firm, Mr. Banish, built that in the 80s. And again, in both projects that you see that top right and bottom right, you can press your nose at the glass and pretty easily, you know, see what's going down there, what the ones that you elected are doing, hopefully what you wanted them to do. The one that we owe the picture to Semi, who took this for us, the picture off to the left at the Reichstag. And it was what Met praised this for good reason, the culture of architectural competitions, which we need more in America, which are not that common. It's more kind of invited competitions, if at all, or direct commissions, but Banish, are rightly so pride itself of always telling them themselves in anonymous competitions, so they're not selected for their name, but for their best idea. And so was the Reichstag, you know, remodeling was a competition that Baron Foster, Lord Foster, the previously Lord, now Baron, won. And he did the best he can, again, to sort of, I guess, the, I don't know how you want to call it, the Nazi fight, because just before the show, you shared with me the horrid thing of DeSantis. You want to just quickly mention that? Well, just something that's in the news today, this morning, right before we did the show, was that Ron DeSantis, the governor of Florida, who is in the running for president of the United States, even though the election is quite a long way from now, one of his staffers created a video and put it online, which ended with a symbol of, that was used, an ancient Germanic symbol that was reused by the Nazis during the 1930s and 40s, which is called Sun and Rod. Is that correct, Sun and Rod? In any case, it's, and he used a symbol of, he used this, reused this symbol in a setting that was reminiscent of a Nazi rally and put Ron DeSantis right in the center of it. And so the Nazi symbols, even though they have been suppressed since the 1940s and are known for their evil associations, still show up to the amazement of me and millions of other people. Scary, scary. And of course, you know, the Reichstag played a role in Nazi era. And so it was a touchy subject and fostered, you know, tried the best to denazify it, to democratize it again and replace the stone dome by this glass dome here. And while it's quite spectacular and thanks, Semi, to having taken the equally spectacular picture here, we see him down there in the choke boat at the center in the middle with Jay, our founding uncle of Tink Pek, Hawaii at the annual Christmas celebration and Governor Waihi.  So no one will deny it. So I know it. And Joey, my oldest son, who you know well, you know, had a Holocaust survivor come into high school, break out in tears. So he knows firsthand. But of course, the coming generations, you know, need to be reminded in different ways, right? And so we have to, you know, keep doing that. And as far as the point, the other point about, you know, watching your representatives here, it's not quite as direct as in the other cases of Bonn and Honolulu, because as you see at the bottom, the chamber where the representatives work is actually below under that second layer of glass on that sort of elevated ground floor, which is actually the roof of the building, but, you know, the ground of the dome. And you see there is a guardrail with illustrations on there and that keeps people away. So they're more separated and more disconnected than in the other cases, which is a little bit of disappointing. Again, it is what it is, but... Yeah, but it also is a different world in that we need more security now. And when the Hawaii State Capitol opened in 1969, it was not as likely that there would be needs for security. And that is something that is far more prevalent now, that is very, and that's very important in architecture as well. And one of the things that we are very familiar with now is that government buildings are surrounded by bollards or very deeply implanted protections so that you can't drive a vehicle up into them to blow it up. And you have to go through metal detectors and so forth. So it's a different world than it was when our state capital was built. Yeah, sadly so, but that's true, sad but true. Next slide, we hold another illustration of things we already talked about last time, but again, I was digging out pictures that I took in 2008 when I was there with our host, Dan Kubrick, who you see his pickup truck in the center picture. This is the Crow Island School that we mentioned. That was the beginning of the firm Perkins, they were still called Wheeler and Will, now they're just called Perkins and Will, together with the Sarenons, Father Alial and the Sun Arrow. And it was a collaboration that they did. And that actually made Perkins and Will become an expert in school design. And we read that they after this built the large amount of 500 schools because this was so successful. It was an inspiration for us for our decarbonized preschool that we see at the show quote at the top center. And we only knew it from literature. And then when that was completed in 2008, I had the chance to see it. And in particular, the kind of broken up courtyard-y sort of syntax of classrooms and outdoor spaces. The other inspiration was at the top right. This is the preview of our soon to be continued auto architecture show. And this is the Solar Hemi cycle by Frank Lloyd Wright, officially called the Jacobs House too, because of its passive solar performance and sudden exposure. So both projects were very informative for us. And it's amazing that from 1940, also reminded me a little bit of talking Ossipoff house that you are in. You also shared with us a couple of shows ago, the preschool or I guess the kindergarten, right? Or was it elementary school that you went to at Punahou, which was also by Ossipoff, but is not existing anymore, unfortunately. This one here still exists. And you can read in the articles is just as fresh as it was back then. I try to find out what, it's a very sort of low profile, one story building, but the entrance is marked by the slab. And there's some articles who basically say it's just a markation for the entrance and others basically say it's a chimney, maybe it's both. And if it is, it's obviously, a signal or a symbol for the fossil, while again, the work of these days should be a symbol of the post fossil. And I thought you particularly amongst other things, I'm sure the soda reminding you of your childhood in your Ossipoff, but I also thought the ones at the top left, these ceramics you might like, right? Am I right? Yes, absolutely. And what interests me is I'm seeing these touches that look very much like they're from the Doko Momo mid-century, 20th century period, the use of the stone, the use of the ceramic tiles, and also that central sort of pylon that is, you just said, is either some kind of an architectural statement of a marking device or it's a chimney. But regardless, it is very typical of that post-war, post-World War II architecture, a lot of commercial buildings, even in Honolulu, used that same type of element to make their buildings stand out, particularly for commercial buildings, you wanted people to be able to see you when they were driving by or looking for your building. And that's one way to make your building distinctive. Absolutely. Okay, so now we put pictures to the words of the Crow Island School, you know how that looks like. And it has to do with Chicago again, because it's up the lake, Lake Michigan, in this town called Winicka in Illinois. So it's up the coast of the lake. So drive up there and enjoy it. And it's not in the book because again, the book is just Chicago, but we're kind of broadening the horizon here. Next slide is also a preview of how we closed, but I put in show quotes at the top to illustrate the points that you made, the solar rightly so, because we're looking at a double facade here, where we have basically an inner thermal threshold, it's triple pane glaze, likely. Then we have a cavity space and then we have an outer glass cavity. But we have at least, I mean, first illustrate your criticism again of similar situations in HoloLunu with a choke bolt at the top. Yeah, we're talking about the gold bond building or the gold bond building as it now looks after it was remodeled. It was originally a plain 1960s box pretty much with a facade that was the same on all four sides, which in one sense is not that good because it's not taking any climatic variations into account from the direction of the sun or the direction of the prevailing wind. But what they did to jazz it up and make it look more 21st century was to create an exterior sort of glass facade over a section of the existing facade. And all that does particularly because the afternoon sun hits it is create a pocket of air that's going to get a lot more heated up. And here in HoloLulu, we don't need to be creating pockets of warm air to help insulate or keep people inside warm. Our problem is that we're often too warm inside. So in a cold climate, there can be a use for such an architectural element. But here in the Hawaiian islands, unless particularly if it's facing the sun, it is not only not of any use, it's actually detrimental because you have to be using more energy to cool your building because you've created a pocket of hot air within your structure, which you don't need. Absolutely. And that way, ironically, they wanted to make it look 21st century and they made it a form like early 20th century puzzle where now you need to AC that more to be able to be livable inside. And they should have done what this building down there which is by Perkins and Will and it's a biomedical research center. They put the shading louvers, the jellices that are retractable behind and the reason or the way to sell this to your client because it's gonna cost you twice as much because it's a double facade, so it doubles the cost. But it also minimizes the cost of maintenance for the very filigree and delicate louvers who might be damaged by the wind and they can if they're protected by the outer shell of glass. And that's how it's basically sold. And it's facing, the lake is, you see on that center picture on the right, you see this historic building in the center, this is facing the lake, the lake is east. So we know that this building in that picture on the left is facing south. And if it's facing south and the louvers are down as you see in the detail to the left of that, it's good because it keeps the thermal threshold where the sun could get through and be converted into heat, keeps that sun out. While in the winter you move the louvers up and you get past the solar gain.  So they actually, if you would wanna do something like that, which we still say is a little bit like, okay, you wear a hat with a wind turbine on and thin film PV layer to run the fan that drags the heat out of your too much clothing. So it's a little around the corner so you could do it easier and just be cold, easy breezy. So we're not saying you should, but if you would wanna do it, then that would be the way to as to learn something. So here comes the shock that I warned you in the audience for because until a little, not that long ago, there was something else on that side. And that we see on the next slide. And that's very shocking to you because you as a preservationist, you love to keep old stuff, especially good old stuff. So this here, we talked about, you probably can figure who the architect is because that was sort of his signature style to go up against his mentor and master, Mies van der Rohe, who was the rectangular boxy guy. And there was Bertrand Goldberg who reveled against his master and said, no, architecture needs to be organic. And which are the most iconic twin towers of his that we already covered in a lot of times? These are the corn cobs, right? Yeah, they're the corn cobs in there. That's the Marina City development that's very famous in Chicago, which was created in the late 1950s, early 1960s. And they're very distinctive cylindrical buildings with rounded lanais on them. So they really do stand out.  So they've maintained a good reputation and desirability even for many decades after their construction. Yeah, and so had this building here because there aren't that many of Bertrand Goldberg and he's certainly an institution in architecture in Chicago. I saw the building the last time, the picture on the right in the middle as I labeled it here, 2010. I was there with my Perry students and I took a picture of it, that picture, not having any idea that next time took me 10 years to come back, shame on me, but it wasn't there anymore. It was replaced by the building that we just saw that in fact they wanna even add onto it. And it would then look like in that little picture at the center of the top as a rendering. And interesting is we have Ginny Gang in Chicago, right? With her Huula Tower. And she was, and this is Michael, if you can get the bottom right. This is someone that Jay likes a lot who is Mr. Kimmelman. And Mr. Kimmelman is an architectural critic as we, I guess, take on the role to be that here in Honolulu. And he was very critical about the plans to demolish that and he reached out to Studio Gang, Ginny Gang and her firm. And they made a proposal, and that was, I labeled, I looked it up, that was in 2012. So when I took the picture and saw it, it was very timely to soon to be scrutinized. And they basically made this proposal to basically keep the building while still satisfying the needs of expanding and they put this tower that looks very sort of scandalous and almost visionarily impossible. But they say they consulted engineers and they said there is enough structural integrity there that you could do that and let's believe that. But obviously, the clients didn't wanna listen. This is an institutional client. We know this very well from our age when we had told them a couple of times not to tear things down as just a cross-architecture hall, which we will get to also either in this show or the other one with Met. And they basically knocked it down and built absolutely new one. This is rather sort of bad these days because everywhere people getting increasingly aware of that we have to keep things in the life cycle. Whenever we can, we need to keep things. And only if we cannot, and I'm sure they had arguments, but again, Kimmelman and Studio Gang basically said, okay, we understand you need more spaces. You need more program. Here you are. So is that shocking enough for you, disorder? Yeah, it's very unpleasant. And when you say we need to keep things, there are two reasons for that. The first being, from architectural standpoints, there are buildings which are never going to be replicated, that are never going to be designed that way or built that way again. And buildings like that need to be preserved simply because we need to keep our history and things that you'll never replicate. But secondly, every time you tear something down and build something new, you're using a lot more resources. You're using a lot more energy. If you can not do the whole demolition and rebuilding process, but make use of things which have already been through the construction process, you're benefiting way more than starting from scratch and you're saving money too in many cases. Yeah, and why we threw in the show quote at the top right, which is the Royal Hawaiian Center, that's why we do this on air every week for many, many years to raise, as Jay calls it, raise public awareness and to inform and educate clients, slash owners of buildings because sometimes they don't even know what gyms they have. Right? Absolutely. And so, I mean, the Hawaiian Center is still there, but it's very, very altered. It's very cheesecake and Tiffany's to a degree that you don't recognize, but it's luckily still there. And as we did the show about the Hermes store that some people from the other side of the world, some French people saw that potential and basically retrofitted it back to its original tropical brutalist nature and added louvers and added vegetation, did a really good job. For that reason, it's gonna be in our book. And so we would encourage clients to do that more. And certainly again, Kimmelmans and Jeanie's proposal had validity and certainly again, that's why the Royal Hawaiian Center is brutalism. This is certainly brutalism. And you can, the point is one of the excuses is, oh, it's not up to energy code. Well, you can bring it up to energy code. You can use vacuum insulation, panels that you put on the inside of concrete. In fact, even in new buildings by Zaha, for example, in Wolfsburg, which is the headquarters of your favorite BWs, she built a science center, the Fano, which is a new building that they basically used insulation on the inside to show off the baton bruit on the outside, which is what's the other. So let there not be excuses. This is what we basically say. Let's work with what one has. And they call this gray energy. By the way, the gray energy is the energy used in the whole life cycle, in the circular economy and ecology of a building. And it's kind of absurd when you have discussions with developers who say, oh no, to keep it would be too expensive. How can this be? It's often an excuse and a lame excuse. Let's just go for the last remaining minute to the next slide, because if you build new, you get things like this here, which again, we would say, hey, where is the double facade? There is only a single facade and how is that gonna perform in both winter and summer? And here they basically, I did the boat tour, I said at the beginning of the show sequence, right? And the very knowledgeable guy basically had the intel here that he said, oh, this building prided itself to have something super cool that reminds us in Honolulu, they call this ocean or seawater air conditioning, that's how they call it. Here they said it's river air conditioning. So they use the walk, the temperature of the river to run it through the thermal mass of the building and then loop it. Well, in the river, it basically caused the fish to die and algae to grow because you were warming up. Hans Crock, the great partner of Alfred Yee that we did a show about who is fostering OTEC told me that in the ocean, it's not the case because the mass of the ocean is so huge that if you inject it back into the zone where the water already has that temperature, you can actually do that. I was skeptical about that, but when Hans told me it works, I believe him. But here it didn't work out. They actually have to stop that. And since the whole thermal conditioning of the building is based on that, they're pretty screwed, right? So here we go, right? You tear down a building like the one by Goldberg and then you build something like this here that doesn't really work that well. Oh, well, I guess. So that's a kind of a sad and bad closing note, but it is because we're at the end of the show. So more to come and discuss next time. You, however, will be on air for the next three weeks with some interesting Doko Momo shows that I, and I'm sure you too, look forward to. And until then, stay happy and healthy. Thank you so much for watching Think Tech Hawaii. If you like what we do, please click the like and subscribe button on YouTube. You can also follow us on Facebook, Instagram, and LinkedIn. Check out our website, thinktechawaii.com. Mahalo.


Hi y'all, she was said that I'm back So what I want to talk about today is okay, I see Lately there are a lot of guys upset for women who You know don't pay them any attention. Don't show them very much interest Ignore them like if they agreed to go out on a date with them they're on their cell phone or texting other people or taking phone calls and a Lot of these men are getting offended and a lot of these guys are like, you know Mad and upset because women are not Catering to them and giving them all their attention even though, you know They're either on a date or friends or just in a conversation. These women are distracted now I know like when when a woman really likes a guy She's not going to be looking at other things. She's not going to be on her phone all the time She's not going to be talking to other people. She's not going to be ignoring him or paying more attention to the Conversation at the next table. She's not going to be you know Not listening to the words that's you know coming out of his mouth She's going to be totally into him hanging on to every word Trying to impress him as well as letting him impress her If a woman is not doing this from the start You know You shouldn't even be asking her out and she doesn't pay you any attention Wherever you are if she doesn't you know if she's not genuinely interested in you as a person And you still pursue her to ask her out She's going to be the same way afterwards because and she's not into you She's just not into you, you know If you're not her type if you don't have enough money if you don't look a certain way if you this if you're that and You keep nagging her and begging her and being thirsty over her She's going to be like, you know what I am hungry today.  I'll do him a favor and accept his 53rd attempt to ask me out. Okay. This is how women think We're not trying to a lot of women aren't trying to use men.  She's like, okay, that food was good What's the name of this place again so I can come back with my friend You know fixing their lipstick. Um, yeah, that was really good. Thank you for taking me to lunch And she's just like all distracted. She's like, mm-hmm. I can't wait to bring My friend or my sister here are you know, this is a really nice place for a real date You know, they're thinking in their mind this because you have pursued them so long after You know that they don't really want to go out with you and finally, you know You got them at a weak moment at a hungry moment at a bored moment and they decide to take you up on your offer and plus felt as though they were doing a good deed by going out with you because a Lot of men will say a woman women are entitled women feel like they're all this and they're the prize And we are the prize and if a man is constantly after someone who's not really interested in them And they're constantly up in their inbox and they're constantly up in their face and they're constantly saying well We should go out sometime just as friends as friends, you know They try to throw in the friends things so you don't think it's a date But then when you go out as friends and then you end up being ignored and dissed and you know Such and such like that You can't be bad because that woman did not you know Tell you in any type of way That she was interested in you Okay, she thinks she's doing you a favor. I mean It's a pity date, you know a lot of women go out on pity dates And and a lot of guys mistake these pity dates as women use of them and then they get offended. No That was a pity date. Okay. A woman can go out a lot of women can go out and just go find anybody to take them to lunch I mean, um, it's very easy, you know Even if the per even if they're not interested in the person They have contacts in their phone right now where they can just call up some random person and say hey You want to go to lunch and that person will take him out to lunch. It's not it's not You know rocket science Women have too many options and I'm not saying this to be conceited Because I know men have lots of options too Um, but the type of women that the men want Are there the type of women that the men really want don't really want them and then the type of women who offer men, you know All of them those men don't want them, you know, they might use them up every now and then And you know, don't call them for weeks and then try to pursue the woman that doesn't even want them They might do that But they they bypass the women who will give them the time of day who will appreciate a date a lunch Who would want to be asked out they pass those women up daily They don't even see them But they're so busy trying to date the women who don't even want them and then they're gonna get mad when they're dissed and then they go on this Rage about i'm not paying for any more dates. I'm not paying for any more dates because women are used No, you just try to date out of your league Tone it down a notch and like if you went for a nine or an eight or a seven Choco for four you might have better luck and i'm not trying to be rude.   You got too many people watching tv seeing all these women on tv and Look a certain way and the men all think these are the type of woman they're supposed to have But you got to have a certain lifestyle to have those type of women. Okay A lot of men are like Going to brazil so they can get the type of women that look the part Of the type of women they see on tv that they think they're supposed to have And they spend all the little money on a plane ticket Take the little vacation from the from their little job and then go down there and they just meet a brazilian prostitute because you know It is legal in brazil prostitution And they feel like the man, you know, they feel like oh I got me one But then reality sets in they got to get back on that plane Come back here and be a loser in their own country again You know, um, and I I really wouldn't call them a loser I would just say they they'll be in their own country again And they're not going to be able to get the type of women If they see on on tv or in videos, they're going to get the type of women who are in their league You know and just like women we see all these guys on tv these ballers these rappers these uh sports athletes and We think that's what men are supposed to be, you know, we think all men are supposed to cater to their woman pay for everything this this this and that and Lavish them with gifts and we we see that on the media. So that's what we want to so if you're coming at us thinking you can have us For free and we're looking at you like where is your money? Where is your uh status? You can't be mad at us for doing the same exact things y'all are doing to us You can't be mad for having higher expectations and higher standards for women out of your league You can't be mad at us for having higher expectations and wanting men to have money You can't because you're doing the exact same thing.  You're looking for someone who is attractive Who looks good who looks like the women you see on tv who looks at least close to it. Um, You're not looking for the ones that Are looking for you So you can't be mad and the thing about women the difference between women and men is that women can get a man with money because they have Skills they have brain power They have games They know how men think They know what leads men.  Okay You you can't do that men can't do that. They can't go and fake A bank account.  They just hope that when they take the makeup off They look at least close to what they look like with the makeup on This is just how it's always been women have always done this. So It's easy to be accepted, you know, as long as you look good in public. It's a status Uh, it's a status for a man to have a an attractive woman All of all his friends like her and think she's pretty if his mom and family likes her and and if she's gonna make pretty children That's what men look for Men don't look for women to have, you know, all this money and stuff women look for men for that Okay, and if you don't read if you don't meet the standards of women, of course, you're gonna be upset, you know But you're only upset at yourself because you settled for your mediocrity. You got to go and do better You know, you got to go do better if I were a man I would I wouldn't even try to date or talk to a woman until I have my whole entire life together Yeah, I might have friends I might have friends with benefits But I'm not going to try to date or seriously be In a relationship with someone that I know I'm the man and I'm supposed to be doing this and that I wouldn't even bother wasting her time And if I saw a woman that was totally out of my league and out of my price range.   Okay, you you'll see a woman that you will take you Any type of way and let you take her to a through a drive-through But you don't want that type of woman you want this one and then you're mad when she Does it call you back Or does it bother with you after the first date?  She felt like she was doing you a favor.  Y'all have to know this There are plenty of women going out on pity dates and y'all think y'all are being used, but it's really pity When I was young I used to do pity dates with guys and then I was like, you know what I feel guilty because I know they're tapping into their rent money to pay for this meal So I'm not even going to go on pity dates anymore. Y'all need to date in your own range By amount, you know, so That's just the honest truth and if you can't deal with that then you shouldn't be dating Or you shouldn't be trying to date certain type of women and I know I'm older now But when I was young and in my prime You know I was I was something to be reckoned with. Okay was very believe me Something to be reckoned with and I wasn't conceited back then. I really wasn't I was very shy actually and that's probably what the turn on was. It's like, you know, I was I was like still believed in true love Oh, I'm gonna marry the prince charming. Oh Yeah, I was like that, but I wised up quickly and I understood Lots of things That most young girls don't understand But man, if you are looking for someone who is attractive and who will date you With no, you know Need for you to be a baller go for younger women You got to go for the younger women because they're they still believe in love and that you'll do the right thing and that You're not gonna break their heart and all that's that they they believe in that still Young ones, okay So that's my advice if you if you can't find a woman your own age To date or to deal with you go for a younger woman who still has Trust faith and uh, you know a little bit gullible because those are the only ones that are gonna fall for y'all Okay, so I'll see y'all later. Bye


Hi, I'm Clarence from St. Paul, Minnesota, and you are watching Trucker Josh vlogs on YouTube. Pace. You have said more diesel or you're out.  He's absolutely right. Who was it? Who was it? Dean Tate, more diesel or I'm out. I need to set a smiley face. Done. Diesel. Whoa. I almost dropped the camera. I'm so excited. Diesel. What can I say, man? People just love me. Look at me. They're so good looking. Look at me. Can you blame them?  They're looking at me, too. Because I'm good looking. I'm a good weasel, man. He's over there.  Literally right in front of him. He's just sleeping there. So he got here late at night. The parking lot was full, obviously. So he decided just to pull the brakes right there. And that's where he's sleeping. Everyone's been honking at him as they go past him. He's blocking half the pumps. They won't wake up and move. I don't know if he can even move because it's E-Log. I mean, usually you can move a little bit like very, very slowly, but I don't know what he's doing over there. But hey, it's another new day. We're here near Nexington, Lexington, Lexington, Kentucky. And we're heading up to Ontario. We're going to be there late tonight and we're going to deliver tomorrow. And then I may have to do a reset up there before I continue, but they'll probably have another load for me going down south and then maybe home. But we'll figure that out later for now. Let's get going. Why not? Let's feed the weasel first. He's hungry. Cincinnati, what are you doing to me? What are all these orange things on the road? What is that?  All those cones there. It's disgusting. Clean up your. Yeah, we had a lane close on us there, and it just messed up traffic for the longest time. We're finally moving because I guess we're past that. It's been a good day so far, though. We've only driven about what are we looking at here? Ninety five kilometers. We're driven about 60 miles. We have another 550 to go. Woo, you excited diesel? I love driving.  I just I just love the driving part. I don't know what's going on here. So we're going around. We're going into Cincinnati here right now, right on the way north on I 71 here. So we've got to go down the hill around the corner. And then we go underneath that by level bridge, right? Suddenly, I want a Coca Cola. That's weird. But it seems that traffic is having a hard time getting through here. So I have a sneaky suspicion that there is another lane closed up ahead. That sign over there says stop and go traffic next four miles. You don't say I didn't even notice. I'm glad they warned me, though, all the way over here and not five miles back where it started. That I'm glad they put that sign right there. Let's people know next four miles, stop and go. I'm in a good mood today. It's been a really good day. Looking forward to going home again, though, but hey, I look forward to going home every day. First day I leave, I look forward to going home. But you can't do everything at the same time. I also love driving this truck and I also need to pay bills. You got to try and balance.  You're always at work way more than you are at home. And it's something that you've got to have a strong marriage for. You've got to have a wife that understands it or a husband, if you're out on the road and I'm lucky to have the wife I have.  And we make it work. You know, it's the it's 2020 with all the technologies we have out there. We video chat multiple times a day. It's not like we don't talk to each other. We talk all day. We're not video chatting. We're messaging each other. She's at work right now, so the phone is quiet. That's my chance to talk to you guys, right? Always someone to talk to. And it's sort of exciting at the same time, sharing what I see with the Internet. That's something that people didn't have just a couple of a decade ago or two decades ago. It's something that's pretty neat, because. I get carried away in my video sometimes.  And you get off in all these different topics. It's very easy to get sucked into all the main topics going around the world today. But, you know, you hear about that enough in the news already. I'd like to give you a relief from all of that. But sometimes I'm not the best at that because I get sucked into it also. You know, I listen to the news, too, with a. Skeptical ear, but I listen to it.  And then I'm just like anyone else, you know. The news is designed to make you mad. So I guess they achieve their goals sometimes with me. But, yeah, I just want to share my life with you out here. Here and on the road. I mean, pardon me, on the road here and at home. We always have plans for stuff to do at home. You know, we've been doing a lot of work on the house and property. Making it all nice. It's nice to it's nice to live, you know, life is a gift. Gotta make the most of it. You only get one shot. Or so they tell us. I mean, if they really had something you could, like, inject into yourself to stop the aging process, we're going to go on this conspiracy tinfoil hat.  Would they really tell us if they had a way to stop the aging process to make you live, like, let's say, like two, three hundred years or longer? They wouldn't tell us that the earth is already overpopulated. They'd keep it to themselves. There's your conspiracy for the day.  Don't worry. I'm no Alex Jones, but I like a good conspiracy. Turns out it was an accident. They had a lane block for accident and another lane block for construction. So here now we're getting to that Cincinnati bridge. You know, I've just realized that the Kentucky side of Cincinnati is called Covington. I've gone through here, like, probably hundreds of times in the last decade. I never knew that. Covington, Kentucky. And we'll cross this bridge and we'll be in Cincinnati, Ohio. We want to be in this lane, right? Yeah. Look at this. Diesel luck. We're driving on the water over the water. We're flying, man. Look at this. Where's the sign? Oh, their slogan is find it here. This road for six kilometers. We don't know what it is, but it's here in Ohio somewhere. Find it. It's like, where's Waldo? Except you don't know who or what Waldo is, but it's here. Find it here. That guy's in a hurry. He's tired of waiting. Bridge to Canada. This way. We're about to go over, I believe, what? It's Blue Water Bridge from Port Huron and to Sarnia, right? Or often than I have in the past, eh? I was on Plotbed for those few years. Well, you hardly ever came out here. I missed this area. We'll get across into Ontario. We'll switch ourselves onto Canadian hours of service because I only have 33 minutes on my 70 hours in the United States. So we're cutting it pretty close. And once I get into Canada, I think 15 or 10 or 15 hours available because it's a different system. So I'll be okay once I get there, but I'll have to reset. Before I come back to the United States, we'll go up there with Deliver. We'll reload and then we'll go and sit probably until Sunday. And then it's a one day drive to, uh, I think my next load is going to Rockland, Illinois. I only got to be there for Monday and today is Wednesday. So tomorrow's Thursday. We got lots of time. Lots of time to reset. Let's pay for our bridge. This is a bit of a lineup here. What in the world? What in the hunt? What's this? Why is the lineup all the way back here? This is just the toll to get on the bridge. Okay. That's moving. I thought they might have closed it or something. Maybe they were changing attendance or something. That's when they, uh, they usually close it while they do that. Maybe it built up a little bit of a lineup. I don't know. I don't know. They got at least two lanes open here. Do you want to go? What does that say? Prepaid credit card. I think I can go here. Yes. Okay. Yeah. They're not accepting cash anymore because of the because of the sickness. So you got to pay with credit card unless you have a prepaid pass. Oh, maybe I should have gone in that lane. I thought this lane would be quicker. Just about our turn here. I believe it's about 16 bucks or something. 18 bucks Canadian. What does it say here? Trucks, 3.25 US per axle or 4.25 Canadian per axle. It's $16 and 25 cents US. You got to cross the bridge. I wish they had easy pass here. So strange to see these bridges so quiet because there's only trucks going across the border, right? Very few cars. This bridge used to be just packed end to end with cars, especially in the summertime and on the long weekends. Now it's just a ghost town. I can't wait till this is all over. I can't wait till November is over. It's going to be so nice to go back to some normalcy a little bit. But who knows? I guess it depends what happens in November. I get better, might get worse. You never know. Fitting that they have blue lights on the sides of the blue water bridge. No, no, no. First we got to talk to the good border people there, Karen. I know you don't want to talk to them. Don't argue with them. Okay. Don't embarrass me. Are we in Canada yet? Passing border, entering Ontario. There it is. Sign here says welcome to Canada. Slide right on. Lake Hume on Circle 2. Highway 402 and then turn left and do 100 meters. Bienvenue, oh Canada. For the Quebecers. Wandering outside their borders.