So, in today's lecture we look at the notion of a vector space almost all of us have done vector calculus, calculus involving two variables, three variables. For example, look at vectors in the plane, we have done vector addition and then multiplying a vector by a scalar. For example, this is called a linear combination of vectors. We have also encountered linear combination of functions especially when we have discussed the notion of solutions of differential equations. If y1 and y2 are two solutions of a linear differential equation then a linear combination alpha times y1 plus beta times y2 is also a solution of the differential equation. The structure that formalizes this notion of linear combination etc is called a vector space. This is a basic object of linear algebra. So, what I will do is today give the definition of a vector space and then give a variety of examples just to tell you that vector space is abundant in the whole of mathematics. So, these examples for instance will include the differential calculus example that I mentioned and also other examples. So, let us look at first the notion of a vector space. So, here is the definition. A vector space consists of the following, consists of the first object as field. So, we say vector space over a field. The first object is f, f field. We say that it is a field of scalars. In this entire course of linear algebra we will restrict our attention to mostly r and in some examples the complex field also. Then we have v, a set of elements, v of course is non-empty. Set of elements, these elements are called vectors. These elements are called vectors. Then third aspect is a rule called vector addition. This is a binary operation which for x, y in v assigns a unique vector which we will denote as x plus y. This x plus y belongs to v. So, there is a binary operation on v. It is called the vector addition. This binary operation satisfies the following conditions. For every x, y in v there is a unique x plus y which also belongs in v. So, it is close to the respect to this operation such that the following conditions are satisfied. Such that let us write these conditions here. First it is commutative. x plus y equals y plus x. Second condition it is associative. x plus y plus z equals x plus y plus z. So, that is associativity. Condition 3, there exists an element denoted by 0. That is in v such that, so this 0 acts like additive identity. So, we have this condition to be satisfied such that x plus 0 equals x. See, all these are for all x, y, z in v. So, such that this holds for all x in v and the last condition, fourth condition is that for every x in v there exists an element which we denote as minus x. This minus x belongs to v such that this is like the negative additive inverse negative element such that x plus minus x equals the additive identity. So these conditions hold for all x, y, z in v. So, this is with respect to the vector addition operation these four conditions must be satisfied. So, we have vector space has a field. So, there is a field over which the vector space is defined. There is v a set of vectors satisfying these four conditions. Now what is the interaction between the field elements and the vector elements that is given by scalar multiplication. So, I will say for a rule called scalar multiplication, a rule called scalar multiplication which for every x in v and a scalar alpha in f which for every x in v and alpha in f assigns a number alpha times x in v. So, for every x in v, alpha in f this product I mean the scalar multiplication alpha times x must be in v and this scalar multiplication must satisfy the following four conditions. So, what are those conditions? Remember that the underlying remember that f is a field. So, f for instance has 0 as well as 1, 1 is the multiplicative identity. So, the first condition relates use as the multiplicative identity. So, I will again write four conditions for the scalar multiplications 1 into x equals x. So, for all x y in v and for all alpha beta in f. So, I have this condition 1 into x equal to x condition 2 alpha into x plus y equals alpha x plus alpha y. I should actually write a dot here but we will follow the convention of not using this dot. Alpha into x plus y is alpha x plus alpha y condition 3 is if you look at alpha beta of x it says you do it repeatedly. This is alpha into beta x and finally condition 4 is you take the scalar alpha and beta look at alpha plus beta of x thus this must be alpha x plus beta x. So the scalar multiplication must satisfy these four conditions the vector addition must satisfy these four conditions the underlying set f must be a field then v is called a vector space over f. So, in this case we say that v plus dot is a vector space over f. So, this is a formal definition of a vector space alright. Let us look at some examples now before we proceed with the examples let us understand that the two examples that I told you in some sense as motivating examples satisfy these conditions example of two solutions coming from a differential equation and the behavior of vectors in the plane for instance. So, let us look at some examples I will give an abundance of examples just to illustrate that this is an absolutely fundamental object in mathematics not just in linear algebra. So, let us look at examples let us start with the simplest example you take v as R also the underlying field as R then we know that then addition is what x plus y is the usual addition usual addition of real numbers scalar multiplication is also the usual multiplication then this is a vector space v over f is a vector space that is we say that R is a vector space we will follow this notation only for this today's lecture we will simply say R is a real vector space C is a complex vector space etc. So, R over R is a vector space as I said we can also say that R is a real vector space. So, this is the basic example similarly one could define C as a vector space over itself in general any field over itself is a vector space over that field second example is also trivial take v to be single term 0 and f is any field you have to define C must define vector addition scalar multiplication vector addition is defined like this scalar multiplication alpha times 0 equal to 0 for all alpha and f then this is a vector space v over f is a vector space over the field f okay these are trivial examples let us look at the first non-trivial example look at f 2 f 2 is what f 2 is the set of all set of all vectors x written in this manner x equal to x 1 x 2. So, I will write it as a column vector set of all x 1 x 2 such that x 1 and x 2 both belong to the underlying field. So, this is f 2 okay set of all column set of all columns which set of all columns which have just 2 coordinates which have just 2 components what is vector addition scalar multiplication once I specify that we can verify whether it is a vector space. So, let us take 2 elements x y in f 2 then x can be written as x 1 x 2 y can be similarly written y 1 y 2 let us also take alpha in the underlying field then x plus y we define x plus y to be the column vector the natural way to define addition is x 1 plus y 1 that is the first coordinate x 2 plus y 2 is a second coordinate this is obviously a binary relation scalar multiplication alpha x you do it for each coordinate multiply the scalar alpha to each coordinate alpha x 1 alpha x 2 then it is an easy exercise to verify that f 2 over f that is v equals f 2. So, v over f is a vector space with respect to this addition operation and this scalar multiplication okay now we can do a little more general we can do a little more general. So, let us look at f n f n this time is the set of all x that can be written as x 1 x 2 etc x n. So, there are n coordinates this time such that each x i comes from the underlying field this simply extends f 2 that was given in the previous example how do you define addition as before coordinate wise x 1 plus y 1 x 2 plus y 2 etc x n plus y n. So, x is this y is y 1 y 2 etc y n addition is defined in this manner scalar multiplication similarly for alpha in the underlying field then again we can show that f n over f is a vector space with respect to these operations okay. So, this is a little extension of the third example we can do a little more general let us look at f m cross n f m cross n now this m cross n should suggest the objects that we are considering here. So, what is the definition here this is the set of all x such that x has m into n coordinates that is one way of looking at it which will give rise to the same vector space as above a slightly different way of looking at f m cross n which is the set of all a which can be written as a 1 1 a 1 2 etc a 1 n.  So, these numbers a i j they all come from f 1 less than or equal to i less than or equal to m 1 less than or equal to j less than or equal to n such an object is called a matrix. So, you collect the set of all matrices with the property that the matrices have m rows and n columns the entries of the matrix come from the underlying field f then again the operations of vector addition and scalar multiplication is natural by the way here it is addition of matrices but still we refer to it as vector addition that is each element in f m cross n will be referred to as a vector okay now this is the terminology that we will adopt. So, a vector will not denote any more objects on the plane or objects on the three dimensional space they can now denote matrices we will do a little more general we will use vectors to denote polynomials we will use vectors to denote solutions of differential equations etc okay. So, let us look at vector addition scalar multiplication. So, take two matrices a and b a return in this manner b return in a similar manner then a plus b is coordinate wise addition.   So, this is how scalar multiplication is defined again one can verify that f m cross n over f is a vector space. See for instance the 0 element will be the 0 matrix the matrix all of whose entries are 0 the negative element of an element a will be the matrix whose entries are the negatives of the corresponding entries of the matrix a etc. So, this is another vector space example of vector space let us next look at p n r for instance what is p n r p n r equals the set of all polynomials the set of all polynomials in a real variable in a real variable t with degree of the polynomial not exceeding with the degree not exceeding the number n that is used as a subscript as the subscript here. So, it is set of all polynomials of degree less than r equal to n polynomial in what polynomial in one single real variable t that is what is denoted by this r ok. Now for example if p q belong to p n of r then these are polynomials I can write polynomials in the following manner p can be written as p naught plus p 1 p plus p 2 t square etc plus p n t to the n remember some of these constants p naught p 1 p 2 etc p n some of these constants could be 0 because it is a set of all polynomials of degree not exceeding n the degree could be less than n in which case p n is 0 for instance similarly q q naught plus q 1 t plus q 2 t square etc t is real variable. So, I have two elements two vectors now so vectors we are using the name vector for a polynomial here.  So, this is a definition of addition vector addition scalar multiplication also you would simply bring it to the coefficients.  So, p n r is a real vector space vector space over the field of real numbers. Let us do a little more general let us now look at the space of continuous functions the space of real valued continuous functions on a closed interval a b the space of real valued continuous functions on the closed interval a b we should again so I am saying the set of all real valued to continuous functions. So, the underlying field will be r we should again define vector addition scalar multiplication now that is similar to what we have done earlier for the case of polynomials this is so called point wise addition. So, we will do a similar thing here for p q in c of a b define p plus q of t. So, p plus q is a new function whose definition is p plus q at the point t is p of t plus q of t and scalar multiplication is defined similarly alpha p is a new function it is just alpha times p of t is most natural way of defining it for all the alpha in r for p q in c of a b. So, it can again be shown that c of a b is a real vector space if you replace the underlying field by c and then consider complex valued continuous functions then c of a b will be a complex vector space we can do a little more general. Let us denote c k this time open interval to be the set of all functions real valued functions. Let me say f with the property that let us say that I will use the variable t for any point on the interval a b. So, it is f of t for instance with the property that d k f by d t k the kth derivative of f is continuous the property that d k f by d t k the kth derivative of f with respect to of course there is only one variable here the independent variable is t this is continuous in a b collect all such functions real valued functions having this property again vector addition scalar multiplication as before vector addition and scalar multiplication as before then this is a vector space is a real vector space we can do a little more general one looks at c infinity of a b this is now the set of all functions real valued functions that are infinitely many times differentiated in the interval a b open interval a b same operations of vector addition scalar multiplication will tell you that this is a real vector space that c infinity a b ok that c infinity a b let us consider a more let us consider another example I will give it as example 10 let us consider let me use this notation ok let me use the notation f of let me use the notation f of a b a b, r to be the set of all functions f from a b to r set of all real valued functions depend on the open interval a b collect all those functions that is my capital f a b, r now in this look at v as the set of all f in f such that integral a to b f of t dt exists that is f is Riemann integrable that is set of all functions that are Riemann integrable then with respect to again usual addition and scalar multiplication we can show that v is a vector space then v is a real vector space now what is the meaning that this is a vector space if f is Riemann integrable g is Riemann integrable then f plus g is Riemann integrable if f is Riemann integrable alpha is a real number then alpha times f is Riemann integrable ok example 11 suppose that we have a matrix a of order m by n so there are m rows n columns entries are real numbers define v as the set of all vectors x and r n such that a x is 0 a x equal to the 0 vector now remember a is an m cross n matrix x is n cross 1 any vector standing alone is a column vector so when I write a vector as it is it is a column vector so the product here is m by n into n by 1 so the usual product of matrices so the resultant is m cross 1 this so this 0 vector on the right hand side is a 0 vector which has m coordinates now this is a vector space with now this is a subset of r n the subset of r n r n already has vector addition scalar multiplication defined it can be shown with respect to those operations that v is a vector space again close with respect to addition means what if x and y satisfy ax equal to 0 and ay equal to 0 then x plus y satisfies a of x plus y equal to 0 because a of x plus y is ax plus ay similarly if x belongs to v then alpha times x also belongs to v because alpha comes out here of this equation alpha comes out of this equation so this is a vector space we also have another example this the example that I gave as a motivating example let us look at the operator L defined on a function y which is at least n times differentiable y is a function of the independent variable let us say t define L of y so I am looking at this equation L of y equals L of y is defined by d to the n y by d t to the n plus a 1 d to the n minus 1 y by dt n minus 1 plus a 2 d n minus 2 y dt n minus 2 plus etcetera plus a n minus 1 plus a n minus 1 d y by d t plus a n y of t let us so let L be defined in this manner L is a differential operator consider v as the set of all y such that set of all y such that L of y equals 0 okay see I must mention here that a 1 etcetera each of these is a function of the independent variable t each of these is a you can take them as constants in particular but in general they are functions of t then look at v as a set of all y so this you need y to be at least n minus you need y to be at least n times differentiable in order for this to be defined so collect all those functions then that satisfies then collect all those functions then such a set is a vector space if you consider real valid function then it is a real vector space let me give you a couple of more examples some intuition coming from geometry so example 3 v is the set of all x in R2 so this is a subset of R2 in fact it is a vector space in its own right set of all x in R2 such that x2 equals alpha times x1 alpha is a fixed number x2 is alpha times x1 alpha is a fixed number okay so which means you collect the set of all vectors which have the property that the second coordinate is alpha times the first coordinate okay geometrically what does this set represent this set represents the set of all points passing through the origin this set represents the set of all points lying on a straight line passing through the origin okay it can be shown again that this is a vector space the set of all points lying on a particular straight line passing through the origin the slope is given by alpha a little more general I will call that as example 14 v is the set of all vectors in the 3 space that satisfies something like x1 plus 2 x2 plus 5 x3 equals 0 geometric interpretation this is the set of all points lying on a certain plane passing through the origin okay this can again be shown to be a vector space okay so I hope the abundance of these examples illustrate the importance of the notion of a vector space let me conclude the lecture by giving 2 by giving at least one example of a subset of R2 which is not a vector space example 15 let us look at v as the set of all x in R2 such that x2 is x1 plus 1 x2 is x1 plus 1 the second coordinate is you add 1 to the first coordinate collect all such vectors this is again a straight line the only thing is that the straight line does not pass through the origin you can show that this is not a vector space okay now you must observe that vector space must have 0 0 must belong to the vector space now this one does not have 0 okay so this is not a vector space one final example non example so this is not a vector space one final example let us say v equal to the set of all polynomials set of all polynomials of degree precisely 2 I do not say degree less than or equal to 2 degree precisely 2 I leave it for you to show that this is not a vector space it is not even close with respect to addition v is not a vector space okay okay let me stop here in the next lecture I will discuss the notion of subspaces examples of subspaces etc.


What's going on guys welcome to another video today me and Connor are actually out on a pond bank fishing and we're gonna be doing LTV challenge, but before I even start this video I'm gonna start making a lot more palm videos just because I know you guys can relate to him I know you guys like him, but if you guys want to see a lot more palm videos Please smash that like button and leave a comment below I want to see you guys as feedback on these videos And if you want to want to see me do more challenges LTV challenges palm videos Definitely leave a like on this video if you guys haven't already please subscribe to my channel and hit the little Bell for post notifications, but we're gonna go ahead and get this video started Um, we're gonna pick all we're gonna both pick a bait We're gonna fish it for a little bit. I'm see what we can do and hopefully we can catch a fish on every single bait in this box Let's go ahead get this video started little combos I got going There's just baits I've had on just had a little chug bug a shaky hit a little duo spin I might throw that duo spin around here in a little bit, but I'm gonna start off with a lucky tackle box baits and We're gonna tie this on My habits up and now it's feeling like a mood. All right guys before I even start this LTV challenge I looked over I was doing some b-roll. There's too bad sitting right here. See if I can snack He broke me off. He broke me off My line must have been afraid. There's one more. There's one more. I Cannot believe he just broke me off and I had him first first pitching There's another one sitting right here. Let's see if I can get one over he went one swam backwards Well guys to be honest I'm really glad I didn't catch that fish because usually if I catch one on the first cast that means it's gonna be really good or really bad and Sadly he was about two pounds. I mean I hooked him.   We're gonna use those last cuz I'm pretty sure we can catch fish on those fairly quick So we got this little Monster bass it's kind of like a frog, but it's a little fish a little bit bait fish So we got that we got a spinner bait. Which one do you want to throw? You want to throw a spinner bait? We are going to get this started look look oh Throw in there Connor throw in there behind my top water Do he just nail it? Yeah, that's all oh He came off It's cuz they're not I mean this bait I Yeah Mmm That's that's a tough one. Yeah, he was just so far out So my I mean it's hard for me to get a good hook set guys a little tip with these frogs and hollow body Baits like this, you know, they're gonna get a little bit of water in Anyone that you really buy is gonna get some water and just squeeze it out If you ever have a problem with you know walking it or you can kind of see he's going under water I'm sure you grab it squeeze all the water out of it I promise you to It'll run better again Look at this fish right here. Look at him. Oh My god, dude, I had a feeling one was gonna smack it right there No, he just he was actually pretty good size, too I threw up on that grass up there in the back and he came out and just hit it There's not committing to it. Not a bad one Let me see it Pretty fish Little thick one Spinner race done keep casting a little bit and we'll both walk over and tie on this soft plastics There you go, man First fish on the spinner bait. Yeah, that one's done Look at that fish swimming right there. See that? Yeah. Oh my gosh Dude that one that hit me a minute guys pretty good be challenge. Oh There you go spinner base getting it done. I'm not bad at all. Oh All right guys, I'm gonna put this down for a minute I'll pick it up and throw it later, but me and Connor are about to tie on some of the soft plastics I'm gonna tie on the worm. He's not to tie on the curl We're about to whack some fish already. No, it's about to happen We're gonna sit here and sell lambs. I'm without a doubt.   There he is. There we go Warm's done.   I'm gonna cast this duo spinner out first first cap That was a bluegill Look at that.  Oh, he came off dude The sad part is I only brought one And I have a feeling they're just gonna slamer this thing How sight fish that I got man. Look at that guys Let's do spins getting it done. Nothing big at all, but you know, it's a fish. I had tons of bites and It's a good finesse approach for these fish. Oh, he just nailed me. Oh, there he is. Oh Dang Oh, he was a good one dude He's about he was two and a half pounds. I didn't think he was that big dude when he jumped up next to me I realized how big he was. Uh-uh. He was pretty good man missing all the Sun I My this bait is crazy, dude. Look at him coming. I'm not a bad one at all It's one thing about this lake. They're all just you know Really pretty Guys my trick to get these key bites of this duo spin is just keeping it You know most of the ponds that we fish have all this grass in it And he really just got to keep this bait kind of out of that stuff and keep it at the top of the water column And make sure none of that stuff gets on there because this there is Wow, I'm about one Dude is crazy, but what I was saying guys With all this grass It's probably the best one yet With all this grass you got to keep this bait at the top of the water column and keep it out of it Because this is one of those baits that doesn't have much movement at all So if there's something a little bit of something that gets on it Then it's just gonna mess the bait up. So That's that's the thing. So you got to keep it out of that grass if you get a good cast It's almost every cast I'm getting a bite on this thing. It's been absolutely amazing these fish are just tearing it up It's so subtle. It's so it's one of those baits that you know, it's a it's an action bait, but at the same time It's not it's really slow and you can trigger those fish the bite, but what a pretty bass It's a bluegill.  They got big as bluegill Actually, he's not as big as I thought Look at him trying to twitch, but he can't Tiny brim. Connor got my papar I threw off the other night I literally was when I was night fishing and broke my line and chucked it out here and we just saw it out there Sitting out there Good ol river to see papar There we go guys. Oh Sight fish this guy. This is on the worm that came in the lucky tackle box Strike keen worm Hyper guy Little tiny one The way I'm rigging this bait is just a simple, you know, Texas red weightless. I'm just running it weightless Right here Simple as that. Let's run in this bait weightless and trying to get the most out of this one Where I'm actually bit it off a little bit made a little bit shorter, but there it is right there The way I'm actually throwing it But this bait out there There's a lot of fish right here. That's why they're so short. I'll throw it out there Let it sink and just let it sit for a minute. You know this bait I'm gonna work really slow, especially these weightless ones and sometimes before you even move it They'll actually have it and you need that one thing one key tip when you're throwing like a weightless bait like this weightless sink oh finesse worm is watch your line Because that could tell you if a fish grabs or not see I could tell it's sinking right now I don't know if you guys can see that but Sometimes that you'll see something with your line where to start taking all for you to see him kind of bang it or and Like that. He just had it. It's actually right when I did that Yeah, yeah Tips and tricks right there guys tips and tricks Pretty fish a striking worm. I Actually did a video a while back where I was just hitting my rod just making the bait jump And I actually caught a lot of fish doing that maybe I should try that a few times Let's put a new worm on but as I was saying just throw your like I said I mean that was that was something good to look back on and You know watch your line jump That's the biggest thing just watch your line jump And most likely it fishes on it But be slow when you're fishing these weightless worms and most conditions You know if I want to be fishing this I mean it's because these fish are a little slow or they're old finicky And they're not really wanting to chase chase bait too much and as you can tell I'm just watching my line It's kind of going out right there, but I think it's just sinking. Yeah It's just sinking see my line right there. It's just Sinking if I see something something weird happen if I see it kind of jump or anything Then I mean there's a fish on it because I can tell you guys how many times I've actually You know saw my line kind of jump and I'm not even really feeling anything. I'm just sitting there, you know Watching my line and I'll hook set before I even fill the fish or fill anything tug on my pole You got it guys Plays bigger than that It's all pretty fish. Oh, he's got it He's got it. Yeah, it's not bad They're good guys Look at that guy Teeth are sharp Pretty little fish or you get a little thumbnail with this fish and release them. I'm not sure. Oh, he's got it That's a better one. They're good guys Plays bigger than that, but it's all pretty fish All right guys, that's gonna end off today, but you know had a great time me and Connor came out here You know put a hurt on some fish, you know, we got to give it a good amount of fish and He got some on that spinnerbait. I got some on that worm and that that top water fish You know, I got a lot of bites out of a couple hook that were actually pretty decent It's just didn't have them all the way and mostly these fish in this pond are just you know smaller fish So what happens really there used to be a lot a lot a lot a lot of big fish in this lake and what happens in these Ponds is all these little fish get in here They start eating all the bait and all these big fish are just left off the dry and they end up dying so Yeah, that's what happens. Just a lot of small fish But overall hope you guys enjoy this video if you want to see more pond fishing videos and LTB challenges Be sure to hit the thumbs up button if you haven't subscribed be sure to subscribe And if you are subscribed or haven't subscribed be sure to hit the little bell next to the subscribe button Which will send you push notifications every time I release a video and you guys know and also leave a comment below I want you guys to think about the pond fishing videos, but thank you guys so much for watching. I'll catch you guys


OK, so this is my last lecture. And I just covered half of the lecture notes, but this is fine. So let me try to at least wrap up many things that I want to tell you in one hour. Unfortunately, what we are going to do now also is the most recent developments. It's quite elementary, and I think it's good because it actually covers some nice elementary statistical mechanics. So what I'm going to do is the M-theory expansion of ABJM theory, because what I was reviewing for you this morning was the top expansion of this ABJM matrix model. So you learn a lot of stuff, but you don't really learn what happens in this M-theory regime. Remember that in this regime, you want to take n go to infinity and k fixed. So what I'm going to do is to introduce a new point of view on this matrix center, which turns out to be quite useful and actually has been generalized for many models. And actually, it points out to a more abstract point of view on this problem, where we're going to reformulate the gaze theory in terms of the spectral theory of an operator, and this turns out to have applications also to some other models, like topological history in theory and so on. But the way to do this is quite simple. So what I'm going to do is to introduce what is sometimes called the Fermi-Gas approach to these matrix models. So this was done first by myself and my student Putrov. And essentially, it's a reformulation of the matrix model.  One is in terms of thermodynamics of a Fermi-Gas, of a free Fermi-Gas. So this is a quite simple system. And equivalent, you can think about this. You can reformulate this in terms of the spectral theory of an operator. So these two things are equivalent, and you can change your piano view according to what you need. So how does this go about? Well, let me not write again the matrix integral for ABJM theory that I wrote this morning. Hopefully, you can check it on your notes. And then the first thing you do in order to introduce this Fermi-Gas approach is to use some non-trivial identity, which has been quite used in the analysis of these Chern-Simons matter theories in the matrix integral of these Chern-Simons matter theories. So remember that in this ABJM matrix model, you have terms involving a shinch, actually a pair of shinch. And then you also have a term involving a cosh. Well, it's a fact of the life of polynomials. This is actually a polynomial identity, essentially, that this can be written as a determinant. So this is the determinant of a matrix whose entries are just the inverse cosh of mu i minus mu j over 2. Now, determinants can be expanded using the permutation group. And this turns out to be a sum over permutations in which you have the sign corresponding to the signature. So you see already here the fermionic nature of the problem. And then here you have a product over twice the cosh of mu i minus mu sigma i divided by 2. So this identity is non-trivial. You can find proofs in many places, for example, in the book on representation theory by Fulton and Harris. And this is just the development of the determinant. So remember that this was one piece of the integrand of our ABM matrix model. And then it's an exercise. I can propose this as an exercise. And if you have doubts about how to proceed, you can go to the paper by Capustin, Willet, and Jacob, where this was first done. So what these people show, they show it only for k equal 1, but k is not important, is that you use this identity. You can actually write the matrix integral in this form. So you write it. You have to do more stuff here in order to just apply this identity that is not sufficient. You have also to use some sort of clever use of Fourier transforms and so on. But the final result is that you get such an expression where rho of x prime is a function of the form 1 over 2 pi k, 1 over twice the cos of x over 2 to the 1 half, 1 over twice cos of x prime over 1 half, and then 1 over twice to the 1 half, the cos of x minus x prime over 2k. OK, so this is already quite complicated. So the integrand is just this kind of orgy of hyperbolic functions. But this expression, you have taught the statistical mechanics I have to do in order to gain my living, it's actually something that you can recognize. This, you have studied the book by Feynman on statistical mechanics. This is the canonical partition function. When I mean canonical, I mean the canonical ensemble of a free Fermi gas with n particles. And rho plays the role of the density matrix for this Fermi gas. So the density matrix, just let me remind you that the density matrix, usually you write it like this, where beta is the inverse temperature. Here I'm going to fix beta equal to 1. So there won't be temperature, but it's actually very good to think about this as an exponent in Hamiltonian. So this is, as I said, relatively elementary statistical mechanics. And then this opens the way to actually analyze this problem with the tools of statistical mechanics. Now, why is this useful? Well, first of all, if you think about the large hand expansion, it's always good to think about the large hand expansion as a limiting with the theory simplifications in way. Now having a thermodynamic analogy for the large hand expansion is always useful, because we know that in thermodynamics, life gets simpler when it is large. This is called the thermodynamic limit. So we should expect that the simplification that happens in the large hand limit that we want to analyze can be actually put in a more intuitive basis by having a thermodynamic analogy. OK, now let's go back to statistical mechanics one on one. So I'm going to refresh some formula for you for statistical mechanics. Of course, you learned in statistical mechanics that the way to deal with a Fermi gas, an ideal Fermi gas, is not to use the canonical ensemble, right? Because in the canonical ensemble, things get relatively complicated. So what you were taught to do is to actually do the calculations in the grand canonical ensemble, right? In the grand canonical ensemble, essentially you get rid of all these sums over permutations. That's why you have to do that. Now the way you do this is you introduce this grand canonical partition function, which is going to depend on the fugacity mu. And k is there with us from the very beginning. And k only enters here through this density matrix. So the density matrix depends explicitly on k. OK, now what is the definition of this guy? Well, this guy is actually a generating function for all the canonical partition functions. And here you use e to the n to the n mu. And sometimes I will denote e to the n mu by kappa. Kappa, I'm going to have trouble in distinguishing these two guys. Kappa is a Greek letter, is the fugacity. Well, mu is the chemical potential. And k, I will try to put it this smaller. This is the coupling of terms I must hear. So it is the parameter appearing in m theory. So this is the definition. And usually you introduce also the grand potential, which is just the log of this guy. And it's a very interesting exercise that you can find also in the book by Feynman. So I propose this as an exercise. These are really good exercises. These are things that one maybe has not studied in the undergrad years, but are good to know. So this can be written in this way. So this is the fugacity. And here we have dzl. And dzl is actually very simple. It's just the trace of rho to the l. So that's why you introduce the grand canonical ensemble. You introduce the grand canonical ensemble because in the canonical ensemble you have a crazy sum over permutations, which is difficult to deal with. While when you introduce the grand canonical ensemble, you get a sum in which you only actually look at cyclic permutations. Trace of z rho to the l can be regarded as a cyclic permutation, just coming from a cyclic permutation. Actually, the way you derive this in this exercise is by the composing the permutations here in cycles. And then we assemble things in such a way that you get this sample thing. You do also this in classical statistical mechanics when you do the virial expansion. This is actually also in the virial expansion you want to go to the grand canonical ensemble to simplify life. So this is in a way a quantum virial expansion for thermogases. Very good. Now, of course, ensembles are equivalent. Sometimes they tell you that ensembles are equivalent in only the thermodynamic limit. And of course, the large limit is only, they are only extremely equivalent in that way. But you can always get one information from the other, information from the grand about this function for this function of isa versa. But here, we have one relationship. You know all the canonical partition functions for all n, the grand canonical partition function. But this can be inverted by a well-known formula. Maybe I should start right in there. Which is this one. So you just take a contour integral over the fugacity divided by 2 pi i. And you have this guy. And then you take kappa minus m minus 1. So this is a triviality while writing. The only thing I'm writing is that you can always extract the coefficient of an infinite power series by considering it's the residue with an appropriate power of k divided, kappa divided. This is really completely true. Now, why this is related to spectral theory? Well, it's related to spectral theory because the Hamiltonian, if you think about this matrix as the exponential of an operator Hamiltonian, you can actually look for this eigenvalue problem n goes from 0, 1, and so on. And actually this eigenvalue problem is very well defined because this operator, you can think about this as an integral operator, an integral kernel. And this actually has a discrete spectrum because this operator is what is called, in mathematics, a trace class operator. So this means, essentially the trace class means that all these ZLs are well defined. Actually, notice that this can be computed from the spectrum by summing overall energy levels. So you can think about this as e to the minus l en. And this quantity is well defined for any l, 1, 2, and so on. And it's a theorem that is an operator of trace class. And in this case, it's also positive. It has a discrete spectrum. So you can think about this as the spectrum of fermions. You have three fermions with a Hamiltonian, which is still didn't write for you. But you can think about this as defining a spectral problem. So you have fermions with energy levels. And this is a free Fermi gas. So you know very well how to do thermodynamics. You just have to feel. You use polyprinciples and feel the levels until you put the last fermion with label capital N. Now, it's also a relatively elementary result that the grand canonical partition fashion of this Fermi system can be written as a determinant of this operator kappa. So you have this equation e to the mu minus en. So this is the famous Fermi factor that appears in the statistics of Fermi gases. And this is sometimes called a first home determinant. So it's a beautiful fact that you can rephrase all your statements. You can rephrase everything in terms of elementary statistical mechanics. But if you are fancy and you want to do some things like more generally, you can think in terms of spectral theory. You have a very well-behaved operator, rho, which has a discrete spectrum. And then this guy is called the freth on the spectrum determinant of this guy. OK, now, so this is very good. But what is it good for? This is very fancy. But maybe it just tells me nothing new. Well, in order to make progress, let's try to see what is the Hamiltonian that appears here. Let's try to write this Hamiltonian. Now, the Hamiltonian can be in principle red because here, I know the exact kernel of this guy. The exact kernel of this operator was given to me by the matrix integral. So I probably can write this operator. And it turns out that rho can be written as follows. And here, you see, this is a one-dimensional gas. Because here, I'm integrating over the one-dimensional variables. So each xi is the coordinate of a fermion in one dimension. So it's a fermi gas in one dimension. Now, in one dimension, I can write this kernel in this way where x and p are canonical commutation, canonical Heisenberg operators with commutes to h bar. And h bar is not any quantity. What is h bar? h bar, I have to be able to read it from here. And it turns out to be 2 pi k. So it's a very excellent exercise that I propose to you that you can write this in this way. And the function ux is log of 2 cos x over 2. And t of p is the same function variable for p. And notice here that you haven't written this is not. So you can think about this as a potential. And you can think about this as a kinetic term. Now, this is very easy to see what it is. This looks like complicated. But if x is large, this is just x over 2. And if p is large, this is just p over 2. So you think about this as a kinetic and a potential term. You have a very simple physical picture for this system. You have here a potential which goes like this, goes asymptotically to x over 2. So you have something like this. And this is a linearly confining potential. So there will be energy levels in this potential. And then you put your thermions here. Now, the thermions are ultra relativistic in the sense that their kinetic term is a linear term in p. Of course, these, they will get corrections. So this is actually not very difficult system. It's a system of linearly confined fermions, which are ultra relativistic in one dimension, free fermions. So the thermodynamics of this system, at least in this approximation, in which you approximate this by this, you could do this in elementary thermodynamics or statistical mechanics. And this is what we're going to do. Now, you have to be careful because thinking about this potential and kinetic term is slightly subtle because this is not the exponential of minus ux minus tp. You see, there are some commutation issues. Actually, sorry, there is no half here. There is half here and there's a minus here. There are some commutation issues. This is not exactly like this. But this will differ of these in H bar corrections. So if you want to do, for example, a semi-classical analysis of this Fermigas or the thermodynamic limits of this Fermigas, this commutativity issue will be not crucial. It's only crucial when you look at sub-leading corrections. But there is something very interesting going on here. What is H bar? H bar is 2 pi k. Remember that k, k is inversely proportional to gs3. So this means that we have a quantum Fermigas and H bar is small when string theory is actually strongly coupled. So this is, in a sense, a dual version of the T2A string theory because then the easy regime for this Fermigas is the semi-classical regime. As always in quantum mechanics, when H bar is small. But H bar is small. It's a strongly coupled T2A string. So this picture of the matrix model opens the way to study strongly coupled T2A theory, at least for this observable, which is the partition function. By using just the WKV approximation to a one-dimensional Fermi problem. Now we'll see that this is very useful. So this is a little bit of a miracle that this is working this way. But now let's do this exercise. Let's do this exercise of studying this Fermigas in the thermodynamic limit. So as I said, this is an exercise that you can do in elementary thermodynamics. So let's do it. Let's assume, in my data, I give you this problem. I give you fermions in one dimension with this potential, this dispersion relation. And I ask you to calculate the free energy of this system at large n in the thermodynamic limit. How would you go about this? Well, there are many ways to do it. But there is a very easy way to do it, which is to consider some sort of classical Hamiltonian, which I'm going to put like this. So this is the approximation to this Hamiltonian. Also, of course, let me actually define the classical Hamiltonian. The classical Hamiltonian will be just ux plus tp. As I said, this is almost what you get, except for h-bar corrections. And on top of that, I can actually approximate these at large energies by this. Very good. So how do you actually go about analyzing this system? Well, there is a very easy way, which is very intuitive, which is looking at the Fermi surface of this gas. So the Fermi surface is the surface of Fermi which have the same energy. And this you can plot very nicely. So this is the available volume of phase space for a fermion with such an energy. So if you look at the Fermi surface, you get a square. Because you have this thing here. So this is 2e, 2e, minus 2e, 2e, minus 2e. So the Fermi surface of the system is this. So this is the Fermi surface. And you know semi-classically, this something I was knowing in almost 1900, that semi-classically, you must have one quantum state for every cell in phase space. So the number of fermions should be approximately given by the volume of phase space, the volume of e, divided by the volume of elementary cell, which is 2 pi h-bar. Now, this is very easy to calculate. So this is 8e squared divided by 2 pi h-bar. On the other hand, I know from elementary thermodynamics that this thing is n where you just change e by mu. So this is one of the relations that you get here. So this means that j mu is approximately given by 2 mu cube. Just have to integrate this, right? Divided by 3 pi squared k, when I'm already using the fact that the h-bar is given by 2 pi k. Well, so now I have computed the grand potential of this Fermi gas at large n. This is a large n approximation because I use semi-classics. And semi-classics is good. What happens here is that when you have many fermions, n is large, since you have a confining potential, the energy of these fermions will get bigger and bigger. So you are in a pruning which, effectively, the energy is large. And the semi-classical approximation, no matter what the value of h-bar, it's always both if the energy is very large. So this is a semi-classical calculation. OK, now you know that this relationship here can be approximated. And this is actually what you have done in statistical mechanics. You haven't used this equation. You have used an approximation to this equation which says something that you have used this approximation. And then you have said that a large n in the thermodynamic limit, so you have some sort of Laplace transform.  The thermodynamic limit f of n is given by j of mu minus n mu, where mu is the derivative of j. Sorry, n is given by the derivative of j with respect to mu. So this is what you learn in the thermodynamics. This is how you relate in the thermodynamic limit the canonical ensemble to the grand canonical ensemble. So it's very easy now. You know j of mu. So you do this Legendre transform. And then you find, first of all, you find that mu, mu is determined by this equation, actually. And n of mu is equal to this. So you derive immediately that mu is given by a square root of 2 pi k to the 1 half n to the 1 half. Another thing that you learn from this relationship is that n large, the thermodynamic corresponds to mu large. The fugacity has to be large. And now you get f of n. And what do you get? Well, you plug it in here, and you get precisely, exactly, pi square root of 2 divided by 3 with a minus sign, n to the 3 half k to the 1 half. So this is the n to the 3 halfs of Clevanoff and Chaitlin on the back of an envelope, just using elementary physics. Let me just check that all the factors are OK. Yes. So you see, this n to the 3 halfs here is completely demystified. This is the scale that you would find in elementary statistical mechanics if I give you a thermi gas with a linearly confining potential and a linear dispersion relation. Actually, depending on the power here, if you change the powers here, alpha and beta, you get any power of n that you want, just by adjusting alpha and beta. So the n to the 3 halfs is actually completely expected from this behavior. And not that you even get the coefficient right. So the coefficient is exactly on the spot that you get from any SCFT. So OK, this is successful, but this is not really what we want to do. This is nice to do this, but we would like to actually go beyond that. So what's the next step? Well, remember that this result is really what I call, sometimes, the strict large n limit. So the strict large n limit means that you really take n to infinity. And this is the leading term in expansion at large n. And by the way, the study of this still larger limit that Kfix was first done by Herzog Klevanov, Pufu and Sabdi, essentially using this matrix model directly. Now, this Fermi-Gasaprost that I'm trying to sell to you is good in driving this in a very limited way. But its power is that it makes possible to actually get the full 1 over n expansion in the m theory limit. So how does this go? How do you do this? I'm going to summarize the calculation. The details are actually fun, because this is, as I said, elementary statistical mechanics. So you can do whatever you want very explicitly. So how would you go about improving this result? Well, if you want to improve this result, you have to go slightly away from the thermodynamic limit. And this means going slightly away from the leading semi-classical limit. Now, where did we make approximations? Well, we make approximations in various ways. We neglected here commutators. We just here took the leading behavior of these functions when x and p is large, et cetera, et cetera. And this can be actually, so we were not very systematic in one word, right? So you want to be systematic about this. You can be. And you can write down some much more precise formula. And the way to actually improve from this is to actually do a systematic WKB approximation. So here we took very large challenges as compared with h-bar. If we want to go beyond that, we have to incorporate corrections in the h-bar approximation. So first of all, let me actually explain how you would go about it. Well, what you find is that this potential of this Fermigas has a WKB approximation. This is by general reasons in thermodynamics. So you would have something like this. So let me use k as my expansion parameter. So k is h-bar. The leading approximation to j was what I computed here. It goes like 1 over k. But then you will have higher order corrections. So what we computed, what I just compute is j0 mu at large mu. So this is just what I did. I just computed this function at large mu. It turns out that you can't do much better. You can compute all these functions, jn at all n, if you want, because you don't get a closed formula. But you can push it as far as you want. And you can get the exact dependence on mu. So what do you get when you do this? I don't have much time to tell you the details of how you have to do this. But it's always playing in the notes. And if you have questions, you can ask me later on. So how this gets corrected? Well, I have to say that this is a very easy calculation as well. And the reason, the way you do this calculation, the easiest way to do this calculation, is to combine this formula, is to use this formula, and combine it with this formula. So how does this go a little bit? So let me just write capital O. Now the Zl's are just the traces of e to the minus Lh. Now if you want just to compute j0 exactly, what you have to do is to compute the exact semi-classical approximation. But in the semi-classical approximation traces, you can just do classical statistical mechanics. So how do you do classical statistical mechanics? Well, you integrate over phase space with the right one-to-measure. And then you use this integral here, the classical of xp. And now you don't approximate this by its polygon, but you use the exact log cost that I was using. When you do that, so by using this, this is really the exact, this will give you the exact leading approximation. So the details, as I said, are in the notes. You calculate this at L in closed form. You plug in there. You have an infinite series. You resummit, you do tricks from the trade. And then you get the following function. j0 of mu is equal to mu cubed divided by 2 mu cubed divided by 3 pi squared. So the k, I don't include it because it's here. Then you get the linear piece, which was sub-leading. So we didn't see this piece before because we were just leading, getting the large mu approximation to this guy. We get a constant involving, again, z of 3 for those of you who like number theory. And then we get an infinite series of sub-leading corrections. Let me write them like this. a0L b0L mu plus c0L e to the minus 2L mu. And these coefficients can be computed explicitly at any L. Now, what is this telling me? Well, this is telling me that this function is more complicated, but it's very interesting to actually look at these exponentially small corrections. Why? Well, because if you remember, I can get from j0, I can get f of nk by doing a Legendre transform. So at leading order, what I have to do in order to see what kind of correction this gives me in f and k is just to substitute the value of mu given by the saddle point approximation. But what is this? Look at this. These are exponentials of minus 2 mu. And then, if I put here the value that I just obtained from doing this trivial calculation of the Fermi gas, what I get is minus square root of 2 times pi k 1 half n to the n 1 half. So you have seen this before this morning. This is exactly the way of a membrane instanton. We saw this. I wrote this this morning. This actually is e to the minus L over Lp to the cube. So in the same way that the tough expansion allowed me to calculate all these nonpretative corrections in terms of what it is, this procedure in which I go to this other picture, in this procedure, remember, the natural regime is the strongly coupled string, super string, because h bar is small, is g string large. And then what I see is that the natural nonpretative corrections that appear here, the natural exponentials that appear here are not the Walsh instantons, but the membrane instantons. In other words, this is a formalism in which I can compute the membrane instantons among theory systematically. Systematically. You see? Here they are. All these guys are membrane instantons. And of course, this is an expansion around k equals 0. So this is still not yet in theory. In theory, I want membrane instantons, but I want them for finite k. But this is a good starting point. We get them for free. They appear here by doing this calculation. So you use the thermodynamics of this gas. And here you have a systematic series expansion in membrane instantons. OK. That's already very good. So what else can you do? Well, the next thing you can do is to try to go. I mean, this is very systematic, right? So every step gives me a gif turn, right? Very, very leading order. I get into the three hulls. When I do the first WKB expansion with corrections of finite mu, I get the membrane instantons. So now what kind of gif I'm going to get next? Well, we are going to see what you get by doing the next WKB approximation. This is the next natural thing to do. I did the leading order and a very serious worker. And then I go to the next order. So I go to the g1 of mu. Let me write it here. I have a question, right? Yes. So let's see what happens for j mu. This is a more complicated calculation. Here you have to use more technology. But essentially, all the technology was introduced by somebody called Bickner in 1930. Bickner was interested in how to calculate quantum corrections to thermodynamics. In order to do that, he invented the Bickner formalism, the Bickner formulation of quantum mechanics was invented to actually do exactly this kind of calculation. So it's not strange that the best way to do these calculations is Bickner's formulation of quantum mechanics. So this is really elementary quantum mechanical calculation. And then you get mu over 24 minus 1 over 12. And then you get essentially a series like this, but with new coefficients. So you get something that goes like this. And then you can now compute the n of mu. It's given by a coefficient. And then something which is already exponentially suppressed. So what happens is the following. You see, there is a diminishing of the intensity of these corrections as you go farther and farther to WKB expansion. If you are interested in polynomial contributions to these guys, already with the leading order and the next to leading order, you are done. Because everything else will be a number plus exponentially small terms. So this is a sort of non-renormalization theorem that tells me that, essentially, the polynomial piece of this function is exactly one loop. Because this is a one loop calculation in quantum mechanics the next to leading WKB approximate calculation. So this has been very useful. So what you get from here is that this JWKB has the following structure. It's given by a perturbative piece of mu.  And this guy here is given my mu cube to mu cube over 3 pi squared k plus mu of 1 over 24 k plus 1 over 3k. And then there will be a function just of k. Why? Because there is this guy which has a 1 over k. There is this guy which has a k squared. There is this guy who has another power of k. And I put them all together in just a function which doesn't depend on me, only depends on k. So this is very nice. And now I remembered that I have a formula that I erased, which gives me, now I can repeat what I did before, I have a formula that gives me the free energy or the partition function when I know something about the grand potentiometer. Remember that when we had Joe's this piece here, I was able to compute these reading piece here. And now let's try to see what we get by including this new factor here. You can tell me, well, you are not going to get much, right? But actually, the answer is not true. This is actually not true. You get a lot of stuff, and let me explain you what. So I use this formula here. And here, approximation means that this formula is true up to exponentially small corrections. But it's going to be exact at all orders in 1 over n. So I get this thing here. Actually, there is a 1 over 2 pi i that I forgot. And here I have a new. Now, what's the function that when you do a Laplace, what's the function that you think when you do the Laplace transform of a cubic polynomial? Well, you have a quadratic polynomial. I get one function, but you get essentially a Gaussian again. But if I have a cubic polynomial, and I choose the integration contour in such a way that this integral converges. So before the integration contour correctly, what I get here is precisely an 80 function. So I get e to the ak. I get a function here, set k to the minus 1 third. And then here I get an 80 function. c to the minus 3 of k, n minus b of k. And these functions here, so this function here, so this coefficient here is c of k divided by 3. So you want c of k is 2 over pi square of k. And this guy here is b of k. Now, notice something very interesting. The 80 function is known to go. This goes to e to the sum coefficient minus n to the 3 halves. So the n to the 3 halves, that's clever enough, and Chaitlin found, is exactly the leading order, exponential behavior of the 80 function. But the 80 function has an infinite series of corrections, which actually scale like 1 over n to the 3 halves. So you have something like this of this form. And this is the full 1 over n expansion of the n theory partition function. So if you neglect exponentially small terms, so terms which go to 0 exponentially with n large, this is the full 1 over n expansion. And what is this doing for me? Remember that f g's, in the tough expansion, the f g's have a constant plus a polynomial piece. Now, this function a of k is taking into account all these constant map contributions c of g. And this 80 function, the sparsing of this 80 function, reproduces all the polynomial pieces of the tough expansion. So in one single strike, and resumming the genus sparsing of type 2 a super string theory, n theory is doing, for me, something remarkable. It's taking pieces of f g. It's taking pieces of this divergent expansion. And it's putting them together in an entire function, in a function which has no divergences whatsoever. The 80 function is a perfectly entire function on the complex plane. So you see, n theory is getting rid of these divergence of perturbation theory. And it's readjusting things for me in such a way that I get things which have analytically much better properties. Actually, it's very interesting that this function has an overall power here, which I remember correctly. So I guess this is 1 over n to the 1 fourth. This exponent here is a typical exponent of the 80 function. And you can see that when you see what it gets in the partition function, it gives you a sub-leading logarithmic correction in n to the partition function. Now, this 1 over n to the 1 fourth, this logarithmic correction. So this gives me a logarithmic correction to the free energy. These logarithmic corrections have been much studied by Asok Sen and collaborators in the context of logarithmic corrections to the entropy of black holes. But mathematically, the problem is very similar here. I mean, the way Asok and his collaborators are trying to reproduce these log corrections is by doing one-loop calculations in supergravity. So together with Asok and my student Alba Grasi and Sajantani Bhattacharya, we actually try to see if one-loop 11-dimensional supergravity reproduces this guy, and it actually does. So here, not only then to the three halves, which is the leading term, it's just the pure classical gravity result is reproduced by supergravity, but also the one-loop correction is actually accounted by 11-dimensional supergravity. Moreover, if you see what is an expansion into the three halves, remember my basic dictionary. n to the 3 is l over lp to the 6. So n to the minus 3 halves is proportional to lp over l to the 9th. And a power series expansion in lp over l to the 9th is what you should expect for the quantum corrections to a gravity theory in 11 dimensions. So all these series of corrections that the 80 function is producing for me are coming from loops in supergravity, at all orders in the Newton constant. So this is really a surprising result, because it's giving me an exact result in all order quantum supergravity. The only thing I'm missing here are non-pertorative corrections. But all the pertorative corrections coming from 11-dimensional supergravity are accounted in this calculation. And you notice that we didn't have to kill ourselves to this calculation. We just took this Fermi gas, and we did the next to leading order calculation. So this is the magic of the Fermi gas. By doing one-loop calculation in elementary quantum mechanics, well, not that elementary, but quite simple, we get an all-order result in quantum supergravity. Actually, it has been argued recently by Gomez, Davolkar, and Drucker that this 80 function, the full 80 function, they argue that you can actually calculate it, represent from a supergravity calculation using localization. So they really can argue that this result is also compatible. The all-order result is compatible with supergravity. Well, you see the power of these approach is that we are really using ADS-CFT to go really beyond the classical supergravity approximation. So we can get results at all orders in quantum supergravity. OK, now, what's next? What should you do next? Well, we have already noticed that n theory is reorganizing the perturbative expansion in a different way. So we don't have to longer worry about the divergence of the FGs because these pieces here are being taken into account in an entire function. So we should ask ourselves, OK, now it's clear that n theory is taking care of this perturbative part, this quantum gravity loops in a nice way. But what happens with the membrane instantons? What happens with all these contributions, which are exponentialism, or what happens to them? Let me give you at least a hint of how this works. It turns out that the exact n to function, this exact contribution of membranes, can be resummed in closed form. So remember that the leading order was a constant, AOL. But actually, if you include all orders W, K, B, you can resum them order by order. And then you get here a function of K mu squared plus BL of K mu plus CL of K. And this, it's multiplying E to the minus 2L mu. Now it turns out that these coefficients can be determined in closed form. This is highly non-trivial. I'm telling you that you can resum the full perturbative WKB expansion. There are just three or four quantum mechanical problems where you can do this. And this is maybe the fifth one, I don't know. But this is not common. It's not common that you can resum the full WKB expansion. And why is this so? Well, remember that in order to get this tough expansion, we use the fact that this matrix model could be related to topological string theory on local p1 and test p1. It turns out that these coefficients are also determined by topological string theory on the same kalabiyahu. But now it's not the standard topological string theory. It's something called the refined topological string theory in what is called the Necker-Sofs-Satashvili limit. You don't know what is this. It doesn't matter. The only thing that you have to know is that these coefficients can be completely in closed form as far as you want, as explicit forms of K. And the reason for those of you who know a little bit about this, let me just spend a couple of sentences, is the following. In the Necker-Sofs-Satashvili limit, it's not a refined topological string theory. It's related to the solution of spectral problems. It solves spectral problems for you. Now, it turns out that the spectral problem that appears here, the spectral problem suited to this operator can be matched in such a way that it's identical to the spectral problem typical of local p1 and test p1. And then you just can use all the technology developed for this model to actually read out the coefficients. So some of them are listed there. And then you are very happy. You are very happy. You are computing this full quantity. And notice that by inverse Laplace transfer, I can read from these main-bring instanton contributions. And then you are very happy.  And suddenly, you notice something bad. You notice that this guy diverges. And this guy also diverges for physical values of K. When K is equal to 1, this guy diverges. When K is equal to 1, this diverges and so on. So you did something wrong. Because my matrix integral doesn't diverge. So for example, b1 of K is proportional to 1 over sine of pK over 2. So you see when K is equal to 2, this diverges badly. So actually, whatever rational value of K, you can always find one coefficient of diverges. So this has an infinite number of poles in the K plane. And then you are really depressed and you don't know what to do. Because this means that your calculation is missing something.  Because this guy is only taking into account main-bring instantons. But remember that there were all these worst instantons that we computed already, actually. We already computed them. So we should add them. So it turns out that this JWKB is not the exact answer. Because this only takes into account perturbative corrections in WKB. But there are non-pertorative corrections in WKB. So the full JMU, and simplifying things a little bit here, the story is slightly more complicated. But let me just provide the idea, because I have just five minutes to go. The idea is that the full grand potential is the polynomial piece, which is responsible for the n to the 3-halve behavior and for the quantum gravity corrections, plus the m2-brain thing. And then you have to add a piece, which I want to call Walsh's new K. And it turns out that when you do this correctly, and again I'm glossing over the details here, the poles here cancel against equal poles but with opposite residues here. So poles cancel. And you find a function which is an analytic function at mu at infinity. So this function has a finite radius of convergence at infinity. So what is going on here? Well, this concentration of poles was found by a true Moriyama and Okuyama, based on the pre-issue that we had done. And it's a very remarkable phenomenon, because it's telling you that if you try to construct a quantity in m theory just by adding membrane instantons or just by adding Walsh's instantons, you are going to fail. This is another way, this is the way in which m theory sees that a theory based only on Walsh's strings and membranes is incomplete. The way tight-to-air super-stream theory sees that Walsh's a theory based on finite stream is incomplete is through another mechanism. It's through the factorial divergence of the series and the necessity of including non-pertoactive terms.  But its coefficients, the price you pay if you don't include the Walsh's instantons, is that all these coefficients have singularities. Now once you have the singularity, once you have both contributions, the theory is completely consistent and no longer you have cancelled these poles, but you also have obtained an analytic function at least in some domain of the mu space. But in terms of analyticity, this is much better than the FGs. Remember that the FGs, the sum of FGs was a factorial divergence series. This is really a convergent function. For any fixed value of k, this function has a finite range of convergence. So this encodes, some of the results here are conjectured, but this encodes all the information about the matrix model BGM at all orders in 1 over N and including, as far as we know, all exponential and small corrections. For example, you can check, remember that at the end of the day, you have to check that you recover the finite gauge theory results. After all these should be a way of computing ZN of k. Now you can actually see that this conjecture gives exactly this, this you have to do it numerically. We have verified this with 500 decimal digits in some cases, okay? So it's clear that there are, well I don't think we are missing any non-perturative contribution here. Everything is encoded here. Okay, now let me make a final comment. Now notice that in the large expansion of TOFT, what is natural, what comes to you naturally are the world city instantons. And then the membrane instantons, you have to add them as non-perturative effects. Here is the other way around. You look at this Fermigas problem and then this standard WKB expansion, so the perturative expansion, what it gives you are the membrane instantons. And the world city instantons, actually, if you look at how it depends on mu, it depends on mu in this way. And this is what it should be because if you look at mu here, it has a K over one half. So it's only when you divide by K you see that you can't trade this by a square root of lambda. So this is mu over K. But in this formalism, something that has K here is non-perturative in H bar, remember that K is in H bar. So the world city instantons, which were appearing for free in the TOFT expansion, here are due to quantum mechanical instanton effects in this Fermigas. So this Fermigas has the perturative WKB story and then you have to add quantum mechanical instantons and these are the world city instantons of the original theory. So if you want the Fermigas, it's a sort of S-dual version of the original string or the original matrix model or the original type of super string. So now just my last sentence because I still have maybe 30 seconds. This formalism has been extended recently in the following way. Remember that here we started with a problem with what's based on this ABJM matrix model. So we, you know, people do localization, got this matrix model for us, we started analyzing it and then we found in the process of analyzing this problem that we could use all these results of topological string theory first to calculate these world city instantons and then also to calculate the membrane instantons. So this was very nice. But then now you can ask the following question. Can we actually use insights from this thing to actually understand better topological string theory? The topological string theory is a simpler model of string theory but it also has the same problems of a standard string theory in the sense that it's perturative expansion is a purely divergent series and there is no known object that actually encodes from which you can obtain the standard topological string amplitudes by using, by doing an asymptotic expansion. So in some cases there are proposals for non-perturative definitions of topological strings but there has no completely coherent point of view and it turns out that this approach to ABJM suggests a way to deal with this problem topological string theory. So remember that here one key thing in our problem was to use this operator to define this Fermi gas picture. And this operator you can think about this as the quantization of the Fermi curve. Now topological strings come with curves in the non-compad case and these are called the spectral curves or the mirror curves. It turns out that these mirror curves can be quantized and this was guessed by people a long time ago and there was a few years ago there was a paper, there were a couple of papers where people did the WKB expansion of this quantized curve and they found this refined theory. Now it turns out that you have to do more than this. You have to take the mirror curve and produce an actual operator, not just a formal WKB expansion. When you do that you find that this operator is of trace class and it has a well-defined spectrum, a discrete spectrum and then it turns out that the the retorative WKB piece of this spectrum gives the refined topological string but then there are also wall-city quantum mechanical corrections to this spectrum and this quantum mechanical corrections give you the standard topological string. So by quantizing the mirror curve you get in the single strike both the refined topological string in the necrosis of the limit and also the standard topological string. In particular you can write down matrix models, explicit matrix models whose one of an expansion gives you the topological string partition functions FGs as an asymptotic expansion. And this you can do for essentially all examples in with low genus. So not only this formalism is useful for understanding the non-pretative completion of super extreme theory but you can also use it to give a non-pretative view on all the interesting models of extreme theory and the picture is always the same. You have asymptotic expansion and the asymptotic expansion you won't eventually realize it as the expansion of something well-defined. And this something well-defined in the case of ABGM theories, this matrix model and in the case of topological strings is an operator associated to the mirror curve. Okay, let me finish here. Thank you.


Hello and welcome to the session. In this session we will discuss a question which says that the dimensions of a window are 2.65 feet by 1.75 feet. A scale drawing of window has dimensions 3 by 4 inches by 1 by 2 inches. Find scale and scale factor. Now before starting the solution of this question we should know some results. And the first result is scale drawing is a proportional two dimensional drawing of an object and scale is equal to drawing dimension upon actual dimension. Now a scale can be written without units when units are same in joint and in actual and a scale without units is called scale factor. Now these results will work out as a key idea for solving out the given question. Now let's start with the solution of the given question. Now we have given a window and its scale drawing and here the dimensions of window are given as 2.65 feet by 1.75 feet and dimensions of its scale drawing are given as 3 by 4 inches by 1 by 2 inches. Now we know that scale drawing is a proportional two dimensional drawing of an object. This means these two figures are similar. Now first of all we have to find the scale. Now from the key idea we know that scale is equal to drawing dimension upon actual dimension so as the scale is equal to drawing dimension upon actual dimension. Now here in these two figures you can see that actual width of this window is 1.75 feet whereas the width of this window in this drawing is 1 by 2 inches. So scale will be equal to drawing dimension which is 1 by 2 inches upon actual dimension which is 1.75 feet. Similarly we can find the scale by comparing the length that is the length in actual and length in drawing. Now the actual length is 2.65 feet and length in scale drawing is 3 by 4 inches. So scale is equal to drawing dimension which is 3 by 4 inches upon actual dimension which is 2.625 feet. Now when we convert this scale in same units then both of them will give us same results. Now we have to find scale factor and we know that a scale without units is called a scale factor. So first of all let us write this scale without units. Now we know that 1 foot is equal to 12 inches so 1.75 feet will be equal to 1.75 into 12 inches which is equal to 21 inches. Now 1.72 inches upon 1.75 feet will be equal to 1 by 2 inches upon now 1.75 feet is equal to 21 inches further this is equal to 1 by 2 upon 21 which is equal to 1 upon 2 into 21. Now 2 into 21 is 42. Now here this scale is written without units so here scale factor is equal to 1 by 42 or we can write it as 1 is to 42. Now if we will write this scale without units then again we will get the same result. So let us take this scale which is 3 by 4 inches upon 2.625 feet further this is equal to 3 by 4 inches upon now converting feet into inches this will be 2.625 into 12 inches further this is equal to 3 by 4 inches upon 31.5 inches which can be written as 3 by 4 upon 31.5 which is equal to 3 upon 4 into 31.5 and this is equal to 3 upon 126 now 3 into 42 is 126 so this is equal to 1 upon 42. So again we are getting scale factor is equal to 1 is to 42. Now we know that these 2 figures are similar and we know that in 2 similar figures corresponding sides are proportional. So from the figures n is equal to upon ABQR upon VC. So this is the solution of the given question and that is all for this session. Hope you all have enjoyed the session.


What if I do this though? No, no, no, no, no, no, no, no, no, no. Ugh. What if I do this? Dodget, you can't. You're in the air. It was already too late. Chatsu! How's it going, everybody? Dodged it all a year back with some more dragonball fighters online matches. And if we are once again facing Syriac, so it's going to be another pretty intense first of three. Let's team turn around. Hey, hey, I'm just really curious about what happened last time. Let me ensure play out. Chatsu's here too. Oh, you want to talk about what happened last time? I'm just curious just in case you know somebody missed it. All right. Well, I guess I could explain it if anybody missed it. So I'll explain through these hands. I might have lost 3-1 last time. It was a bad set for me. But you know what? I didn't tech like that last time. That's the difference maker. Dude, could you stop? Stop. Wait, stop. You knew already. Chatsu, hold him down. Wait, what? Wait, what? I was not hanging 10 when I said that, bro. Uh-oh. Wait, I'm not blocking. Hold on. My TN is supposed to be more. Why is this doing so much damage? I've asked a lot of questions. How's your day going? The spark? Stop. I thought you'd expect the 6M, but you didn't. I don't expect anything. Listen, so here's the deal. I got sparks. Oh, no. No. The sparking right out. This is going to hurt me more than you. Oh, gosh. What do you mean?  That hurt me way more than it hurt you. Your scarf, I'm dead. It's sparking. That doesn't drain my health as much as it just did. And now I'm really a- Come on, Twitch of the God, why not? I am down right now. Hold on. I have a plan. Homing energy orb. Yeah, that's what everybody does. I was going to do the most obvious thing, and I was going to see if you were ready. Come on. You know that was sick. Bro, where's my second spark? I was not expecting that to hit you at all. God, I haven't played Goku Black in so long. Dude, you just keep jumping. I keep excited. From a mix, I wonder if Goku Black's going to get buffed. I started being inquisitive. That was really the downfall of me. This guy needs to get buffed. Gets it by a slow-ass command grab. I don't like all of that. I just like it. Wait, I'm not dead. Oh, in fact, I'm really not dead. I didn't do the right thing. Drop it. All right, whatever you want. He is so loud. He yelled it so loud. I thought he'd hear it, actually. Don't do it, man. I just I know I'm predictable. But geez, it's like, hey, Goku, come fight. Oh, hold on. Let me skedaddle. Thank you for your years. Nothing. All right. Stop this, though. No, no, no, no, no, no, no, no. What if I do this? Dodge it. You can't. You're in the air. It was already too late. Wait, hold on. Can I block that? No. Oh my god. OK, I was like, I thought I was blocking it. All right, but you know, we've been here before.  Voli, voli-muffist. What about my kick? Voli-muffist, voli-muffist. Well, Goku versus Vegeta. I'd say this is a pretty canon way to end it. Bro, you have blue hair. I am yellow hair. Shocker. I can't. I know it's a classic opener, but I still can't believe it. You're going to do it again, too. Nah. I'm not.  I know you want to. I'm really not, though. Mixups. Same sign. Oh no, I missed. I'm going to. Oh, I missed. You're going to want, though? Nothing. Oh, that is just unfortunate for you. Homie drop. That is just unfortunate for you. All right, homie drop. I'll drop you to the ground. I didn't like this variation of homie drop. I did not. I had a much more pleasant thing in mind. I'm more of a fan of the classics, but I'll take that. The old rough first game, but, you know, we won. It was close, but now. Volleyball hands, these hands. Volleyball hands. I did say volleyball hands, huh? You're not ready for my volleyball hands. Wow, his hands flowed so good. How does he sound like he never.  I don't even know what God did. It's not even like I was hitting you that much. He's just kept doing it all the time. I just had these hands, dude. That's all I have. That's all I've ever had. Oh, that would have looked so cool. And now here I am. That's a good TN. You got three eyes, can't see anything. Whoa, bro. That's a low blow. How is that a low blow, bro? You should be able to see more. People with three eyes are going to be really offended that way, you just said. Not everyone with three eyes is 20-20. Kiddy it.  I missed a nimball already. Stop. I too wait. Stop lying, game. Demon of them. Bucky. But the ground counter. Don't, don't, don't. Orb. I kind of forgot to hit the counter. My whole plan was ruined. Are you going to ruin the day you did this? Yeah. What? That was so fast, frame one. Nerf that. Oh, I got eye lasers.  Bro, surprise if you're really about it. I'm not about it.  Yeah, well, it was more of a threat than something I'm actually capable of doing. I've been sick. Oh my god. All right, I can still make this come back. You can. Do it. You know you want to, so just do it. You know you want to just do it. You know exactly what I'm talking about. I was about to do it right.  That's crazy. I RSVP'd in everything. All right, I'll do it. Uh-huh. You promised me that you would let me get away with it. I promise no such thing. Dude, why is this combo still going on? Embarrassing. That's interesting. I'm going to do the level one variation just because the level three wouldn't have hit both of you guys. And I want, oh, that's interesting how that happens sometimes. Oh, the whole energy just changed up in here. Don't use Keyblast to start it off. That's crazy. Buff that move. That move is trash. Don't use that move. I don't like anything that's occurring. For this anime? Anime? Dude, stop. Oh, God, I'm almost dying. OK.  I was so afraid to do any of those moves.  So bad, bro. Is that just one one right now? No, it's two. Are you up to it? No, I won one. You did it. Did you? No, I don't think so. It's only been eight minutes. This is the way. Wait, did you? Wait, did you win one? No, I'm tripping. I don't know. Maybe I lost all of them, but it felt like I won. I might have lost everything. I think it's too low. I'm confused, though. Oh, God, well, this is not a good look. It's not a good look at all, actually. It's a terrible look. OK. I'm dead. My God, Tien didn't even get to do anything. He just kind of existed. I'm just genuine. OK, so I've lost everything. All right, I'll just buy it. I believe you. I'll just accept it. Don't drag her. All right, I'm not looking too good right now. Let me sit up. Let me turn on my switching chairs, switching to a better gaming chair. That's a nice chair. Yeah, DS Racer, you know how it is. Your back is going to hurt, man. Yeah, it improves my gaming ability tenfold, so it's kind of worth. Not a perfect. Who really won? Who really won, then? You did, OK? Just get off me. Relax.  All right, I could make this comeback. West City will never. I'm literally the 13th highest BP ranked yoncha. It says so under my name. That's my turn. You do have full ability to mash there. Oh, my God. Oh, my gosh. Oh, my gosh. Oh, no, no, no, no, no, no. Dude, I'm not going to make a comeback if I don't start now. I mean, I should have started like years ago. The fact that you're still doing this is just an embodiment of how childish you'd be right now. You're just going for it.  That's why. Yeah. I'm like, that's not my turn. That wasn't my turn for a baby, it was my turn. I don't know. This is not good. You're just swinging. And you have enough of an advantage to swing. Do I even live this? I hope not. Oh, I'll take it. I was so worried about another. I just want to say that that last one was a miracle. Oh, my God. All right. I could still make this comeback. It's just a little less likely than it ever was before. And now it's. Oh, my God. How do you even get this? All right. Bad game, bad game. OK, all right. So I guess I have one, allegedly.  I probably did really lose the first two. I just felt like I needed to win that second one so bad that, like, when I lost, in my head, it was registered as a win way earlier. I had already won. Stop. That was an interesting strategy. Start. I've really lost a lot of momentum. That imaginary win you gave yourself really propelled your momentum. Stop that. That is, that is threatening. Damn, that is annoying. All right. You can finally calm down. We can be free again. Freedom.   That was aggressive. That wasn't really a combo, more like a demonstration of what not to do. Oh, no, no, no, backlash. All right, just before my spark runs out, just stop time so I can maximize the mage. I dislike. I almost barely registered that was a command grab. I'm not super nasty. I'm not trying that though. Chouts, you hold them down. Oh, Chouts, you're my air movement. Oh, that was sick. That was unfortunate. Air movement. Nice.  You got hit by the 2M. 2X that. OK. RUN. Ha, ha, ha, ha.  I'm not winning this game. Bro, I have coffee butts. I could feel it in the air when it's not going my way, dude. Hold on. I actually have a plan now. And the plan is the exact same plan as the last three times. I'm going to win. I'm going to win this with Goku. Why? So Tien's not. I don't know. I just think he's been doing the most work. Chouts, you hold him out.    Chouts, you don't let this man get up. Oh, one too many reflexes.  You want to call out a hit? OK, beautiful. I'll take that. Still for an eye, I guess. Stupid. All right. All right. Now, be misses. No. All right. Can I kill him? No, I can't. Yes. That's cool. I'll take it. I'll take it. A little extra meter. Yeah, we cost a little extra, but I got to make sure to get the kill, you know? All right, now to win with one TOD. I don't know any TODs. All right, I was hoping that would connect. Oh, no. That's not good. All right, we still have Sparks Park. Just hold out. We still got like eight minutes on this spot. What is happening? Just let me end this game, bro. Oh, gosh. This is not good. What are you doing? I thought you're going to.  I was trying to go to the other side in the dragon rush, but you didn't know. That's why I look really dumb. I was like, God, all right, this is the last game. It's two to two. Everything's on the line.   All of the pride bet. OK, I didn't get the five MPART. All right. You got to take those. You got to take those. That's not good. That's not good at all. Didn't mean to do the freeze before my spark runs out. I'm dead regardless. I know, but I just don't want to lose as much life. Yeah, there's just things live. Yeah, yeah, yeah. Oh, that one could have been sick, but I'm bad at the game. Me drop. Oh, God. Back dash. Yeah, you got you got me. You got me. Oh, OK. No sliding at the quarter. My great.  I like the old match. I took a real long time to land. I guess T.N. is dead here. Yes. Wow, I didn't think you'd actually take advantage of it. That's a little rude. All right. All right. Pick a low. We got reads. Oh, you almost did. Almost. Almost had to read. This is bad.  You're not wrong. Oh, you weren't. You're supposed to block that. Oh, I love not knowing how to play defense sometimes. It has its perks. Is that even going to hit? Don't drop it because I'm down tech mashing. Don't mash. I told you exactly what I'm doing. I know I try to be. I don't have to well, but I couldn't get it. Oh, God. I'm still up. That was unfortunate. But hey, sometimes not getting hit is a move. Why are you over there? I don't know. Oh, I was matching level three because I was like, maybe I have enough time to do this. All right. Get out of here. You blocked that. Same side. I too wish that's not blind. All right. That wasn't going to do it clean. I have to get Vegeta back in here. I didn't fully realize that I didn't fully realize that you've got to be kidding me. You've got to be kidding me. We dislike it here immensely. You've got to be kidding me. Dude, stop running. Stop running. Don't run. Come on. I lost everything. I'll do it. Oh, OK. Oh, what? Oh, you're so lucky, man. Now I got the worst character to rely on.  That was kind of sick. I can't believe it's going to be decided by Piccolo or Goku Black. Oh, God.   No, but I'm not going to be. I'm a little bit. I'm a little bit.  Dude, you're just trying to maximize damage. Oh, is this going to hit? No. Oh, don't.  Oh, that's it. No, not a super dash. No, not a super dash. Please. No. I was not doing anything else that required. I can't. We're just so lucky. I brought the combo, bro. And I thought I had the imaginary win earlier. Oh, my God. Imagine we watched the footage back.  I hope not. But that's the end of the video, regardless. Hopefully you guys enjoyed it. If you did, leave a like. Let me know your favorite moment down below. Check out the link to Deon, Shadow, should be in pinned comment. Check out the videos on your screen. Other than that, I have been dotted with it. And I will see you in the next one. Goodbye.