 So this is the continuation of the previous lecture and my aim is to explain why the implicit function is function theorem is important namely that it helps you to think of the zero locus of a function of two complex variables as a Riemann surface. So what I told in the last lecture was what the idea of a Riemann surface is namely it is a you may call it a device or it is a structure that allows you to do complex analysis on a surface okay a surface that you can for example imagine in three space like the cylinder or the sphere or the torus and the method is that you try to define when a function on at a point of the surface defined on a neighbourhood at a point on the surface is holomorphic or analytic and you do that by composing the function with a coordinate chart at that point and then to make sure that the resulting definition of an analytic function is consistent you only use charts collection of charts which have the property that whenever two charts intersect the corresponding transition function is holomorphic which is equivalent to saying that the transition function is a holomorphic isomorphism okay it is a holomorphic function which has an inverse which is also holomorphic okay and the moment you give x your surface x a collection of charts that covers every point and which are pairwise compatible then you make x into a Riemann surface okay and then you can study you can define and study holomorphic functions analytic functions on the Riemann surface and you expect that by studying the properties of analytic functions on the surface you will be able to get some information about the geometry of the surface or you expect the geometry of the surface to be reflected when you study the analytic functions on the surface or the holomorphic functions on the surface. So of course the application that I have in mind is to look at the implicit function theorem okay for a function of two variables so let me do that right away application of the implicit function theorem okay. Suppose f from C2 to C is holomorphic I mean which is the same as analytic in each variable okay what this means is that you write f what it means is that you write f is equal to f of z, w z is the first complex variables w is the second complex variable and f is of course you know of course you are assuming f to be continuous right and for each fixed value of w the resulting function of z is analytic and for each fixed value of z the resulting function of w is analytic you assume that okay so it is analytic in each separately in each variable right and we want to look at the locus f of z, w equal to 0 in C2 okay. So you are looking at this the points in C2 which are zeros of this function you are looking at the zero locus of this function okay. So that is it is just f inverse of 0 it is a inverse image under f of 0 all those points which go to 0 under f okay the first thing that you should realize is that if you think very nively heuristically C2 is two dimensions and if you look at the locus f equal to 0 okay you are looking at you are cutting down by one equation so the locus must be one dimension it should be one dimension less from a space of two dimensions you are looking at solutions of one equation so the dimension has to come down by one so the resulting should be one complex dimension okay so it should be a surface so you expect it to be a surface which it is provided f is good enough okay. So now what I am going to do is suppose we look at a point a point in the locus say z not, w not so this is a point where f is 0 okay suppose you are looking at a point like this okay and then what I do is that I look at the conditions that I need to apply the implicit function theorem okay and what is the implicit function theorem say the implicit function theorem says that your function the explicit the implicit function can be solved to give an explicit relationship for a variable in terms of the other variable provided the partial derivative with respect to that variable is not 0 okay. So if do f zeta, eta by doh zeta alright at z not, w not it is not 0 suppose it is not 0 alright this means that you know you are looking at f as a function of the first variable keeping the second variable equal to w not okay and then you are taking its derivative and then you are evaluating that derivative at z not okay this is the same as d by d z of f of z, w not at z equal to z not alright and if that is not 0 the implicit function theorem says by the implicit function theorem there exists a delta greater than 0 such that so let me write that down basically it says that since the partial derivative with respect to the first variable is not 0 you can solve for the first variable explicitly as a function of the second variable at that point okay. So in other words there is a delta greater than 0 such that for all w with not w minus w not less than delta we get a function E z as a function of w satisfying f of E z, w namely f of E w, w equal to 0 this is what the implicit function theorem says okay this is what the implicit function theorem says. So you know I am going to draw schematic graph I mean schematic in the sense that I am going to draw a graph which is well which you really cannot draw in 3 space okay but then you will have to imagine it with some imagination. So here is C and here is C okay so each I have drawn only a line okay but I wanted to think of this as C and this also as C alright and then I am actually drawing the locus f of E z, w equal to 0 okay so this is C2 this is C2 this is the z coordinate and this is the w coordinate okay this is the C2 plane, this is the z coordinate, this is the w coordinate if you take any point on the plane it will have 2 coordinates z0,w0, z0 will be the first coordinate which is what you will get when you project under P z projection under the first coordinate you will get the first coordinate as z0 and then if you project under the second coordinate P w you are going to get the second coordinate okay so every point has 2 coordinates and I am taking a point on the locus of this equation so that means I am taking a point such as f of z0,w0 is 0 right and now I am looking at now I am assuming that the first partial derivative at z0,w0 is not 0 so let us interpret this so I think this spelling is wrong it should be t,h,e okay so you know I have a neighbourhood of w0 okay so you must think of this neighbourhood of w0 as a disk on the complex plane the w plane centred at w0 radius delta okay so this is that neighbourhood which is given by mod w-w0 is less than delta it is actually a disk okay but I cannot show that here right because I am thinking of this as an axis single line and what is happening is that I have a what is the implicit function theorem say it says that you know I have function w going to gw so there is a function like this so there is a function like this which goes from this neighbourhood okay such that if I draw the graph of this function the graph of this function will be this locus namely if I take any small w in this neighbourhood and I take the point w, gw then rather gw,w the way I have written it because w is second variable for me if I take the corresponding point gw,w that point lies on this locus that is what it means to say f of gw,w equal to 0 for every w in this neighbourhood so in other words what I am saying is that locally the implicit function theorem says that the locus where f vanishes is actually the graph of a function okay it is a graph of the function and now the beautiful thing about a graph is that the graph of a function if you project it to the free variable okay that is always an isomorphism okay so in other words you know so let me draw this correctly so this is this is a dotted line and this rounded arrow is g which is a map from here to here and of course this is the projection this is p,w okay look at look at the set look at the subset gw,w such that w mod w-w0 is less than delta look at this subset okay actually I am just writing I am just this is just the graph of g okay this is just the graph of g only thing is that normally in the graph you write the variable and then the function but I am writing it the other way I am writing the function and then the variable normally you write the graph of f of x,y equal to f of x is x,fx okay so ideally the graph of g I should write as w,w but I am writing it as gw,w because of this diagram okay so this subset mind you this is inside this locus f inverse is 0 right because gw,w satisfies f of z,w equal to 0 so it is in this locus alright then I want to say that this is the image of this disk under so I just want to say that this is isomorphic to the disk actually because of g so let me say that I am just using the fact that you know if I apply the projection so you know from so I have f inverse 0 that contains this so you know this subset I am writing here is this portion of the locus which is the graph of g over this disk right except that I have switched the order of the variables right so this is the set of all so let me write that here gw,w such that mod w-w0 less than delta this is a subset of this and now what you do is you apply projection onto w what you will get is the set of all you will get the set mod w-w0 less than delta because you know if I if I project it under if I project it under pw I will get back my disk and what I want to tell you is that this map I claim it is a homeomorphism is a homeomorphism note that this map is a homeomorphism ok I just want to say that whenever you take a graph whenever you take the graph of function from the graph to the free variable ok that is always a homeomorphism right. So I just want to say that this pw is a homeomorphism which means of course this is a projection restricted to this piece of the curve ok I claim it is a homeomorphism to show that I will have to show that this is both injective surjective ok and then I have to show it has an inverse alright and what you must understand is that it is it is it is bijective because it has an inverse the inverse is actually g it is just it is given by g namely it is given by w going to gw,w ok. Because it has the inverse w going to gw,w this is just the graph this is just the graph map except that I have written the dependent variable first and then the independent variable normally in the graph you write first the independent variable and then the dependent variable you write x,fx for the graph of f ok. So I should ideally write w, gw but does not matter I am writing it as gw,w ok in this case it is very clear that this is a this is a continuous function ok and because this is the inverse of p,w it follows that p,w is bijective with this as the inverse and well so the moral of the story is that for the point z0,w0 I have found you know I have found an open set mind you this is an open set now the reason why this is an open set is because it is homeomorphic to an open set this is an open set on in the complex plane and this is a homeomorphism ok therefore this is also an open set a homeomorphism will always carry open sets to open sets. So what is going to happen is that this piece of the graph that I have marked ok is actually an open subset of this locus ok and it is homeomorphic to this disk ok it is homeomorphic to this disk and it is so it is a disk like neighbourhood of the point z0,w0 what is a disk like neighbourhood it is an open subset which is homeomorphic to a disk. So you see I have this open this piece of the graph which is an open subset containing the point z0,w0 and that is homeomorphic to the disk under the projection and you know the moment I have something like this I have a chart because I have identified a point on this locus along with the neighbourhood of that point with a disk ok therefore this gives me a chart so moral of the story is we take the pair g,w,w such that mod w-w0 less than delta,p,w so I will use some notation I will use gamma g,p,w receded to gamma g I am using gamma g because I mean by capital gamma g the graph of g ok. So capital gamma sub g is the graph of g except that mind you have switched the order of the variables so this is the graph. So you take this graph along with the projection receded to the graph that is a chart is a complex coordinate chart at z0,w0 so what is the moral of the story the moral of the story is if you look at the locus where f vanishes at a point of that locus where f vanishes if the first partial derivative with respect to I mean if the first partial derivative with respect to the first variable is not 0 then I can get a coordinate chart ok at that point. Now the same argument will tell you that if instead see it may happen that I may not be lucky perhaps the first partial derivative with respect to the first variable might vanish ok but then there is still hope if I have you know that the second I mean the first partial derivative with respect to the second variable does not vanish then again the implicit function theorem will tell you that I can solve for the second variable as a function of the first variable ok and then what I will get is there also I will get a chart ok. So the moral of the story is throughout this locus at every point where either the first way first partial derivative or the second does not vanish I will always get a chart ok so you know if I put the condition on this function f the condition called smoothness which is that there is no point on the locus of 0s of this function where both partial derivatives vanish I mean that is a having such a point is very bad it is called a singular point it is called singular because I cannot apply the implicit function theorem to apply the implicit function theorem I should have at least one of the partial derivatives non-vanished if both partial derivatives vanish that is a singular point and I do not want to consider functions which whose 0 loci have singular points does not vanish ok. So you know if I am working with a smooth function ok namely a function such that there is no point on the 0 locus of the function for which both partial derivatives vanish then at every point I will get a chart because of the implicit function theorem the chart may be projection on to the second coordinate if the first partial derivative does not vanish and it will be a projection on to the first coordinate if the second does not vanish ok. So I can cover it by charts now the beautiful thing is these charts are automatically compatible ok these charts are automatically compatible and therefore they make this locus into a Riemann surface that is a beautiful thing. So the moral of the story is if you are looking at a smooth function ok of two variables ok then it is automatically a Riemann surface it is a Riemann surface which is sitting inside C2 it is a Riemann surface which is embedded inside C2 ok. So let me also write let me write that down so you know so this is the importance of the implicit function theorem I can when I am studying 0 locus the 0 locus of a function of two variables if it is smooth function I am already looking at a Riemann surface ok I am already looking at a Riemann surface there are some technicalities which I will try to explain very soon but let me let me let me say the following thing suppose we had a point z1,w1 in f inverse 0 that is f of z1,w1 is equal to 0 with doh f sorry doh f zeta by doh eta at z1,w1 not equal to 0 ok. So I am considering another point where the you know this first partial derivative with the second variable does not vanish and that point is again on this locus the 0 locus of the function ok. So if I draw another diagram it should look like this so here is my complex plane this is z coordinate this is another complex plane this is the w coordinate and here is my locus this is f of z,w equal to 0 which is otherwise f inverse of 0 and of course all this is happening in C2 the whole space is C2 that is where everything is happening C cross C ok and now I am having a point z1,w1 alright if I use projection onto z I get the point z1 if I use projection onto w I get the point w1 ok and I have assumed that the partial derivative with respect to the second variable does not vanish ok at that point. Now again the implicit function theorem by the implicit function theorem there exists lambda greater than 0 such that for z with mod z-z1 lesser than lambda ok we have we get a function of z we get a function omega in terms of z omega is equal to h of z ok such that such that f of z,h of z equal to 0 ok that is in other words you are just saying that if the implicit function theorem just says that you know if the partial derivative with respect to the particular variable is not 0 then you can solve for that variable as a function of the other variable. So the partial derivative with respect to the second variable w is not 0 so I can solve for w which is a second variable with respect to the partial derivative is not 0 in terms of the other variables is the first variable ok so I can get a function w is equal to h of z for z in a neighbourhood of z1 ok which satisfies this equation so I get an explicit solution to this implicit equation ok. Now what does it mean if you draw a diagram similar to that now I will get the neighbourhood here I will get this neighbourhood here this neighbourhood here will be mod z-z1 less than lambda ok I will get a neighbourhood namely a disc in the complex plane the z plane centred at z1 reduce lambda and I will get a function of z so I will get a h I will get a function h like this ok and if you draw the graph of that h I am going to get this piece of the graph ok so I am going to just get this piece of the graph ok. So again what will happen is that we will have again you will have that the set of all the points in the graph of h gamma h the set of all points z, h of z such that mod z-z1 less than lambda this will be a subset of the 0 locus and if you take projection to the free variable z ok onto this neighbourhood at this disc centred this surrounding z1 reduce lambda is a homeomorphism this will be a homeomorphism because it will have inverse z going to z, h of z ok so it will be a homeomorphism. So what this tells you is that I get this so in this case I get this graph of so this this piece of the of this locus is actually graph of h it is a graph of h and that is an open subset it is an open neighbourhood of the point z1, w1 because it is homeomorphic to an open set in the so it is a disc like neighbourhood and the homeomorphism is given by the projection onto z restricted to that open set alright so that is a chart. So I get a chart at the point z1, w1 ok so we can take gamma h, pz restricted to gamma h as a chart at pz1, w1 ok. So in this case this is the graph of g ok you got a function g from the w from the disc on the w plane and this is this becomes graph of g this is what happens if the partial derivative with respect to z does not vanish and if the partial derivative with respect to w if it does not vanish then this piece of this locus will become the graph of h where h is a function from a neighbourhood of the point with the first coordinate alright. So in any case if the curve is smooth ok namely if the function f is smooth then you get you automatically you get charts like this and all the charts come because of implicit function ok. So if f is smooth that is for every point in f inverse 0 either dou f by dou zeta or dou f by dou eta does not vanish then f inverse 0 has collection of charts because of the implicit function theorem because of the implicit function and now the natural question you will ask is that well does this collection of charts which come naturally because of the implicit function theorem does it make it into a Rayman surface and what is it that you have to check to say that it may it makes it into a Rayman surface you have to only check compatibility ok. And all I want to tell you is that the compatibility is trivial ok what is the compatibility the compatibility is that the transition function should be holomorphic ok. If you take a point if you take 2 nearby points whose charts are in this direction ok ok then the transition function will go like this and come back ok and that will be identity map on W which is of course holomorphic ok. So if you have 2 charts of this type which overlap then the identity function then the transition function is just W going to W which is of course holomorphic as a function of W. Similarly if you have 2 charts of this type then the transition function is z going to z that is the identity function here that is of course holomorphic so you get compatibility the only thing you have to case check is you if you have a chart in this direction intersecting the chart in this direction ok and if you have a chart in this direction intersecting the chart in this direction you know if you go like this what you will get is z will go to you see if you go like this ok you will either get g or h ok which are both analytic. So then also the transition functions will become holomorphic so for example you know if a chart like this overlap with a chart like that ok and if I took the transition function like this then I am first going by so I am going z to z, h of z ok and then if I project on to W I will get h of z. So the transition function will become h and h is of course holomorphic why is h holomorphic because that is because of the implicit function here. Similarly if I go if I take the other transition function namely if I go like this and then come down via that then it will be W going to gw,w and then if I take first projection I will get gw so it will be w going to gw which is simply the function g and g is also holomorphic again because of the implicit function theorem. Therefore you see automatically all charts are compatible all the charts are automatically compatible just because of the implicit function theorem. So automatically this becomes a Riemann surface ok it automatically becomes a Riemann surface that is the beautiful thing. So let me write that it is clear that the transition functions are either identity on z or identity function on w or g or h which are all holomorphic the transition functions are of course holomorphic because identity function identity functions are holomorphic and the g and the h that you get their functions that you have gotten by the implicit function theorem they are holomorphic ok. So the moral of the story is that f of z,w if f is a smooth function f of z,w becomes a Riemann surface ok. So let me write that thus if f is a smooth function namely that either the first or the second part either the partial derivative with respect to the first variable or the partial derivative with respect to the second variable does not vanish at each point where f is 0 ok. Then f inverse is 0 in C2 automatically becomes a Riemann surface this automatically becomes a Riemann surface ok. So now what I am going to do is I am going to tell you a little bit of technicality ok. So this is of course I have conveyed the main point that one very important application implicit function theorem is that you can look at the 0 locus of a smooth function as a Riemann surface ok. So you can look you can do complex analysis on this surface ok very naturally alright. So the technical point is a following see when we define a Riemann surface if you want to give a abstract definition of a Riemann surface you have to first define what an abstract surface ok. So the definition so let me quickly recall these facts which you can try to understand if you do a little bit more of reading. So what you need is basically you start with a topological space x ok which is house dwarf ok and which is second countable ok namely you assume that it has a countable you know it has a countable basis ok. So you start with the topological space which is house dwarf and which is second countable alright and which locally looks like the plane alright the complex plane or the real 2 plane such a topological space is called a surface it is called a real surface ok. So you know the sphere the torus the cylinder they are all real surfaces ok and we also put the extra condition that you work only with connected topological spaces ok. So when I defined Riemann surface here I told you a Riemann surface is a surface x with a complex atlas namely with a collection of compatible charts but I did not tell you what that x is I told you for example x could be you know you can think of x as a sphere or the torus or the cylinder but in general what can x be the answer to that is x should be a topological space which is connected which is house dwarf which is second countable and which locally looks like looks like the plane the fact that it locally looks like the plane is what tells you that every point of x has a neighbourhood which looks like a disc ok and that is the only way of saying that it is a surface a surface is something that should locally look like the plane ok. So this is a technical definition and the point I want to make is that if I want to really with respect to that definition if I want to say that this 0 locus is a Riemann surface I will have to verify that this is house dwarf I have to verify this is connected I have to verify this is second countable ok and the truth is that I mean the house dwarfness and the second countableness are not so difficult to verify ok the slightly more technical thing is the connectedness ok to say that for example if you take a polynomial if f of z,w the simplest kind of function in two variables that you can think of as a polynomial in two variables ok and if you want to ensure that the 0 locus of that polynomial is connected that is this graph that I have drawn here is actually a connected set in C2. One nice condition is that the polynomial should be irreducible that is a polynomial cannot be factored into a product of two different polynomials two non constant polynomials ok. So the proof of this fact is not so easy but you can take it as a statement as a theorem that if f of z,w is a polynomial which is irreducible then the 0 locus of f is actually connected ok. So you get connectedness you and I told you that it is house dwarf and second countable is something that you can verify ok because that is already there for C2, C2 is of course house dwarf C2 is of course second countable alright so Euclidean space. So the moral of the story is that in the with this formal definition of a Riemann surface also if you take for f an irreducible polynomial in two variables which is smooth then the 0 locus will be actually a Riemann surface ok and so this tells you that the formal side of the picture is also correct ok if you want to think of Riemann surfaces the formal sense as house dwarf second countable connected topological space is endowed with a complex atlas. This locus is automatically one such Riemann surface ok and all this is just a beautiful corollary of the implicit function there ok so I will stop here.