 So, what I am going to do now is to try to tell you about one of the important applications of the implicit function theorem which is to look at the zero locus of a function of two variables as Riemann surface okay. So, of course with suitable conditions on the function of two variables okay. So, of course here I mean function of two complex variables okay because that is the kind of function that we dealt with when we did the implicit function theorem. So, this requires you know trying to tell you what Riemann surface is so that is something that I will have to quickly recall for you okay. So, let me explain that. So, you see so I must say the following thing there is in this same program under which we are having this course there is another set of lectures for another course on Riemann surfaces which I have given and there you will find detail exposition about Riemann surfaces okay. So, one can I mean you can refer to that when you want more details but I will try to be as brief as possible and to just give you an idea of the kind of things that we need okay. So, you see what is the idea of Riemann surface? So, what is that? So, the aim is to take a real surface say like the say cylinder or torus or sphere in R3 and try to do complex analysis on it okay. So, the idea is well you have for example a cylinder of course by cylinder I mean infinite cylinder which is which for example you can think of as S1 cross R okay S1 is the unit circle unit circle cross a copy of the rail line will give you a cylinder okay and then you can also think of the two sphere S2 so which is I mean this is the two sphere in three space alright or you can think of also the torus which is T2 okay which is actually well this is same as S1 cross S1 by that I mean it is homeomorphic to S1 cross S1 okay. So, these are all you know surfaces that we can imagine sitting inside R3 three dimensional real space okay and the point is you if you take any point on such a surface and take a small disc like neighborhood then you can it is homeomorphic to a disc in the plane okay it looks like a disc on the plane. So, you know if I take a point here and then I draw I cut out a disc like a neighborhood from it and flatten it I will get a disc on the plane I can do the same thing to a point here okay and I can do the same thing to a point here okay. So, of course the surface is curved but if you take a point on the surface and you take a small neighborhood about that point and you flatten it okay then it looks like disc or a neighborhood of a point on the plane alright and what is the aim the aim is I one would like to do complex analysis on the surface okay what do I mean by doing complex analysis okay well by doing complex analysis I mean define the notion of what a homeomorphic function is define the notion of what an analytic function is and then study properties of analytic functions try to prove lots of nice theorems about analytic functions and more importantly try to see how the properties of analytic or homeomorphic functions change as the object changes. So, you expect homeomorphic functions on this to have something to do with the fact that the cylinder you do not expect the homeomorphic functions on this it will be the same as a homeomorphic functions on this at least the underlying objects are different. So, you expect some you expect this difference to show up somehow when you study homeomorphic functions. So, the aim is that you in trying to do complex analysis you try to define and study homeomorphic functions on each of these surfaces and then you try to see whether analyzing such functions gives you some information about the geometry of these objects and thus for example distinguishing these objects okay. So, the sphere has among the three these two are special when compared to this in the sense that they are compact okay because these two are both closed and bounded subsets of R3. So, they are compact alright whereas this is an infinite cylinder okay it is not bounded and you know a subset of Euclidean space is compact if and only if it is closed and bounded so it is certainly closed but it is not bounded okay. So, this is not compact and therefore you know so these are topological properties then you have many other properties for example some other topological properties for example among the three this is the surface that is simply connected the notion of simply connected surface is the property that you take a loop on that surface then you can continuously shrink that loop to a point without going away from the surface okay and you can see that I can do this on the sphere I cannot do this always on a torus because you know if I take a loop that goes around the tube like thing okay that makes of the torus then I can never shrink it to a point okay similarly if I take a loop like this I can never shrink it because of the hole in between okay so this is not simply connected whereas this is simply connected and similarly this is not simply connected because if I take a loop that goes around because of the hole in between I can never shrink it to a point on the surface without leaving the surface. So this has these two are compact this is not compact this is simply connected this is not simply connected this is not simply connected okay so there are so many topological properties and then and then there are I mean you can go on to do some kind of differential geometry okay where you try to study the curvature of your object okay then you know that if I draw the sphere has a constant curvature alright and then if you look at the torus it depends on you know what kind of curves I take on the torus alright. So there are lot of things that geometrically you can do on these things on these objects and you the question is always to analyze to see how when you analyze functions on these objects how properties of functions reflect these geometric properties see that is the whole quest of doing analysis with geometry in mind okay. So we always try to do analysis with geometry in mind namely you try to do analysis on an object which means study functions on an object and then try to see how properties of these functions somehow bring out the geometric properties of the object by geometric properties one means of course topological and much more complicated structures on the object okay fine so the aim is to do complex analysis on these objects or more generally you know I could just think of some surface in 3 dimensional space okay you can think of a surface like the surface of the wave if you want okay a smooth wave and then well if I take a point on that surface and then you take a small disc like neighborhood about that point then the question is you can flatten this out and it will again look like a point on the plane and then you can ask how to do complex analysis on this so the aim is I will have to define what is meant by an analytic function at a point okay. So to begin with I have to define the notion of when a function is analytic at a given point and then if I then I can use that definition to say when it is analytic on an open subset okay and then I can also use that to say when it is analytic on the whole surface in this way I will get analytic functions on the whole surface and then I study all these analytic functions I study their properties and the final hope is that when I study the properties of these functions that should bring about that should bring out some geometry hidden in the surface okay so some of the geometry should be captured by that. So the aim is therefore if you give me a surface like this how do I define an analytic function at a point of the surface okay so let us look at the natural way to do this okay the natural way to do this is well so here is my surface X alright X you could think of X to be either the cylinder or the sphere or the torus okay and there is a point P there is this point P on this on the surface alright maybe I will use small x because the surface is denoted by capital X and then I have this I have this I will call this as D because this is a disk like neighbourhood and what do I mean by saying that it is a disk like neighbourhood it is actually homeomorphic to a disk on the complex plane okay namely if I translate the statement that it is a disk like neighbourhood into formal mathematics I am just saying that I have chosen a homeomorphism namely a topological isomorphism of this neighbourhood okay with say the unit disk in the complex plane with if you want the point X going to this the origin okay or more generally I could of course chosen it to be homeomorphic to any disk finite radius and with X going to the centre of the disk okay so basically what I have is I have I have a I have a homeomorphism so let me call it as Phi okay which is defined on this D okay into the complex plane okay so Phi Phi of D is a disk in the complex plane and Phi from D to Phi of D inside C is a homeomorphism so what I have done is I have just translated the statement that D is a disk like neighbourhood of the point small X on the surface okay which means that I have chosen a homeomorphism of D with a disk in the complex plane and you know of course I am not writing it Phi of X goes so X under Phi goes to Phi of X and if you want you can think of Phi of X to be the centre of the disk okay so you know with Phi of X centre of the disk Phi of D okay and so this is what I mean by a disk like neighbourhood of a point and now suppose on this disk like neighbourhood suppose I have a function f suppose I have a function f defined on this with values in complex numbers alright so I have a complex valued function defined on this disk alright and what is my aim my aim is to tell when that complex valued function is analytic at the point small X okay remember our aim is to do a complex analysis on the surface which means I have to define and study analytic functions on the surface okay but to define an analytic function on the whole surface the first step is to define analytic function at a point okay once you define analytic function at a point then you define analytic function on a set an open set to be a function that is analytic at each point. So the problem is firstly to define analytic and analytic function at a point and so what does it mean it means suppose you are given a function in the neighbourhood of a point okay when will you say it is analytic at that point that is the question so here is my point small X on the surface capital X I have this function f defined on this neighbourhood of this point small X which takes complex values okay and my aim is I want to I want to define when f is analytic at small X okay and the see the it is very easy to see how so what I am going to do is I am going to just say for example draw a diagram here so here is my complex plane and this is my phi D this is my disk the image of the disk the image of this disk like neighbourhood D here is a disk and it goes to this point which is well this point is phi of X that is the centre of this disk that is where that is what phi maps small X2 okay so of course this phi is this phi actually okay of course please do not confuse it with an L set the symbol for the L set okay. So you know now what am I going to do see the complex analysis that we have studied only allows us we have done complex analysis on the plane right if you have a function defined in a neighbourhood of a point on this plane then I know what it means to say that the function is analytic okay so I can use that now to tell you when f is analytic at X because you see if I take phi inverse mind you phi is a homeomorphism it is a topological isomorphism which means phi is you know bijective it is continuous and the inverse of phi is also continuous that is what homeomorphism is so this so there is this map in this direction also which is phi inverse and that is a continuous map that is also topological isomorphism the inverse of a homeomorphism is also a homeomorphism so you have a topological isomorphism from here to here and then I can follow by this function f so what I will get is I will get a composition like this and this composition is phi inverse followed by f okay so I go by phi inverse and then I apply f and why is that helpful because that is a function from an open disk in the complex plane to the complex plane and for such a function I know how to define when it is analytic so it is very simple so what I will do is we define we define a function f from D to C defined given on a disk like neighbourhood P to be analytic at x belonging to D okay if after choosing a homeomorphism phi from D to phi of D inside C the resulting function phi of D to C given by first go by phi inverse then go then apply f is analytic at phi x so you see it is a very simple definition I want to say when f is analytic at x okay but this point x and this point D I mean this point x and this neighbourhood this disk D disk like neighbourhood D are identified with the point phi x which is a centre of the disk phi D okay so instead of saying f is analytic at x I will say that this composition function is analytic at this point phi x which corresponds to x it is a very natural definition okay so this is a nice way of deciding that when a function is analytic at a point okay but there is a small issue the issue is that you know the only ambiguity is that I use this homeomorphism phi I have chosen a homeomorphism phi of this disk like neighbourhood with a disk in the complex plane okay but there could be many homeomorphisms of this disk like neighbourhood with other disk like neighbourhoods or other neighbourhoods in the complex plane okay and then the question is that if I change phi okay my function f circle phi inverse will change phi of D itself will change okay and phi of and the point phi of x will change therefore and my function f circle phi inverse will change okay and then if my definition is sensible then I should either always get that this is analytic at phi x for every phi that I choose or it is not analytic at phi x for every phi that I choose of course but I of course I want only to choose phi such that it is always analytic so there is a condition for that the condition is that you know instead of if you change phi okay then the change in phi should induce a holomorphic isomorphism between subsets of the open subsets of the complex plane that is the condition okay we technically we use the word that the coordinate charts they differ by transition functions which are holomorphic so let me explain that see if you take an open subset of the surface and you give me a homeomorphism of that open subset with an open subset of the plane okay that is called a coordinate chart okay it is called a complex coordinate chart that is because you are because by giving this isomorphism of this open subset homeomorphism of this open subset is an open subset of C what you are doing is that for every point here you are giving a complex coordinate namely the coordinate on the target so it is called a coordinate chart okay. So specifying this phi specifying this pair phi and D is what is called a coordinate chart okay more generally you can specify a pair consisting of a homeomorphism defined on an open subset this the D I have chosen his disc like neighbourhood but I could have chosen an open subset alright and such a pair is called a coordinate chart and the whole problem is that my definition of analyticity of a function at a point should not change if I change the coordinate chart if I choose different coordinate charts and the fact that I mean the condition that will ensure that this does not that this happens is the condition that the two coordinate charts they differ by a holomorphic isomorphism. So let me explain that more generally so let me say this so the only ambiguity in this definition is that phi could vary okay phi could vary and we want to consider only those phi for which f circle phi inverse is holomorph is analytic or holomorphic at phi x always. So so let me so let me write a little more a pair u, phi where u inside x is open and phi from u to c to phi of u inside c is a homeomorphism onto an open subset phi of u okay phi of u of c okay is called a complex coordinate chart is called a complex coordinate for each point for each point of u okay. So the example is if you take this disc like neighbourhood surrounding the point small x and you take this homeomorphism of this disc like neighbourhood with a disc in the complex plane then this pair d, phi that is a complex coordinate chart it is a complex coordinate chart for every point of d because for every point of d it gives you a point in the complex plane which has coordinates okay and it does it in a manner which is topological isomorphism because phi is a topological isomorphism it is a it is a homeomorphism okay. So instead of choosing a disc an open disc like neighbourhood the point and this homeomorphism and more generally choosing any open set okay and calling such a thing as a chart alright and then the question is so you know the more general picture will look like this the more general picture is that I will have I will have this I will have the surface x see I will have this point small x okay and well I will have some open set u alright of x and I will have a coordinate chart which consists of u and a homeomorphism phi into the complex plane. So this will go into the complex plane and it will map this open set into some open subset of the complex plane so this will be phi of u okay. So this is an isomorphism topological isomorphism of an open subset containing this point x with an open subset of the complex plane right. The special case that I wrote down there was when this looked like a disc and this was actually a disc okay but it can be instead of disc just an even an open set okay and the reason for that is well analytic functions are not necessarily defined only on discs there the most general set on which you can define an analytic function is an open set okay and of course if it is an open connected set I mean if it is an open set it is a union of discs anyway okay. Nevertheless now the point is you know suppose I have suppose I have now suppose I have another chart okay so I take another chart so I have something like this. So here is here is a so you know let me let me number it or okay let me call this as u1 let me call this as phi1 this is phi1 of u1 okay. Now suppose I take another open subset of x say u2 suppose I take another open subset of x and suppose I give you another chart namely I give you another homium of some phi2 from u2 into a subset of the complex plane this is phi 2 of u2 this is also an open subset of the complex plane and well this phi2 is also a homium of some. So this is another identity so u2 is another neighborhood of the point the same point small x and phi2 is identification of that neighborhood open neighborhood u2 with an open subset phi2 of u2 of the complex plane. So you know what is happening is that the neighborhood of that point common to these two sets they have a pair of coordinates because you see if I take this intersection namely u1 intersection u2 okay if I take this intersection then for every point in the intersection including point small x I get a if I take its image under phi1 I will get a point in the complex plane so that will give me a coordinate alright and if I take its image under this chart okay then I will get another point so I am actually giving you two coordinates okay and well this shaded region the intersection okay this is u1 intersection u2 this of course you know the intersection of two open sets is again an open set okay and under a homium of some an open subset is again identified it is mapped homomorphically onto an open subset of the target when you have a homium of some of one space into another that is a topological isomorphism of one space into another then an open subset of the source space is carried by this homium of some onto an open subset of the target space and if you restrict the homium of some to that open subset then it is again a homium of some from that open subset to its image. So if you take this u1 intersection u2 what will happen is that is an open subset of u1 and of u1 and under this map phi1 which is a homium of some it will go to well it will go to some open subset of phi1 of u1 so it will go to some shaded region here okay and this shaded region will be just phi1 of u1 intersection u2 it will just be it is just like phi1 gives you an homium of some namely a topological isomorphism of u1 with phi1 of u1 it will also give you when restricted to u1 intersection u2 a homium of some of u1 intersection u2 and its image which is phi1 of u1 intersection u2. In the same way if I take the image of under under phi2 of u1 intersection u2 I will get another open subset here I will get an open subset here and what is this this is just phi2 of u1 intersection u2 it is just the image under phi2 of u1 intersection u2 which will carry u1 intersection u2 isomorphically topologically isomorphically that is homomorphically on to the subset okay so what will happen is you see therefore I have a map like this so I have a map from so I can define a map like this okay and this map is you first apply so it is this map is from phi1 of u1 intersection u2 to phi2 of u1 intersection u2 okay and this map is you first apply phi1 inverse okay and then apply phi2 right. So just for notational reasons let me let me look at the direction so let me call this as g12 g12 is well yeah let me call this is phi1 circle phi2 inverse okay and therefore it has to go I have to apply phi2 so it is going to be it is like this not like this but it is like this. So I apply phi2 inverse and then I apply phi1 okay mind you when I write phi1 circle phi2 inverse it is not defined everywhere okay phi2 inverse this I am considering it only on this shaded region which is the image under phi2 of u1 intersection u2 okay. So what I mean by g12 is g12 is actually phi2 inverse restricted to phi2 of u1 intersection u2 followed by phi1 restricted to u1 intersection u2 this is what it is but if I write it like this it looks terrible okay. So I am applying I am taking this shaded set which is phi2 of u1 intersection u2 I am applying phi2 inverse to that mind you I can apply phi2 inverse to that because phi2 is a topological isomorphism so its inverse is also a topological isomorphism so I apply phi2 inverse so I am restricting phi2 inverse to this set that is what I have written here okay. So this vertical line denotes restriction of a map to a subset right and then what I am doing is that so if I take phi2 inverse and restricted to this subset it will carry me in carry it will carry this into this shaded region which is u1 intersection u2 and then there I apply phi1 so which means I am applying the map phi1 restricted to u1 intersection u2 that is what I have written next this is the map phi1 restricted to u1 intersection u2 and this is called a transition function it is called a transition function and you know what is the importance of this transition function it is the following see suppose I have a function defined in a neighbourhood of the point okay suppose I have function defined in a neighbourhood of the point you know to say that the function is analytic at that point I have to use a chart that is what my definition says alright so you know if I have a function defined on this u1 intersection u2 with values in the complex plane okay then to say that the function is analytic at the point x okay I can either choose this chart or that chart alright because to say that the function is analytic at the point I have to use a chart that is how I do it alright and the point is I want to do it in such a way that it is consistent okay namely I do not want it to be analytic with respect to one chart and not analytic with respect to another chart that means there is some restriction on the charts when the overlap and that restriction is that this transition function should be a holomorphic isomorphs okay so see so let me write that down and explain why f is analytic at the point x with respect to with respect to the chart u1, phi1 well if what is the condition I so you know this point this point x goes to a point here okay which is phi1 of x and what is the definition of f being analytic at x I compose f with phi1 inverse and I say phi1 inverse followed by f is analytic at phi1 x now that is my definition here so the definition is f circle phi1 inverse is analytic at phi1 x okay but then f is also analytic at x with respect to the chart u2, phi2 if f circle phi2 inverse is analytic at phi2 x what does this mean this means I am trying to define f to be analytic at the point x using this chart which means what I do is that I take this point x I take its image here I will get a point phi2 x and now I am saying is now you look at this composite function which is first apply phi2 inverse then apply f now that composite function is defined from an open subset of the complex plane and it is a complex valued function so it makes sense to say it is analytic alright so you see what is happening is now I had 2 charts I had 2 charts I am getting 2 definitions of the same function being analytic at the same point okay so what you can continue I mean you can see from this that as many charts as I can find at that point okay I will get as many definitions of the function being analytic at that point but you see the property of function being analytic at a point is intrinsic it should not depend on anything okay all nice properties like continuity differentiability okay analyticity they are all intrinsic to the function they should not these are all properties that should not change if you change coordinates if you make a nice change of coordinates these properties will not change so the property of function being analytic should not change so that means all these definitions should be consistent with one another it should not happen that you know my phi1 and phi2 are charts I mean these 2 charts are such that you know with respect to this chart the function is analytic at x but with respect to this chart the function is not analytic at x such things should not happen okay that tells me that the kind of charts that I can consider they should be restricted in some way and the restriction is the following it is the following both are equivalent if g12 is a holomorphic isomorphism which happens if g12 is holomorphic okay so you see so let me make a couple of statements you see first of all I am saying that look at the nice condition that this transition function is holomorphic okay mind you this function is from this shaded region to that shaded region so it is a function from an open subset of the complex plane to another open subset of the complex plane okay and it is actually a homeomorphism because it is a composition of homeomorphisms this way from this shaded region to that intersection is a homeomorphism because it is given by phi2 inverse okay and from that shaded region to this shaded region is again a homeomorphism because it is given by phi1 so this is a composition of homeomorphisms so it is a homeomorphism okay but in particular it means that it is one to one mind you okay. So if you put the condition that g12 is holomorphic then you have 11 holomorphic function okay but we have seen in the previous lectures that if you have one to one holomorphic function it is a holomorphic isomorphism just putting the condition that a holomorphic function is one to one will ensure that the inverse function is also holomorphic okay that this is what we saw when we studied the inverse function theorem okay. So the condition that g12 is holomorphic is sufficient to guarantee that g12 is actually holomorphically isomorphic name be that the inverse of g12 inverse that is holomorphic okay and why does that help to say that these two definitions are equal that is because of the simple calculation you see if I take g12 and then apply f circle phi1 inverse okay I will get f circle phi2 inverse. See because you see g12 is phi1 phi2 inverse okay forget the restrictions do not worry about the restrictions because I mean writing them also down will make the notation complicated. G12 is phi1 composition phi2 inverse okay if I plug it in here I will get f composition phi1 inverse composition phi1 composition phi2 inverse and that will and because phi1 inverse composition phi1 is identity I will get f composition phi2 inverse. So what this tells you is that this function f circle phi1 inverse it differs from the other function f circle phi2 inverse by a function which is holomorphic isomorphism so you know if this is holomorphic and then because this is holomorphic the composition will become holomorphic so this will become holomorphic conversely if this is holomorphic I can move g12 to the other side by operating by g12 inverse because mind you g12 inverse is also a holomorphic isomorphism and that will tell you that this will be holomorphic. So this is the same as saying and this is equivalent to you know f circle phi1 inverse is f circle phi2 inverse composition g12 inverse and g12 inverse is if you look at it carefully it is g21 the way we have defined it and g12 is holomorphic if and only if g21 is holomorphic that is because an injective holomorphic function is a holomorphic isomorphism okay. So the beautiful condition is that if you are only restricting to deciding the analyticity of a function by using charts which whenever they overlap they satisfy this condition that the transition functions are holomorphic then for all these collection of charts the holomorphic the analyticity of a function the holomorphicity of a function will not change okay. So the moral of the story is that to do analysis on a surface you just give me a collection of charts okay which covers every point and in order that the notion of a holomorphic function at a point does not depend on your chart make sure that whenever two charts intersect they are compatible in the sense namely that the corresponding transition functions are holomorphic okay. So if you therefore if you start with the surface and cover it by such a collection of charts which are compatible to one another then you will be able to concretely tell when a function on that surface is holomorphic at a point and therefore when it is holomorphic on an open set and so on. So you can study you can define and study holomorphic functions. So what this discussion tells you is that if you want to do complex analysis on a surface you have to endow that surface with a collection of charts which covers all of the surface which covers every point of the surface and with the condition that whenever two charts intersect the corresponding transition functions are holomorphic such a collection of compatible charts is called an atlas okay and a surface given an atlas is called Riemann surface and it is called Riemann surface because it was first studied by Riemann and basically that is the structure that you need to define and study holomorphic functions on the surface okay. So let me write that down. He say we see that if x is given a collection of charts which cover every point of x and which are compatible which are pairwise compatible that is whenever two charts intersects intersect the transition function the transition function are holomorphic then we may define and study holomorphic functions of course sometimes I use the word analytic sometimes I use the word holomorphic but they are one and the same okay. So we may define and study holomorphic that is analytic functions on x without ambiguity such a collection such a collection of charts on of compatible charts on x is called the complex atlas on x and x together with the complex atlas atlas is called Riemann surface and the Riemann surface is the correct structure that allows you to do complex analysis on surface okay. Of course the important point is that you could have different sets of you could have different atlases giving different Riemann surface structures on the same set x okay and then it becomes a question of geometry as to how many such structures are there whether they are isomorphic to one another or not and how these structures reflect the geometry of and the topology of x and so on and so forth it is a very interesting area of research and study okay. So I will stop this lecture by just saying that the reason why I did all this is to tell you that if you take a function of two complex variables and if you put some nice smoothness conditions on the on that function namely you put the condition that at least one of the partial derivatives does not vanish at each point then the locus of zeros of that function becomes a Riemann surface and why it becomes the Riemann surface is because of the implicit function theorem. So the implicit function theorem allows you to look at the zero locus of a complex function of two variables okay as a Riemann surface and therefore on the set on that zero locus you can do you can do complex analysis okay that is the that is an important application of the implicit function theorem I will explain that in detail in the next lecture.