 Alright so we are we are discussing meromorphic functions okay and we were looking at meromorphic functions on the extended plane in the last class and in the last lecture and you know and we proved that the a function which is meromorphic on the extended plane is none other than quotient of polynomials okay namely a rational function okay. So what I need to what I want to tell now is about the collection of meromorphic functions on a domain okay you take a domain in the extended plane and look at the set of all meromorphic functions defined on that domain okay then that set has a has a nice structure in fact algebraic structure it is a field okay and in fact it is an algebra over the complex numbers okay and so the so it is a field extension of the complex numbers and the and the properties of this field extension algebraic properties of this field extension they have got they capture a lot of topological and geometric properties of the domain okay. So this is how there is a link from you know from the complex analysis side to the algebra side okay so you know geometry involves an interplay of various areas of mathematics. So studying something geometrically will involve studying it from the analysis view point okay and studying it from the topological view point also studying it from the algebraic view point but when you are looking at a particular nice object okay when you study it analytically it will have some properties okay some special properties and then when you studied algebraically it will have some properties special properties when you study topologically it will have some special properties and the fact is that these properties are interrelated it it is there is some beautiful relationship hidden relationship between these the analytic the algebraic and the topological properties of a nice object and that that relationship is what you may call as geometry, okay. So if you want to really understand the geometry of an object you have to analyze it using all the three you know viewpoints algebraic, analytic, topological, okay. So in that sense you know how do I do geometry on a domain in the complex plane or in the extended complex plane, okay. What I can do is of course the analysis is there the analysis will worry about what kind of functions you can define on the domain, what are the holomorphic functions or analytic functions on the domain, what are the meromorphic functions on the domain and so on that will that will be the view point from analysis but then how do you go to algebra, okay. The point is that you take the set of meromorphic functions that forms a field, okay and that is a field extension of the complex numbers and you study the properties of this field extension, okay. So in field theory you have lot of you would have come across in a course in algebra in field theory that field extensions have there are of so many types, okay there are algebraic extensions, there are transcendental extensions and then there were there are normal extensions, there are splitting fields, there are Galois extensions, okay. There are of course separable and non separable extensions and we study all these things and of course the most important thing here in general is the study of the nature of Galois extensions because that connects up with group theory which is it connects up with the so called Galois groups so you see the moment you look at the field of meromorphic functions you get an extension of the complex numbers and then you can do algebra, okay and somehow these things are all connected and I will try to give a couple of examples, okay. So first of all let me begin by first saying that if I take a domain in the extended complex plane then this set of all meromorphic functions on the domain is actually a field, okay. So let me write that down the field of meromorphic functions. So let D inside the C union infinity be a domain so you are taking a domain in the extended complex plane C union infinity. So in particular mind you it is a non-empty open connected set, okay and the advantage of taking a domain in the extended plane is that you can also look at the neighbourhood of infinity, okay that is the advantage. So you are also including the point of infinity, right. So let so here is the notation I will put script M of D be the set of meromorphic functions on D, okay. So what is this script M of D? Script M of D is the collection of all meromorphic functions on D and you know what a meromorphic function is. We have defined a meromorphic function to be a function which is analytic at all points but except for points on an isolated sets which have to be pole singularities, okay. So it is analytic except for poles and the moment you say analytic except for poles it means the singularities can be only poles and that in particular means that the singularities can only be isolated because poles are isolated singularities by definition, okay. So you take all the meromorphic functions on the domain, okay. Now the fact is that this is a field, okay. See so let us see that let f and g be meromorphic functions on D, okay. Then you see then you can notice the following things, number 1 if lambda is a complex number, okay then lambda f is also a meromorphic function, okay. Meaning a meromorphic function by a constant is going to keep it meromorphic, okay. Of course if the constant is 0 you will get 0 and the 0 is a constant function and of course you know when you say meromorphic analytic is also included, okay. So the definition for meromorphic is that it is analytic except for poles that does not mean that it has to have a pole. It can be it can have no poles and it can be analytic everywhere. So holomorphic functions are also included in the set of meromorphic functions, okay. So you know so the first so this statement is obvious if I take a meromorphic function multiplied by a constant if the constant is 0 of course I am going to get the 0 function which is holomorphic which is analytic because it is a constant function, okay. But if lambda is not 0 lambda times f will also be meromorphic and it will have the same poles, okay by multiplying by lambda you are not going to change the poles and you are not going to change the order of the poles. Essentially you are just multiplying by a constant, okay. So this is one obvious thing then the second thing is that if you take the sum of these two meromorphic functions this will also be a meromorphic function, okay the sum of f and g will also be meromorphic why because you see the fact is that f is meromorphic so it has some fine it has a collection of poles, okay at an isolated set of points. Then g is also meromorphic so it has also poles in another isolated set of points and then you take the union of these two isolated sets that is again an isolated set, okay and these are the only points where f plus g will have problems, okay. So at a point where f does not have a problem and g does not have a problem f plus g will not have a problem that is at a point where f is analytic and g is analytic f plus g of course will be analytic, okay. So the only problems for the function f, analyticity of the function f plus g will be at the points where f and g have problems, okay and it is possible that some of that could be some cancellations okay. So you know for example f may be 1 by z minus z not g may be minus 1 by z minus z not so if I take f plus g I will get 0 which does not have a pole at z not okay. So some poles see some poles can cancel out also and sometimes the order of a pole can come down okay when you add of course when you when I say add it also includes subtraction because subtraction is just adding with minus 1 multiplied by the second function okay. So the moral of the story is that sum of 2 meromorphic functions is again a meromorphic function it could very well be analytic okay some poles may cancel out all the poles may cancel out for example if you take the function you take its negative and add it you will get 0 and that is clearly holomorphic it is a constant function. So sum is meromorphic so you know the moment you look at the first 2 things this will tell you that you know M of D is a vector space over complex numbers see because it is you see so there is a scalar multiplication if you think of complex numbers as scalars then there is a scalar multiplication and there is addition so this becomes a so M of D is C vector space so you get that immediately okay. Now let us look at f times g look at f times g see f multiplied by g will also be a meromorphic on D this is also very very clear because it just from the fact that you know what are the problem points the problem points are the points where f has problems and g has problems okay so if you take out those problem points then f times g will be analytic so at a point where f is analytic and g is analytic f times g will be analytic and the only place where f times g will fail to be analytic is probably on the set union of the set of poles of f and poles of g okay and so you see and of course if you want you can write out always the principle parts and see that you know if 2 functions have poles at the same point they have common pole then if you multiply the product function will have a pole with higher order in fact it will have order equal to some of the orders that is obvious if you write out the principle parts okay so in the Laurent expansion alright so I mean the point is that you know all these algebraic operations of adding, subtracting, multiplying by a constant and just multiplying and of course we are going to see division all these things they do not change the meromorphic nature okay so by adding or subtracting or dividing or multiplying or multiplying by a constant you cannot change a meromorphic function into a non-meromorphic function you know if you are only working with meromorphic functions you will get back again meromorphic functions okay so fine so you have f and g are the product f times g is also meromorphic of course by product one means point wise product okay so f g is a function which at each point z is defined by f of z times g of z alright then of course I can say the same of f by g this is also a meromorphic function provided g is not identically 0 okay of course I should not divide by 0 so the fact is that see when I take f by g okay what are the problem points the problem points will be poles of f poles of g and now you will have extra problem points at 0s of g because they are 0s of the denominator they become the 0s of g will become poles of f by g they are likely to become poles of f by g and of course you know it might happen that some 0s of g may cancel out with some 0s of f because the 0s of f are on the numerator the 0s of g are on the denominator some 0s might cancel but the set of problem points are just the poles of f the poles of g and the 0s of g and these are this and you know the 0s of an analytic function are also isolated you know that that is a theorem okay in fact that is a that is another version of the identity theorem if you have seen it in a first course in complex analysis so therefore the set of points where f by g will have problems is still an isolated set of points okay and at each of those points you can only get poles you cannot get anything worse okay so it is so therefore f by g is also meromorphic in particular I could have taken 1 by g I can put f equal to 1 I will get 1 by g is also meromorphic and that means so what will I get I will get 1 by g is meromorphic if g is not identically 0 so that means every non-zero meromorphic function every meromorphic function which is not identically 0 has an inverse okay and that is what you require for a field okay a field should be basically a group under multiplication if you throw out the 0 element from the set okay so well if you look at all these things these things will tell you that M of D is a field and you know you put it together with this fact that we saw earlier we have seen just above that M of D is also a complex vector space of a field extension with a field which is also a vector space is an algebra okay so basically you can very well see that M of D contains complex numbers because the complex numbers sit as constant functions okay you take any complex number lambda you think of it as a constant function lambda constant function lambda is analytic it is defined everywhere so it is analytic on any domain okay and it is meromorphic because mind you when I say meromorphic I am allowing also analytic or holomorphic meromorphic means that it can either be analytic and if it is not analytic that is if it has singularities the singularities must be only poles that is what it says so meromorphic does not say that it should not be analytic so in particular M of D contains the complex numbers as a sub field you know the complex numbers of course form of field and therefore M of D is a field extension of the field of complex numbers so M of D is a field so let me write that M of D is a field extension field extension of the field C of complex numbers and the beautiful thing is that the geometry on geometry on the domain D is done by the lot of topology of the domain D is connected to and a lot of analysis on the domain D namely the behaviour of existence and behaviour of meromorphic functions on D is connected with the algebraic properties of this field extension okay that is the geometric content okay so I mean this goes back to the work of the classical giants like Riemann and Clifford and Weierstrass and Abel and Jacobi and you know all these people who developed theory of Riemann surfaces okay of course principally from Riemann so let me write that down so let me write it out as a diagram so you have M of D and this is over C so I am using this I am using this field theory you know notation you write a field a bigger field on top and you put a smaller field in below and then you put a vertical line saying that the thing that comes above is a field extension this the things that come below are subfields okay so this so the algebraic properties of this field extension the algebraic properties of this field extension they are connected so there is a so the analysis on D which is existence of special existence and properties of meromorphic functions on D that is the analysis on D okay and this part is algebra on D the algebra on the domain is actually studying you may think of it as studying the this field extension and then there is the topology of D topology of the domain D and you know so I am not trying to be very particular or trying to go in detail but topology at the minimum for example D is of course connected but one of the simplest things that you can look of look at is whether D is simply connected or not okay and then or if it is not simply connected you can see you know if it is multiply connected and how many holes it has and so on and so forth okay after all D could be something like a something like an amoeba something that looks like an amoeba okay with some holes okay after all an open set can look like that and then the topology worries about whether it is simply connected if it is not simply connected how many holes are there and so on and so forth okay so all this topological information for example is encoded in the in the fundamental group of D and so on and so forth okay and in fact more precisely I should say that you have to study the theory of covering spaces of the domain D that is what the topology of D means so let me write this here the theory of covering spaces of D related which is actually related to the fundamental group in fact subgroups of the fundamental group of the fundamental group of D of course you know the so you know at this point let me tell you that if you have done a decent second course in topology see there is something called covering space theory which takes a topological space with decent properties for example something that is a house of locally house of connected locally path connected locally simply connected okay and then you study what are called as coverings of topological coverings of D and then there is a there is a Galois theory of coverings which says that you know there is a Galois correspondence between the coverings and subgroups of the fundamental group of your topological space okay and in fact under this under this Galois correspondence the you know the so this Galois correspondence is an analog of the Galois correspondence that you have in field theory see the Galois correspondence in field theory gives you a correspondence between field extensions of a given field and subgroup of the Galois group subgroups of the Galois group okay and there is an analog so the Galois correspondence in field theory is a correspondence on the one side between field extensions and on the other side between subgroups of a group and in this case it is the Galois group okay so it is a connection it is a connection between field theory and group theory okay and it is very useful because lot of field theory problems can be translated to group theory problems and lot of group theory problems can be translated to field theory problems. In the same way the covering space theory is very very similar what it does is it translates topological coverings which is topological data into subgroups of the fundamental group. So it also connects to topological side the topological side to the group theory side okay so that you can use some algebra in your topology okay so that is why usually this is a part of usually a first course in algebraic topology okay. So of course all this is very uninteresting if D is simply connected because if D is simply connected then your you know your the fundamental group is trivial okay but then it is still not so easy in fact that is the you know I will explain why you see see this whatever I have written here the algebra the analysis in the topology of D I have written it for a domain D in the extended plane okay but I am what the philosophy is that this is this holds for any Riemann surface okay now there is something called a Riemann surface a Riemann surface is something that locally looks like the plane okay but globally it may be a different surface. So for example it may be a cylinder in 3 space okay it may be it may look like a torus alright or it may look like an n torus so it may be it it might look like several tori which are stuck together by removing discs and open discs and you know pasting the boundaries of those open discs okay. So these are called Riemann surfaces and these are studied by Riemann okay and Riemann was fascinated to note that on these Riemann surfaces you can put many complex structures there you can put non-isomorphic complex structures and you must think of a complex structure as a structure which allows you to decide whether a function on that surface is holomorphic or not okay. So Riemann found that you know see Riemann try to do what we do in complex analysis on the plane on the plane what do we do we take a domain okay and ask when a function is analytic at a point okay and if it is not if it is analytic then of course you know if it is not analytic then you see whether that point is an isolated singularity and so on that is how you do the analysis. Now what Riemann wanted to do was he wanted to do this on a surface so he wanted to say that suppose I have now a function on the on a torus okay or say even an open subset of the torus alright when I say open subset you take the induced topology from R3 in which the torus sits okay and then suppose I have a function which is complex value to define on an open subset of the torus when can I say it is holomorphic when can I say it is analytic okay. So you are trying to study when a function defined on an open subset of a surface is analytic the answer to this is that you should define what is called a Riemann surface okay and there are different Riemann surface such as you can put and Riemann found that he was fascinated by these different Riemann surface structures and the most beautiful theorem in moduli theory is that you actually take the set of all these Riemann surface structures that itself becomes a nice object it becomes an analog at least on an open set it becomes an analog a higher dimensional analog of Riemann surface which is called a complex manifold and of course it could have a boundary which could have some singular points but it is a very beautiful object okay. So the moral of the story is that I am trying to say that whatever I am writing here for D a domain the extent plane it also works for it also works for a domain it also works for a Riemann surface okay and so for example in that context you know it is really it is really amazing that you get lot of so you know if you ask this so let me ask you a fundamental question the fundamental question is suppose you have a simply connected Riemann surface okay so the moment I say simply connected the topology seems to be very trivial because in the sense that in the fundamental group is trivial so you do not expect anything special but then you can ask how many simply connected Riemann surfaces are there which are not isomorphic to each other okay. Now you more or less know the answer partially because you know the Riemann mapping theorem tells you if you have seen it in a first course in complex analysis which you should have done that you know any simply connected open subset of the complex plane which is not the whole plane has to be holomorphically isomorphic that is biholomorphic to the unit disc okay so if you take domains in the complex plane okay simply connected domains in the complex plane there are only two types up to holomorphic isomorphism one is the whole plane the other one is unit disc okay so now it is an amazing fact that well even before that let me say look at the Riemann sphere okay which is you know we use that to study the point at infinity where the stereographic projection the Riemann sphere is also a nice surface of course and is compact okay and you can actually make it into a compact Riemann surface okay now the fact is that it is also simply connected the sphere is simply connected so that is also another simply connected Riemann surface now it is a very deep theorem that you take any simply connected Riemann surface it has to be it has to be isomorphic holomorphically isomorphic to one of this one of these three any simply connected Riemann surface has to be either it should either look like the whole plane okay or it should look like the unit disc or it should look like the Riemann sphere there are no other possibilities it is a very deep theorem it is called uniformization so even in the simply connected space even in the simply connected case you get a very deep theorem and the theorem is very hard to prove because you have to use lot of techniques from analysis to prove it okay so it involves lot of analysis it involves study of harmonic functions meromorphic functions etc okay and it has it involves lot of reasonable amount of functional analysis and measure theory okay you have to do you have to do all this to get that theorem okay so anyway so the fact I want to say was that now given all these three aspects of view putting them together is what geometry is all about okay so let me write here so geometry of D is you know the interplay between these three the geometry of a domain is actually the interplay between the analysis on the domain the algebra on the domain and the topology on the domain and I have given you a rough idea you know the analysis on the domain is the complex analysis part okay the algebra on the domain is to really study the field of meromorphic functions the algebraic properties of the field extension given by the field of meromorphic functions of the domain and the topological part is to study the covering space theory of the domain okay and it is and it so happens I mean as the great classical giants like Riemann and Clifford and Abel and Jacobi and we are stress have found and Clifford for example that you know all these properties all these various points of view they are all interrelated okay so it is an amazing fact it is an amazing fact and discovering that is what doing geometry is all about okay so you should not think that you should not think at high school level the geometry is just about drawing triangles and circles and you know measuring of angles and arcs and things like that but it is really higher geometry in the higher sense is actually looking at the interplay of all these things okay so I well now you see I want to give you a couple of examples so the so here is the first example so here is an example you take the domain to be the extended plane itself you take the domain to be the extended plane itself okay so after all we are studying domains in the extended plane so take the whole extended plane that is also domain alright it is in fact it is simply connected because it is you know it is homeomorphic the Riemann sphere the Riemann sphere is simply connected so it is simply connected it is compact okay so it is a very nice thing now what are the field of meromorphic functions on D okay what are the field of meromorphic functions on the extended plane so you know this is an extension of the complex numbers as we have seen this is an extension of complex numbers but you know what is it that we proved last time a function which is meromorphic on the extended plane is none other than a quotient of polynomials it is a rational function okay and therefore this is exactly equal to the this is exactly equal to C round bracket Z this is the algebraic notation C round bracket Z and the C round bracket Z is actually the field of fractions or quotient field of C of square bracket Z and C of square bracket Z is standard notation is the ring of polynomials in the variable Z with complex quotients okay and C of Z is the field of fractions which is quotients of such polynomials and so you take quotients of polynomials but of course you do not put in the denominator 0 anything other than 0 you put okay so the moral of the story is that you have very nice description of this field extension in the in the case of the C union infinity which is extended plane and usually you know extended plane is thought of as a Riemann sphere you know they are isomorphic but you can make them also isomorphic in a holomorphic sense by giving the Riemann sphere a Riemann surface structure okay so often people do not use if you see the literature you will see that people often use C union infinity instead of C union infinity they keep saying Riemann sphere all the time okay. So now you can see that what are the properties of this field extension you see this field extension is actually it is purely transcendental and has transcendence degree 1 okay it is purely transcendental and has transcendental degree 1 well the transcendence degree is actually the number of algebraically independent variables that generate the bigger extension okay so the bigger extension C of z is generated by a single variable z and that is the only algebraically independent variable okay that one variable is enough so the transcendence degree is actually 1 okay and it is purely transcendental because there is no element in C z which is algebraic there is no element in C z which is not in C and which is algebraic or C and that is you know why that is because complex numbers are algebraically closed they are all algebraically closed so this is an exact so field theoretically this is so this is called this is what is called as a function this is a simplest example of what is called a function field in one variable okay and the beautiful thing is that now if you take any compact Riemann surface okay then if you take the field of monomorphic functions on that compact Riemann surface what you will get is a function field in one variable but the only thing is that it may not be purely transcendental above the transcendent above there may be an algebraic part okay so it will be first a transcendental extension purely transcendental extension of degree 1 just like this and then be and above that you will have an algebraic extension which will be a finite extension okay so that is how it looks in general okay and well I will give you another example for that so let me write this here this is a purely transcendental extension of transcendence degree 1 okay so see the so the picture that is that is associated with this is the Riemann sphere okay so this is the picture that is associated with this and for all practical purposes you think of the extended plane as a Riemann sphere okay. Now let me let me tell you more generally what is it that happens with something so let me give you an example of a more complicated case the so called the case of so called elliptic functions or W periodic functions okay so here is so here is what I am going to do what I am going to do is I am going to take so I am going to define what a W periodic function is see a W periodic function so this is the topic of what are known as elliptic functions so this is also an example okay W periodic function is a function f of z with f of z plus omega 1 is equal to f of z and f of z plus omega 2 is equal to f of z where omega 1 by omega 2 is not real and of course omega 1 omega 1 omega 2 are non zero okay so I am just defining what a W periodic function is so the see the definition is very very simple for example sin theta you know is periodic with 2 pi because sin of theta plus 2 pi is the same as sin theta so the idea is that to the variable of the function or the argument of the function you add the period the function value should not change okay so what the first equation says is that f of z plus W1 is equal to f of z actually tells you tells you that W1 is a period okay and the second equation f of the second requirement f of z plus W2 equal to f of z tells you that W2 is also a period so W1 and W2 are periods and the fact is that these we want these periods to be linearly independent over R in other words what you want is that if you take these two complex numbers W1 and W2 then you join the origin to them okay namely you take the vectors that they represent in the plane then these should be different these vectors should be linearly independent they should be in different directions you know they will be linearly dependent if and only if the quotient W2 by W1 or W1 by W2 is a real number okay so this condition that W1 by W2 is not real is just to tell you that these two vectors are two different vectors they will form therefore a basis of C over R okay C the complex numbers over R is a two dimensional vector space and they will form a basis so this is equivalent to saying that W1 W2 form a basis for C over R okay and we are putting this condition in order to make sure that essentially you have two distinct periods okay which are in two so it is periodic in two directions okay the fact that f of z plus W1 is equal to f of z tells you that you know if you translate along the direction of W1 by integer multiples of W1 the function value does not change okay so you must remember that when I say f of z plus W1 is f of z it follows that f of z plus n1 W1 is also f of z for all integers n1 okay because I can just use induction f of z plus W1 is equal to f of z so f of z plus 2 W1 is f of z plus W1 plus W1 which is f of z plus W1 which is f of z and so on and so forth okay so what it tells you is that the moment something is a period then all its integer multiples are also periods okay and similarly you also have for the other so but what is adding W1 adding W see addition of a complex number is this translation along the direction along the vector that is represented by that complex number okay so you know basically if I have a point z what is what is z plus W1 it is actually this vector so this will be z plus W1 I am just translating z by the vector W1 okay and then similarly what is z plus W2 I am just translating z by the vector W2 alright and that is if I add W1 but if I add minus W1 you know I am translating in the other direction if I add minus 2 W1 I am translating in the direction opposite to W1 two times and so on and so forth okay so these are all so the moral of the story is that you know I basically the function values do not change if you translate along two different directions okay that is why it is called W periodic it is periodic and the period there are two different periods okay and such functions are called actually now you have to put some more condition on these functions the condition you put on these functions is that you know to make them very interesting these points W1 these points which are given by integer multiples of W1 added to integer multiples of W2 okay that they will form a lattice a grid in the plane okay and the function becomes very interesting if the function is meromorphic exactly at those points okay and such functions are called elliptic functions and believe it or not they are exactly the functions which are the functions meromorphic on a torus at a single point okay and this is the beginning of the so called Weierstrass Fe theory there is something called the Weierstrass Fe function okay which is a fundamental model of this kind of function and the beautiful thing is that every torus the complex structure on any torus can be controlled by prescribing such a function okay and so the Weierstrass Fe functions completely give you so if you want to study the various complex structures you can put on a torus what you will have to do is you have to study various W periodic meromorphic functions which are otherwise called elliptic functions the reason they are called elliptic is because this is beautiful this complex the moment you put a complex structure on the torus it becomes believe it or not it becomes a cubic curve it becomes a cubic curve and therefore it becomes an algebraic geometric object okay so algebraic geometry also comes in geometry also comes in algebra comes in a beautiful way okay and and this is also part of a very deep theorem which says that you know you take any compact Riemann surface it is algebraic it can be it is just given by a common 0 set of a bunch of polynomials okay and that is an amazing theorem okay. So I have so what I want to tell you is that I have given an NPTEL video course on Riemann surfaces and all these things are explained in detail throughout the course you can you can when you find time you can have a look at that and the other thing that I want to tell you is that there is this book that I have written and it reads an introduction to families deformations and moduli this book is basically available as a free freely downloadable copy in the form of a navigable PDF file and it contains a lot about the geometry of Riemann surfaces so at least the first chapter so that is also something that can be advanced reading material for people who are interested in pursuing this so let me write let me continue so I have also f of z plus n to w2 is equal to f of z for all n to in z so in totality what I will get is I will get f of z plus n1 w1 plus n2 w2 is equal to f of z for all n1, n2 in z if I put both these together okay and what are these points n1 w1 plus n2 w2 they are the vertices of a grid of parallelograms okay. So in fact you know if you draw this if I draw a diagram it is going to look like this so I have this this is my complex plane and you see I have w1 here I have w2 here okay and you know then you know if I draw this parallelogram then you know pretty well that this is w1 plus w2 okay by the parallelogram law of additional vectors if you want and then you know if I extend this parallelogram below then I am going to get this point is going to be you know it is going to be w1 minus w2 okay and this point is going to be minus w2 and if I extend it like this this point is going to be w2 minus w1 and this is going to be minus w1 minus w2 and this is going to be minus w1 okay and more generally if I draw if I look at all these points that go on that I get as the vertices of the parallelograms that I get by simply starting with this parallelogram and simply displacing it by either plus or minus w1 or plus or minus w2 okay that is by translating it with plus or minus w1 or plus or minus w2 I will get so many the whole plane is covered by these parallelograms okay and the vertices of the parallelograms are precisely the points which are of the form n1 w1 plus n2 w2 okay and that is called a lattice okay and the fact is that you see where is the you know just to give you an idea of what is going on where is the topology coming in here. So the fact is that what you do is you divide by the equivalence relation equivalence relation Z1 is equivalent to Z2 if and only if there exists n1 n2 such that Z1 is equal to Z2 plus n1 w1 plus n2 w2 okay so see this is the plane this is the complex plane and I am defining an equivalence relation on the plane the equivalence relation is two points are equivalent if one of them is a translate of the other by one of these grid points okay and what is the advantage of this advantage of this is that if two points are related like that then the w periodic function will have the same value at both points because f of Z1 will be equal to f of EZ2 plus n1 w1 plus n2 w2 but that is also equal to f of Z2 so f of Z1 will be equal to f of Z2 okay because when I apply f to this equation okay on the right side I will get f of Z2 because of periodicity of f so I will get f of Z1 equal to f of Z2 okay what it means is the function does not the value of the function does not change if you change the point by a translate by a vector which belongs to one of the grid points okay so if you divide by this equivalence relation what you will get is you will get the torus you will get a beautiful torus and you can see that very very easily you just take this fundamental parallelogram okay this fundamental parallelogram will if you take the interior every point in the interior will be a unique representative in the in its equivalence class but for points but then you will have to only identify the boundaries see if you identify the top boundary with the bottom boundary you will get a cylinder okay and then you will have to identify these two which will in the cylinder look like circles so if you identify them you will get a torus okay so the moral of the story is you take the plane divided by a lattice like this you get the torus okay and all the points in the grid including the origin the origin is here they all will go to a particular special point on the torus okay and the beautiful thing is that the function that you defined on the complex plane will go down to a function on the torus okay and if the function is meromorphic exactly at all these grid points it will go down to a meromorphic function on the torus and you know you may wonder why should I worry about why should I not consider holomorphic functions of the torus and you very well know the answer there will not be any non constant holomorphic function of the torus because the torus is compact since the torus is compact if you define a holomorphic function on the torus okay you are going to get something that is holomorphic you can use Liouville kind you can use the Liouville theorem if you have a holomorphic function on the torus if you compose it with this map that is that goes from the complex plane to the torus you will get a holomorphic function on the plane but since it is defined on the torus which is compact its image is compact therefore the image is bounded so I get an entire function which is bounded and that is going to be constant by Liouville theorem okay and this picture also explains why the only functions on the torus are exactly the functions on the plane which are W periodic with respect to the periods W1 and W2 and the if you take this unique point P which is the image of the grid the meromorphic since you know holomorphic functions are not available they are constants the only things that are available are the meromorphic functions and then if you look at meromorphic functions on the torus at the point P you will get the they will be the same as W periodic functions okay and therefore the moral of the story is that you know if you look at the if you look at the field of meromorphic functions on the torus okay they are the same as the collection of meromorphic functions on the collection of W periodic functions with these two periods okay and that is a field of course okay mind you what is the domain now the domain is the whole plane okay and I am looking at meromorph I am looking at functions which are meromorphic with poles at points of the grid okay possibly at points of the grid alright and then what I get is I get a field and what is that field that field is nothing but this field of meromorphic functions on the torus which are meromorphic at a given point okay and so let me call the torus as T okay mind you the torus it depends on the choice of W1 and W2 okay and it is a different story that there is a lot of geometry there but what I want to tell you is that well I want to tell you the following thing simplest meromorphic function function on C that is W periodic that is W periodic with respect to W1 and W2 is the Weierstrass phi function and this is the phi function so there is a pretty symbol for that very special symbol so phi of z is so there is a formula for that basically it is a formula that will tell you that it is a meromorphic function which has a double pole with residue 0 at each of those points of the grid okay so you know what you are going to get let me write that down here you can find this in any standard book for example Alphos book on complex analysis which is a classic so it is 1 by z square plus summation over n1, n2 belonging to z minus 0, 0 z cross z in fact so I should write I should write it carefully n1, n2 ordered pair belonging to z cross z minus 0, 0 it is 1 by z minus n1, w1 minus n2, w2 the whole square minus 1 by n1, w1 plus n2, w2 the whole square so this is the expression for the Weierstrass phi function which was discovered by Weierstrass and of course if you go through in detail the lectures of my video course you will see how this comes about but you can see something immediately you see that this 1 by z square is the principle part at the origin and that will tell you that you know origin is a double pole and residue is 0 because there is no 1 by z term okay and then you look at each of these other terms 1 by z minus n1, w1 plus n2, w the whole square tells you that n1, w1 plus n2, w2 is a point of the grid is a general point of the grid okay and if you when I write 1 by z minus that point the whole square actually I am looking at a pole of order 2 at that point okay and again the residue there is 0 alright. So as a result this already gives you a you know it gives you a meromorphic function which is having a double pole at each of these grid points okay and this extra term that is added here is for convergence because you know I have added infinitely many poles okay I have added poles at every point of the grid I have made every point of the grid into a double pole and I am getting a huge series I want you to converge and it is only for this convergence that this extra constant term is being added okay and therefore I get this fair function and here comes the amazing here comes the amazing theorem. The amazing theorem is the following that if you take the complex numbers and you take the meromorphic functions on d on the complex plane with respect to w1 and w2 okay you look at the meromorphic functions which are w periodic at with periods at w1 and w2 okay and mind you this is the same as the meromorphic functions on the torus okay which are meromorphic at that unique point which I will call it star which is the image of the grid the whole grid the whole grid goes to a single point on the torus because all the points in the grid are equivalent to each other okay and they all define a single equivalence class so it is a single point on the torus that mind you the torus is a set of equivalence classes okay topologically and you give it the quotient topology alright. Now the beautiful thing is that what is this set of meromorphic functions you know it is a field what is that field you know you know what that field is that field is just the field gen it is just the field of fractions of Phi fz and its derivative it is beautiful and how does this extension break up it breaks up as the first Phi fz this is again a transcendental extension is a purely transcendental extension of transcendence degree 1 and then from here to here from here to here this is an algebraic extension this is an algebraic extension because the derivative of the Phi function Phi prime satisfies a polynomial relation with respect to Phi and that is expressed as a differential equation it is a very famous differential equation and that differential equation interestingly it comes from analysis but it tells you that the torus is algebraic it tells you that the torus is nothing but a cubic curve okay which is an amazing illustration of the fact that in general a compact remand surface is given is algebraic it is given by algebraic equations okay so all these details you can have a look at in detail in more detail in my video lecture course but this is to tell you that a lot of geometry is involved by looking at the field extension given by the field of meromorphic functions okay so I will stop here.