WEBVTT
Kind: captions
Language: en
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Alright so we are discussing Meromorphic functions
okay and we were looking at Meromorphic functions
00:01:52.200 --> 00:01:59.999
on the extended plane in the last class in
the last lecture and we prove that a function
00:01:59.999 --> 00:02:06.189
which is Meromorphic on the extended plane
is none other than quotient of polynomials
00:02:06.189 --> 00:02:14.180
okay namely a rational function okay, so what
I need to what I want to tell now is about
00:02:14.180 --> 00:02:20.310
the collection of Meromorphic functions on
a domain okay, you take a domain in the extended
00:02:20.310 --> 00:02:25.650
plane and look at the set of all Meromorphic
functions defined on the domain okay then
00:02:25.650 --> 00:02:31.610
that it has a has a nice structure in fact
algebraic structure it is a field okay and
00:02:31.610 --> 00:02:40.370
in fact it is an algebra over the complex
numbers okay and so it is a field extension
00:02:40.370 --> 00:02:46.700
of the complex numbers and the properties
of this field extension, algebraic properties
00:02:46.700 --> 00:02:53.319
of this field extension they have captured
a lot of topological and geometric properties
00:02:53.319 --> 00:02:55.340
of the domain okay.
00:02:55.340 --> 00:03:06.379
So this is how there is a link from the complex
analysis side to the algebra side okay, so
00:03:06.379 --> 00:03:13.750
you know geometry involves an interplay of
various ways of mathematics, so studying something
00:03:13.750 --> 00:03:19.680
Geometrically will in all studying it from
the analysis viewpoint okay and studying it
00:03:19.680 --> 00:03:24.680
from the topological viewpoint also studying
it from the algebraic viewpoint but when you
00:03:24.680 --> 00:03:29.540
looking at a particular nice object okay when
yesterday the analytically it will have some
00:03:29.540 --> 00:03:36.190
properties okay, some special properties and
then when you study it algebraically it will
00:03:36.190 --> 00:03:39.859
have some properties, special properties.
When you study topologically it will have
00:03:39.859 --> 00:03:46.519
some special properties and the fact is that
these properties are interrelated.
00:03:46.519 --> 00:03:51.769
There is some beautiful relationship, hidden
relationship between the analytic, the algebraic
00:03:51.769 --> 00:03:57.959
at the topological properties of a nice object
and that relationship is what you may call
00:03:57.959 --> 00:04:01.389
as geometry okay. So if you want to really
understand geometry of an object you have
00:04:01.389 --> 00:04:10.019
to analyse it using all the 3 viewpoints algebraic,
analytic, topological okay, so in that sense
00:04:10.019 --> 00:04:15.279
know how do I do geometry on a domain in the
complex plane or in the extended complex plane
00:04:15.279 --> 00:04:23.520
okay. What I can do is of course the analysis
is there, the analysis will worry about what
00:04:23.520 --> 00:04:29.219
kind of functions you can define on the domain,
what are the holomorphic functions or analytic
00:04:29.219 --> 00:04:35.770
functions on the domain? What are the Meromorphic
functions on the domain? and so on that will
00:04:35.770 --> 00:04:45.150
be the viewpoint from analysis but then how
do you go to algebra.
00:04:45.150 --> 00:04:50.979
The point is that you take the set of Meromorphic
functions that forms a field okay and that
00:04:50.979 --> 00:04:56.180
is the field extension of the complex number
and you study the properties of this field
00:04:56.180 --> 00:05:02.039
extension okay, so in field theory you have
lot of you would have come across in a course
00:05:02.039 --> 00:05:10.830
in the algebra in field theory that field
extensions are of so many types okay there
00:05:10.830 --> 00:05:14.213
are algebraic extensions, there are transcendental
extensions and then they were normal extension,
00:05:14.213 --> 00:05:21.409
there are splitting fields, there are extension
okay there are of course separable and non-separable
00:05:21.409 --> 00:05:26.529
extension and we study all these things and
of course the most important thing here in
00:05:26.529 --> 00:05:33.190
general is the study of the nature of extensions
because that connects up group theory which
00:05:33.190 --> 00:05:45.210
is it connects up with the so called groups.
So you see the moment you look at the field
00:05:45.210 --> 00:05:49.020
of Meromorphic functions you get an extension
of the complex numbers and then you can do
00:05:49.020 --> 00:05:54.520
algebra okay and somehow these things are
all connected and I will try to give couple
00:05:54.520 --> 00:06:01.880
of examples. So 1st of all let me begin by
1st saying that if I take a domain in the
00:06:01.880 --> 00:06:08.350
extended plane then the set of all Meromorphic
functions in the domain is actually a field
00:06:08.350 --> 00:06:09.350
okay.
00:06:09.350 --> 00:06:30.200
So so let me write that down the field of
Meromorphic functions, so let D inside C union
00:06:30.200 --> 00:06:39.120
infinity be a domain, so you are taking a
domain in the extended complex plane C union
00:06:39.120 --> 00:06:48.000
infinity, so in particular mind you it is
a nonempty open connected set okay and the
00:06:48.000 --> 00:06:53.570
advantage of taking a domain in the extended
plane is that you can also look at a neighbourhood
00:06:53.570 --> 00:06:58.160
of infinity okay that is the advantage, so
you are also including the point at infinity
00:06:58.160 --> 00:07:14.839
right, so let so here is the notation I will
put script m of D be the set of of Meromorphic
00:07:14.839 --> 00:07:19.350
functions on D okay.
00:07:19.350 --> 00:07:27.169
So what is this script m of D? Script m of
D is the collection of all Meromorphic function
00:07:27.169 --> 00:07:31.789
on and you know what a Meromorphic function
is? You have defined the Meromorphic function
00:07:31.789 --> 00:07:41.579
we are function which is analytic at all points
but except or points on an isolated sets which
00:07:41.579 --> 00:07:47.740
have to be always singularities okay, so it
is analytic except for poles and the moment
00:07:47.740 --> 00:07:51.890
you say analytic except for poles it means
the singularities can be only poles and that
00:07:51.890 --> 00:07:55.669
in particular means that the singularities
can only be isolated because poles are isolated
00:07:55.669 --> 00:07:58.390
singularities by definition okay.
00:07:58.390 --> 00:08:03.089
So you take all the Meromorphic functions
on the domain okay now the fact is that this
00:08:03.089 --> 00:08:14.110
is a field okay, so let us see that let f
and g be Meromorphic functions on D okay then
00:08:14.110 --> 00:08:22.380
you see then you can notice the following
things number 1 is lambda is a complex number
00:08:22.380 --> 00:08:30.370
okay then lambda f is also a Meromorphic function
okay multiplying a Meromorphic function by
00:08:30.370 --> 00:08:35.110
a constant is going to keep it Meromorphic
okay of course if the constant is 0 you will
00:08:35.110 --> 00:08:42.149
get 0 and this 0 is a constant function. Of
course when you say Meromorphic, analytic
00:08:42.149 --> 00:08:47.190
is also included okay, so the definition for
Meromorphic is that it is analytic except
00:08:47.190 --> 00:08:53.490
for poles that does not mean it has to have
a pole it can be it can have no poles and
00:08:53.490 --> 00:09:01.200
it can be analytic everywhere, so holomorphic
functions also included in the set of Meromorphic
00:09:01.200 --> 00:09:05.870
functions okay.
00:09:05.870 --> 00:09:09.630
So this statement is obvious if I take a Meromorphic
function multiply by constant, if the constant
00:09:09.630 --> 00:09:13.959
is 0 of course I am going to get the 0 function
which is holomorphic which is analytic because
00:09:13.959 --> 00:09:20.300
it is a constant function okay but if lambda
is not 0, lambda times f will also be Meromorphic
00:09:20.300 --> 00:09:25.440
and it will have the same poles, okay by multiplying
by lambda you not going to change the poles
00:09:25.440 --> 00:09:31.750
and you are not going to change the order
of the poles. Essentially you just multiplying
00:09:31.750 --> 00:09:43.260
by a constant okay, so this is one obvious
thing then the 2nd thing is that if you take
00:09:43.260 --> 00:09:50.630
the sum of these 2 Meromorphic functions,
this will also be a Meromorphic functions
00:09:50.630 --> 00:09:57.139
okay. The sum of f and g will also be Meromorphic
why because you see the fact is that f is
00:09:57.139 --> 00:10:04.920
Meromorphic so it has some it has a collection
of poles okay and an isolated set of points.
00:10:04.920 --> 00:10:10.391
Then g is also Meromorphic so it has also
poles in another isolated set of points and
00:10:10.391 --> 00:10:16.760
then you take the union of these 2 isolated
set that is again an isolated set okay and
00:10:16.760 --> 00:10:24.029
these are the only points where f plus g will
have problems okay so at a point where f does
00:10:24.029 --> 00:10:27.960
not have a problem and g does not have a problem,
f plus g will not have a problem that is at
00:10:27.960 --> 00:10:33.360
a point where f is analytic and g is analytic,
f plus g of course will be analytic okay,
00:10:33.360 --> 00:10:39.980
so the only problems for the function analyticity
of the function f plus g will be at the points
00:10:39.980 --> 00:10:46.311
where f and g have problems okay and it is
possible that some ofůthere could be some
00:10:46.311 --> 00:10:47.980
cancellations okay.
00:10:47.980 --> 00:10:54.080
So for example f may be 1 by Z minus Z naught
g may be minus 1 by Z minus Z naught, so if
00:10:54.080 --> 00:11:00.580
I take f plus g I will get 0 which does not
have a pole at Z naught okay, so some poles
00:11:00.580 --> 00:11:07.480
can cancel out also and sometimes the order
of a pole can come down okay when you add
00:11:07.480 --> 00:11:13.820
course when I say add it also includes a subtraction
because subtraction is just adding with minus
00:11:13.820 --> 00:11:20.010
1 multiplied by the 2nd function okay, so
so the moral of the story is that sum of 2
00:11:20.010 --> 00:11:25.890
Meromorphic functions is again a Meromorphic
function. It could very well be analytic okay
00:11:25.890 --> 00:11:29.530
some poles might cancel out all the poles
may cancel out for example if you take the
00:11:29.530 --> 00:11:34.390
function and you take its negative and add
it you will get 0 and that is clearly holomorphic
00:11:34.390 --> 00:11:40.710
is a constant function, so some is Meromorphic
so you know the moment you look at the first
00:11:40.710 --> 00:11:46.070
2 things is will tell you that you know m
of D is a vector space or complex numbers
00:11:46.070 --> 00:11:53.170
see because it is you see so there is
a scalar multiplication.
00:11:53.170 --> 00:11:57.620
If you think of complex number a scalars then
there is a scalar multiplication and there
00:11:57.620 --> 00:12:09.230
is addition, so this becomes aůso m of D
is C vector space so you get that immediately
00:12:09.230 --> 00:12:21.290
okay. Now let us look at f times g look at
f times G, see f multiplied by g will also
00:12:21.290 --> 00:12:27.510
be Meromorphic on D this is also very clear
because just from the fact that you know what
00:12:27.510 --> 00:12:31.330
are the problem points on Mark the problem
points are the points where f has problems
00:12:31.330 --> 00:12:40.000
and g has problems okay, so if you take out
those problem points then f times g will be
00:12:40.000 --> 00:12:45.580
analytic, so at a point where f is analytic
and g is analytic f times g will be analytic
00:12:45.580 --> 00:12:52.250
and the only place where f times g will fail
to be analytic it is probably on the of the
00:12:52.250 --> 00:13:04.030
set of poles of f and poles of g okay and
so you see and of course if you want you can
00:13:04.030 --> 00:13:16.480
write out always the principal parts and see
that you know if you functions have poles
00:13:16.480 --> 00:13:22.670
at the same point they have common pole then
if you multiply the product function will
00:13:22.670 --> 00:13:27.720
have a pole with higher order in fact it will
have order equal to sum of the orders that
00:13:27.720 --> 00:13:33.770
is obvious if you write out the principal
parts okay, so in the Laurent expansion alright.
00:13:33.770 --> 00:13:41.390
So I mean the point is that you know all these
algebraic operations of adding, subtracting
00:13:41.390 --> 00:13:46.950
multiplying by a constant and just multiplying
and of course we are going to see division
00:13:46.950 --> 00:13:54.780
all these things they do not change the Meromorphic
nature okay, so by adding or subtracting or
00:13:54.780 --> 00:14:00.160
dividing or multiplying or multiplying by
a constant you cannot change a Meromorphic
00:14:00.160 --> 00:14:04.000
function into a non-Meromorphic function.
If you are only working with Meromorphic functions
00:14:04.000 --> 00:14:11.180
you will get back again Meromorphic function
okay, so fine so you have f and g are the
00:14:11.180 --> 00:14:15.950
product f time g is also Meromorphic of course
by product 1 means point wise product okay,
00:14:15.950 --> 00:14:23.200
so f g is a function which at each point Z
is defined by f of Z times g of Z alright
00:14:23.200 --> 00:14:33.500
then of course I can say the same of f by
g this is also a Meromorphic function why
00:14:33.500 --> 00:14:43.840
did g is not identically 0 okay of course
I should not divided by 0, so the fact is
00:14:43.840 --> 00:14:49.339
that see when I take f by g okay what other
problems points?
00:14:49.339 --> 00:14:55.360
The problem points will be poles of F, poles
of g and now you will have extra problem points
00:14:55.360 --> 00:15:00.420
at 0 so g because they are zeros of the denominator
they become the zeros of g will become poles
00:15:00.420 --> 00:15:05.800
of f by G, they are likely to become poles
of f by g and of course you know it might
00:15:05.800 --> 00:15:11.050
happen some zeros of g may cancel out with
some zeros of f because the zeros of f on
00:15:11.050 --> 00:15:16.000
the numerator the zeros of g are on the denominator
some zeros might cancel but a set of problem
00:15:16.000 --> 00:15:25.779
points are just the poles of F, the poles
of g and the zeros of g and you know zeros
00:15:25.779 --> 00:15:31.620
of an analytic function also isolated, you
know that, there is a theorem okay in fact
00:15:31.620 --> 00:15:35.520
that is another version of identity theorem
if you have seen it in the 1st course in complex
00:15:35.520 --> 00:15:41.880
analysis, so therefore the set of points where
f by g will have problems is still an isolated
00:15:41.880 --> 00:15:47.390
set of points okay and at each of those points
you can only get poles you cannot get anything
00:15:47.390 --> 00:15:52.540
worse, so therefore f by g is also Meromorphic
in particular I could have taken 1 by g I
00:15:52.540 --> 00:15:58.351
can put f equal to 1 I will get 1 by g is
also Meromorphic and that means so what will
00:15:58.351 --> 00:16:02.300
I get I will get 1 by g is Meromorphic if
g is not identically 0.
00:16:02.300 --> 00:16:06.860
So that means every nonzero Meromorphic function,
every Meromorphic function which is not identically
00:16:06.860 --> 00:16:13.790
0 as an inverse okay and that is what you
require for a field okay. A field should be
00:16:13.790 --> 00:16:17.680
basically a group under multiplication if
you throughout the 0 element commonly set
00:16:17.680 --> 00:16:25.589
okay, so well if you if you look at all these
things, these things will tell you that m
00:16:25.589 --> 00:16:33.870
of D is a field and you know you put it together
with this fact that we saw earlier we have
00:16:33.870 --> 00:16:39.680
seen just above that m of the D is also a
complex factor space of field extension with
00:16:39.680 --> 00:16:49.110
the field which is also a vector space is
an algebra okay, so basically you can very
00:16:49.110 --> 00:16:54.580
will see that m of D contains complex numbers
because the complex numbers sit as constant
00:16:54.580 --> 00:16:59.240
channels okay you take any complex number
lambda you think of it as a constant function
00:16:59.240 --> 00:17:02.890
lambda. Constant function lambda is analytic
is defined everywhere.
00:17:02.890 --> 00:17:09.169
So it is analytic on every domain and it is
Meromorphic because mind you when I say Meromorphic
00:17:09.169 --> 00:17:16.909
I am allowing also analytic or holomorphic.
Meromorphic means that it can either be analytic
00:17:16.909 --> 00:17:20.409
and if it is not analytic that is if it has
singularities, the singularities must be only
00:17:20.409 --> 00:17:24.699
poles that is what it says. So Meromorphic
does not say that it should not be analytic,
00:17:24.699 --> 00:17:31.780
so in particular m of D contains the complex
numbers as a subfield you know the complex
00:17:31.780 --> 00:17:37.050
numbers of those form a field and therefore
m of D is a field extension of the field of
00:17:37.050 --> 00:17:38.660
complex number, so m of D is aů
00:17:38.660 --> 00:17:53.120
So let me write that m of D is a field extension
of the field C of complex and the beautiful
00:17:53.120 --> 00:18:03.549
thing is that the geometry on the domain D
is done by a lot of topology of the domain
00:18:03.549 --> 00:18:11.380
D is connected toů and a lot of analysis
on the domain D namely the behaviour of the
00:18:11.380 --> 00:18:16.220
existence and behaviour of Meromorphic functions
on D is connected with the algebraic properties
00:18:16.220 --> 00:18:23.600
of this field extension okay that is the geometric
content okay. So if this goes back to the
00:18:23.600 --> 00:18:30.860
work of classical giants like Riemann and
Clifford and Weierstrass and Able and Yakobi
00:18:30.860 --> 00:18:37.200
you know all these people who developed theory
of Riemann surfaces okay of course principally
00:18:37.200 --> 00:18:41.019
from Riemann, so let me write that down.
00:18:41.019 --> 00:18:49.230
So let me write it out as a diagram so you
have m of D and this is over C, so I am using
00:18:49.230 --> 00:18:56.760
this field theory notation you write a field,
a bigger field on top and you put a smaller
00:18:56.760 --> 00:19:02.110
field below and then you put a vertical line
thing that the thing that comes above is a
00:19:02.110 --> 00:19:21.500
field extension and things that come below
are some fields okay, so
00:19:21.500 --> 00:19:31.350
the algebraic properties of this field extension
they are connected so there is a, so the analysis
00:19:31.350 --> 00:19:51.890
and D which is existence
of a special existence and properties
00:19:51.890 --> 00:20:05.760
of Meromorphic functions on D that is analysis
on D okay and this part is algebra only, the
00:20:05.760 --> 00:20:10.950
algebra on the domain is actually studying
you might think of it as studying this field
00:20:10.950 --> 00:20:20.580
extension and then there is the topology of
D, topology of the domain D.
00:20:20.580 --> 00:20:29.490
So I am not trying to be very particular or
trying to go in detail but topology and the
00:20:29.490 --> 00:20:33.950
minimum for example D is of course connected
but one of the simplest thing that you can
00:20:33.950 --> 00:20:39.430
look at is whether D is simply connected or
not okay and then or if it is not simply connected
00:20:39.430 --> 00:20:45.230
you can see if it is multiply connected and
how many holes it has and so on and so forth
00:20:45.230 --> 00:20:57.770
okay after all D could be something like an
amoeba with some holes okay after all an open
00:20:57.770 --> 00:21:03.130
set can look like that and then the topology
worries about whether if it is simply connected,
00:21:03.130 --> 00:21:07.350
if it is not simply connected how many holes
are there and so on and so forth okay, so
00:21:07.350 --> 00:21:12.580
all this topological information for example
is encoded in the fundamental group of D and
00:21:12.580 --> 00:21:21.150
so on and so forth and in fact more precisely
I should say that you have to study the theory
00:21:21.150 --> 00:21:26.170
of covering spaces of the domain D that is
what is the topology of the means.
00:21:26.170 --> 00:21:38.170
So let me write this here the theory of covering
spaces of D related which is actually related
00:21:38.170 --> 00:21:55.280
over the fundamental group, in fact subgroups
of the fundamental
00:21:55.280 --> 00:22:05.020
group of D, of course you know the so you
know at this point let me tell you that if
00:22:05.020 --> 00:22:10.980
you have done decent in topology see there
is something called covering space theory
00:22:10.980 --> 00:22:16.170
which takes the topological space with decent
properties or example something that is house
00:22:16.170 --> 00:22:22.620
of locally house of connected locally path
connected, locally simply connected and then
00:22:22.620 --> 00:22:27.180
you study what are called as coverings of
topological covering and there is a Galva
00:22:27.180 --> 00:22:32.120
theorem of covering which says that you know
there is a Galva correspondence between the
00:22:32.120 --> 00:22:37.860
coverings and subgroups of the fundamental
group of your topological space and in fact
00:22:37.860 --> 00:22:51.480
under this Galva correspondence so this
Galva correspondence is an analog of the Galva
00:22:51.480 --> 00:22:53.590
correspondence that you have in field theory.
00:22:53.590 --> 00:22:58.559
See the Galva correspondence in field theory
is you correspondence between field extensions
00:22:58.559 --> 00:23:06.800
of a given field and subgroup of the Galva
group okay and there is an analog, so the
00:23:06.800 --> 00:23:12.090
Galva correspondence in field a really is
a correspondence on the one side between field
00:23:12.090 --> 00:23:17.400
extensions and on the other side between subgroups
of a group and in this case is the Galva group
00:23:17.400 --> 00:23:23.070
okay, so it is a connection between field
theory and group theory okay and it is very
00:23:23.070 --> 00:23:26.620
useful because a lot of field theory problems
can be translated to group theory problems
00:23:26.620 --> 00:23:32.600
and lot of group theory problems can be translated
to field theory problems. In the same way
00:23:32.600 --> 00:23:36.669
space theory is very very similar, what it
does is it translates topological coverings
00:23:36.669 --> 00:23:41.780
that is topological data into subgroups of
fundamental group, so it also connects to
00:23:41.780 --> 00:23:47.990
topological side, the topological side to
group theory side okay so that you can use
00:23:47.990 --> 00:23:50.000
some algebra in your topology okay.
00:23:50.000 --> 00:23:55.460
So that is where usually this is a part of
usually 1st course in algebraic topology okay,
00:23:55.460 --> 00:23:59.669
so of course all this is very uninteresting
if D is simply connected because if it D is
00:23:59.669 --> 00:24:09.559
simply connected then the fundamental group
is okay but then it is still not so easy in
00:24:09.559 --> 00:24:21.410
fact there is I will explain why. Whatever
I have written here, the algebra, the analysis
00:24:21.410 --> 00:24:26.830
and the topology of D I have written it for
a domain D in the extended plane okay but
00:24:26.830 --> 00:24:32.760
what the philosophy is that this holds for
any Riemann surface okay. Now there is something
00:24:32.760 --> 00:24:38.230
called Riemann surface, the Riemann surfaces
something at locally looks like a plane okay
00:24:38.230 --> 00:24:43.331
but globally it may be different surface,
so for example it may be a cylinder in 3 space
00:24:43.331 --> 00:24:50.540
okay it may look like a torus alright or it
may look like n torus, so it might look like
00:24:50.540 --> 00:24:58.190
several tori which are stuck together by removing
disk and open disk and pasting the boundaries
00:24:58.190 --> 00:24:59.940
of the open disk okay.
00:24:59.940 --> 00:25:07.860
So these are called Riemann surfaces and these
were studied by Riemann and Riemann was fascinated
00:25:07.860 --> 00:25:13.931
to know that on these Riemann surfaces you
can put many complex structures there you
00:25:13.931 --> 00:25:17.580
can put non-isomorphic complex structures
and you must think of a complex structure
00:25:17.580 --> 00:25:22.190
as a structure which allows you to decide
whether a function on that surfaces is holomorphic
00:25:22.190 --> 00:25:30.970
on not okay, so Riemann found that you know,
see Riemann try to do what we do in complex
00:25:30.970 --> 00:25:37.530
analysis on the plane. On the plane what we
do, we take a domain and ask when a function
00:25:37.530 --> 00:25:41.919
is analytic at the point okay and if it is
analytic then of course you know if it is
00:25:41.919 --> 00:25:46.470
not analytic then you see whether that point
is an isolated singularity and so on that
00:25:46.470 --> 00:25:47.480
is how you do the analysis.
00:25:47.480 --> 00:25:52.010
Now what Riemann wanted to do was he wanted
to do it on the surface, so he wanted to say
00:25:52.010 --> 00:25:57.799
that suppose I have now a function on a torus
okay or say even an open subset of the torus
00:25:57.799 --> 00:26:04.300
alright, when I say open subset you take the
induced topology from R3 in which the torus
00:26:04.300 --> 00:26:09.950
sits okay and then suppose I have function
which is complex valued defined on an open
00:26:09.950 --> 00:26:15.659
subset of the torus, when can I say it is
holomorphic, when can I say it is analytic?
00:26:15.659 --> 00:26:19.070
So you are trying to study when a function
defined on an open subset of a surface is
00:26:19.070 --> 00:26:23.830
analytic, the answer to this is that we should
define what is called Riemann surface okay
00:26:23.830 --> 00:26:28.730
and there are different Riemann surfaces structures
you can put and Riemann found thatůhe was
00:26:28.730 --> 00:26:32.299
fascinated by these different Riemann surfaces
structures and the most beautiful theorem
00:26:32.299 --> 00:26:36.340
in Modley theory is that you actually take
the set of all these Riemann surface structures
00:26:36.340 --> 00:26:39.480
that itself become a nice object.
00:26:39.480 --> 00:26:46.450
It becomes an analog at least on an open set,
it becomes an analog higher dimensional analog
00:26:46.450 --> 00:26:50.900
of Riemann surface which is called a complex
manifold and of course it could have boundary
00:26:50.900 --> 00:26:55.200
which could have some singular points but
it is a very beautiful object okay, so the
00:26:55.200 --> 00:26:59.460
moral story is that I am trying to say that
whatever I am writing here for D, D at domain
00:26:59.460 --> 00:27:07.570
in the extended plane it also works for a
Riemann surface okay and so for example in
00:27:07.570 --> 00:27:18.340
that context it is really amazing that you
get a lot ofů so you know let me ask you
00:27:18.340 --> 00:27:23.620
a fundamental question, the fundamental question
suppose if you have a simply connected Riemann
00:27:23.620 --> 00:27:28.240
surface okay, so the moment I say simply connected
the topology seems to be very trivial because
00:27:28.240 --> 00:27:32.200
in the sense that the fundamental group is
trivial so you do not expect anything special
00:27:32.200 --> 00:27:37.520
but then you can ask how many simply connected
Riemann surfaces are there which are not isomorphic
00:27:37.520 --> 00:27:39.250
to each other okay.
00:27:39.250 --> 00:27:45.230
Now you more or less know the answer partially
because the Riemann mapping theorem tells
00:27:45.230 --> 00:27:49.410
you if we have seen it in the 1st course in
complex analysis which you should have done
00:27:49.410 --> 00:27:54.550
at you know any simply connected open subset
of the complex plane which is not the whole
00:27:54.550 --> 00:28:00.500
plane has to be holomorphically isomorphic
that is by holomorphic to the unit test okay,
00:28:00.500 --> 00:28:05.919
so if you take domains in the complex plane
okay, simply connected domain is in the complex
00:28:05.919 --> 00:28:11.530
plane there are only 2 types up to holomorphic
isomorphism, one is the whole plane the other
00:28:11.530 --> 00:28:22.480
one is the unit disk okay, so now it is an
amazing fact that even before that let me
00:28:22.480 --> 00:28:28.020
say look at the Riemann sphere okay which
is you know we use that study the point at
00:28:28.020 --> 00:28:31.870
infinity as the stereographic projection,
the Riemann sphere is also a nice surface
00:28:31.870 --> 00:28:36.710
of course and is compact okay and you can
actually make it into a compact Riemann surface
00:28:36.710 --> 00:28:41.700
okay. Now the fact is that there is also simply
connected, Spear is simply connected so that
00:28:41.700 --> 00:28:43.039
is also another simply connected Riemann surface.
00:28:43.039 --> 00:28:49.559
Now it is a very deep theorem that you take
any simply connected Riemann surface it has
00:28:49.559 --> 00:28:54.970
to be isomorphic, holomorphically isomorphic
to one of these 3. Any simply connected Riemann
00:28:54.970 --> 00:28:59.370
surface has to be either it should either
look like the whole plane okay or it should
00:28:59.370 --> 00:29:03.610
look like the unit disk or it should look
like a Riemann sphere there are no other possibilities.
00:29:03.610 --> 00:29:08.990
It is a very deep theorem there is called
uniformization, so even the simply connected
00:29:08.990 --> 00:29:14.960
case you get a very deep theorem and the theorem
is very hard to prove because you have to
00:29:14.960 --> 00:29:19.520
use a lot of techniques from analysis to prove
it okay, so it involve a lot of analysis,
00:29:19.520 --> 00:29:26.850
it involves study of harmonic functions, Meromorphic
functions, et cetera and it involves a reasonable
00:29:26.850 --> 00:29:33.880
amount of function analysis and measured theory
you have to do all this to get that theorem
00:29:33.880 --> 00:29:42.720
okay. So anyway so the fact I want to say
was at now given all these 3 aspects of putting
00:29:42.720 --> 00:29:52.429
them together is what geometry is all about,
okay. So let me wright here so geometry of
00:29:52.429 --> 00:30:00.360
D is the interplay between these 3.
00:30:00.360 --> 00:30:06.870
The geometry of a domain is actually the interplay
between the analysis on the domain, algebra
00:30:06.870 --> 00:30:12.980
on the domain and the topology on the domain
and I have given you a rough idea, the analysis
00:30:12.980 --> 00:30:18.040
on the domain is the complex analysis part
okay. The algebra on the domain is to really
00:30:18.040 --> 00:30:21.669
study the field of Meromorphic functions,
the algebraic properties of the field extension
00:30:21.669 --> 00:30:24.710
given by the field of Meromorphic function
the domain and the topological part is to
00:30:24.710 --> 00:30:31.080
study the curving space theory of the domain
okay and it so happens I mean as the great
00:30:31.080 --> 00:30:37.960
classical giants like Riemann and Clifford
and Able and Yakobi and Weierstrass have found
00:30:37.960 --> 00:30:45.059
and Clifford for example at you know all these
properties, all these various points of view,
00:30:45.059 --> 00:30:52.850
they are all interrelated okay so it is an
amazing fact and discovering that is what
00:30:52.850 --> 00:30:54.419
doing geometry is all about okay.
00:30:54.419 --> 00:31:00.549
So you should not think that high school level
the geometry is just about drawing triangle
00:31:00.549 --> 00:31:05.890
encircles and you know measuring of angles
and arcs and things like that but it is really
00:31:05.890 --> 00:31:10.770
higher geometry in the higher sense is actually
looking at the interplay of all these things
00:31:10.770 --> 00:31:18.310
okay. So I well now you see I want to give
you a couple of examples, so here is the 1st
00:31:18.310 --> 00:31:22.290
example so here is an example.
00:31:22.290 --> 00:31:31.620
You take the domain to be the extended plane
itself okay see after all we are studying
00:31:31.620 --> 00:31:36.260
domains in the extended plane so take the
whole extended plane that is also domain.
00:31:36.260 --> 00:31:40.400
In fact it is simply connected because you
know there is homeomorphic the Riemann sphere
00:31:40.400 --> 00:31:45.720
and the Riemann sphere is simply connected,
so this simply connected, it is compact okay
00:31:45.720 --> 00:31:53.720
is a very very nice thing. Now what are the
what are the field of Meromorphic functions
00:31:53.720 --> 00:32:00.909
on D okay, what are the fields of Meromorphic
functions on the extended plane, so you know
00:32:00.909 --> 00:32:05.850
this is an extension of the complex numbers
as we have seen this is an extended of complex
00:32:05.850 --> 00:32:10.159
numbers but you know what is it that we proved
last time.
00:32:10.159 --> 00:32:15.240
A function which is Meromorphic on the extended
plane is none other than a quotient of polynomials,
00:32:15.240 --> 00:32:22.960
it is a rational function okay and therefore
this is exactly equal to the, this is exactly
00:32:22.960 --> 00:32:28.750
equal to C round bracket Z this is the algebraic
notation C round bracket Z and the C round
00:32:28.750 --> 00:32:47.409
bracket Z is actually the field of fractions
or quotient field of C of square brackets
00:32:47.409 --> 00:32:52.669
and C of square bracket Z is standard notation
is the ring of polynomials in the variable
00:32:52.669 --> 00:33:00.190
Z with complex coefficients okay and C of
Z is the field of fractions which is quotient
00:33:00.190 --> 00:33:05.059
of such polynomials, so you take quotient
of polynomials but of course you do not put
00:33:05.059 --> 00:33:12.370
in the denominator 0 anything other than 0
you put okay, so so the moral of the story
00:33:12.370 --> 00:33:19.919
is that you have very nice description of
this field extension in the case of the C
00:33:19.919 --> 00:33:23.370
union infinity which is the extended plane
and usually you know extended plane is thought
00:33:23.370 --> 00:33:29.360
of as Riemann sphere you know they are isomorphic
but you can make them also isomorphic in a
00:33:29.360 --> 00:33:33.929
holomorphic sense by giving the Riemann sphere
a Riemann surface structure okay.
00:33:33.929 --> 00:33:38.870
So often people do not use if you see the
literature you will see that people often
00:33:38.870 --> 00:33:44.309
use C union infinity instead of C union infinity
they keep saying Riemann sphere all the time.
00:33:44.309 --> 00:33:49.950
So now you can see that what are the properties
of this field extensions you see this field
00:33:49.950 --> 00:33:57.390
extension is actually is purely transcendental
and has transcendence degree 1 okay. It is
00:33:57.390 --> 00:34:05.900
purely transcendental and has transcendence
degree 1. Well the transcendence degree is
00:34:05.900 --> 00:34:10.869
actually the number of algebraically independent
variable is that it generate the bigger extension
00:34:10.869 --> 00:34:16.049
okay, so the bigger extension C of Z is generated
by a single variable Z and that is the only
00:34:16.049 --> 00:34:20.010
algebraically independent variable okay that
one variable is enough, so the transcendence
00:34:20.010 --> 00:34:28.029
degree is actually one okay and it is purely
transcendental because there is no element
00:34:28.029 --> 00:34:33.760
in C Z which is algebraic there is no elements
in C Z which is not in C and which is algebraic
00:34:33.760 --> 00:34:37.840
and that is you know why that is because Comcast
numbers are algebraically closed they are
00:34:37.840 --> 00:34:39.250
all algebraically closed.
00:34:39.250 --> 00:34:47.040
So field theoretically so this is what is
called as a function this is the simplest
00:34:47.040 --> 00:34:52.190
example of what is called a function field
in one variable okay and the beautiful thing
00:34:52.190 --> 00:34:58.380
is that now if you take any compact Riemann
surface okay then if you take the field of
00:34:58.380 --> 00:35:02.160
Meromorphic functions on that compact Riemann
surface what you will get is a function field
00:35:02.160 --> 00:35:07.630
in one variable but the only thing is that
it may not be purely transcendental about
00:35:07.630 --> 00:35:13.630
the there may be an algebraic part okay
so it will be 1st of transcendental extension,
00:35:13.630 --> 00:35:19.990
purely transcendental extension of degree
1 just like this and then about that you will
00:35:19.990 --> 00:35:23.470
have an algebraic extension and which will
be a finite extension okay. So that is how
00:35:23.470 --> 00:35:30.410
it looks in general okay and well I will give
you another example for that, so let me write
00:35:30.410 --> 00:35:44.309
this here this is a purely transcendental
extension
00:35:44.309 --> 00:35:58.369
of transcendence degree 1 okay. So the picture
that is associated with this is the Riemann
00:35:58.369 --> 00:36:05.640
sphere okay, so this is the picture that is
associated with this and for all practical
00:36:05.640 --> 00:36:10.010
purposes you think of the extended plane as
the Riemann sphere okay.
00:36:10.010 --> 00:36:23.270
Now let me tell you
more generally what is it that happens with
00:36:23.270 --> 00:36:30.900
somethingů so let me give you an example
of a more complicated case is the case of
00:36:30.900 --> 00:36:38.609
so-called elliptic functions or doubly periodic
function okay, so here is what I am going
00:36:38.609 --> 00:36:45.330
to do? What am going to do is I am going to
take, so I am going to define what doubly
00:36:45.330 --> 00:36:58.830
periodic function is? A doubly periodic
function so this is the topic of what are
00:36:58.830 --> 00:37:14.109
known as elliptic functions
this is also an example okay w periodic function
00:37:14.109 --> 00:37:31.970
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
00:37:31.970 --> 00:37:50.349
2 is equal to f of Z where Omega 1 by Omega
2 is not real and of course omega 1 omega
00:37:50.349 --> 00:37:58.170
2 are nonzero okay so I am just defining what
a doubly periodic function is?
00:37:58.170 --> 00:38:03.930
So see the definition is very very simple
for example sign beta will you know is periodic
00:38:03.930 --> 00:38:09.770
with 2 pi because sin of theta plus 2 pi is
the same as sin theta, so the idea is that
00:38:09.770 --> 00:38:13.790
to the variable of the function or the argument
of the function you add the period the function
00:38:13.790 --> 00:38:18.940
value should not change okay, so what the
1st equation says is that f of Z plus w 1
00:38:18.940 --> 00:38:24.650
is equal to f of Z actually tells you that
w 1 is a period and the 2nd equation f of...
00:38:24.650 --> 00:38:31.860
The 2nd requirement f of Z plus w 2 equal
to f of Z tells you that w 2 is also a period,
00:38:31.860 --> 00:38:38.030
so w 1 and w 2 are periods and the fact is
that we want these periods to be linearly
00:38:38.030 --> 00:38:45.810
independent over R in other words what you
want is that if you take these 2 complex numbers
00:38:45.810 --> 00:38:53.380
w 1 and w 2 then you join the origin to them
okay namely you take the vectors that they
00:38:53.380 --> 00:38:59.430
represent in the plane then these should be
different the vector should be linearly independent
00:38:59.430 --> 00:39:03.410
they should be in different directions you
know they will be linearly dependent if and
00:39:03.410 --> 00:39:09.310
only if the quotient w 2 by w 1 or w 1 by
w 2 is a real number okay.
00:39:09.310 --> 00:39:13.720
So this condition that w 1 by w 2 is not real
it is just to tell you that these 2 vectors
00:39:13.720 --> 00:39:20.440
are 2 different vectors. They will form therefore
a basis of C over R, the complex numbers over
00:39:20.440 --> 00:39:25.580
R is a 2 dimensional vector space and will
from a basis so it is equivalent to saying
00:39:25.580 --> 00:39:32.310
that w 1 w 2 form the basis for okay and you
are putting this condition in order to make
00:39:32.310 --> 00:39:39.540
sure that essentially have 2 distinct periods
which are in 2 so it is periodic into directions
00:39:39.540 --> 00:39:44.869
okay. The fact that f of Z plus w 1 is equal
to f of Z tells you that you know if you translate
00:39:44.869 --> 00:39:50.820
along the direction of w 1 by integer multiples
of w 1 the function value does not change
00:39:50.820 --> 00:39:55.990
okay, so you must remember that when I say
f of Z plus w 1 is f of Z it follows that
00:39:55.990 --> 00:40:06.680
f of Z plus n 1 w 1 is also f of Z for all
integers n 1 okay because I can just use induction
00:40:06.680 --> 00:40:14.910
f of Z plus w 1 is equal to f of Z, so f of
Z plus 2 w 1 is f of Z plus w 1 plus w 1 which
00:40:14.910 --> 00:40:18.420
is f of Z plus w 1 which is f of Z and so
on and so forth okay.
00:40:18.420 --> 00:40:23.460
So what it tells you is that the moment something
is a period then all its integer multiples
00:40:23.460 --> 00:40:30.190
are also periods okay and similarly you also
have for the other but what is adding
00:40:30.190 --> 00:40:36.720
W1, see addition of a complex number is just
translation along the direction along the
00:40:36.720 --> 00:40:41.230
vector that is represented by a complex number
okay, so you know basically if I have a point
00:40:41.230 --> 00:40:49.240
Z, what is Z plus W1? It is actually this
vector, so this will be Z plus W1. I am just
00:40:49.240 --> 00:40:57.670
a translating Z by the vector w 1 okay and
then similarly what is Z plus w 2 I am just
00:40:57.670 --> 00:41:03.140
translating Z by the vector w 2 alright and
that is if I add w 1 but if I add minus w
00:41:03.140 --> 00:41:07.830
1 you know I am translating in the other direction.
If I add minus 2 w 1 I am translating in the
00:41:07.830 --> 00:41:13.010
direction opposite to w 1 2 times and so on
and so forth okay.
00:41:13.010 --> 00:41:17.609
So the moral of the story is that you know
basically the function value do not change
00:41:17.609 --> 00:41:23.589
if you translate along 2 different directions
okay that is why this called w periodic, it
00:41:23.589 --> 00:41:28.400
is periodic and the period there are 2 different
periods okay and such functions are called
00:41:28.400 --> 00:41:32.849
actually now you have to put some more condition
on these functions, the condition you put
00:41:32.849 --> 00:41:42.089
on these functions is that you know to make
them very interesting these points w 1 these
00:41:42.089 --> 00:41:50.510
points which are given by integer multiples
of w 1 added to integer multiples of w 2 okay
00:41:50.510 --> 00:41:56.260
that they will form a lattice, a grid in the
plane okay and the function becomes very interesting
00:41:56.260 --> 00:42:02.500
if the function is Meromorphic exactly at
those points okay and such functions are called
00:42:02.500 --> 00:42:07.970
elliptic functions and believe it or not they
are exactly the functions which are the function
00:42:07.970 --> 00:42:17.260
is Meromorphic on a torus at a single point
okay and this is the beginning of the so-called
00:42:17.260 --> 00:42:23.720
Weierstrass phi theory there is something
called as Weierstrass phi function which is
00:42:23.720 --> 00:42:30.859
fundamental model of this kind of function
and the beautiful thing is that every torus
00:42:30.859 --> 00:42:35.940
the complex structure on any torus can be
controlled by prescribing such a function
00:42:35.940 --> 00:42:44.099
okay and so the Weierstrass phi functions
completely give you.
00:42:44.099 --> 00:42:49.390
So if you want to study the various complex
structures you can put on a torus what you
00:42:49.390 --> 00:42:53.990
will have to do is you have to study various
doubly periodic Meromorphic functions which
00:42:53.990 --> 00:42:57.910
are otherwise called elliptic functions, the
reason they are called elliptic is because
00:42:57.910 --> 00:43:04.590
this is beautiful the moment you put a complex
structure on the torus it becomes believe
00:43:04.590 --> 00:43:09.849
it or not it becomes a cubic curve, it becomes
a cubic curve and therefore it becomes an
00:43:09.849 --> 00:43:14.170
algebraic geometric object okay, so algebraic
geometry also comes in, geometry also comes
00:43:14.170 --> 00:43:20.599
in, algebra comes in in a beautiful way okay
and this is also a part of a very deep theorem
00:43:20.599 --> 00:43:29.050
which says that you know you take any compact
Riemann surface it is algebraic it is just
00:43:29.050 --> 00:43:35.840
given by a common 0 set of a bunch of polynomials
okay and that is an amazing theorem okay.
00:43:35.840 --> 00:43:43.960
So what I want to tell you is that I have
given an NPTEL video course on Riemann surfaces
00:43:43.960 --> 00:43:49.570
and all these things are explained in detail
throughout the course when you find time you
00:43:49.570 --> 00:43:54.700
can have a look at that and the other thing
that I want to tell you is that there is this
00:43:54.700 --> 00:44:02.960
book that I have written and it reads ôAn
introduction to families deformation and moduli.
00:44:02.960 --> 00:44:11.180
This book is basically available as a freely
downloadable copy in the form of a navigable
00:44:11.180 --> 00:44:17.750
PDF file and it contains a lot about the geometry
of Riemann surfaces, so at least the 1st chapter
00:44:17.750 --> 00:44:22.500
so that is also something that can be advanced
reading material for people who are interested
00:44:22.500 --> 00:44:33.599
in pursuing this. So let me continue so I
have also f of Z plus n 2 w 2 is equal to
00:44:33.599 --> 00:44:46.079
f of Z or all n 2 in Z, so in totality what
I will get is I will get f of Z plus n 1 w
00:44:46.079 --> 00:44:57.270
1 plus n 2 w 2 is equal to f of Z for all
n 1, n 2 in Z if I put both these together
00:44:57.270 --> 00:45:04.290
and what are these points n 1 w 1 plus n 2
w 2? They are the vertices of a grid of parallelogram
00:45:04.290 --> 00:45:09.150
okay in fact if you draw this if I draw a
diagram it is going to look like this.
00:45:09.150 --> 00:45:17.200
So I have this so this is my complex plane
and you see I have w 1 here I have w 2 here
00:45:17.200 --> 00:45:24.690
okay and you know then if I draw this parallelogram
then you know pretty well that this is w 1
00:45:24.690 --> 00:45:32.650
is w 2 okay by the parallelogram law of additional
vectors if you want and then you know if I
00:45:32.650 --> 00:45:37.540
extend this parallelogram below then I am
going to get this point is going to be you
00:45:37.540 --> 00:45:46.710
know it is going to be w 1 minus w 2 okay
and this point is going to be minus w 2 and
00:45:46.710 --> 00:45:57.599
if I extend it like this, this point is going
to be w 2 minus w 1 and this is going to be
00:45:57.599 --> 00:46:06.190
minus w 1 minus w 2 and this is going to be
minus w 1 okay and more generally if I drawů
00:46:06.190 --> 00:46:14.000
If I look at all these points that go on that
I get as the vertices of the parallelograms
00:46:14.000 --> 00:46:23.790
that I get by simply starting with this parallelogram
and simply displacing it by either plus or
00:46:23.790 --> 00:46:29.680
minus w 1 or plus or minus w 2 okay that is
by translating it with plus minus w 1 of plus
00:46:29.680 --> 00:46:35.310
or minus w 2 I will get so manyůthe whole
plane is covered by this parallelograms okay
00:46:35.310 --> 00:46:40.190
and the vertices of the parallelogram are
precisely the points which are of the form
00:46:40.190 --> 00:46:47.390
n 1 w 1 plus n 2 w 2 okay and that is called
the lattice okay and the fact is that you
00:46:47.390 --> 00:46:55.210
seeůjust to give you an idea of what is going
on where is the topology coming in India,
00:46:55.210 --> 00:47:08.970
so the fact is that what you do is you divide
by the equivalence relation Z 1 is equal in
00:47:08.970 --> 00:47:19.440
to Z 2 if and only if there exist n 1 n 2
such that Z 1 is equal to Z 2 plus n 1 w 1
00:47:19.440 --> 00:47:22.670
plus n 2 w 2 okay.
00:47:22.670 --> 00:47:27.991
So see this is the plane this is the complex
plane and I am defining an equivalence relation
00:47:27.991 --> 00:47:32.210
on the plane, the equivalence relation is
2 points or equivalent one of them is a translate
00:47:32.210 --> 00:47:38.950
of the other by one of these grid points okay
and what is the advantage of this? The advantage
00:47:38.950 --> 00:47:43.200
of this is that if 2 points were related like
that then the w periodic function will have
00:47:43.200 --> 00:47:51.540
the same value at both points because f of
Z 1 will be equal to f of Z 2 plus n 1 w 1
00:47:51.540 --> 00:47:58.150
plus n 2 w 2 but that is also equal to f of
Z 2, so f of Z 1 will be equal to f of Z 2
00:47:58.150 --> 00:48:04.710
okay because when I apply f to this equation
okay on the right side I will get f of Z 2
00:48:04.710 --> 00:48:09.210
because of periodic city of f so I will get
f of Z 1 equal to f of Z okay. What it means
00:48:09.210 --> 00:48:15.260
is that the value of the function is not change
if you change the point by a translate by
00:48:15.260 --> 00:48:20.180
a vector which belongs to one of the grid
points okay.
00:48:20.180 --> 00:48:30.070
So if you divide by this equivalence relation
what you will get is you will get the torus,
00:48:30.070 --> 00:48:33.710
you will get a beautiful torus and you can
see at very very easily you just take this
00:48:33.710 --> 00:48:39.369
fundamental parallelogram okay, this fundamental
parallelogram if you take the interior every
00:48:39.369 --> 00:48:46.700
point in the interior will be a unique representative
in its equivalence class but for points but
00:48:46.700 --> 00:48:52.210
then you will have to only identify the boundaries,
see if you identify the top boundary with
00:48:52.210 --> 00:48:57.859
the bottom boundary you will get a cylinder
okay and then you will have to identify these
00:48:57.859 --> 00:49:04.130
2 which will in the cylinder look like circles,
so if you identify them you will get a torus
00:49:04.130 --> 00:49:05.560
okay.
00:49:05.560 --> 00:49:10.221
So the moral of the story is you take the
plane divided by a lattice like this you will
00:49:10.221 --> 00:49:17.100
get the torus okay and all the points in the
grade including the origin, the origin is
00:49:17.100 --> 00:49:23.109
here they all will go to a particular special
point on the torus okay and the beautiful
00:49:23.109 --> 00:49:29.569
thing is that the function at you defined
on the complex plane will go down to a function
00:49:29.569 --> 00:49:35.940
on the torus okay and if the function is Meromorphic
exactly at all these grid points it will go
00:49:35.940 --> 00:49:41.089
down to a Meromorphic function on the torus
and you know you may wonder why should I worry
00:49:41.089 --> 00:49:46.880
about why should I not consider mophet functions
in the torus and you very well know the answer
00:49:46.880 --> 00:49:51.750
will not be any non-constant holomorphic function
on the torus because torus is compact. Since
00:49:51.750 --> 00:49:56.690
the torus is compact if you define a holomorphic
function on the torus okay you are going to
00:49:56.690 --> 00:50:04.579
get something that is holomorphic you can
use Liouvilleĺs theorem if you have a holomorphic
00:50:04.579 --> 00:50:10.110
function on the torus if you composite with
this map that is that goes from the complex
00:50:10.110 --> 00:50:15.740
plane to the torus you will get a holomorphic
function on the plane but since it is defined
00:50:15.740 --> 00:50:19.440
on the torus which is compact its images compact
therefore the image is bounded.
00:50:19.440 --> 00:50:22.950
So I get an entire function which is bounded
and that is going to be constant by Liouvilleĺs
00:50:22.950 --> 00:50:28.480
theorem okay and this picture also explains
why the only functions on the torus are exactly
00:50:28.480 --> 00:50:32.329
the functions on the plane which are doubly
periodic with respect to the periods w 1 and
00:50:32.329 --> 00:50:38.920
w 2 and if you take this beginning point p
which is the image of the grid the Meromorphiců
00:50:38.920 --> 00:50:42.860
since you know holomorphic function is not
available they have constants, the only things
00:50:42.860 --> 00:50:48.460
that are available are the Meromorphic functions
and then if you look at Meromorphic functions
00:50:48.460 --> 00:50:54.680
on the torus at the point p they will be the
same as doubly periodic functions okay and
00:50:54.680 --> 00:51:06.329
therefore the moral of the story is that you
know if you look at the field of Meromorphic
00:51:06.329 --> 00:51:12.990
functions on the torus okay we are the same
as the collection of Meromorphic functions
00:51:12.990 --> 00:51:18.790
on the collection of doubly purity functions
with these 2 periods okay and that is the
00:51:18.790 --> 00:51:22.150
field of course. Mind you that what is the
domain now?
00:51:22.150 --> 00:51:29.000
The domain is the whole plane okay and I am
looking at functions which are Meromorphic
00:51:29.000 --> 00:51:37.480
with poles at points of the grid okay possibly
at points of the grid alright and then what
00:51:37.480 --> 00:51:42.580
I get is I get field and what is that field?
That will is nothing but the field of Meromorphic
00:51:42.580 --> 00:51:53.550
functions on the torus which are Meromorphic
at a given point okay and so let me call this
00:51:53.550 --> 00:52:02.670
torus as T okay mind you this torus depends
on the choice of w 1 and w 2 okay and it is
00:52:02.670 --> 00:52:09.760
a different story that there is a lot of geometry
there but what I want to tell you is that
00:52:09.760 --> 00:52:15.230
well I want to tell you the following things.
00:52:15.230 --> 00:52:35.410
Simplest Meromorphic function
on C that is doubly periodic
00:52:35.410 --> 00:52:48.550
with respect to w 1 and w 2 is the Weierstrass
phi function
00:52:48.550 --> 00:52:55.900
and this is the phi function so there is pretty
symbol for that very special symbol so phi
00:52:55.900 --> 00:53:03.579
of Z is so there is a formula for that basically
it is a formula that will tell you that it
00:53:03.579 --> 00:53:09.640
is a Meromorphic function which has a double
pole with the residue 0 at each of those points
00:53:09.640 --> 00:53:16.900
of the grid okay, so you know what you are
going to get let me write that down here you
00:53:16.900 --> 00:53:23.930
can find this in any standard book example
book on complex analysis which is a classic
00:53:23.930 --> 00:53:44.040
so it is 1 by Z Square summation over
n 1, n 2 belonging to Z minus 0, 0 in fact,
00:53:44.040 --> 00:53:56.190
so I should write it carefully n 1, n 2 ordered
pair belonging to Z across Z minus 0, 0 it
00:53:56.190 --> 00:54:13.230
is 1 by Z minus n 1 w 1 minus n 2 w 2 the
whole square minus 1 by n 1 w 1 plus n 2 w
00:54:13.230 --> 00:54:14.819
2 the whole square.
00:54:14.819 --> 00:54:19.630
So this is the expression for the Weierstrass
phi function which was discovered by Weierstrass
00:54:19.630 --> 00:54:25.010
and of course if you go through in detail
the lectures of my video course you will see
00:54:25.010 --> 00:54:28.650
that how this comes about but you can see
something immediately you see that this 1
00:54:28.650 --> 00:54:33.170
by Z square is the principal part at the origin
and that will tell you that you know origin
00:54:33.170 --> 00:54:38.081
is a double pole and residue is 0 because
there is no 1 by Z term and then you look
00:54:38.081 --> 00:54:42.280
at each of these other terms 1 by Z minus
n 1 w 1 plus n 2 w the whole square tells
00:54:42.280 --> 00:54:48.170
you that n 1 w 1 plus n 2 w 2 is a point of
the grid is a general point of the grid okay
00:54:48.170 --> 00:54:54.119
and if you when I write 1 by Z minus that
point the whole square actually I am looking
00:54:54.119 --> 00:54:58.940
at a pole of order 2 at that point and again
the residue there is 0 alright.
00:54:58.940 --> 00:55:05.160
So as a result this already gives you were
you know it gives you a Meromorphic channel
00:55:05.160 --> 00:55:09.430
which is having a double pole at each of these
grid points okay and this extra term that
00:55:09.430 --> 00:55:15.700
is added here is for convergence because you
know I have added infinitely many poles okay
00:55:15.700 --> 00:55:19.640
I have added poles at every point of the grid
I have made every point of the grid into a
00:55:19.640 --> 00:55:23.520
double pole and I am getting a huge series
I want it to converge and it is only for this
00:55:23.520 --> 00:55:27.800
convergence that this extra constant term
is being added okay and therefore I get this
00:55:27.800 --> 00:55:30.750
phi function and here comes the amazing theorem.
00:55:30.750 --> 00:55:37.500
The amazing theorem is the following that
if you take the complex numbers and you take
00:55:37.500 --> 00:55:48.421
the Meromorphic functions on D on the complex
plane respect to w 1 and w 2 okay you look
00:55:48.421 --> 00:55:57.530
at the Meromorphic functions which are doubly
periodic periods at w 1 and w 2 okay and
00:55:57.530 --> 00:56:03.650
mind you this is the same as the Meromorphic
functions on the torus okay which are Meromorphic
00:56:03.650 --> 00:56:10.530
at that unique point which I will call at
star which is the image of the grid the whole
00:56:10.530 --> 00:56:14.360
grid, the whole grid goes to a single point
on the torus because all the points on the
00:56:14.360 --> 00:56:17.971
grid are equivalent to each other okay and
they all define a single equivalence class
00:56:17.971 --> 00:56:22.170
so it is a single point on the torus, so mind
you the torus is set of equivalence classes
00:56:22.170 --> 00:56:25.650
okay topologically and you give it the quotient
topology alright.
00:56:25.650 --> 00:56:31.730
Now on the beautiful thing is that what is
this set of Meromorphic functions? You know
00:56:31.730 --> 00:56:38.490
it is a field, what is that field? You know
what that field is, that field is just the
00:56:38.490 --> 00:56:47.730
field of fractions of phi of Z and its derivatives
it is beautiful and how does this extension
00:56:47.730 --> 00:56:55.609
breakup, it breaks up as the 1st phi of Z
this is again a transcendental extension,
00:56:55.609 --> 00:57:10.410
it is purely transcendental extension of transcendence
degree 1 and then from here to here this is
00:57:10.410 --> 00:57:17.212
an algebraic extension. This is an algebraic
extension because then the derivatives of
00:57:17.212 --> 00:57:22.900
the phi functions free prime satisfies the
polynomial relation with respect to phi and
00:57:22.900 --> 00:57:24.369
that is expresses a differential equation.
00:57:24.369 --> 00:57:29.910
It is a very famous differential equation
and that differential equation interestingly
00:57:29.910 --> 00:57:34.579
it comes from analysis what it tells you that
the torus is algebraic, it tells you that
00:57:34.579 --> 00:57:38.790
the torus is nothing but a cubic curve okay
which is an amazing illustration or the fact
00:57:38.790 --> 00:57:42.460
that in general a compact Riemannĺs surface
is given is algebraic it is given by algebraic
00:57:42.460 --> 00:57:50.690
equations okay. So all these details you can
have a look at it in more detail in my video
00:57:50.690 --> 00:57:58.430
lecture course but this is to tell you that
a lot of geometry is involved by looking at
00:57:58.430 --> 00:58:03.079
the field extension given by the field of
Meromorphic functions, okay. So I will stop
00:58:03.079 --> 00:58:03.089
here.