Okay so what we are discussing now is trying to understand the behaviour of a function as a mapping in a neighbourhood of a critical point. So if you recall a critical point is a point where the derivative of the function vanishes okay and of course you have already treated the case when the derivative does not vanish, we have already seen that when the derivative does not vanish then there is a small neighbourhood about the point where the function is even one to one and you can invert the function okay. So we come to the case when the at the case of a point where the function has vanishing derivative and to study the behaviour of the function at a point where the derivative vanishes we needed to understand the idea of branches of a function especially branches of a logarithm and that is what I explained in the last lecture. So let me recall so we recall that we had the complex plane and we had the function f of z is equal to e to the z is the exponential function and this takes values in the complex plane and the point is that we wanted to think of an inverse to this function and the inverse function would be well the logarithm log z okay and of course how this log is defined first of all we should understand in a set theoretic sense that e power z is not a many I mean it is not a one to one function so it does not have a set theoretic inverse okay it is a many to one function okay and in fact if you take any value of z which is translate of z by an integral multiple of 2 pi i the exponential function will take the same value because e to the 2 pi i is 1 okay. So therefore it is not a one to one function and in principle one should not think about one should not say that one wants an inverse function okay because it is you talk about an inverse function only if a function is bijective is injective okay but we want a function which is inverse in the functional sense namely a function that can undo the effect of exponentiation so the logarithmic function is suppose to be a function that under that cancels the effect of exponentiation okay but the fact is that when you try to write out a function in this direction okay then you get not one function you get several functions okay and these are called branches of the logarithm and if you recall how we define the branches was like this what we did was we took we defined first of all what is meant by the principle branch of the logarithm okay and so well let me not say let me call this plane as the z plane and well then this becomes the omega plane or the w plane where w is f of z and well the inverse function should therefore be log w okay and so if I follow that convention here this is the principle branch of the logarithm so this is from the complex plane which corresponds to the w plane and what we are going to do is to define the logarithm first of all you cannot define the logarithm of a z of this of the 0 complex number so you remove this okay. So what you do is you look at c star which is c minus the origin you throw out the origin because the exponential function misses out the value 0 it never takes the value 0 there is no logarithm for 0 so you throw that out and then what you do is well how do you define the principle branch of the logarithm well you say the principle branch of the logarithm is given by take the natural logarithm of the non the positive real number mod w is the modulus of w and add to it that is the real part and the imaginary part is the principle argument of w and what is the principle argument of w it is this angle from minus pi less than or equal to argument of w strictly less than pi okay and if you define it like this log w makes sense as a function and takes values in complex numbers and if you and this is z plane and well the fact is that this is an inverse functional inverse to e power z because you take z and then you take e power z and you plug e power z for w and you will get back z okay so in that sense well it is an inverse function alright now well the problem is that we want an inverse function which is you know we want an inverse function which is not only just a function we want it to be continuous we want it to be analytic because we are interested only in analytic functions now the problem is that this log w it is defined on the whole punctured plane okay that is the plane minus the origin but it is not continuous because the continuity the problem with continuity is at this negative real axis and that is because the continuity is with the imaginary part of the function which is the argument function is not continuous because the argument function on the negative real axis and just at points below the negative real axis the argument is close to minus pi and just above the negative real axis it is plus pi and this is a jump in the argument of 2 pi nearly and this jump tells you that the argument is not continuous across the negative axis. So what you will have to do is well you cut this out okay there are two ways of doing it one is you cut out this and by cutting this out I am declaring that the portion of the negative real axis above the negative real axis is far away from the portion that consists of the negative real axis and the points below it and when I say they are far away then I do not have to check continuity across the negative real axis okay. So in that sense the cutting helps okay so if you cut the plane so as to separate the portion of the negative real axis above it from the portion of the from the negative real axis and the portion below it then it is called a slit plane okay and on that plane log is of course continuous and continuity and continuity on points above the negative real axis is means continuity in a small disc about such a point and for continuity the same will hold for continuity of at a point below the negative real axis but for a point on the negative real axis it will be continuity on the boundary okay. So the neighbourhood will only be a disc only the neighbourhood will consist of an open disc only the lower half of the open disc centred at that point on the negative real axis okay and the upper part of the disc will not be considered as part of the neighbourhood because it has been cut off okay or it has been separated alright. So the point is that if I take log w and slit this plane okay mind you when I slit this plane I am not throwing out the negative real axis I am keeping the negative real axis okay only I am separating the negative points on the negative real axis with all the points above it okay with that kind of slit plane okay log becomes a continuous function alright but then I am not happy with that we would like to have log as an analytic function but you see but to have a function which is analytic you know the definition of analyticity is that it is defined at a if a function is to be analytic at a point first of all the function should be defined in a whole neighbourhood of the point okay. So what happens is that if I take a point above the negative real axis or below the negative real axis certainly there is a small disc surrounding that point where the log function will be analytic okay because that you can verify by any number of means for example you can check the easiest way is you know that these two the real and imaginary parts you know for example it satisfies the Cauchy Riemann equations if you want you can verify that and verify that the first partial derivatives are continuous and so on and so forth. So it is very easy to check that this defines an analytic function of w if you throw out the points on the negative real axis okay and but for points on the negative real axis is a problem because they are boundary points okay for the slit plane the points on the negative real axis is a boundary okay and at a boundary point you never define analyticity because analyticity is always defined at an interior point because a function is analytic at a point the definition already requires a function should be defined in a open neighbourhood containing the point. So you never define analyticity at a boundary point okay but then so you know if you want to think of log w as not just a continuous function but as an analytic function then I will have to take I will have to delete the negative real axis I will have to delete the negative real axis and I will have to and then log becomes you know it becomes an analytic function alright. So I will have to throw out the origin along with the negative real axis then it becomes an analytic function and the point is that well this is not the only functional inverse to the exponential function there are many others and the other branches of the logarithm are given by well other branches other branches are given by taking actually the various branches of the argument function which differ by a fixed integer multiple 2pi okay. So the other branches are given by the principle branch plus 2npi i so I just add 2npi i and get the other branches okay. So all these put together will give you the various branches of the logarithm alright and well now there are two issues that one has to worry about one issue is that I have so many branches and I am not able to see them as it will be nice if I can see them as one function okay. The other issue is that I have problems with defining logarithm on the boundary I mean the negative real axis that I have thrown out okay. So I mean well you know the negative real axis had to be thrown out because I chose my argument to be from minus pi to plus pi and that creates a problem at minus pi alright. But instead of there is nothing special about the angle pi okay I could have chosen any ray okay and I could have called that angle as theta and I can replace this side by theta and that side by theta plus 2pi alright and then I would get then I will get different branches of the logarithm which for which the slit will be along the ray alright. So there is nothing special about the points on the negative real axis okay. So the moral of the story is that I would like to really if necessary I would like to have even a branch an analytic branch of the logarithm on the negative real axis with which I can work okay and how is all this possible all this is possible by looking at the Riemann surface for log z. So I told you that what we do is that well you know we paste together as I explained in the last lecture we paste together you know so many copies of the plane carefully along the well along the slit negative axis. So what I do is that I have I have several copies of the slit of the slit plane and mind you I am not throwing out the negative real axis okay. But what I am doing is that well I take I identify the upper part and the lower part and you know I stick it in such a way I take 2 such copies of course you know this is I am just throwing a boundary but there is no boundary this is just to tell you that this is a piece of the plane which looks like a well disc right. But it is a whole plane that I want you to imagine and I have cut it across the mind you have not deleted the negative real axis. I just cut along the negative real axis in such a way that I have separated the negative real axis and the portion below it from the portion above the negative real axis and the reason is because I want to join the portion above the negative real axis to the negative real axis in the portion below it in another piece okay. So this thing which I have denoted with plus signs on one piece well I will you know attach them to this piece okay and well the portion on the negative real axis and below it will be attached similarly to the portion above the negative real axis in the piece above okay and these are the various pieces which correspond to the different branches of the logarithm okay and therefore the real the resulting picture will look like this resulting picture will be that of a rayman surface it will be so it is going to be well I have one piece like this and well I have attached yeah it is a little difficult to draw it yeah so it is something like this. So if I keep joining it along the negative real axis I will get something like this okay I will get a diagram that looks like this and in this diagram what has happened is that I have taken so many copies of the plane which is just so many copies of the punctured plane I have not thrown out the negative real axis but I have done the curtain paste along the negative real axis in such a way that I get a continuous surface alright and if I do this continuously what I get is rayman surface okay in the sense that it is a surface namely it is locally homeomorphic to the plane okay because it is consisting of various sheets as you may think of it each sheet being a copy of the punctured plane okay so it is locally homeomorphic to the punctured plane alright and it is certainly housed off and it is second countable and therefore it is immediate that what you get here is a by definition rayman surface rayman surface is supposed to be well topologically at least it is supposed to be something that is locally homeomorphic to the plane and which is housed off and second countable and of course on top of that you also need a collection of compatible charts to be able to do complex analysis on that to be able to decide when a function defined at a point neighbourhood of a point of the surface can be called an analytic function but then you get the charts because these homeomorphisms will allow you to identify the certain sheet with the punctured plane so there is a projection like this so here is a projection this is a projection map so we say that this is a covering of the plane of the punctured plane so you know I can remove the origin here and what happens is that this is a covering which we say is a countable covering because there are as many copies as there are integers and because each copy corresponds to a choice of n which is the domain of the branch of this logarithm alright and the moral of the story is that once you take this so this rayman surface which is the rayman surface for log z then the fact is that from this rayman surface if you take the function log I can define a function log w on this surface which is a single valued function okay mind you this surface extends in infinitely in both directions I have just drawn of three sheets okay but it extends infinitely in both directions and on each sheet I have the corresponding branch of the logarithm alright so but the point is because I have pasted it in a nice way okay the argument function the argument function on one piece if I take only for example if I take the principal argument the principal argument function is on this on the piece on which it is defined it is discontinuous at points on the it becomes well if I did not separate the if I did not cut the plane just for points on the negative real axis and below the negative real axis the argument function takes values minus pi and below and just above it takes the value plus pi but then what I do is I join this portion of the negative real axis and the points below it with another function for which the argument above is again values close to minus pi and lesser than minus pi and by doing this what I have done is I have extended the argument function continuously so the moral of the story is that once I do this bluing process the various argument functions okay from each of these which correspond to the various imaginary parts of the various logarithm functions the branches of the logarithm functions they all fit into one continuous function. So what happens is that you have one single valued function log function okay and that single valued function lives on this ringman surface on each sheet it corresponds to a particular branch of the logarithm on all put together it is a single logarithm and it is an inverse to the exponential function okay. So in fact I should also write argument of say argument of probability right. So the moral of the story is that the moral of the story is the following the moral of the story is that if you have a function which is an analytic function okay and then you can think of an inverse function you will not get one inverse function but in general you will get several inverse functions they will be called branches of the inverse functions and these branches will live on a ringman surface and the ringman surface will be got by doing cutting and pasting along what are called the branch loci okay or the branch points of the function. So for that is the reason we say that the negative real axis we choose as the branch points for the branch cuts for the so called logarithmic function. And of course the branch point is usually called the origin okay because the branch point is a point from which the branch cut originates. So the branch point is origin the branch cut is the negative real axis and you know by doing this cutting and pasting of various copies along the branch cut you are able to get hold of a ringman surface on which all the branches agree as a single function they all show up as a single function. You are able to stitch them all together to get a continuous function not only continuous function log w will actually be now an analytic function on this ringman surface mind you this is a ringman surface. This is a ringman surface because from every piece or every sheet you have the projection which identifies it with the complex plane okay and if you calculate the transition functions that the transition function will only be a translation by multiple of 2 pi i and you know translations are of course holomorphic. So the moral of the story is that these homeomorphisms that identify each sheet with the punctured complex plane they give charts and the compatibility of charts is automatic because if you calculate the transition functions they will simply be translations by you know multiple of some multiple of 2 pi i but you know every translation is holomorphic and the condition for charts to be compatible is that you know well the condition for charts to be compatible is that the transition functions are holomorphic okay. So the moral of the story is that you get a ringman surface so on this surface you can ask whether a function defined on the surface is holomorphic and the answer is that if you take log w on the surface it is holomorphic okay and it lives throughout the surface and it is holomorphic okay and that is the that is what happens here and that is what happens for a general function also sufficiently good function you will see that function will when you try to write out the inverse function you will get several branches these branches will be well defined and holomorphic if you make branch cuts along the so called branch points and if you do branch cuts properly and do cutting pasting process then you will get a ringman surface on which all the branches will define a single valued function okay. So this is the general philosophy alright and the point that you will have to understand is that well one important thing is the non-vanishing nature of the derivative of the function that you sort to invert. So in this case I am trying to invert e power z its derivative is e power z that never vanishes okay so the non-vanishing of the derivative is something very very important alright and so what happens in more in more generality is the following. So well the more general situation is like this so here is a theorem which we often would use the theorem is that so let me state it let me state it like this let f of z be an analytic function that is never 0 on a simply connected domain D okay then there exists an analytic branch of log f defined on D okay. So see here I am what I have done here is I am trying to get the logarithm of a complex number alright that means I am trying to invert e power z right. But more generally suppose I have an analytic function that never vanishes alright suppose I have an analytic function that never vanishes on a domain and assume that the domain is simply connected that means the domain has a property that it has no holes geometrically it means that given any loop in that domain that is a simple closed curve in the domain you can continuously shrink it to a point without going outside the domain okay which essentially means that you draw any simple closed curve simple closed curve of course means simple means that there are no self-intersections okay. So a closed curve is a continuous image of an interval with the initial point equal to the final point okay and you take any such simple closed curve at the fact that it can be shrunk continuously to a point is the same as saying that the region that it encloses is also inside the domain there is no part of that region which goes outside the domain which means that puts a hole in the domain so there are no holes that is what simply connected means and the point is that if you have a function which is nowhere 0 not that which means it is not 0 at any point of a domain is simply connected then you can find an analytic branch of logarithm of the logarithm of that function on the domain okay. So the proof of this is well pretty easy probably you have already seen this in a first course in complex analysis but I can recollect it for you so you see so here is my well here is my simply connected domain I have drawn a bounded domain but it need not be but the domain is the this region which has no holes okay and well and you know and what is given is that there is a function w equal to f of z which is never 0 on this domain alright. So what I do is I do the following thing what do you mean by saying that I have a logarithm of f of z and analytic branch of the logarithm it means it is an analytic function which is a logarithm of f of z namely that function is what if you exponentiate that function you will get f of z that is what it means alright. So you have to find a function h of z which is analytic in this domain okay and such that h of z is log f of z but what does that mean it means that e power h z should be f z okay. So if you exponentiate that function that you are trying to find it should give you f that is what it means. So well how do you construct such a function it is very very easy what you do is you fix a point z0 in the domain alright take any other point z and it is very simple take any curve gamma from z to z0 and you define an integral fix an z0 in D let for gamma for z in D let gamma be a curve from z0 to z of course by curve we mean a continuous image of an interval alright define h of z to be well integral from z0 to z along gamma so I will put this in brackets and of course I am going you know what I am going to write I am going to write D log f z okay which you know where of course you know what D log f z means where D log f z is supposed to stand for f dash of z f dash of z by f of z d z okay. So I define this function alright this mind you f of z does not vanish okay and f is analytic alright therefore f dash is also analytic and f does not vanish so f dash by f is also analytic alright therefore the integrand is an analytic function alright therefore this integral is well defined. So first of all you must understand that first of all for an integral to be well defined I mean the integrand should be continuous function alright and the integrand here is f dash by f okay the continuity will not be a problem for f dash because f dash is a derivative of an analytic function you know derivative of an analytic function is also analytic. So it is also continuous but the problem is there is an f in the denominator and that should not vanish okay but then we have assumed that f is never 0 on the domain therefore there is no problem about f being the denominator therefore you have a quotient of continuous function therefore it is continuous. So in particular it is also continuous on the path gamma therefore this integral is well defined there is no problem about this integral okay. Now what I want to say is that this integral is independent of the path okay see if I choose why is that true because if the integral is dependent on the path then this h of z will not be properly defined because to define h of z I have chosen a path okay which is a path that connects z0 to z of course inside d I am not going outside d alright but this h of z should not depend on the path if it depends on the path then it is not a well defined function but I claim it does not depend on the path why because you know if I choose some other path say gamma prime alright then if I you know that if I go along gamma and if I come back along the reverse of gamma prime alright then I get a closed loop and on that closed loop if I integrate this if I look at the integral of the d log fz that is f dash of z by fz I will get 0 my Cauchy's theorem because f dash by f continues to be analytic there because you see f dash is already analytic f is also analytic the quotient is analytic wherever the denominator does not vanish but the denominator never vanishes the denominator is f it never vanishes therefore the moral of the story is because of Cauchy's theorem this integral does not depend on gamma it does not depend on what path you choose to connect z0 to z so long as you make sure that the path lies inside your domain okay so by Cauchy's theorem theorem you know integral over gamma from z0 to z d log fz is the same as integral over gamma prime from z0 to z d log fz well actually you know I should be a little careful because I have used z here I should not use z in the integral you know maybe I should use zeta alright it is better to use zeta and then I will also change this to zeta so that it is easier because z is supposed to be the fixed value the fixed point where I am trying to define the function h alright. So let me change this to zeta let me make a difference between the variable of integration and the limit of the integral so well here it is this is better notation okay and by Cauchy's theorem this is equal to this because the difference is 0 by Cauchy's theorem okay. So this tells you so h of z is well defined so it tells you that h of z is well defined now you see now look at h of z now again let us apply another theorem namely Morera's theorem to conclude that h is actually analytic you see if you look at the function so you know Morera's theorem is the kind of converse to Cauchy's theorem okay what does Cauchy's theorem say it says you take a function as analytic if you integrate it on a closed curve simple closed curve and assume that a function is analytic inside the curve and also on the curve then the integral is 0. Now the converse to Cauchy's theorem will be if you have a function for which whenever you take a closed curve the integral is 0 then is that function analytic that is the expected converse but for the expected converse which is Morera's theorem you need to put the extra condition that the function you are starting with is already continuous. So you see in this case you see I am integrating the function that I am integrating is f dash by f okay and f dash by f is continuous throughout the domain f dash by f is continuous throughout the domain right and its integral over any closed curve is 0 because it is analytic therefore it satisfies the conditions of Morera's theorem and therefore this integral is an analytic function okay. So let me write that in the proof of Morera's theorem we see that h of z h of z is differentiable with derivative f dash of z by f of z okay and so the Morera's story is so in particular the proof of the theorem tells you that h is analytic okay. So thus h dash of z is f dash of z by f of z and h is a branch h is an analytic branch branch of log f in D okay. So it is easy to define I mean the looking at the logarithmic derivative it is easy to define a branch of the log okay and branch of the log which is analytic and a simply connected domain. All you need is a simply connected domain alright and okay I need this now what we will do is that we will try to study the function z going to z power m at the origin okay and what we claim is that you know this is how any function will behave at a critical point up to you know translations and up to some simple transformations okay. Any function at a critical point will behave like the mapping z going to z power m at the origin okay that is what I wanted to explain. So the function f of z is equal to z power m and I will assume that m is greater than 1 okay. So look at the function of course if m is equal to 1 it is the identity function there is not much to say but you want to look at this function and if you look at it then the derivative f dash of z is m z to the m minus 1 m is at least 2 and this is 0 only at z equal to 0 namely the origin so it has only one critical point. So the origin is a critical point is the only critical point with critical value 0 the origin is the only critical point and the critical value is 0 okay and of course I would like to study the function this function at the in a neighbourhood of the critical point namely in neighbourhood of the origin and the reason I am doing this is because I have this theorem about the availability of a logarithm for a non vanishing function on a simply connected domain I can use this to study the behaviour of any analytic function at a point which is a critical point okay. So this is the foundation so let us understand this case alright so well the first thing that you that you can see is well I have so I have this so I can draw a diagram similar to the one here okay that where I have constructed a Riemann surface for log z okay namely a surface which covers the punctured plane on which log is all the branches of the logarithm live together as one function okay. So I am going to do the same thing here so what I am going to do is I am going to just take the function ow is equal to f of z is equal to z power m okay and this is a function that goes from the complex plane to the complex plane and this is the z plane and this is the w plane and well of course the critical point is the origin alright. So I will single that out by putting a circle a small dot circular dot here so the origin goes to the origin and you know that the inverse function if I think of the inverse function it has to be z is equal to w to the power of 1 by m this is what the inverse function should be it should be the mth root of the variable that should be the inverse function alright because this way the function raises the variable to the power m the function in the other direction should take the mth root but the point is that you do not have one root you have in fact m of them and these are different branches and the branch point is the origin and the branch cuts are all raised the branches are defined on sectors alright. So let me draw the diagram for let us take m equal to 3 okay if I take m equal to 3 then you know pretty well that you know if I take this ray which is well it is 2 pi by 3 is this angle alright then as z for this whole domain namely this whole sector with angle 2 pi by 3 is going to be mapped on to the whole plane alright that is because if I write it in polar coordinates z power m if I write if I write z as r e to the i theta then you know z power m will become r to the m e to the i m theta and if m is equal to 3 I am going to get e to the i 3 theta. So as theta varies from 0 to 2 pi by 3 3 theta will vary from 0 to 2 pi so this whole so sector okay is going to be mapped to the whole plane okay and similarly you know if I take the other sector namely if I take this which is going to be 4 pi by 3 okay then this sector beginning from here to here that is an angle of 2 pi by 3 and that is again going to be mapped on to the whole w plane okay if you remove the origin then the origin there is also removed alright and well similarly this sector which again is an angle of 2 pi by 3 alright that is also going to be mapped on to the w plane alright and of course so you know this whole copy of the z plane is being mapped thrice on to the w plane when m is 3 and you know when it is m it is going to be mapped m times and it means that you know if you take the image of a for example if I take the unit circle here I take the unit circle here and I travel along the unit circle and you know I look at its image there what I will get is well I will get the unit circle but it will be traversed 3 times if m is 3 and it will go around m times for a general m okay and this also tells you something for example you know if I have so you know if I take this point 1 the unit circle will of course go to the unit circle only thing is it is going to go here 3 times as I go there 1 around 1s you get actually if you look at the point 1 it goes to 1 but then if you try to take the inverse function then you get the cube roots of unit right because you are taking the when I take m equal to 3 I am trying to base I am trying to look at values of 1 to the 1 by 3 which are the cube roots of unit and so what I will get is I will get these 3 values I will get one value if you call it as omega the other yeah so the problem is this omega conflicts with the w there so probably I will use so I will use a funny omega which is the omega with a dash on top of it so this is an omega with a dash on top of it which is a complex cube root of unity and of course you know the other one is omega squared okay and all the 3 points 1 omega and omega squared all the 3 will be mapped to the point 1 okay. So you know what is going to happen you know that there are going to be 3 branches for this function alright there are going to be 3 branches for this function and the 3 branches again are going to how are you going to write them down so that they are analytic they are continuous so we make use of this theorem okay we make use of this theorem so let me go back to so let me rub this part of diagram and go to some heuristics when you define powers okay so you define z1 to the power of z2 how does one define this in complex analysis or more generally analysis you define it as e to the log z2 log z1 I mean this is how you define it okay so because if you know rules z2 log z1 should become log of z1 to the power of z2 okay and e and log has to cancel this how you define for example if a and b are real numbers okay you would define a to the b like this the only problem is that when you define this log this log has several values and therefore if you take z1 to the power of z2 you get several values okay and all those values come because the various values of this log. So in the same way you know if I want to follow that example and I want to define I want to define the function w to the power of 1 by m okay then how would I do it I would define in the same way w to the power of 1 by m to be e to the 1 by m log w this is how I would define it okay and now the question is how do I define log properly you know that there are so many branches of the logarithm and so and you know all these branches of the logarithm they all differ from the principle branch by adding multiples of 2 pie i okay. So what you do is that you define one branch using the principle logarithm okay so what you do is so this tells you that you define w to the 1 by m okay and when I say this maybe I will put here I will put a lower round bracket 1 to say this is the first branch this is one branch I am defining and I am defining it as e to the 1 by m log w okay I can define it like this where log w is the principle branch of the logarithm where log w is defined as ln mod w plus i times principle argument of w the argument of w of course varying from I make a well branch cut from minus pie e to this okay and you know that if I throw out the negative real axis you know that principle logarithm is you know an analytic function and therefore I get an analytic function which will be one branch of w to the power of 1 by m okay. So on this on the cut plane on the on the plane on the cut plane c minus negative real axis which is well minus infinity to 0 okay I throw out the negative real axis w to the power of 1 by m subscript 1 in brackets is a branch of w to the 1 by m okay it is a branch of the mth root function alright and now so once you have this one branch then how do you get the other branches the other branches are given by well I need some notation so I think I will get let me look at it for a minute w to the 1 by m so you know you are going to get m branches alright in the case of w is equal to z cube you get 3 branches okay one branch and under each of those branches the image of one will be under one branch it will be one and the other it will be w which is a complex omega which is a complex cube root of unity and under another branch yet another third branch it will be omega square okay so you get 3 branches. So if it is m you will get m branches and what are how do you write those branches there you can write them as e to the 1 by m log w plus 2 pi 2 okay maybe I should not use i because i is always a complex cube root of unity I mean square root of unity so let me use j or matter I use even n alright so then I will have 2 n pi i okay these are the various branches of the logarithm alright and you will see that essentially you will get only n distinct ones you will get only m distinct ones okay you will get only n distinct ones because well because you know if I start with so if you know if I start with so I will start with w to the 1 by m if I take and here if I put n is an integer okay then these are the various branches of the logarithm and therefore I should get the various branches of the m through function but mind you there are infinitely many branches of the logarithm there are as many branches of the logarithm as there are integers alright but then if you take the m through function there are not so many there are only m branches and the reason is because there is a 1 by m that is appearing outside this n has to be read mod m because it will repeat n mod m will repeat okay so you will get only m branches so let us write it down see for example when I put w0 1 by m is my original notation 1 so this is really bad so I just change it so that my w1 is actually w1 so which means I will have to put okay so I will have to make the following adjustment I just put it as 2 into n minus 1 by i if I do this alright and I put n equal to 1 then w1 is going to be well my w1 it is going to be e to the 1 by m log w alright this is what I get when I put n equal to 1 alright then I put n equal to 2 I am going to get e to the power of 1 by m log w plus 2 pi i alright and this will go on up to w to the power of 1 by m d well 1 to m when I put m what do I get I will get e to the power of 1 by m log w plus 2 into m minus 1 pi i I will get this and mind you all these are distinct they are all distinct because there is e to the 2 pi i by m which is getting multiplied alright this function is this function multiplied by e to the 2 pi m every successive function is the previous function multiplied by e to the 2 pi i by m okay so these are all distinct functions they are all mind you they are all analytic functions they are all branches of the root function alright but then if I put m plus 1 I will get back w1 okay because if I put m plus 1 you know w1 to the well if I calculate w1 to the m m plus 1 what I will get is I will get e to the 1 by m log w plus 2 pi 2 m pi i okay and then you know if I simplify e to the what 1 by m into 2 pi 2 m pi i will give me e to the 2 pi i and that is 1 so this will this will again go back to w to the power of 1 by m the first branch okay so you will get a repetition. So the moral of the story is that if you calculate w to the power of 1 by m n what you will just get is you will get w to the power of you will get the branch that corresponds to n mod m this is what you will get as n repeats I mean as n varies over all integers so essentially you get only these you get only these m branches you get only one you only get m branches and of course the problem with all these branches is that at the wherever you have cut it on the that is on the negative real axis they are all not going to be analytic okay. And again if you want to think all of them and the reason why they are not going to be analytic is because well the even if you try to make the argument continuous by separating the lower the negative real axis and the portion below it from the portion above it if you do that you can actually make the argument functions continuous but even then you know you are not going to make the function analytic on the negative real axis the only way to do it is to construct the Rayman surface for the root m root function okay and the Rayman surface will be what you will get is the Rayman surface will be as many sheets as there are branches okay. So in the case of the when you considered the logarithmic function which is the inverse function to the exponential function then the logarithmic function all the branches of the logarithm could be realized as single function on the Rayman surface for log z which is which is which has as many sheets infinitely many sheets as many sheets as there are integers because these are the as many branches of the log that you have but here you have only m branches of the of the mth root function. So what you will get is you will get an m sheet at Rayman surface I will be content you draw it like this and well and you will get on this Rayman surface the function w to the 1 by m will make sense as a single valued function okay which is analytic and you can see all the m branches of the mth root function living here as a single analytic function okay and of course I should tell you that this is of course not these are not just m disks and again I should tell you that this is a this m sheet at Rayman surface is an m sheet at cover of the punctured plane okay it is what is called a covering space okay in the sense that if you take any point here if you take any point here and you take a sufficiently small neighbourhood then its inverse image will be m points with m neighbourhoods which are homeomorphic to this under the projection map and if you call this as a projection map and for example in this case you can see them as you know you can see them as three you can see them as three points here for a point there corresponding to the images of the three branches okay and this is what show up as three we should show up as three separate neighbourhoods above in the covering space okay which is the Rayman surface for function w to the 1 pi m okay. So now I will end with the statement that this behaviour of the function f of z at the critical point z equal to 0 is will be the basis for the behaviour of all of any analytic function at a critical point the behaviour will be like this up to conformal change okay that is what I will explain in the next lecture.


Preface. A ghost story first led me to the edge of Chinatown. One crisp morning, I dodged the crowds in Union Square and walked past a pair of stone lions up a hill. I had an address, 920 Sacramento Street, and a description. I was looking for a five-story structure built with misshapen red bricks, some salvage from the earthquake and firestorms that raised much of the city in 1906. Passing a church and a YMCA, I came to an old building with metal grates on its lower windows. Above the main entryway, I peered up at century-old raised lettering that read, Occidental Board Presbyterian Mission House. On a plexiglass sign mounted onto the bricks at eye level, I read, Cameron House established 1874. I climbed the steps and pressed an intercom button. A lock clicked. I pushed one of the tall doors open and walked into a dark wood-paneled foyer. I first visited Cameron House in 2013. By then it was mostly famous for being haunted. The staffer would later tell me he once sensed a ghost in its musty basement. There he said terrified former slaves, nearly all of them Chinese girls who'd been sold or kidnapped, had hidden in one of its dark corners from their former owners. The girls and young women who lived there under the care of two remarkable immigrant women, Donaldina Cameron, the youngest daughter of a Scottish cheap farmer, and Tin Fu Wu, a former household slave sold by her father to pay his gambling debts. Cameron served as legal guardian to most of the girls and teens living in the home. They called her Lomo or mother. She, in turn, often referred to them as her daughters.  We'll end there. Thank you so much for that. So how many of you here have been to San Francisco's Chinatown? Every single person. Every single person, I would hope. I mean, it remains a huge tourist attraction and draw for locals, right? Absolutely. And as you mentioned in your book, it's the oldest in North America. But yet I think sometimes we go for the dim sum or the tachkis, but we don't know the history exactly. And something that I found so wonderful about your book was how it sort of uncovered this place that we've been to, shopped at, love, and yet reveals some stories that I think otherwise have not been as visible. And I was hoping you could kind of talk about that. The role of, you know, what Chinatown meant for these women, you know, both as a place where it was like a landing pad, but also there are elements of it that they felt they wanted to escape. Sure. Yeah, and that they kept Cameron House in Chinatown and not somewhere else. Yes. Let me ask a related question. How many people have been to Cameron House who are in this? Oh, many, many people. Yeah, okay. A few of you haven't seen it, though. Probably walked past it. For those of you who haven't been there, it's literally a block and a half down from the Fairmont Hotel, and it's right below the University Club. So from Nob Hill, you look down on the roof of Cameron House. But shall I start in the 1870s when this project really began? Exactly. Would that be okay? Yeah, okay. So Chinatown in the 1870s was not what we see today by any means. It was wood tenements for the most part. It was nearer to the waterfront, wasn't it? It was much closer to the waterfront. The landfill hadn't been built yet. It was extremely dense. It was roughly eight square blocks.  But within that small area, 12,000 people lived. It was by far the most compact part of the city. And I actually think it remains the densest neighborhood in San Francisco. Yes. And just like it is today, of course, it's a place where immigrants first came. And it continues back in the 1870s. Of course, the predominant language was Cantonese at that point. But probably the most striking thing about Chinatown in those days at the time the house was founded was that there were so many more men than women. There was a real gender discrepancy. And there was something the census takers said, ten men for every one woman in that area. And of course, that gender discrepancy created a huge demand for sex, both from Chinese men who were not without their wives or families for the most part, but also white men in Chinatown during that period of time had joined the Barbary Coast. Yes. So very similar to today, it remained an area where new immigrants would land, just as you mentioned in River of Stars. That's a key part of your book. Thank you. But actually what also struck me about your book was the fact that it was a tourist attraction from the get-go. Yes. It was that place where you could see exotic sights and walk on the wild side and get some good food. Yes. That was one of the most surprising things I dug up is the thriving tourist trade, and particularly in the 1880s and 1890s, but there were just enormous number of travel descriptions from that time. People who had been led by guides to see supposed opium dens, they may have actually been actors. Right. We don't know. Yeah. And then, so what was it, I guess, just to back up, what led you to the doorstep of Cameron House in the first place? Yes. I was interested in writing a book about the history of San Francisco and came across Donaldina Cameron's first person account of the 1906 earthquake. Which is a really gripping section of the book. Thank you, Vanessa. Yeah. And it was such a riveting account of that disaster and what happened. And she was responsible for a household of somewhere between 50 and 60, mostly girls and young women, even a few babies. And she and the other staffers of the home had to lead all those people to safety that night after the earthquake hit and then back across the city to get onto a ferry and then head to Marin County where they spent several months. Well, and then she had to run back in the midst of like all the danger to get their documents. Yes, because she was concerned that they had no documents that travelers could try to take them back or steal them back. So, yes, she made her way past the policemen who were guarding the area at that point. So, but how did you even come across that account? Like what was the thread that sort of first drew you to that? Hmm. You know, it was funny because I think I found it on FoundSF, which is a kind of a digital history of the city. And sometimes they post primary documents like this one. So I was, you know, I was just zooming around the web. I mean, I don't know how I found it other than that. When you're a writer, even surfing the web gets to count as work or a business expense. So just, you know, and I think what was so important, like, I think we were talking about how this is 150th anniversary of the Chans Continental Railroad, which is also covered in your book a bit. And just the fact that I know so many of the descendants or people within the Chinese American community have been pushing to have for the story to be told, because as they put it, these Chinese pioneers that, that often Chinese are considered perpetual foreigners, no matter how deep their roots. And what really struck me was stories of people like Tian Fu and other women who came up through Cameron House, just how inspiring they were. And it just, but also that they were, they were a part of San Francisco history. And is that something you thought about as, while writing this book? Absolutely. It was a part of the history that I think the wider community of San Francisco didn't know. And one of the wonderful things that's happened in the last few weeks is somebody came up to me who works for the city and said, I think we should have a statue of Tian Fu up there, too. Yes. Wouldn't that be great? That would be totally, that would be worthwhile and inspiring. Yeah. Yeah. So, and certainly, you know, I always loved journalists. I was a journalist for many, many years. And I just loved discovering the life story of Ngyung Poon Choo, the minister turned kind of crusading civil rights journalist who ran the Chung Sae Apol, the Chinese Western Daily. So that was such a wonderful story to follow him through that long career as well. Yes. In the book, she chronicles how he kind of came up through the ministry, but then thought he felt that journalism might reach more people. Yes. Well, he, and it did, he probably did, but he ended up marrying a woman who came out of the mission home. And so he was kind of a natural, he was an advocate for it over the years. And many of the issues confronting the larger Chinese American community, of course, he wrote about in his role as editor of this paper. It was just a wonderful way to follow the larger evolution and legal battles and issues confronting the community. Yeah. I thought you did a wonderful job sort of, sort of looking at these larger social issues but told through personal stories. Thank you. But though with the, that journalist, what I also remember was during the quake, he had a whole book. Yeah. That he'd written, unpublished about the history of Chinese, but the soldiers wouldn't let him back in and it burned. Yes. And it was very frustrating and a very strong contrast to Donaldina Cameron who was able to retrieve her book and he was not. Right. So, yeah. She used her, her white privilege. Yes, white privilege. Yeah. Yeah. We're something. Exactly. Exactly. So what was it, the other thing that really struck me was that both Donaldina and her sort of protege aid, Tian, you know, they, they remained unmarried their whole lives. And what was it that, you know, maybe they didn't find that special someone or what, do you think the fact that they remained unmarried given sort of the roles and expectations for women at the time, that's what enabled them to take on these sort of like lives of, of fighting for justice that otherwise they might be saddled down with other expectations? Well, certainly all of the women I write about who came through the home were living outside the Victorian woman norm. Right.  Or they were women like Donaldina and Tian who decided, I don't want to marry. I don't want to have children. I want to work for justice. Yeah. So absolutely that was one path for them at a time when women couldn't vote, didn't have very much political or financial power and had very few options. Yeah. And yet in some ways the sort of happy ending for girls through, it was a matchmaking bureau in a way. Can you talk about that? Yes. It was kind of a de facto marriage bureau and in fact there's quite a Cameron House diasporas. I know many of you know there are families all over the country where a grandmother, a great grandmother came through Cameron House and I probably visited half a dozen archives all over the country and at Yale was a particularly interesting archive to go to. And there was a scholar there who had tracked the families particularly in the Midwest that had come out of Cameron House. So that was interesting. Well I think it just goes back again to the point that Chinese have been here. We're not all recent immigrants that part of the American story. Well I love this story too because it's a cross-cultural story. It's not just a story about the Chinese American community. It's a story about women bonding together. Across culture. Across cultures. And in 1874 in the first annual report of this project it was described as women's work for women. Which was kind of in my view a very early feminist statement. Right. Well even to consider that someone from a different background was a full woman as well deserving of all the women's rights. And so do you think the fact, can we talk about religion? Sure. That was part of the book where there's a question like did someone truly convert or they quote unquote a rice Christian. Meaning they wanted the bowl of rice in return for saying the Lord's prayer. So what was that something that you grappled with while reporting and writing the book. Just like how to examine issues of faith and good deeds that come with strings attached. Well there were metrics to measure conversion. So the women who ran the home kept very detailed records. Those records are kept at the San Francisco theological seminary. And each year they would note how many women who converted, residents of the home had converted. And in a typical year there might be two or three women who converted whereas 60 women would come through the home. Something like that. So it was not a condition that you convert to live in the home. But there were expectations that you would go to church and that you would study the Bible for sure. Grappling with faith. I'm a nonfiction writer and so it's one of the great frustrations I think is you can only see so far into the interior life of people to the extent that they left a written record behind. Right. And I had a lot of written, official written written records from the Presbyterian church. But I, you know, I had fragments of diaries and things that might offer more kind of less guarded revelations of what people were really thinking at the time. The closest I came were letters and there were letters that I found all over the country between, for example, Dolly Cameron, Donaldina Cameron, and Tian. And those did give me glimpses into what they really felt. And I certainly think Tian was an absolute person of faith. I think she really did believe. And so that same thing with Dolly. Yeah. The letter that you close the book with is between Donaldina and Tian is so tender. It is tender, isn't it? Yeah. It is so touching. And one of the most tender things I thought was traveling to Evergreen Cemetery in Los Angeles, which is one of the oldest cemeteries in the city. And for many, many years, it was segregated. So Chinese people would not be allowed to be buried in the same area as white people. But I tracked down the Cameron family plot. And sure enough, there's Tian Fu Wu buried in that same plot. Side by side. Wow. And the other thing to remember is that they lived fairly recently. Donaldina only died in 1968. And Tian died in the 70s. So it wasn't that long ago. Right. Well, like all things that we think is historic or distant actually have contemporary relevance. Yes. And so maybe we could talk about that as well. Another aspect of the book that was so heart-wrenching was that sometimes women would get rescued, but then they would get deported. And back to war-torn China in some cases. And I just know with what's been happening at the border and families being separated and children being kind of kept in these detention centers, have you been thinking about the sort of way the past is resonating today? Absolutely. I mean, on the one hand, if you take a step back, this book is a story about how we treat immigrants. And this was a very compassionate way to treat immigrants. In fact, when the Angel Island immigration station was opened, the house sometimes would be a place that was a better place for women to stay rather than spending months in that immigration center, which was the conditions were not very good. I also think it's so interesting the periods of history when we've seen surges and empathy. Certainly at the beginning of the British abolitionist movement to try to stop the transatlantic trade in Africans. And this was an example of that surging empathy because certainly the white world around Chinatown in the 1870s when this project began was extremely hostile. And so those women were acting against the larger prejudice. So I think, you know, one last point I would make about it is that individual actions can be important. They can make a difference to people's lives. And one of the things that I found most interesting in researching this book was to try to track the lives of some of the women who came through the home. One of them was Dr. Bessie Zhang, who became the first Chinese-American woman to graduate from your alma mater at Stanford University. Oh, right. Is she the one who went to the private girl school in Philadelphia? No, that was Teen who did that. Actually, reading your book, I kept thinking I could turn this into a novel and I could turn this into a novel. Someone's already making a novel out of one of the characters. Oh, OK. I just had a walk with her. Oh, OK. Which is to say this book is stuffed with interesting characters that you will want to follow everywhere. So maybe we could talk a bit about the research. Just what sort of you said the research librarians here were great and you come through archives, but kind of go through that process with us. How do you even keep track of everything? Well, Vanessa and I are both members of the San Francisco Writers Grotto and I think at the beginning of this project, I sent out a note to everybody and I said, is there any easy way to scan documents? And I quickly got a message back saying try turbo scan. Oh, OK. Which just rocked my world as a researcher because I could go into archives, take pictures of my iPhone. And now my iPhone's not right here, but it's close by. I have all of these searchable PDFs. I have tens of thousands of documents. Oh, they're not JPEGs or PDFs. Oh, that's perfect. Yeah. So that was interesting. And what I usually tell people too is chronology is your friend, especially as a nonfiction writer. And that was the very first thing that I did, which is do a very, very detailed chronology and figure out where characters were at certain points at time. But I'm a serious research fall down the whole kind of person and spent a lot of time upstairs in the fifth floor of the San Francisco History Center, which is an absolutely wonderful place. Has everybody been there? A few of you. Oh, it's fantastic. I can't tell you how much I enjoyed working there and also the librarians I work with were wonderful. There's a woman named Christina Moretti who was in the San Francisco Chronicle not long ago. She's known as the photo detective because she'll track down things. She's very, very good at it. I mean, I think we're fortunate to have sort of, that these are public documents and that you or me or anyone has access to it. Well, I would say that one part of the book involved private documents. Oh, yes. So Cameron House has records that date back to 1874, and mainly the oldest and the most important one is the log book in which every single girl or woman who came through the home was described. So there was the date, her name, and why she was there. And it's quite an exceptional document. And I had the great good fortune of working with the former executive director of Cameron House, Doreen Durham-Cloud, who organized those records over the years. And for certain key people and cases, she was very generous and let me look at those case files. That's wonderful. Yeah, how did you, given that you're not Chinese yourself, can you talk about what it means to sort of gain the trust and access into a community that isn't your own at first and then just sort of what it means to write this book at all, I guess? Well, I work very closely with Doreen and with other people involved with Cameron House and met some of the directors. I see people that I've met here over the years. The late Bob Kwong was extremely helpful. He was a longtime supporter of Cameron House. But I would say it took a very long time. I did not see the actual room in which all the records that Cameron House were kept until four years into the project. So it took a long time for, I think everybody to feel I would treat the materials and the stories with the respect they deserved. Did you ever sense a ghost? Yes, I did. Absolutely. So the basement is very, very spooky and you can kind of see where girls probably hid. Might have hid. Might have hid. And there really is a tunnel, which is probably an old coal chute going up to the street. Generations of Cameron House kids probably looked up at that tunnel and wondered if they could get out that way. Just a little bit more about this ghost. Did you sense a chill, a presence, a door slam? What happened? The house hasn't changed very much. It was rebuilt right after the earthquake in 1907, 1908. It was retrofitted back in the 1970s, 70s I think, maybe early 80s, but really it hasn't changed in about a hundred years. And the basement especially feels, I just, maybe I have too much of an imagination, but the first tour guide I think that took me around, he believed there was a ghost. The air felt heavy with something. It did, novelist. Well I think we have time for a few questions. Yes. Absolutely. I'm sorry if I wasn't clearing about that before. The house was founded in 1874. It was really a charitable project. It was a group of Presbyterian church women who saw the plight of Chinese girls and women in Chinatown and wanted to create a safe house for them. And initially they rented an apartment and then that got too crowded. They bought a house across the street. Then that was destroyed in 1906 and rebuilt what we now know as Cameron House. The girls got there in all sorts of different ways. The traditional rescue would be that staffers from the home, usually accompanied by a detective or a policeman, would go to the brothel or go somewhere that they'd heard the girl was, was being kept and tried to take her. So that was a traditional rescue. In many instances, in fact, the first story I tell in the book is a young woman who herself runs across Chinatown, finds the house because she had heard about it and asks for help. So she effectively rescues herself for saves herself. A pregnant teenager. A pregnant teenager, yeah. So thank you for that question. And so some 3,000 perhaps went through. Maybe other, what are some of the, and it sort of expanded. Maybe you can talk about how it kind of changed over time to involve boys or, yeah. Absolutely. So best estimate, 2,000 to 3,000 women and girls came through the home from the period 1874 to the mid to late 30s. After that period of time, it changed its mission. Initially, the women and girls had been trafficked. So many of them were forced into prostitution. As time went on, there was more of a situation where it was almost a social services agency. So young girls who were orphaned might end up there. Or women who had been abused by their husbands might come to the home for refuge. So there were all kinds of different circumstances. In fact, some of the photographs and probably the most famous photograph taken by Louis Stelman, which is in the California State Library, many of the girls in front of the house are extremely young. They're not sexually mature at that point. Yes. Yes. So this image, it came from Cameron House. Cameron House was very generous in providing us with historic images. We don't know the story behind this girl other than she probably was a resident at the home at some point. We don't know her identification. No, it's not... We used a number of Arnold Gemfes photographs. This is not one of them, though. It does look similar. So we don't really know what the origins of that image are. And if anybody has any ideas, I'd love to hear it. It looked like one of our characters, and we did compare a number of photographs to try to decide whether this might be her, but we don't know. In the background are the records of the trafficker from the National Archives in San Bruno. And so these ended up in the court case that brought down a very famous slave girl ring in the 1930s. The Wong C Duck. Yes. I wouldn't say rescues. The traffickers almost... The traffickers tried to get the girls back, and there were bars on the outside of the windows to keep them out. The doors were always guarded 24 hours a day. The girls, for the most part, weren't allowed outside unaccompanied because of the fear that they would be snatched back, essentially. And the traffickers also employed lawyers to try to wrangle them back through the court system. Was it solely Chinese girls and women, or were also non-Chinese women, females, and also males rescued by Cameron House? The quick answer to your question is yes. So yes meaning it was also Japanese women. At least there were a number of Japanese women, particularly around 1900 to 1914. There was a Syrian woman. There were boys.  I think, do you have another question? I was just... Which was called Chung Mai. Yeah. The former Chinese orphanage in El Cerrito. Yes, that's Chung Mai. Was that also founded by Donalina Cameron, or Cameron House? Yeah, no, that's a great question. It was founded by a Catholic missionary at the urging of Donalina Cameron. She was saying, I can't take care of these boys. We need a boy's home. So that's how that started. Likewise, there was a girls home started in Oakland, and the building was designed by Julia Morgan, and it's now on the Mills campus, and it is the Julia Morgan School for Girls, and there's a Donalina Cameron room, which is kind of cool. Right. That school opened in the wake of the quake, right, because they were looking for some place to house them afterwards. Yeah, yeah. I'm a Cameron House alumni, and my daughter went to the Julia Morgan School for Girls. Oh, wow. And I was wondering if you could talk more about the Mingkwang home, and what's your research from? Sure, so I did write a couple chapters on that home and the founding of that home. It was funded by the man who owned the biggest steamship company between China and San Francisco, and he was a Presbyterian, and he underwrote it, and it was mainly again that there were so many young girls and boys coming to the home that the staffers, including Donalina, decided it was much better to separate the younger from the older women. And the original place where the young ones were housed in Oakland was absolutely awful. It was unhygienic. The girls were getting sick. There were all kinds of problems, and Julia Morgan's building is quite a beautiful one if you ever have a chance to see it. Donalina seemed very skilled at getting wealthy patrons to, you know, the Carnegie's or this steamship guy. How was it that she was able to sort of make a pitch that there should be empathy and humanitarian aid? Well, I think she was very striking. She was a tall woman. She had an auburn hair. She had a Scottish lilt when she spoke. Her family had fallen on fairly hard times, but she came from quite a prominent and moneyed family in the UK and in Scotland originally. And so she had this ability, I think, to go into the drawing rooms of the wealthy of the time and also to work in extremely modest circumstances herself. One of the details that really struck me was that she lived in a room for many, many years where it was, she had such a small closet, there was only room for two pairs of shoes. So she lived herself extremely modestly, even though she was a good fundraiser. There was a very sweet moment in the book where she goes away and the girls, as a surprise, redecorate the room for her. She's in her fifties. Yes. Yes.


Bemidji now has a place where you can stretch your dollar when it comes to food and it will only cost you $20. Haiti Clotter shows us how Ruby's Pantry is feeding the community. It happens the third Wednesday of every month. You'll see a line that wraps around the block as people wait at Mount Zion Church for Ruby's Pantry to open. For anyone that eats, there is no financial obligation or criteria that they have to meet, so everyone is welcome to come. This is the second month Ruby's Pantry has distributed food in Bemidji. Some have waited as early as noon although food isn't issued until five. Was it worth the wait? Oh yeah, that's a lot of food, I'm going to have to have a big party. It's called a hand up and not a hand out and companies donate the food to Ruby's Pantry. Guests pay $20 and get food like pizza, chicken and potatoes and household items that would usually cost almost $120. Sometimes a paycheck's not that large and they can cover the first two weeks and this is something that we hope will take care of the last two weeks at least. The food is brought into the gym where volunteers help divide the food into 400 shares. Two large boxes equal one share and each person will leave with two boxes worth of food. Joanna Downey heard about Ruby's Pantry a few weeks ago and decided to stop by. She says Bemidji is a small town and appreciates any help it gets. Instead of just going to like a food stall, there's, you can contribute to it and so that kind of makes it neat so you feel like you're giving a little bit back too. I think it's great. Driver Patrick Trent has made trips to Detroit Lakes and Monago but this is his largest haul. Today 400 people were served. It says that we're doing something right. There's a lot of needy people and it's a good deal. Last year Ruby's Pantry provided food for more than 100,000 families. In Bemidji, Haiti Claude are Lakeland News. And Ruby's Pantry has 65 locations in cities across Wisconsin and Minnesota including Little Falls and Brunard. If you've enjoyed this segment of Lakeland News, please consider making a tax deductible contribution to Lakeland Public Television.


There's no point in fighting a war that you're not interested in winning. There's no point in fighting a war that you're going to lose. There's no point in fighting a war where you care more about the civilians of the enemy than you do about your own people. Now, I'm not, I'm not urging Donald Trump to go fight a war against Iran. God forbid, I'm just telling you what should happen.  Yeah. Iraq and Afghanistan had been disasters and I told you they would be disasters in 2002 when they started. I am on record. You can find videos of me. You can go read my book. Winning Than Winnable War. Winning Than Winnable War is a book I wrote three essays in. Ilan Jona wrote the rest of the essays. One of the essays I wrote is with Ilan. He edited it. It's his book. That book articulates a proper foreign policy, a proper way of winning a war with Iran, with Iraq, with Afghanistan. And by the way, it predicts the rise of ISIS. It predicts the defeat in Afghanistan. It predicts Iranian influence over the entire Middle East. It predicts a defeat in Iraq. All of this is predictable. All of this was predicted by me and my colleagues at the Iran Institute. A lot to talk about today. There's a lot of stuff going on. And of course, the main thing in the news is the assassination, the killing by the United States of Soleimani, whatever his name is, Soleimani, the Iranian leader of the Kurds Brigade, one of the leading generals, maybe the most important leading general within the Iranian military, one of the most feared figures in the Middle East, one of the most influential, one of the most influential certainly Iranians, members of the Iranian regime in the Middle East and one who has had a long and by Iranian standards, very successful military career. And so let me just say, good riddance. I'll say that the killing of Soleimani is both a good thing, a commendable thing.  And at the end of the day, fairly meaningless. So in terms of the long run, in terms of the Middle East, in terms of US strategy, in terms of 4D chess, will have very little impact on where things go, very little positive impact on where things go. And is at the end of the day, not a strategic decision. So let's get into it and let me try to explain and let me try to justify why I think that even though this is a good thing, killing of an evil guy is always a good thing, particularly one with American, the blood of Americans on his hands as some commentators have said today, this should have been done years and years ago. It is really a sign of American weakness that it has taken this long to happen. And at the same time, it is insignificant in the big picture in terms of, you know, Yuan and in terms of the threats that the United States and the civilized world faces in the Middle East. So who is the Soleimani guy? He was born in Iran. I think it was born in 1957. So that would make him 62, 62. What am I talking about? Yeah, 62, 63. He joined the Iranian military with no formal training during the Iran-Iraq war. He distinguished himself during the Iran-Iraq war as a committed soldier. Remember, this is the war immediately after the Iranian Revolution. He actually actually, sorry, he joined the military before the Iran-Iraq war. He joined the military during the Iranian Revolution. He joined the Revolutionary War Guard. The, you know, the special forces, the portion of the Iranian military that is responsible for the preservation of the revolution, that is responsible for the preservation of the Islamic Republic. He excelled during the Iran-Iraq war, even though he had no formal training, and became the commander of the Kurds force, QDS, you know, the elite force within the Revolutionary Guard in the late 1990s. He was then post-911 responsible for, once the regime of Hussein was destroyed by the Americans, he was basically responsible for, you know, working with the Shiites within Iraq, establishing Shia militias, Shia militias that then went on under his command to kill Americans, Shia militias that were instrumental in partially working with Americans and partially working against Americans in defeating the Sunni Intifada, the Sunni kind of uprising, and instrumental really in gaining political control over Iraq for the Shiite majority in the post-Iranian invasion. He was then, he has been throughout the 90s and the 2000s, he was Iran's representative in bolstering the Hezbollah. Hezbollah is Shiite terrorist organization based in Lebanon that really dominates and holds hostage, if you will, Iranian politics has been at war Hezbollah has with Israel since 1983, and Hezbollah with the Iranian support was responsible for the death of, I can't remember any more, how many? I think it was 144 Marines in Beirut in, it was at 444 Marines in Beirut in 1983 when a suicide bomber drove a truck bomb into the Marine barracks in Beirut. So all of that was facilitated, made possible, supported by and orchestrated by the Iranian, the Iranian Revolutionary Guard of which Suleiman was a major commander. He had been rumored to be a candidate for the President of Iran a number of times, but has always stepped back and not run. He has remained a military leader. He has, every time there being student demonstrations in the streets of Iran or any kind of semblance as a move towards more liberalization within Iran through internal revolution, he has been at the fourth he was, he's dead now, he was at the forefront of those arguing for clamping down, for making sure that there was no revolution, making sure that the Islamic Revolution sustained itself. He is supposedly behind, most of the Hezbollah was against Israel. The Hezbollah's attacks against American interests, both in Saudi Arabia and in Iraq. And he is also responsible for Iran's support for the Syrian regime, Bashar al-Assad's Syrian regime, in bolstering the Syrian regime, in supporting it both militarily with weapons and with the support of Hezbollah, of course, in together with the Russians in making sure that Bashar al-Assad remained as a dictator of Syria. I'd say, lastly, that he and his troops are probably more responsible for the defeat of ISIS than anything that the West has done. He was at the front lines of the fight against ISIS. Remember ISIS is Sunni. Remember that ISIS came out of the Iraqi insurgency. It came out of what was called al-Qaeda of the Levant, al-Qaeda of Iraq. Al-Qaeda of Iraq spent much of the insurgency attacking Shiites. Indeed, it was al-Qaeda of Iraq that declared Shiites as apostates and enemies of the Sunnis. Al-Qaeda, bin Laden's al-Qaeda kind of disassociated themselves in al-Qaeda of Iraq because of their attacks on Shiites. Al-Qaeda in Afghanistan and in Pakistan has always tried to be friendly with Shiites. Indeed, it was quite friendly with the Iranian regime before and after 9-11. But al-Qaeda of Iraq targeted the Shiites as part of this long-term big disputes between Shiites and Sunnis. There's no point in getting into what the difference is between them. Anyway, he was instrumental in the defeat of ISIS. Indeed, ISIS was stopped and ISIS was defeated ultimately, probably more through the effort of the Iranians than through the effort of any other force out there. They had an interest, both because they wanted to preserve the Syrian regime, which is kind of not exactly a Shiite, but it is a subgroup within the general Shiite religion. And of course, they're wanting to preserve the Iraqi control of Iraq by their Shiite government. So Iran, by defeating ISIS, preserving Syria, has basically established control over, if you think about it, Iran, Iraq, Syria, and Lebanon. It has a whole arc, and of course, its support for Hamas has significant control over the Gaza Strip. So it has the entire futile crescent is controlled by the Iranian regime, and much of that, much of that was achieved through the efforts of this guy, Suleiman. He was also responsible probably for the arming and support for the Shiites in Yemen who have taken over much of that country and have ultimately attacked, you know, all at war with Saudi Arabia. So Suleiman, he was a major force within Iran, a major force within Iranian military, a very competent military officer who is responsible for the deaths, particularly after the invasion of Iraq, but since then over the last, you know, 18 years of many Americans. And is a major, has been a major force to destabilizing the Middle East and attacking the interests of the United States and the Middle East. Now, given all that, he should have died and good for anybody to have killed him, so good for Donald Trump to have killed him. But it changes nothing. The fact is that he has already been replaced. There's already been an announcement out of Iran for the nomination of the head of the Islamic God. So there's already somebody to take his place. Killing one person never really changes the dynamic. It's not going to change the dynamic in the Middle East. I need the headphones in a little bit because I'm going to show some video of Donald Trump. It's not going to change the dynamics of the Middle East. It's still true that Iran controls Iran, Iraq, Syria and Lebanon and Yemen. And it's still true. Even on Afghanistan front, the East, the Western provinces of Afghanistan are controlled by Iran. So Iran has massive influence throughout the Middle East and throughout the region, throughout the Islamic world. It is the dominant player in the Islamic world. It is the number one sponsor of terrorism around the world. It is the number one sponsor of radicalization throughout the world. It's building mosque, Shia mosque around the world in Saudi Arabia. Are the dominant players in terms of, and again, Saudi Arabia, it's less the regime and more the people and the religious organizations, but they are the dominant players in attacking the West whenever and wherever they can. So Iran, nothing about Iran has changed. Iran now knows that Donald Trump is willing to kill some of its operatives. They'll be more careful next time. They won't be quite as easy a target. It's important to note that the killing was done on Iraqi soil where the United States has a significant military presence and a day after or two days after the American embassy in Iraq was attacked by Iranian supporting militias. It was not an attack on Iranian soil. It was not attack on Iran. I think the leadership, the Iranian leadership still feels fairly secure in themselves and in their ability to command their forces and command their influence throughout the Middle East. I don't think anything has really happened to undercut their confidence in all of that. It is, there's nothing strategic here. I mean, unless this is intended to provoke the Iranians into doing something more substantial that then leads to war, which will lead to taking over Iran, that would be kind of a 4D chess move, if you will. But then why is America playing games? If America wants to take out the Iranians, just take out the Iranians. I mean, there's been plenty of provocation. I have argued many, many times throughout two decades that Iran declared war on the United States in November 4th, 1979, when it took the American embassy in Tehran since then and held hostages for over a year. That is a declaration of war when you take somebody else's embassy. That is a declaration of war.  But they've attempted to do that, again, an act of war. So Iran has engaged in acts of war against the United States for 40 years. And all we can do is assassinate one of their generals.  I mean, they attacked, they mined the Straits of Homoes, they were American tankers, among others, traveled through and oiled to the United States, traveled through. They took a British ship, they seized a British ship, a Britain, which is an ally of the United States. And of course, they attacked the largest refinery complex in the world in Saudi Arabia in a massive attack, no response to any of that. They even shot down a U.S. drone over the Persian Gulf, no response to any of that. So the U.S. has played it weak, whether under Bush, under Obama, or under Trump. This is the first act of any kind of serious act against the Iranian regime that any administration has taken in decades, maybe ever. And it's good, it's good if it was within some kind of strategic context. But Trump has made it clear, he made it clear immediately, today indeed. He has made it clear that his intention is not regime change in Iran, and I'll get to that video in a second. But the fact is that the only way, the only way to put an end to the bloodshed, to put an end to the killing of Americans, to put an end to regime that acts strategically and constantly against American interests is regime change. Now, my preferred methodology of regime change is internal regime change, is to do whatever's possible to being about an internal revolution within Iran. But as a precursor to that, or as assistance to that, it wouldn't harm to take out the entire leadership of the Iranian regime. Starting with the supreme leader, Khamenei, should be assassinated immediately. And indeed, the entire ruling council of clerics should be eliminated as quickly and as thoroughly as possible. Thus encouraging the Iranian people to rise up against them and establish maybe what potentially could be a pro-western regime in Iran that would not be religiously fanatic, and that would not be supporting terrorism all over the world. In other words, the solution here is not to kill a general, the solution here is to destroy the regime, to flatten them, to do as much damage as is necessary. And I'm not going to specify how much, but as much damage that is necessary to bring about regime change in Iran. I don't think that requires occupation. I don't think that requires, given that some way a significant percentage of Iranians want regime change. I don't think you need to occupy it. I don't think you need to send ground troops. Maybe you do. Again, we'll let the generals decide exactly how they want to do it. But regime change is essential. Regime change is necessary. And if a lesson is learned or taught at the same time, if Iran has made an example of what happens when you kill Americans, all the better, all the better. But none of that is going to happen. None of that is going to happen. The Israelis have been playing this whack the mole game of killing military leaders for decades. Every time they kill the Hamas military leader. And guess what? The next day there's another Hamas military leader. And then they kill that Hamas military leader. And the next day there's another one. And they do this over and over again, decades, and nothing happens. Nothing changes. Nothing is fixed. I mean the day when they start killing the political leaders, instead of killing Gisbala, for example, military leaders, it is Nazrallah, the spiritual and political and Islamic leader of Gisbala, which should be assassinated. Then you might get a response. Then people will know you're serious. But just killing generals is weakness, shows no commitment to actually win. These are people who are plotting constantly to kill, in the case of Israel, Israelis, in the case of Americans, Americans. It would be absolutely an act of self-defense. Absolutely an act of self-defense to crush the Iranian regime. And I have been calling for this since, well, at least since 9-11. But really way before that, I just didn't have a voice way before that. Again, Iran declared war on the United States November 4th, 1979. Let me also say this. The Iranian military is unbelievably weak. It would not stand any chance against the United States in any dimension. The casualties, if the war was done right, would be minimal. And yet you would rid the world of one of those evil regimes of the last 50 years. They still don't have nuclear weapons. But one day they will. And when they do, it'll be much more difficult, much harder, much bigger challenge. Look at North Korea. We tiptoe around them. We walk in eggs around them. Because God forbid we can't, we can't take on. We can't take them on because they got nukes. They could turn South Korea into glass. The Iranian military is not. Iranian military fought the Iraqis when they were weaker to a stalemate in a 10-year war that killed well over a million people. Stalemate with Iraq. The Iraqi military that we just walked all over. The Iranian military since then is significantly weaker. It does not have. In 1980 to 1990, when they fought the Iraq war, they had American weapons that America had sold the Shah. Today they have none of those weapons. They have some Russian weapons. They have some self-made weapons. But they are weak.  The Americans walk all over them. The reasons we haven't won the war in Iraq and Afghanistan is because we don't know how to fight. I'm not, by the way, let me just be clear. I am not suggesting we fight a war with Iran today with this president and with the kind of ideas that our military has in their heads. I have no willingness, no interest in fighting a war in Iran under the same terms that we fought in Iraq and Afghanistan. There's no point in fighting a war that you're not interested in winning. There's no point in fighting a war that you're going to lose. There's no point in fighting a war where you care more about the civilians of the enemy than you do about your own people. No, I'm not urging Donald Trump to go fight a war against Iran. God forbid. I'm just telling you what should happen. What we need today, what I call the new intellectual, would be any man or woman who is willing to think. Meaning any man or woman who knows that man's life must be guided by reason, by the intellect, not by feelings, wishes, whims, or mystic revelations. Any man or woman who values his life and who does not want to give in to today's cult of despair, cynicism, and impotence, and does not intend to give up the world to the dark ages and to the role of the collectivist. Using the super chat, and I noticed yesterday when I appealed for support for the show, many of you stepped forward and actually supported the show for the first time. So I'll do it again. Maybe we'll get some more today. If you like what you're hearing, if you appreciate what I'm doing, then I appreciate your support. Those of you who don't yet support the show, please take this opportunity, go to Iranbrookshow.com slash support, or go to subscribestar.com, Iranbrookshow, and make a kind of a monthly contribution to keep this going. I'm not showing the next...


Welcome to the ITU studio in Geneva. We're very pleased to be joined in the studio today by Professor Emanuele Givernetti who is Professor in Economics for the Anglia Ruskin University in Cambridge campus in the United Kingdom. Professor Givernetti, thank you very much indeed for being with us in the studio today. Thank you, I'm delighted to be here. This year I wanted to point out that we're celebrating the 25th anniversary of the ITU development sector. The ICT landscape has changed obviously dramatically in that time. How do you think that this study on ICT for SDGs will contribute in responding to many of today's challenges? I think that this study will be really essential in pulling together different branches of knowledge, focusing on how can we really use and harness ICTs to attain the sustainable development goals. And I think that ICT provides both a wonderful opportunity to attain those goals and at the same time possess great risks. So it is very important to have a very balanced and multi-sided approach to understand these effects. Now there are several studies and debates on how ICTs are key enablers for economic growth and innovation. I'm sure you're familiar with a number of them. I wanted to ask you, can you tell us more on how the main outcomes of this study and its added value to the existing discussions will make a difference and specifically its contribution to the United Nations Sustainable Development Goals? Yes, I think we try to look forward instead of backward. We are not trying to see where are we now, but where can we go? What should we do in order to attain the SDGs? So I think this project is really going to focus on this and this is probably the main difference. But also the fact that there are different experts with different backgrounds, coming from different fields of expertise. And so there is a constant debate if you want from different perspectives to try to capture what is the real possibility of moving forward in attaining the SDGs. Now a number of experts have been asked to contribute to different chapters of this study. I wanted to ask you specifically about yours, which I believe is Chapter 3 and it involves ICT as enablers for innovation ecosystems. Perhaps you could tell us a little bit about what that means. Yes, of course. Yes, Chapter 3. And there is a large literature on innovation ecosystems and this literature is focusing often on developed countries only. So in a way there aren't too many contributions focusing on ecosystems, innovation ecosystems for developing countries, which of course is a very important theme in relation to the SDGs. But even in the literature of focusing on ecosystems, innovation ecosystems for developing countries, again the recent and strong focus on how ICT can be used in enabling the achievements of this innovation, in focusing these innovations towards sustainability and sustainable development goals. So these chapters really tend to focus on the key role which is played by different platforms, by different digital infrastructures that, given the nature of these new ICT technologies, are now not only just a digital infrastructure in themselves, but also provide a lot of scope for human interaction, face-to-face interaction, building capabilities. So there is this merging of a digital and human infrastructure which is an essential component for the ICT-enabled ecosystem to produce innovation and growth and to achieve some of the key sustainable development goals. In your opinion, what's going to be the true value of this study? The true value will be to understand also to provide a roadmap for policy makers to try to understand what are the typical mistakes to be avoided, who are the key players, how activated key components of an ecosystem, because without the involvement of all the key components it will be very difficult to achieve with a good degree of success the SDGs. Well we thank you very much for your contribution today and of course your contribution to this study and we very much look forward to its launch in the very near future. Thank you very much. I was delighted to be invited to work at RTU. It is a very stimulating environment and I'm very happy to be here. Thank you.