Shri Vijay Kulkarni was connected with several educational institutions and organizations and was a recipient of many honours and awards, including the Dr. Govardhan Dandas Parik award in 1985. With his students and colleagues, he undertook many significant research projects, especially for the improvement of science and mathematics education of children from disadvantaged sections of the society. The place of language and education was a major focal point of his research. He wrote extensively, simply and convincingly in Marathi and English, on science and the need for a scientific culture in society. The lover of books, he was a brilliant orator known for his scholarship, mastery of words and a refreshing and original sense of humour. Shri Vijay Kulkarni expired on July 13, 2002. As a tribute term, HPCAC has instituted a series of annual memorial lectures. These lectures, given by eminent scientists and educationists, deal with central issues in science, technology, education and society. Of course, this description about Vijay Kulkarni cannot really do justice to his efforts, his thoughts, his foresight. Be at the centre are really beneficiaries of all the initiatives that he took. And I can certainly say that he continues to be an inspiration, even today, to be a guide as we go ahead and make many decisions. And the priorities that he set for the centre at that time, I think were still continue to be priorities and we admire greatly the foresight that he had. And one of the priorities was to institutionalize the work of science education, which was very unique at that time. And so, the result of that is the institution that you see today. This is what I'd like to say to those who knew Vijay Kulkarni. To those of us, many of my colleagues, younger colleagues who perhaps have, who of course have not had a chance to meet Vijay Kulkarni, I would certainly suggest that you read many of his writings. The project reports that he wrote at the beginning of the centre, the many projects that were started are full of insights and they are very, very educating as well as inspiring. As we remember Vijay Kulkarni, we have the support today of Sri Vijay Kulkarni's family. They of course give us unstinting support in every year they are here present with us on this occasion. And I'd like to request our Dean, Professor Sugrachunawala, to express our love through this token to Mrs. Vijay Kulkarni. We are very honoured today to have an eminent speaker to deliver the Vijay Kulkarni Memorial Lecture. Professor Rohini Godbole was a very eminent physicist. Many of you may know her, she's certainly not a stranger to the Homi Baba Centre. In fact, she just informed us that she is earlier spoken at the Vijay Kulkarni auditorium from this very podium. But we're very happy to have you here and a very warm welcome to you and I request Professor Sugrachunawala to hand over the bouquet. Let me tell you a little bit about our speaker today. Professor Godbole obtained her doctorate from the State University of New York at Stony Brook in 1979. She is currently Professor at the Centre for High Energy Physics in the Indian Institute of Science, Bangalore. Prior to that she worked with TIFR and the University of Mumbai. She is a visiting professor at the University of Dortmund, Germany, at CERN Geneva, at DCD, ESY Hamburg and the University of Utrecht in Netherlands. She has authored more than 280 research publications and many have been cited very highly. There is a description about her work in physics. I'm not the best person to present that to you. I'm sure she'll say something of that in her talk. I'd also like to mention that she is a recipient of the Vander Waals Chair at the Amsterdam Institute of Physics and Astronomy. She has worked extensively on different aspects of particle physics phenomenology over the past three decades. As I said, she has authored many research publications. She has authored several graduate-level monographs and textbooks on advanced topics for PhD students and researchers in particle physics. She serves as a member on various national and international committees. She is an elected fellow of all the three major science academies in India and also the World Academy of Sciences. She has served on the editorial board of several prestigious journals such as Current Science and the Indian Journal of Physics. She has won many awards and distinctions which include the S.N. Bose Medal of the INSA for Theoretical Physics, the Meghnath Saha Award for Physics of the Asiatic Society of India, the DST J.C. Bose Fellowship of the Government of India, Honoris Causa Delet from the S.N.D.T. Bremens University and so on. Additionally, she has worked towards pioneering a variety of programs at numerous fora to raise awareness on the subject of women in science. She has co-edited the book Leela Vathi's Daughters, Women's Scientist of India, containing autobiographical and biographical sketches of about 100 women scientists in India and also a book inspiring girls to a career in science called A Girl's Guide to a Life in Science. This is really inspiring in particular the work that you have done on the life of women scientists in India and I remember Prof. J.S. Ramdas, our former director, mentioning this work, referring to this work many times, all of us have read it and it's truly inspiring and I'm very happy to welcome you once again Prof. Rohini Godburi. So, I think I will rather stand here. Good evening everybody. Thank you very much for this extremely kind introduction. I am truly honored to be given this talk, giving this talk, which is VG Memorial Lecture. As people might know, Jayashree was and has been a very close friend and in fact, I got to know VG as her thesis supervisor. That was my initial introduction with him and I first talked to him in the Nana Chowk office. Of course, I cannot say that I discussed things on science education or apart from Jayashree's own PhD thesis, which we used to discuss quite heartily. I don't think I had many other academic discussions with him, but we were friends in TIFR of the old days. Almost everybody, it is a big family to that extent and because of that I am truly happy to be giving a lecture in his memory. So, let's see what. So, when Sugara asked me to give this lecture, what happens? No, my initial reaction was to suggest a topic in science, but then she said that she would like me to talk about something to do with science, education, society. So, then I said this is my second love. I mean, I still have to say physics is my first love, but this is indeed my second love and I am very happy to be sharing some thoughts on this subject, women in physics and mathematics. Oops, what is this? This is Max. So, let me try to tell you how I actually, the idea of this kind of colloquium originated. So, that will be very brief and I will give some few preliminary remarks and then the second part will be kind of a nice part because I will just tell stories, very inspiring stories of women in physics and mathematics from early history to I will come up to middle of 20th century after which I stopped because luckily for us the numbers have started increasing after that. But, I mean if I have to tell stories I can complete finish two hours and we will still be sitting here listening to the stories, but the point about telling the stories is to next ask the question what lessons do we learn from this story. So, that is where I want to come and then connect it to today's situation in India and then when I ask the question, have the issues changed? I will not betray any secrets if I said that they have not changed and then what is you know what are we to do? What are we doing in India in fact and some lessons that we learnt in the process of trying to do something over the period of last 15 years. So, let us see how we go. So, the origin of this colloquium actually started when I was visiting ICTP and you know talking some of us and some of you who might have gone to the International Centre for Theoretical Physics in Trieste, we will know that on the wall there are various photographs. So, on one wall there is this famous picture and nobody who has you know anybody who has studied modern physics will know this picture. This is a famous Solve conference of 1927 and every single person here is the architect of the physics of this century and there is one woman Madame Mary Curie. The second side of the wall there was another picture and this is actually the picture which was 1960 ICTP was founded and this is ICTP I mean first one of the first conferences and actually again people here are again just like here perhaps not as illustrious they are still the many very very distinguished scientists of that period people like Hugo Amaldi and of course of the Salam and so on and so forth and then there was this one lady at the back. So, I was just curious and I said you know I was talking with the director and I said do you know who this lady is? So, we started hunting and then we found that this lady was not a physicist but she was from the administrative staff. I mean over tea I can share stories where people and I think I can share it and many other women share it very often places like big places like ICTP or CERN sometimes young brash men have mistaken many of us a secretary is wanting to do some secretarial job for them but that is a different story for a different day. Then few days after that actually I went to the German physical societies main building I was invited to give a talk there and here on the door they have a big photograph of M. E. Neuther you know sort of they call that door M. E. Neuther door and then right next to it you go past the photograph and you get in this room and this is their German physical societies lecture hall and for those of you who do not need German it is the opening in November 2003. She is from 1882 to 1935 this is in 2003. So, I was quite you know sort of that here we have this two towering women of physics and mathematics but after that what has happened in this photograph this is German physical society I can find only one. I mean we are not talking of very extraordinarily selective crowd. So, we started talking about this and then one said maybe one should actually talk about this subject with everybody you know. So, therefore, I began sort of the way saying why should we be discussing issues of women in science why should we all be discussing issues of women in science is the question we want to understand. So, some of the people actually have said this yeah we know that the number of women doing science is small yes that is not a surprise it is known but is it so bad and the answer indeed is that yes it is bad and not because you want to give full scope to women's creativity and so on and so forth not as a favor to the women it is as a requirement of science because any creative activity can only gain from diversity it does not have to be only women it can be you know dress it can be creed it can be cast diversity actually always adds to the level of any creative activity and science as we all know is an extremely creative activity. So, by not increasing that number we are not using optimally our intellectual potential and in India at least the changing economic realities also mean that the number of women participating in science is increasing that is also another reason why the number is increasing there is one reason there is more realization but the number is also increasing because as usual whenever there is a vacuum women move in to fill that vacuum but if that is going to happen and if we want science of high level indeed we must make sure that these women can participate in science effectively and efficiently. So, therefore it is very important to discuss what efforts we need to put in. I want to begin with some myths and biases US Supreme Court judge Justice Roberts he is no no more the Chief Justice but in 2015 he went on record in the court while discussing a case he asked the question what unique perspective does a minority student bring to a physics class replace minority by women does not really matter I mean equality equity is the name of the game but here is this person who is Supreme Court Chief Justice in the United States of America and he wonders why is it necessary to increase participation of black people in science what is going to be achieved by that it is no different than women ok. Tim Hunt Nobel Prize winner of 2011 and he in an interview in 2015 said let me tell you about my trouble with girls three things happen when they are in the lab you fall in love with you or when you criticize them they cry. Now I must tell you that I have not seen so often like these occurrences but of course if he is giving this an interview must be his experience. So, these are these biases that we have and it is because these biases exist we must discuss the subject about participation of women in science in a clear fashion. So, why do I focus on mathematics in physics in general we are a long way off from gender equity in science period but some are more unequal than others and physics and mathematics happens to be a little worse off I could add engineering too. So, these are the three areas which are even further down the line. So, the numbers are even smaller in these areas compared to other disciplines not just that perhaps not in India but the world over there is a phenomenon that women cannot do physics and mathematics. In fact, Larry Summers was actually quoting some study when he said that women lack capability in mathematical and physical science. So, when this is happening we one realizes that you cannot let this go by you need to think about these issues. In fact, in 2016 the women's day slogan was pledge for parity. Now, if there is any vague hope of our effort fulfilling such a pledge then actually scientists natural and social scientists all together because this cannot come about by only either of the two you need both such scientists and social scientists. We need to discuss and we look at the issue like we solve a scientific problem and therefore the discussions it is obvious that discussions should not be restricted only to women they should not be restricted only for women's day and they certainly should not be only for women's conferences. In fact, I will share some small story with you when I gave this colloquium at ICTP because after all that discussion then we got into this and Fernando Pevedo said why do not you give a colloquium. So, we gave I gave a colloquium and then I was sitting and preparing my colloquium in the Indian postdoc with whom I was sharing the office. So, he said or maybe he was next door. So, he came in said what are you doing? I said you know I am preparing a colloquium for women in science which I want to give next week. So, he said ah is there some women's conference. So, I was not too far wrong. So, this is the point is that and that is the reason why I very eagerly accepted Sugara's suggestion that I think for in somebody memory of somebody like ah VGK who were looking at science education and new parts I think this is the most apt ah discussion to to discuss namely what about women in science in general and by physics and mathematics in particular. So, I have collected some references and there is these are the few websites where from where I over the years I have collected information plus if I quote any numbers I may not, but if I quote any numbers they are from this report called women scientists in India which I wrote for ah women in science and technology in Asia document that came out. So, we sort of put together a lot of information and for some of you if you are interested in my ah thoughts and on the subject after all this happened this was a editorial that I wrote in ah February issue of current science where I discussed some of these things in little detail. So, let me begin with women in mathematics and physics. So, the earliest the good old Google helps us. So, you will find photographs that are coming here from Google, but it is not just Google that is why I set up those references that I told you there are a very good websites ah we have put this information together. So, the earliest we could think of one can find trace is Hippasia she was certainly one of the earliest non women mathematician and she was daughter of the member of the library of Alexandria and she actually used to give lectures to students and translate books mathematical works and into ah wrote commentaries on the mathematical works and she was also a philosopher. Unfortunately she was killed in the unrest that was happening at that time in this part of the world. So, but she was at what what I am going to do is that I am going to kind of try to bring out what were the things that made these women what they were. So, I want you to follow that trend through this. In India I wanted to be a mathematician or a physicist. So, a scientist. So, I could not unfortunately take Gargi or Maitreya. So, the one thing that we could think of is the fact that the 12th century mathematics in India when it was really flourishing a treatise in mathematics which was written by Bhaskara too which is called Lila Vati and in this he writes most of the problems addressed to Lila Vati. This book is 1150. So, from few hundred B.C. we have already jumped to 1150 and one of the things I have translated here it says finite child Lila Vati tell me how much is the number resulting from 135 multiplied by 12 if you understand multiplication by separate parts and by separate digits and tell me beautiful one how much is that product multiplied. So, we can legend has it that she was his daughter. So, we choose to believe that maybe she was there as the India's first human mathematician. But anyway this particular book actually I found this from the it is a photograph of the original manuscript and this is a famous problem that there is this peacock and it wants to catch the snake. The snake is going at a certain speed the height of the pillar is given and that is how the snake is flying at a certain speed and the question is asked when will the snake the bird catch the snake and of course the answer is in solving this technological problem. So, this is the Lila Vati now from 11th century I come to the 17th century immediately she was the first woman PhD in 1678 in Europe and she was in University of Padua and here I want to share something interesting with you actually she wanted to learn theology but then she was told that women should not learn theology why don't you go and learn mathematics. So, you see whatever is considered as the top most excellent that's not for you you know why don't you take a job right it is the same story all over again. So, she actually studied mathematics because she was not allowed to study theology and she wrote actually a book at that time where she used she learned Latin and she used to lectures at that time. Then in 18th century again these are actually almost all in Europe the names that I have. So, I am writing only names and the one in red are something on the women about whom I will say something. So, Sophie Germain was a French mathematician and an extremely gifted mathematician Mary Somerville was actually the lady who defined what is meant by the process of science. So, she was the one who tried to tell people what is it meant when we do science and Caroline Hershield was the sister of the great Hershield. So, I will tell you a little bit about these three women about other women not because they are not important but I simply will run out of time. So, this is Sophie Germain as you can see she sort of died quite young 56 and she was completely self-taught she was at home in the period when it was the French Revolution and her father had a lot of books at home. So, she read these books she got interested in mathematics and in fact she was completely overawed by the idea that Aristotle was killed when he was trying to work out a problem. So, she said she has written this that she said you know it was amazing to think that you can think of a abstract problem in such rapture that you do not even realize and that is what I would like to do that is what I would like to experience this is what this girl wrote but she was told by her family that it is not proper for middle class girls to study mathematics and physics. So, she was told not to go she had no formal training at that time equal polytechnic had just started. So, women of course were not allowed in equal polytechnic but she was not to be denied so, she actually used the pen name of a friend who used to go to equal polytechnic and she would take the problems from him she would solve the problems and she had a question so, she communicated with Lagrange and Gauss and actually she did extremely important work on theory of elasticity and Fermat's last theory and in fact she was the first women to win a prize from the French academy but this essay that she wrote for this prize had to go back three times because the people who were assessing it said this looks very good but it needs to be formalized it needs to be better but she did not know any better because she has never been taught so, one just wonders you know I mean she actually there are something called German numbers which are also named after so, one wonders what would have been the contribution of this woman but the family had even taken away her candles so that she should not be able to study then she came out and I mean this is really stories which are written down and a really romantic stories almost but I mean not so romantic when you think of it that in the freezing summer this poor girl came out and she was trying to study in the cold room and I trying to write but very important time was already lost in all this not that it affected what she did so, here is Gauss saying this how can I describe my astonishment and admiration on seeing my esteemed correspondent M. Le Blanc metamorphosed into this celebrated person what a woman because of her sex our customs and prejudices encounters infinitely more because then men in fact familiarizing herself with number theory is naughty problem and this is the great Gauss he says further she proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree he tried very hard for her to get an honorary degree I am taking this this is quite important she was drawn to mathematics by equations which were appearing in fashion magazine and she realized that I like mathematics but she was of course her father opposed studying sciences she got married as things would move her husband died and after the death of her first husband she got married again and the second husband was extremely supportive he would go to the library and bring book back to her and he started Laplace's treatise so that we made a commentary and her commentary was so important that it inspired search for nature and this is a woman who hasn't had formal training in mathematics in fact this is where it's very interesting she was in contact with both Babbage and Herschel so you see mentors playing an important role in getting getting a woman ahead she was the first honorary member of the Royal Astronomical Society and as I said she defined what is the process of science actually there is an article by her on that then comes Caroline Herschel she was an astronomer sister of William Herschel she came to England from Germany to help her brother keep house so she used to be called called herself Cinderella of the family but in this case actually William Herschel was a pretty good guy from what I can see he asked the king to pay his assistant so she became the first woman who got paid for her research she actually used to first help him grind the mirrors and so on and then she started making observations in her own right and in fact even she got a gold medal of the Royal Astronomical Society and she contributed to searches of nebulous clusters and comments and even when Herschel died in Germany but she continued doing her research over there and she died sort of a very peaceful death at an old age and had contributed very much to the searches so now comes the as I said 19th or 20th century where we start getting slightly bigger numbers and this is Adelaveless all the computer programs the first algorithm was written by Adelaveless and very few people know it that a woman wrote the first algorithm Sophia Kowalska she was a very celebrated mathematician I am going to tell you the stories of these two Maria Michel I will not tell you but she this is the first time a name from USA makes its appearance she was an astronomer and please remember this name Charles Agnes because this is going to come back later she was a mathematician a British who moved to the US of course the great Marie Curie Amy Northard a equally great person very few people know about her but absolutely one of the greatest scientists that lived and today this year we celebrate 130 years of North's theorem and I will come to you then I will talk about her what the theorem would mean to all of us is Lisa Meitner who was a theoretical physicist CS Wu who was a fantastic experimentalist and Maria Gopatmayer who was a theoretical nuclear physicist I put them three together because in my opinion all these three women should have got a Nobel Prize but only this one got here and that too is an interesting story she got her Nobel she got her first permanent job only two or three years before she got the Nobel Prize so and the other two of course had this one was successful by all accounts she was the president of the American physical society but she didn't get the Nobel Prize and Lisa Meitner I will not have time to cover her story but again it's one of the sad stories and I personally feel that when I studied nuclear physics you know we never realized that she was playing she had played such an important role in fission understanding that nuclear nucleus can be nuclear fission can happen she was the one who worked it out and there is enough evidence that she was the one who was doing it on a postcard she has written a letter to her collaborator of 30 years saying that she was of course she had to leave Germany so she had written him a letter and said please do such such things so that you can test whether what I am saying is right or wrong so these are these women and again my story goes I will pick up stories of only these four women and these two I have information and we can come back to them but we will want to do something more than just hear stories now this is what is interested she is the daughter of she was the daughter of Lord Byron she was very much encouraged by her mother this was very interesting by Mary Somerville so the importance of having women themselves who are enthusiastic about science cannot be under you know under emphasize plus she was interacting with Babbage and he encouraged her to write a commentary on an article by Luigi Manabra from Turin on the analytical machine which had been proposed by Babbage so Babbage brought that article and you write a commentary on this and that commentary contained the first algorithm literature has great effidences some mentions in the literature about the doubts that whether Babbage did the work or she did the work so it seems a recurring theme and I want to share that theme and this is not just old wives tell but this is there is evidence that people have written this now this is one hell of a mathematician this woman grew up on a Russian estate she was taught at home up to the age of 15 a tutor came taught physics and mathematics just like she was taught to sing and dance you know she was being groomed to be a good complete woman along with this she wanted to study mathematics so then she realized that the only way she can get out of Russia is that if she gets married so she stuck a marriage of convenience with a cousin who was also studying geology so these two people traveled to Germany after being married so once you are married you are safely you can be safely sent out and then she actually did and she became the first woman professor in mathematics in Europe in Uppsala she was the winner of the famous French prize called Pre-Bodham and this is again these are these competitions where you have to write essay you have to solve problem and she got this prize and imagine on the day she got this prize she has written this she has been happy as human creatures especially as women so this was a highly successful academic life but extremely depressing personal life this is again a common theme that comes from time to time and again one hears some fair statements about the work that was very important to the force somebody who was a scientist par excellence man or woman no questions asked but even she didn't have it easy and that is clear in this life is not very easy for any of us but what of that we must have perseverance and above all confidence in ourselves we must believe that we are gifted for something and that must be attained saying something about life one has to realize that in spite of the Nobel prize she got her position after the accidental death of Pierre Curie who never failed to emphasize her contribution and in fact it was because of his insistence that she got the Nobel prize that this is again well documented she by the way she never made it to the fellowship of the French academy two Nobel prizes were not enough for that okay then comes the fantastic Emmy Noether she was called a creative genius by Albert Einstein but imagine that this creative genius could not get a degree because universities at that time were not allowing women to come to the university just as luck would have it by the time her father was a mathematician and she was allowed to listen to lectures and as luck would have it by the time she finished listening to the lecture the university decided that they could grant degrees to women so she got her Ph.D. from the University of Erlanga and I think she was the first woman to get this degree but after she got the degree for seven long years she didn't get a permanent job but paid job she was not paid it is in this seven years that she was invited by none other than Albert Einstein the greatest physicist of 20th century perhaps and Hilbert the greatest mathematical physicist and these two people invited her because they wanted her to help them in understanding a basic problem in general theory of relativity which was an apparent non-conservation of energy and her Ph.D. work was on invariances so they said here is an invariance somebody who understands invariances maybe she would be able to help us and they were right she did the most important work which I called Noether's theorem and the importance of Noether's theorem is that the entire structure of theoretical particle physics today would not have been possible if we had not had the deep understanding of Noether's theorem about symmetries of nature I mean seriously it underpins everything that we do and this is not even her most important work this is side by side it is a work that she did because somebody asked her to solve a problem after that she went on and did extremely important things in mathematics and there are conferences which are every year which take place which are devoted new algebraic ideas but in spite of all this for the first 7 years she did not have a paid position during this 7 years she was actually working at Göttingen teaching Hilbert's classes they were announced in Hilbert's name and she used to teach them and he used to pay her for that so she had support from this extremely strong physicists and mathematics people but the university authorities did not allow her to get a job then during this time when she did not even have a permanent position she was invited to give a talk at the international mathematical congress which is one of the biggest so meetings in mathematics and then came the inevitable war she had to move to US at that time remember this Charles Angus Scott she was the principal of a women's college but she knew her mathematics she came to Germany and sort of talked to Emy Nerther and said I know we are a small college but we would like to have you and I can make arrangements for you to come there and again Einstein was trying to get her a job at Princeton but she was travelling from this college which is a women's college but Princeton University did not give her a job but also she did not give them a very long time within a year of moving to USA she actually expired due to some complications but she was immensely successful as a researcher as teacher she had people around her who were called Nerther boys and Nerther girls and the family was extremely supportive mathematical giant in fact she used to be called Der Nerther because in Germany if you know any German Der is masculine so she used to be called Der Nerther and in fact there was a exhibition of mathematicians of the 20th century and in that there were all men men of 20th century Nerther so this is at least telling you the level of her mathematics not just level of physics which we are all very pleased about but something else too and this is a small digression this is for my if there are teachers and younger students in the audience this year actually the physics news contained articles only by women the Indian physics association the picture from about Meninother because there is an article about Meninother her life and her theorems in this IPA in fact we have noticed that in the last two years there hadn't been a single article by a woman author in physics news they are not just physics news but the resonance of March 2017 had articles only by women and so also current science these are the kinds of things that we need to do sorry it's said that we have to do it but that is the kind of thing that perhaps we have to think about and none of this will be you know it will be completely incomplete without this woman Mariam there is a funny the first woman to get a fields medal she came from Iran you must also none of these women who we are talking of came from very liberal societies very few of them actually had that kind of very liberal atmosphere which supported their education so there is something else that these women had within them so it took Meninother got this Akbar Toivner memorial prize in 1932 and fields medal in 2014 rather a big gap but better late than never covering women and I want to end this story with a very interesting story which I had not known this is a story about a lady called who is called mother of mathematical ecology I mean we know all those systems biology is a big thing now but she is the one who started the game and she was actually working completely from her by herself and submitted her thesis to University of London the University of London surprisingly was very accommodating they accepted her thesis and she got her she was born in 1924 she got her first university job in Canada in 1968 but this as I said she started the subject of mathematical ecology and modeling of natural systems and she got a very famous award Eminence Ecologist Award in 1986 now I want to tell you a story she wrote a paper in the early part of her career when she was just beginning and in that she showed that Robert MacArthur who was himself a very eminent ecologist he was wrong an equation in his paper was wrong analysis that he had done was not quite correct because he hadn't been careful about whether you take average after squaring or whether you should I mean something about as simple as this so therefore she wrote an article correction to the MacArthur formula for abundance of species so this is the analysis mathematical analysis of abundance of species and I want to share this with you here is you know this is the article correction to one of MacArthur species abundance formulae science 151592 and this is the article she had written not please see this not on Mrs. Pillou's comments she is Mrs. Pillou she is not Professor Pillou first thing and just read the comment I am happy to agree that Mrs. Pillou's formula is literally correct and ours only drew in approximation the distinction between the processes when applied to a single senses is somewhat vague biologically and almost undetectably small numerically which may right might have been true that's not the point here she was trying to really point out something very critical and therefore he says let us hope that these comments do not withdraw additional attention to what is now an absolute approach to community ecology which should be allowed to die a natural death I mean if anywhere it was wrong you could have been magnanimous enough and said yes I was wrong and she was right but you don't even want to do that so I think this tells you these stories tell us a lot already about the society about the people it might tell us something about Mr. MacArthur for what he was and last but not the least you know I have talked about women who are very famous women who made it big women who won awards women who shone but for every one such woman I think there are tens and hundreds who in their own way persevered and achieved something and the story of these four women which are called three women which are hidden figures and the fourth one I have added myself and these are from similar period as you can see 1910 to 2008 1905 and 1928 she is also still there who were these three women these three were the first African-American ladies she was the supervisor in NASA she calculated the orbits of Senator Glenn's orbit when he was going around the earth and she was the first African-American engineer ever to be allowed to graduate in the state of one of the southern states I am sorry I do not remember but these three were trailblazers in their own right because they were fighting on two fronts they were women and they were African-Americans in the 60s in the this is before the civil rights movement that we have talked about so actually there is a very famous theorem and it is called but in fact the computations were done by me so now people like Fermi did not want to take the credit away from her but the community started calling it Fermi Pasta Ulam and now it is recognized that without her work this theorem would simply fail because she demonstrated numerically a lot of very important stories so these are what I call really the hidden figures so this is the story of women's hand and I want to now take from here on and so I am not going to tell any more stories so now I am going to ask the question what did we learn from all these stories of course we learned one thing a single story these women were supremely confident they were capable and they were confident and they returned each one of them in their own way fought for what they wanted somewhere luckier than the others some had to fight more than the others but the path was not easy for almost anybody because if you remember the stories support of the mentor was absolute essential supporters women mentors played an extremely important role academia was normally normally seems to have bored to supreme achievements not always but by and large I mean when the achievements were like I mean others achievements these big mathematicians people like Einstein appreciated it helped her so academia does go down to really supreme achievements of obstacles like don't study theology middle class girls should not study mathematics and physics so they are illogical obstacles and it's imaginable that even today this can be really a very major cause and as I said even if academic achievements were appreciated there certainly seems to be a bias in the recognition and awards that come with women's way in fact Maria Gopatmayer was not given a job because of the so called nepotism rule because husband and wife cannot be given job at the same institute and that's why she was working at the sort of soft money job for many many years before as I said she got her permanent job two years before she got the number and I would seriously like to ask the question whether the under the nepotism rule two brothers or father and son were disallowed from having a job at the same institute I would really like to do this I mean relations are relations but this is the kind of story that has been happening and we need to this question is you know what is in India now I come back come from the bigger picture because at least we want to see this bigger picture exist how do we compare are the stories any different I talked about this hidden figures that I don't always necessarily need to talk about Nobel Prize winners or you know great achievers so in modern day India there is some few women physicists of note and as I said I stopped in the middle of 20th century so there are many currently working very eminent women mathematicians, physicists whose names I am not going to take but that's not the point there is something about the conditions and I think the conditions that made whether they helped or hurt these women will tell us something quite substantial so I will tell you stories of these three women and story of Hans Gill who retired as a professor from Punjabi University so I want to tell you these four stories I am going to tell you these are written in this book which was already mentioned in 2005 we brought out a book of stories of women scientists who trained in modern India or are working in modern India the purpose was exactly this that I hear all these stories I think of Rosalind Franklin I think of Dorothy Hutchkins Marie Curie but they were there they are up there I cannot never reach there I think not each one of us is going to be a Dorothy Hutchkins and Marie Curie but not every man we have here who is a scientist is going to be Albert Einstein either so we have examples of women closer to us with whom we can relate that's very important and that's why we started this book we called it Lila Vati's daughters because we said that these are all daughters of the daughters of that spirit and we therefore always felt and as you can see in the stories that I have tried to tell you having women who have achieved something their help means a lot so role models at least in modern day India are extremely important and the stories I am going to tell you I have taken from this book and right now this any of you will ever be interested you can buy this book from the web page right now we are actually translating this book into different Indian languages so the first one I want to tell you is Anna Mani she submitted her period in 1945 but somehow never got her degree now the reasons I don't know but Sebe Raman certainly was not very keen on having a woman student and she kind of had to put her foot down and say no this is it I want to work very successful meteorologist and deputy director of meteorological department and also the director general of observatories in India Delhi and she actually remained single by choice she said I want to do science and I don't think I can do it if I have a family so she decided to remain single then so she worked with Sebe Raman she is the first woman PhD in physics in India in fact and her thesis advisor was Satyendra Nath Bose she worked in the centre glass and ceramic research institute all her life and she has daughters and nieces all of whom are very successful physicists today about three of them so this is a story of a person who did some interesting physics with a very you know reputable advisor and went on to do some science which was related generally had her generally was happy if you read her story her story is very happy on the other hand there is somebody like Anna Mani who had to struggle a whole lot then comes the story of Hans Gill she got a PhD in 1965 and she used to go to school dressed as a boy in mathematics and this is a fact of life in fact I was very you know my own mother who was maybe 20 years younger to her older to her she couldn't study mathematics after sixth grade because there was no male teach the female teacher to teach mathematics so these are the realities of the 20th century India so Indian story today is that in academia I think I had one more story what happened to it Indian story I am I don't know maybe it will come I should have had one more story the story is of I think I have it Janaki Amal who was the first one among the founding fellows to do science and she got her PhD in 1932 and she was the first director general of geological survey of India so in her case she also chose to remain single to be able to do science I just want to tell stories and then we will then we will come to some hard truths so what is Indian story in academia we were not women are not perceived as being incapable of intellectual achievement nobody tells you that you shouldn't study mathematics or physics many university prize winners happen to be women we don't seem to have our larry summers but in the end of the day we still don't seem to be exactly inundated with women doing science in IISC for example I don't have more than 10% professors total 10% faculty are more than 10% are not women so the serious the presence of women students in schools and colleges is high their level of achievement is high but participation of women in research in science is rather low and presence in high positions in academics is also quite low so the serious leakage in the pipeline from college to university to scientific careers I can if anybody is interested to really show that this is the summary of data which we have collected so this is the same thing that I said before that participation of women in studying science or further matter teaching science most of our very good teachers who teach science are women in fact why because that's a job that you can combine well with your family everybody is happy you can do your job you can use your as long as you do that you do whatever else you want I think that is still the Indian story but when it comes to doing science it is certainly not commensurate with the participation in studying and teaching science and the numbers are really small when you think in terms of decision making positions when you think that none of our major scientific institutions have ever had a woman director forget about the director have not even had an associate director who is a woman ISI have got its first woman director in 85 years last year so when you say decision making positions this is really true among the secretaries of departments we have had only now secretary of what is it called ICMR that's now a woman and DBT secretary was also a lady for a few years but that's it this is certainly now we realize that in India our problem is not you know development of human resource but deployment of human resource because what's happening is that when you train these women who train themselves finally that training is going down the drain and we cannot afford again the same story as a country we cannot afford this loss so India has this sort of real issue on the other hand some things are much much better because Royal Academy Royal Society which was founded in 1660 elected its first woman fellow in 1945 French Academy founded in 1666 elected its first fellow woman fellow in 1979 same is true of National Academy of Sciences US whereas Janaki Amal was the founding fellow of Indian Academy of Sciences but at the same time the first woman fellow in the physics division was elected in 1975 in INSA and the first woman fellow in Indian Academy of Sciences in physics division was elected in 1992 so we start we start very nicely but somewhere along the way we seem to have lost the path this is Janaki Amal and she then comes Vimla Bhuti again the same story she has worked with an extremely illustrious advisor she did a very important work in plasma physics and she again chose to remain single she achieved quite a lot but she actually chose to remain single so this is the sort of scenario so I just tell you stories but now is there some hard data do we know what is the status of women in science in India so Indian Academy of Sciences had constituted a women in science panel and that is this panel is still functioning and this is the first study of its kind science career for Indian women that we published in 2004 so that was kind of the beginning in India officially to realize that you have to study this subject you have to collect data and then you have to analyze it on top of it what is this about is that I already told you that the data tell us that the drop comes after PhD in many other places the drop comes slowly throughout actually we do not seem to lose too much from BSc to MSc to PhD the first postdoc position the entire drop comes so it is like a precipitous drop and therefore what we decided is to have a survey called loss of trained scientific women power what fraction are we losing and why now what we did is that we tried to locate women who had left science after doing a PhD mind you that was extremely difficult so I have a database of our 2,000 women out of which 200 are women who left science then there are some who do science some who are not doing science but using their science in jobs some who are in industry so on and so forth we also had equally group of men in our database and we gave them a questionnaire and we asked them many other questions but one question we asked them why do you think women who leave science except for those who left everybody else said they choose to leave because they cannot finish family and work simultaneously but the women who left said that they left because they could not find women from society or from institutions to be able to continue very often they got a statement we do not hire husband and wife together instead of being proactive and saying can we find you something to do that would never happen and then they have just given up so this was to me an extremely important lesson of course in addition to that survey and many other surveys that we have done there are two kinds of actions we need to take so there are two types of actions that we need to take one are societal and other are policy so societal I already told you namely for example what we have been doing with Indian Academy of Sciences that is like a role model program I want to now we want to see is there something that we can do at a policy level actually one of the common because they had either the parental or parental in law support that they were strong role models and they had help during early career especially for raising children and almost all of them had mentors and senior colleagues but almost everybody said but for chance they may not have been there so chance played an extremely important role in all of them being able to get where they got they could identify times when it would have gone so if we can do something to remove chance from this game that would be something that we would like to think about so which means that we need to create the means to facilitate negotiation of a science career this is at the social level we must create awareness you know it is not impossible to maintain a career and family balance if people get support from everybody quarters that's important see and we have to sensitize the parents and the cost students alike and we have to gender really address the gender balance from an early age and this is actually happening now that we are people are trying to proactively include young girls in inspire and this is really very very nice but these are the kinds of things that we need to do there are a lot of policy changes that are being thought about and that's a good thing too but till you change the social structure and provide a level playing field to women you will never be successful and very little of that level playing field come from any policy changes it comes from attitudes because somehow I feel that excellence always works is a little bit of a myth because CNRS performed a study that the performance index for women compared to men in the same position as almost always about 1.5 to 2 times higher and this is numbers ok we have you count the number of publications you count number of students you count the invited talks so on and so forth and there is a Swedish study published in nature which also proves this so an awareness of this that this happens without you are knowing there is this unconscious bias that seems to be there and we need to be all aware of that unconscious bias then only we can take the next step so whatever policies we will do whatever flexi timings we will give that of course has to happen but in addition to that the important thing is for us to realize that you know for example a woman doctor she also has children she does not have problems conducting her life with children and her work and life balance she also sometimes has to go in the middle of the night outside but she can manage it why does it not happen for a woman scientist because I believe and this is now where I am giving you my own judgment I believe that we do not attach enough prestige to being a scientist just like as I said everything else if you can be a scientist do that that is a hobby so for a woman I think still in many people's mind science is a hobby and till you know it is your hobby if you cannot do it do not do it that is the attitude and this may not be there obviously not in the audience who is sitting here but this is the attitude that we have to try to get rid of and that is where we have to work quite hard but in of course there have to be policy changes there should be a good crash on every campus there should be high priority to young couples for on campus housing this is all that has come out of our studies as people saying that these are the things we want so proactive hiring policies for helping couples manage dual careers that is actually one of the biggest killer and then the usual this is the elephant in the room we try to very often hide harassment under but we need to be very clear because harassment does not mean I want to be clear here harassment is not sexual harassment harassment what I am talking about is a gender harassment where for example you know if you have to go and pick up your child she has to she can women can never be free they have to take care of this but on the other hand if a man goes and these are experiences I have seen so I am quoting from my own experiences that if a man goes and picks up his child and the women yeah of course you have to do this so it is this and I think this also begins the day I would say these small things are harassment until and unless we all realize this sometimes they happen without knowing say in a conference very often people cut a joke which is a little bit off without realizing that there is a mixed audience and I feel at that time in my opinion at this point it is up to women to say thus far and this please realize that this is a mixed audience what happens we women also very often tend to take it and keep quiet because we say we do not want to make any but this is also an harassment and I think our society is simply not yet alive to these kinds of harassment whenever we speak of harassment we only think of sexual harassment so ask me and the last but not the least I really think time has come where we have to try and see whether we can and we have been talking to government but it has not yet happened we have been trying to say that there should be something called a gender audit which means that each institution I am not saying the number should be 50-50 because by the way in the survey when we ask people do you want reservation nobody wanted reservation nobody wants a job just for women nobody wants what is required what is required is that for each institution to decide given the people the pool the gender division in the pool what can I hope to aim for let's say in 10 years set that as a goal and then move on towards it right now in for example both in England and in Australia such programs are actually going on where the institutions are taking the burden upon themselves I think time really has come where institutions have to think about it for example you know simple thing we don't have on web pages if you have to ask a question what is the fraction of women faculty in your institution you have to take the book and start counting providing this number or what is the fraction of women students these numbers are very simple but they are very useful because numbers begin once you see the numbers I have been sort of personally proposing but I want to go and here say that the lack of numerical representation is a symptom and achieving numerical targets will not mean that the problems are solved problems will be solved when our way we think is changed and that is really all that I have got to say the last word is this this is what I have said in my article let me just assert the following basically consideration of how to increase the tribe needs to be always in the minds of powers that be it should not be it's not a concern of the women who want to increase the numbers it has to be in the higher up that we have to increase the numbers and it should not be just restricted to days like the international day for women and girls in science or international women day then and only then can we come up with solutions which will work for us in India and I hope that a day will come when we will just speak of scientists and engineers and not as women scientists or main scientists or women engineers and main engineers and I think the way to achieve this surprisingly goes through the path of being aware for some time and that's what we hope will happen thank you very much Madam, I am Sripadade Javarnode Vidyalay Satara basically Javarnode Vidyalay Satara I am Sripadade my name is Sripadade Madam, I am from Hyderabad, Andhra Pradesh what you just talked about gender differences throughout my life what I found is totally different from what you said I could not name him now he was a gynaecologist in Hyderabad from last 25 years he could not practice in India because still in this age also no woman offered to go to him so he had to set his practice change and he went to foreign countries what is this then still I am a teacher but as a teacher gender differences are created from our society itself still girl students are not allowed to meet us at any time so not only male gender is responsible for this female gender is equally responsible even in our household also what I found woman is the enemy of woman and not the man no okay may I I mean the point was if it came across as that men are all evil that is not the purpose of my argument that is why I said that you have to you have to train the parents I did not say you have to parents train either the mother is in law or father is in law I am saying we have to train our way of thinking and that means both proposals that I am suggesting in fact in my honest opinion have to be gender neutral proposals for example right now we have proposal women can come back to science after a break if I have my way I would say anybody with a well justified reason for a break should be allowed to come back to science finally it will be women who will come back because men normally if they take a break they take a break they want to leave it that is different but on paper in my mind that would be absolutely the best of the all worlds so having all the things in a gender that is if you notice this is what I said we want to achieve gender equity but the way to achieve that gender equity has to go by being all of us being very aware that there are gender differences and we need to work towards removing it is not just men or just women it has to be a community activity we spoke about physics and mathematicians but what about other disciplines like chemistry as I said I just chosen somewhat arbitrary I mean I just happened to like the physics and the mathematics that these women did I wanted to talk about them no I think I will try to make that point that in India frankly I think there is only a minor subject dependence but there still is in biological sciences still you will find more women but of course that does not mean that the ICMR director or the director of the all India Institute of Medical Sciences have become women only after much much later so in medicine you have 50% women actually a little bit of a joke there is an award called the Arvina award so that is the award for making the most stupendous mistake people who have actually died because of making some stupid mistake and if you know there was an article that 98.5% of the awardees are men you can draw your list no this is just a joke I had a more or less a similar kind of question or a comment if you look at the life sciences research institutes well respected research institutions in India you will find that the number of men and women are roughly comparable even though the number of women in the doctoral stages is actually more I agree with you see that is what I said is that our fraction is not commensurate it is not commensurate that is why I do not want to say whether it should be 50, 50 or 30, 20 or 30, 70 it should be commensurate to their entry in the field but do you think as opposed to the case in mathematics and physics which you have your talks title is on mathematics and physics do you think that this also is due to social biases I cannot tell you I have been trying to think very hard on this to be honest and to be honest I just go back and think with me and people around me when I was going into physics and I do not think anybody said oh my god why are you doing physics whereas in western countries I have actually seen young girls being told that you should go and do pedagogy please do not do physics so in India I do not think entering the subject now why girls do not take physics and mathematics I really do not another point IIT is a different story we can discuss that separately because the point is that somehow admissions to IIT has a artificial there is a story there I mean not deserving even boys can get in there easily you have to go and attend some school in Kota or Hyderabad they know stay closeted for you know I mean even nature writes about India China scram schools and India's schools in Hyderabad so I do not have enough data there are people here who have some data but I think it is there we are trading a little bit on the thing actually I would go in the other way that if I should be happy that if I look at teaching in colleges actually I find that women are doing a great job and they are there if you go to colleges their principles vice principles there is no problem the problem somehow seems to be when we associate at least this is my judgment that things that have a prestige associated with it there seems to be a problem there and therefore I think it is an unconscious bias this is my own thinking that the more the prestige perceived with a position you have less and less I do not know the reasons I mean this is something very hard to figure out the fact is they are low I really do not have much more to I cannot add anything here if anything we need to have more studies which are that is what I meant by saying that you need to have studies with social scientists and natural scientists really to ask this question I cannot have those studies so I cannot really comment nice talk I was just wondering you spoke about many stories one of the most striking stories of discrimination against women is perhaps the case of Joseline Bell Joseline Bell and Rosalind Franklin and now I will add Vera Rubin I would agree with that now Rosalind Franklin I had kept out because I was I was strictly talking of physics and mathematics and Joseline Bell I sort of left out because I was saying that I did not say it in so many words but I was more or less talking about women people who are no longer there quite often so but that was yes but she is part of physics clearly that is a very blatant case Vera Rubin I think even more blatant cases Vera Rubin I cannot think of a more difficult sort of bigger problem for the world of physics forget about anybody else physics, astrophysics, fundamental physics everybody alike and she did not get a Nobel Prize for discovering dark matters so I mean what can we do sorry sir I think we full of insights I think I have been thinking how to do it I just wanted to make one announcement for the Vijay Pratipa first of all I mean I am giving my vote of thanks to Professor Rohini Godwale thank you very much for accepting our invitation and giving such a spontaneous and free willing talk I really enjoyed it and I noticed a point which I want to make that the VGK Memorial Lecture which we are organizing VGK Memorial Lecture and it is given by very illustrious speakers and this is the first woman speaker we have had these are hidden biases can I tell you a small funny story Kings I stood there and I said look I was the only woman speaker so I said look there is only one woman speaker and she is also talking about women in physics as though I don't know anything about physics how can I talk about what physics is going to be done in 2020 and half the gentleman did not understand my irony and he said how could we ask a man to talk about women in science thank you anyway the announcement that I am making for Vijay Pratipa is that at 6 o'clock there will be two documentaries which will be shown these are documentaries about children and education one of them is Chanda's shoes and the other is Jadu's fish documentaries they will be shown in the NIUS G4 and I invite all the teachers to join us there at 6 o'clock after tea and at present I just want to thank the audience Mrs. Kulkarni, Anita and all the people who are involved in making this program a success thank you madam and please join us for the


How's it going guys know if we're kicking their bass TV here with another video and today I'm just here to talk about what we caught those fish on during the last two tournaments We had a tournament on Lake Martin, which is a high school bass open I'm sure you guys have seen the video to that and the weekend after that I had the high school bassmaster classic and that was actually this last Saturday And I'm gonna be making a video over that soon some of you guys on this screen I'm seeing a sneak peek to that with some other cool footage from the video that I've been working on so Stay tuned for that and if you haven't checked out the Lake Martin video go check that one out So now I want to talk about the baits I had two main baits that we caught the fish on during the tournament my partner did as well One was the drop shot which I don't have here because we didn't catch those during the bassmaster classic But that's what my partner caught them on but our main baits were a jig And this is by Chattahoochee Jigs And the second bait was a fish head spin by Swarman Hornet Lures, and this is the fire right now Everyone's talking about it because of Casey. I actually won it on the bassmaster classic This is what everyone's talking about. I'm telling you this is an extremely awesome bait I will not step foot on the lake without it It's pretty much one of my number one baits a jig and a fish head spin So my combo that I had the jig on is a legend extreme by St. Croix Rods and the reel is an MG extreme by Abagar Sia And for the fish head spin, this is just a regular standard MGX also on a legend extreme by St. Croix Rods These are really good rods if you haven't checked them out. Definitely go check them out at St. Croix Rods I'll leave a link in the description over everything I talked about in this video And so if you guys have any questions over the fish head spin or the jig by Chattahoochee Jigs Send me a message leave a comment below And that's about it guys, so please like comment share subscribe. See you guys next time


Alright, so you see I am continuing with this motivation for Zalkman's lemma, alright and so let me recall what we were doing, so I will let me use a different colour. So basically you are trying to understand you know the behaviour of a normal family at a point, okay that is the understanding that is you know the key to understanding the statement of Zalkman's lemma, okay. How does a normal family of meromorphic functions, how does it behave at a point at a given point? So you know the important outcome of this analysis is that you know you can characterize normality at a point and normality at a point is defined as normality in a neighbourhood of that point, in some neighbourhood open disc surrounding that point, okay and then so you see this means that normality can be defined locally and it is in fact a local property like analyticity, okay. So we start with the family script f, these are meromorphic functions defined in a neighbourhood of the point Z0, I am assuming the point Z0 is in the complex plane it could have been also a point in the extended complex plane namely it could have been the point at infinity, okay all arguments will work but you have to make the right modifications you know how to do that because we know how to deal with the point at infinity and taking infinite limits and things like that, okay. So all right, so well what I told you last time is that you start with this family script f and what you can do is given a function small f in script f, okay you want to study that function in a neighbourhood of the point Z0, okay. So what you do is that you know you take this neighbourhood of this point which is actually it is a disc centred at Z0 radius rho, okay you take this neighbourhood and I am also assuming that the boundary of that disc is in the domain where the functions are defined, okay. So and the reason is why the reason why I am including the boundary is because I want a compact set, okay the disc along with the boundary forms a compact set it is a closed and bounded set, okay and well so suppose I want to study a particular function small f in the family script f at the its behaviour very close to the point Z0 then what I do is I zoom into the point Z0 and try to study the behaviour of the function and how do I do this zooming I do the zooming in this way so I define this zoomed function here, okay g. So given f I zoom f to get a new function g, okay and what is this zooming you are zooming the function f centred at Z0 and the zoom factor is 1 by epsilon where epsilon is supposed to be a positive small positive quantity so that 1 by epsilon is large in fact epsilon is to be taken less than rho, okay and then you define this new function which is for this f you define this function g which is a zoomed function g is just you take f look at its behaviour in a small disc around Z0 and then you zoom the behaviour by a factor 1 by epsilon, okay and you so what you do is that so this is how the zooming is defined the zoom function g of Zeta is f of Z0 plus epsilon Zeta which means that actually what you have done is you are actually translating Z0 to the origin, okay and then what happens is that you know this whole disc centred at Z0 radius rho along with the boundary translates to disc centred at the origin with radius rho by epsilon, okay so 1 by epsilon is the scaling factor and mind you you should think of it like this epsilon is small so 1 by epsilon is large so rho by epsilon is greater than rho so you have actually zoomed, okay and so the behaviour of the zoomed function is being studied here, okay and in this disc and well and I am calling this zoomed function g so this is a very very simple thing and the point is that the difference between f and g is just a you know it is a bilinear transformation it is actually it is translation and it involves a translation it involves a scaling, okay so what you have done is you have taken the variable Zeta you have scaled it by epsilon, okay you have multiplied it by epsilon mind you epsilon is a positive real quantity and then you have translated by Z0, okay so the g and f are of the same type of function, okay g is analytic if g is analytic if and only if f is analytic g is meromorphic if and only if f is meromorphic and whatever if g has a pole at a certain point then f will also have a pole of the same order at the corresponding point, okay so and conversely so g and f are literally the same function except that you have made a change of variable alright but the point is that g is a close up look of f it is you are looking at f very close in a neighbourhood of Z0 that is the whole point, okay now what you do is that this is what you do if you have a single function but what you could have done is you could have done this to a sequence of function so the point is if I knew that this family script f is normal then suppose you assume that the family is normal in this close disc which means actually it is normal in an open set which contains this close disc including the boundary, okay and you know suppose this family is normal you know normality is the correct notion of compactness that we require normality actually means normally sequentially compact it means that given any sequence you can find a subsequence which converges normally, okay so converges normally means converges uniformly on compact subsets so assuming that the family script f is normal and I want to study the behaviour of the family at a given point Z0, okay so if I take a sequence fn in this family then you because of normality you can find a subsequence fnk that will converge normally on this disc in fact it will converge uniformly on that disc because that is a close disc, okay normal convergence means it is uniform on compact subsets and this disc centred at Z0 radius rho is you know it is a compact subset and again at the back of your mind you should remember that you know this argument will work even if Z0 is a point at infinity the only thing is that if Z0 is a point at infinity you will have to think of this you should write this disc you know if you want with the Euclidean metric if you want with the spherical metric or you must invert the variable and look at neighbourhood of 0, okay so you can deal with the case when Z0 is the point at infinity also, okay but in any case given this sequence fn I have this subsequence which converges normally so let me call the limit function as f and then we have already seen this whenever you take a normal limit of meromorphic functions the limit is either identically infinity or it is meromorphic, okay and now look at what happens to the zoomed functions so if fnk converges to f normally then the zoomed functions for the sequence will converge to the zoomed function of the limit, okay so this is obvious, okay because the difference is only the difference is just a change of variable given by a bilinear transformation consisting of a scaling and a translation, okay so this gnk will converge normally to g, okay and all this is happening mind you because your zoom factor is 1 by epsilon it is all happening here it is happening in this open disc centred at the origin radius rho by epsilon which is greater than rho, okay and well this is where it is happening now you see the point is that whenever a family of meromorphic functions converges then the family of spherical derivatives will also converge, okay now this is just saying that you know the taking the spherical derivative will preserve the normality the convergence and the normal convergence, okay so fnk converges to f normally so what will happen is that you will get this which is that the spherical derivative of fnk converges normally to the spherical derivative of f, okay and we already know that because fnk converges normally to f g and k converges normally to g, alright and therefore the spherical derivatives of the g and k hash they will converge normally to g hash, okay and but the point is that you see the original family that the family script f you have assumed is normal so this is this family script f is normal and Marty's theorem tells you that you know that a family is normal if and only if the spherical derivatives are normally uniformly bounded, okay and since we are already looking at a compact set, okay there is a uniform bound for all the derivatives the spherical derivatives of the functions in the family mind you have to use spherical derivatives because functions in the family are not analytic functions they are meromorphic functions so you cannot talk about an usual derivative at a pole, okay but then you can talk about spherical derivative at a pole. So by Marty's theorem the Marty's theorem mind you tells you that normal sequential compactness is equivalent to normal uniform boundedness of the spherical derivatives, okay and it is a generalization of the Montel's theorem for analytic functions which says that you know the normal sequential compactness is equivalent to uniform boundedness normal uniform boundedness of the original family, okay and but the point is when you go to Marty's theorem you go to from the uniform boundedness of the normal uniform boundedness of the given family you switch to the normal uniform boundedness of the family of spherical derivatives, okay that is a quantum jump that you make from Montel's theorem for analytic functions to when you go to Marty's theorem which is for meromorphic functions. So there is this uniform bound M which works for all functions in the family script F so it will work also for these f n k's it is a bound for not the functions but it is bound for the derivatives, okay spherical derivatives. So all the spherical derivatives of the f n k's are bounded by M and then well what happens is therefore now if you take the limit so there is a chain rule that is working here, there is a chain rule the spherical derivative of the zoomed functions g and k hash is epsilon times the spherical derivative of the f n k, the spherical derivative of the g n k's is epsilon times the spherical derivative of the f n k's because this so you know the idea is that you are zooming by a factor 1 by epsilon, okay you are going closer to the point by you are zooming into the point by a factor 1 by epsilon, okay but then the spherical derivative becomes smaller. This spherical derivative gets multiplied by the factor epsilon mind you epsilon is a small quantity so multiplying by epsilon makes the quantity smaller, okay whereas 1 by epsilon is a zooming factor so as you zoom closer your spherical derivative is going to become smaller, okay that is what is happening, okay. So for the zoomed function if the original functions have this bound M then the zoomed functions have the bound epsilon M and then because the zoomed functions g the spherical derivatives of the zoomed functions converge normally to the spherical derivative of the limit function which is the zoomed limit function what it will tell you is that the limit function also if you take its spherical derivative is bounded by this epsilon times M, okay. Now the point is that you know all this happens in this domain which is the disk center the origin radius rho by epsilon, right and mind you must remember that as I make a smaller and smaller and smaller I am zooming closer and closer to the point z0 but then the zoomed functions are in larger and larger disk center at the origin because rho by epsilon becomes larger and larger as epsilon becomes smaller, okay. So now the point is that you repeat this game, okay by changing the zoom factor, okay. So what you do is instead of taking single epsilon you take a sequence of epsilon which go to 0, okay a sequence of which go to 0 of course from the right namely that you take sequence of positive epsilon that go to 0, okay. So if you do if you repeat the game with that what you do is that now you know the nth function g n, okay well I think I have not written it correctly last time the nth function g n is the zooming of the nth function f n by the factor 1 by epsilon n center at z0 with the variable zeta so that should have been an f n here, alright. So g n is the zooming of f n, alright but the only thing is that now the zooming factor is also dependent on n, okay. f n is zoomed by a factor 1 by epsilon n and I am calling it as g n, okay just what we did some time ago was keeping all these epsilon n's the same, the same epsilon but now I am making the epsilon n's smaller as n becomes larger, okay well so if I do this what happens is mind you if I make the epsilon n smaller then the domains on which the zoom functions work these domains become bigger, okay so you know so I get these domains, okay mod zeta lesser than well I think there was a row there was a row there was a row there, mod zeta less than row by epsilon n and these are increasingly larger disks as epsilon n goes to 0 these disks cover the whole plane, okay and in fact as n that means as n tends to infinity you are getting a you will cover the whole plane in particular you will cover any compact subset of the plane. So what you can say is that these zoom functions g n you know you can talk about normal convergence on the whole plane, okay. So actually what happens is that the anyway as before the f nk's will converge to f, okay and therefore the f nk hash will converge to f hash where hash denotes the spherical derivative and then the corresponding zoom functions g nk hash will converge to g hash all these convergences are all normal, okay but the only thing is that this when you come to the zoom functions this normal convergence is on all of c it is on all on the whole complex plane because now you have covered the whole complex plane you have covered every compact subset of the complex plane by a sufficiently large disk centered at the origin where that means all the g n's all the g nk's beyond a certain k they are all defined on any compact subset beyond a certain stage, okay. So fine so you have this and the point is that you know the g nk's if you take the spherical derivatives they are bounded by epsilon nk times the bounds for the original functions which is m, okay and therefore this limiting argument will tell you that the limit function g hash the limit function g if you take its spherical derivative it has to be 0 and therefore you get a constant function on all of c, okay. So you see the point is therefore you know as you go closer and closer and closer to a point and look at the zoom functions, okay then the zoom functions they tend to become constant that is what it means, okay the zoom functions become constant. So you know to sum up all this all I am saying is that you take a normal family you take a point where the family is normal and then what happens is that as you take any sequence of in that family you can always find a convergence of sequence if you study it at that point, okay then the zoom functions will you know they will tend to a constant function that is the whole point, okay and Zalkman's lemma is all about you know being able to find zoom functions which converge to a non-constant meromorphic function, okay. So something opposite to this happens, okay so what I want to do first is that I need to you know first of all tell you that I can repeat this argument with not just one point Z0 but I can even take the sequence of points tending to Z0, okay. So now what we do is now take a sequence Zn going to Z0 in say the same disk mod Z minus Z0 is not equal to rho if you want, okay well for that matter if you take any Zn sequence Zn going to Z0 the sequence beyond a certain stage will lie in that disk, okay but I am assuming this whole sequence lies in that closed disk center at Z0 radius rho, okay and what you do is now you zoom at each Zn, okay see all this time we were zooming only at Z0 but now what you do is you zoom at each Zn so you know they so you define Gn of Zeta to be the zooming of you zoom the function Fn but now centered at Zn and you know the zooming factor is 1 by Epsilon n and the new variable is Zeta. So this means that Gn of Zeta is well it is Fn of Zn plus well Epsilon n times Zeta so this is the thing but then I have to worry about one has to worry about where this will what about the domains of the Gn's, okay you will have to worry about that but the point is that essentially as n tends to infinity you are coming closer and closer to Z0 so the domains of the Gn's is going to again cover the whole complex plane, okay. So one can write out the details for that so you know the picture is like this the picture is that you know so I have this Z0 and you know there is so there is Zn and then I have this so inside Z0 there is this big disk with radius rho where I am concentrating my attention and of course even the boundary is included though I am putting a dotted line well and what I am doing is that Zn I am taking you know I am taking a smaller disk radius Epsilon n, okay and well and then you know so Zn is Zn's are tending to Z0 so well Zn plus 1 is closer to Z0 if you want and there is it and then I am taking a much smaller disk and the radius of that disk is Epsilon n plus 1, okay. So you know I am coming I am taking smaller and smaller disks about points which are going closer and closer and closer to Z0, okay and the point is that on this I have this zoomed function, okay and then if I go to Zn plus 1 I have another zoom function it is Gn plus 1 of Zeta and this is Fn plus 1 centered at Zn plus 1, yeah so it is Epsilon n plus 1 Zeta so this is just assuming of Fn plus 1 centered at Zn plus 1 with the factor 1 by Epsilon n plus 1 and the variable Zeta, okay. So I have these zoom functions, okay and well of course you know I will have to assume that the Epsilon n's are chosen in such a way that the disk centered at Zn radius Epsilon n is within my big disk centered at Z0 with radius rho, okay but that will happen eventually even if you did not assume it beyond a certain stage it has to happen, okay but we assume therefore without loss of generality that this happens that the diagram is like we have already shown it so you know for example well you know so if you take so I am assuming that if you take Z1 then I am taking the Epsilon 1 that I am taking is such that the disk centered at Z1 radius Epsilon 1 lies inside this big disk centered at Z0 radius rho, okay and the point is that if this does not work I can replace the Epsilon i's by some Epsilon i's which I can calculate, okay. So the condition I want is that you know the distance of Z0 to any point of the disk should not be greater than rho, okay it should be less than rho that is the condition I want. So well let us assume that the picture is like this, all right you do we do the same calculations as before you see fn the you know you have this fn case they converge normally to f they converge normally to f on normally this is normally and well and what happens is that you will therefore get that the spherical derivatives will converge to f hash normally and you will also get that the zoom functions will also converge normally to g and so will their spherical derivatives, okay. So you will get all this as before mind you the only thing now is that you also change the center of the zooming, okay the centers of the zoomings are all different now, okay but it does not that does not change the calculation for the spherical derivative, okay because spherical derivative for the zoom functions with respect to Zeta and as far as Zeta as concerned the Zn's are all constant, okay. So well again what you will get is that you will get that again you will get that you know the original functions f and k their spherical derivatives are bounded by m and this will tell you that the zoom functions their spherical derivatives are going to be bounded by epsilon the corresponding scaling factor times m I mean not scaling factor the corresponding epsilon times m and if you take a limit as n tends to infinity you will get that the limit zoomed function which is the zoomed function of the limit function that will have spherical derivative 0, okay that is because the epsilon nk tends to 0 and m is a fixed constant positive constant and the moment you and as before the moment you know that the spherical derivative of meromorphic function is 0 it has to be constant. So this will tell you that g is constant on the complex plane mind you this constant could have very well been the constant value infinity that is allowed, okay because we are in the context of meromorphic functions but the whole point is now you have got this important characterization you take a point Z0 where a family is normal, okay then what happens is that you take any sequence of point Zn going to Z0 and you take a sequence of radii which are going to 0 the epsilon n's, okay then given any sequence in the family you can find a subsequence such that if you take the subsequence and zoom it with respect to this sequence the zoom functions converge uniformly on compact subsets of C that is normally unsee to a constant function this is the behaviour of a normal family at a point, okay a normal family in a neighbourhood of a point, okay and the amazing thing is that the whole point is that this characterises normal families, okay this is by this you can actually characterise normal families, okay so let me write this down and the fact that you can characterise normal families like this is actually the philosophy behind Zalkman's lemma and the way it works is it tells you when this is contradicted. So Zalkman's lemma will tell you that such a thing will not happen if the family is not normal, okay so let me write this so proposition is well let script F be normal at Z0 which means that it is a normal family in an open disk containing Z0 and of course all my families of functions are Meromorphic, okay so let me write here normal family of Meromorphic functions at Z0, okay given any sequence Zn tending to Z0 and any sequence of radii epsilon n tending to 0 plus, okay for any sequence fn in the family script F, okay we can find a subsequence fnk such that the zoom sequence of fnk with respect to Zn and epsilon n that zoom sequence converges to a constant function normally on all of the complex plane, okay. Gnk of Zeta is equal to the zooming of fnk with respect to centre Zn Zk radius 1 by I mean scaling factor 1 by epsilon k new variable Zeta which is by definition you just take fnk of Zk plus epsilon k times Zeta this is the zooming, okay now such that if this is this zoomed family then Gnk converges to a constant normally on the whole complex plane. So this is the characterization of this is how normal families behave, okay and the big deal about so this is normality at a point, okay this is normality at a point Z0 which means which by definition is normality in a small open disc containing Z0 that is what it means, okay and if you want to cover this to normality on a whole domain then you know you have to say a family is normal on a domain if it is normal at each point of the domain, okay you can define it point wise. So now let me state Zalkman's lemma which will tell you and you will appreciate what it is all about but you know with all this background that I have given you will see that Zalkman's lemma is something natural to expect, okay. So here is Zalkman's lemma and it is actually well it is called Zalkman's lemma but it is actually theorem. So what is a lemma? The lemma is about normality it is about non-normality, okay see I will tell you in very short words what we just saw is that if you have a normal family in the zoom functions go to a constant, okay Zalkman's lemma tells you that if you have a family which is not normal you can get hold of a zoomed family of functions which will go to a non-constant meromorphic function that is all that is the whole point, okay. So let me write that of course you know you should always keep saying this in as little words as you can so that you grab the main idea but then of course when you write statements you will have to be you know you have to bring in lot of notation you have to worry about lot of notation and then of course proofs are even more slightly complicated but the point is that you should always be able to zoom out you know and say things in just a few words because that is how you will map it in your memory and remember it and so that you do not lose track of the idea, okay. So here is a lemma suppose script F is a family of meromorphic functions on a domain D that is not that is not normal. So here is a non-normal family then what happens is that there exists a sequence Zn converging to Z0 in D and epsilon n going to 0 plus with and further a sequence fn in the family script F such that the zoomed functions gn of zeta is equal to the zooming of the fn's centered at the Zn's scaling factor 1 by epsilon n new variable zeta namely fn of zeta plus epsilon n zeta converges normally on C to a non-constant to a non-constant that is the whole point to a non-constant meromorphic function g on the whole complex plane such that okay well the fact that you know it is a non-constant meromorphic function so you do not expect the derivative to be spherical derivative to be 0. So the spherical derivative at the origin will be 1 and all the spherical derivatives is bounded by 1 okay so this is Zolkman's limit. So it tells you see it tells you that the behaviour is exactly opposite to what you saw for a normal family okay so we will try to prove this in the next lecture.


Everybody forced here ripping a box of 2023 pristine at baseball hobby for Andrew H. Appreciate you Andrew Let's do it remember guys Kyle will be live on the fanatics app tomorrow at a 10 a.m. Eastern about nine hours from now Sam is live for group breaks at five Eastern tomorrow right here fractures of Austin Riley and Josh Young Throw strawberry purple 299 mookie bets. Oh snap Julio Rodriguez Alec Thomas who are redemption popular demand auto relic redemption of Miguel Cabrera nice Go one of the legends right there Miggy relic auto redemption both babies fresh faces of Francisco Alvarez David Bednar Lungo to 75 a Swings of summer short print Francisco Alvarez There we go. Very nice Fractur Goldie or Fractur Kiermeyer Posey Kyle Tucker Gold Jose Canseco Number 33 of 50 gunner Henderson base and they've on aggressive fresh faces In case cards now got Alec Manoa fresh faces auto rookie refractor Nolan Gorman Gold refractor Sandy Alcantara exact Thompson rookie auto Fractur Manny Machado And blue refractor David Bednar out of 75 Well, Miguel Cabrera be nice over a little something nicer in the in case there, but that'll do it Appreciate you Andrew. We'll get these out to you