Please come. Good morning. Please come sit down. Ms. Madhumita. Yes, sir. What is PIEX? You've done BBA from PIEX. What is the screen? Sorry, maybe it's smelling. Oh, sorry. I had something bad. I think it's spiked. Banipat Institute of Engineering and Technology. And what are your subjects in BBA? Sir, all the subjects that are there, there are lots. Not sis, something specific. Sir, semester wise, like 6-6 subjects each. Financial management, accountancy, business environment. But there are two subjects on which you had to do more work than others. No, sir. So it was some kind of a general course. So all the subjects were there. I mean, MBA, B-studies, specialized subjects. But here, just an introduction. Alright. Now, if you finished in 2015. Yes, sir. Now this is 2020. Yes, sir. So these four years, how do you explain the gap? Yes, I was intelligible for two years to appear for civil services. I was what? I was intelligible for two years. So my first attempt was in 2017. Okay. This is your second or third? This is my third attempt. What happened in the first two? Sir, I cleared the prelims in the first attempt. I could not qualify for the interim. And in the second prelims, I could not qualify. Okay. Yes. In the northeast, very recently, there have been two developments. One was the Bodo Accord. Yes. And the other was the Guru Agreement, a recent plan. Can you tell me something about the Bodo Accord first? Sir, I have actually read Bodo Accord, but I can't recall the details as of now. Okay. But about the brews, I don't remember. Yeah, the brews. They were putting up in Mizoram, I guess. You tell me, whatever you remember. The two states in Olyam, Mizoram and Tripura, they were evacuated from Mizoram and they were temporarily living in some shelters in Mizoram. But now there were talks of repatriation. But these people did not go back because they fear that they will be persecuted again. So they were given some sort of compensation to stay put where they are politically. Okay. Have you heard of this Act East policy of the government? Yes. What was the Act East policy? So previously, the policy was look East. We were not acting on what we were wanting to achieve. But right now, Act East means that we want greater integration with the Asian nations in terms of trade and also in terms of soft power like Buddhism. And in that matter, we were actively negotiating for the regional comprehensive economy partnership act, but that did not come through. And after that, we are also trying to have the BCIM, but work on that has also been stopped because of the China sector. But they were, I think Asian nations were jointly, they were the guests of another public day like last year. So we are actively pursuing other issues. There is connectivity issue, improving connectivity. There is something called the Kaladan multimodal vicinity. What it connects from where to where? Why is it called multimodal? So multimodal is because it involves various modes of transport. Such as? Like railways. Railways? Railways also overland. Where? The terminal point in Argan region, Kolkata. It starts from Calcutta. Yes, sir. And then where does it end? Myanmar is, Rakhine state is the one where it actually starts and it terminates in Kolkata. The Kaladan multimodal is from Calcutta to Mizora. Through Myanmar. Yes, sir. Actually, I was saying it is with Myanmar. It is not from Calcutta to Mizora? No, sir. I was saying that the two countries in my life, I did not talk about. So tell me why is it called multimodal? From where to where? Can you tell me? If you don't know, then I am not asking. No, no, sir. I am multimodal obviously because there are various modes of transportation. Where is the railways in that? That's what I am asking you. Okay. Is there a railways? No, sir. Right. I am sorry, sir. I guess. I suggest you look into this. Okay. Thank you. Have you heard of Kulbushan Jadav? Yes, sir. Who is Kulbushan Jadav? So Kulbushan Jadav is present in Pakistan in facing charges of terrorism. And he was sentenced to death. But then India petitioned in the International Court of Justice. Okay. And they put a stay on that. So what is the situation now? Situation now is that we had asked them to grant us consular assess which Pakistan did. And after that we actually, his mother and his wife were allowed. Consular assessor. Indian diplomat was not allowed to have conversation with him or which we protested also. But after that I think they have been no major developments on that. Now, you know the government had announced a policy of aspirational districts. Yes. Have you heard about that? Yes. Can you tell me something about that? Sir, these are the districts where there is low socioeconomic development. Like in terms of education, scale or health issues, etc. So we wanted to have a list of all these districts and pursue them. How many districts are there? I guess 121. 115. Yeah. And who is monitoring this? Neethi Aayu. Neethi Aayu. How do you do that? So Neethi Aayu basically is an advisory body. That is true. But have they appointed anybody? Sir, there are prabharism officers. So who are the prabharis? Government officials. So I think I'm not actually aware of that. Thank you. You've studied history as a subject. Yes. So if I asked you to mention any three turning points from one Indian history, what could you say? Ma'am, the first would be the revolt of 1857. Okay. Because it was the first time that we ever tried to actually expel British from India. Second would be the formation of international congress because that is when our protest assumed a sort of institutional form. And the third would be the arrival of Mahatma Gandhi because his coming into Indian politics was a sort of transformation from the politics of elitist class to that of masses. So do you think Mahatma Gandhi's philosophy is still relevant? Gandhi's philosophy of truth and non-violent is relevant because non-violent protests are I think the most acceptable form of protest. So do you see any non-violent protests happening currently in our country? Yes, ma'am. There's one in Shreyaengbagh but they're not violent and it's been quite a constable period of time they've been staying there. There's also some attempt to rewrite history. Do you think it's current? The statement? Ma'am, it's essentially a question of ideology. What do you mean essentially a question? Is it correct or it is not correct? Yes ma'am. I mean people try to interpret history according to their own ideology. I mean history is something that is very, you know, just according to their perception one can have any sort of, okay your fact is there but one may interpret it in a different manner the other may interpret it in a different manner. Okay. Now you're from Haryana. Yes ma'am. What are these khap panchayats? Khap panchayats are sort of community organization which adjudicate disputes between people. So are they playing a negative role? No, it's both negative also and positive also. In the recent past they have been associated with honor killings but they have done some positive work also. I remember there was one in Rohatak which had put a ban on using caste surnames. Okay. Now we talk about the sex ratio being very poor despite lots of acts. Can you tell me what the acts are which address this problem of poor sex ratio? Acts. One is PCP and DD and the second is... Okay PCP and DD. Yes ma'am. So we have this act yet the sex ratio in the country is poor in Haryana and Rajasthan also very poor. Yes ma'am. What do you think we are not implementing the law? Actually implementation of the law is there but whenever we try to ban something it goes underground. Illegal business flourishes. If there is legislation or enforcement very strict in one particular state then there are other states where this is not as strict as it should be. So do you think that's the only reason that we have a low sex ratio in the country? That's one of the reasons. The multiple reasons. Yes, mention some. Ma'am one is the attitude of society. Yes. And then there is always a fear of sexual harassment of very unsafe public places. And ma'am then the financial burden of having a girl child is also there. The very moment a girl is born we start thinking about the dowry that has to be given. And it is also that people do not feel like investing in a girl's education is a sort of investment because I think it is also going to go to somebody else. Now coming to your subject, you also studied theories of motivation and agro-government theories. Yes ma'am. On governance. So if I asked you to give me one theory on motivation which would help you to motivate your employees. Which theory would you mention and why? Ma'am I had studied master laws. Okay. There will be something for how you would motivate your co-workers. Motivation would depend. First of all I have to understand what motivates them. Okay. It be financial incentives or maybe they get satisfied with rewards or just recognition. That sort of thing has to be identified and after that I will devise a way to motivate them. All right. Thank you. Yes. You are from Paripat. Yes. Tell me about some major battles which have taken place in Paripat. Three battles have taken place. Actually Empire is shaking three battles have taken place. The first one was between Ibrahim Ludi and Babar. And it laid the foundation of Mughal Empire in India. Thereafter the second battle was between Hemu and Akbar. And it kind of re-established the Mughal Empire because it was on the verge of collapse. Hemu was about to win it and had he actually won then we would not have that strong Mughal Empire that we had later on. And the third one was between the Marathas and Ahmad Shah of Delhi. But you of course you could have written the words of Ibrahim. Oh yes. They took place on the same ground. Now what was the common thing in the outcomes of all these battles? So the incumbent power lost. Well, you can say the invaders won. And second thing is that the superior army is lost. The numerically superior army is lost. But now in today's I have seen that Pani was actually decided because of the history of India. Because the invaders were successful. Now today which country poses the biggest security threat to India in your opinion? I think the two countries are immediately involved. China and Pakistan both. Why? Some reasons. Sir, one is because of the history that we've had with these countries. With Pakistan we had three wars, three major wars. The security threat means they should have the capacity to... Sir, they do have the motive. But on capacity front I think they do not have but they keep threatening us with the nuclear war. China also doesn't have the capacity? China does have the capacity but it doesn't have the motivation. Okay, I take your point. Now have India and Pakistan ever participated together in any joint exercise? Joint political exercise? Sir, since we and Pakistan are members of the SCO, there was one exercise that was carried out. And I think we were able to participate. SCO was the execution exercise for the counter-revolution. Now Haryana has been in the news a lot. I'm telling you this last year. On the issue of parody, what are the causes and what has the government helped to address it? Sir, the cause is that actually there are two various varieties of rice that were grown in Haryana. And the one that is grown in Kethal, Kurukshetra, Yamnanagar, etc. I don't know the name actually but that variety is something that is not edible for the animals. And so it cannot also be used as a, what is it, mulch in the field because it is very hard and it does not degradable actually. So they tend to burn it rather than dispose it off in a manner that it actually eliminates them. And government did also, I think one was penalty that whoever burns it would have to pay some fine. NGD also imposed some penalty, I remember. And then there was also an announcement that whoever doesn't burn it would also get an incentive. And sir, there is this happy cedar which has been subsidized. So if a farmer is willing to use it to actually treat the stubble on the field, then we can do that. You have interested Sanjeeva? Exactly. That's a crude human form of a deity. We prepare the various body parts with the help of clay and then colour it and then affix on the world with the help of cow or dog. In which India are they popular? Sir, it's popular in the northern states like in our state it is called Sanjee, in Madhya Pradesh it's called Jhanjee. So there are various names but in northern states it's popular. Okay. You heard of a scheme called Vivaatsya Vishwas? Yes. Give me some more facts about it. Sir, I know this much that it's a direct text dispute resolution scheme. I'm sorry? Direct text dispute resolution scheme. Yes. So the officials are now taking those people into confidence who have text disputes. The officials are taking? Yes sir. There is a bill which has been introduced in power. Yes, in the budget. How can officials take into confidence? No, no sir. They are informing the people who have text disputes that you pay the principal amount. How can they inform unless the bill has been passed? Sir, I read in today's newspaper that the scheme is... I can't read that. So the bill has been introduced. Give the highlights of the bill. What does it... Sir, I know only... There is a benefit of this. I mean, why is the government doing it? Give me a reason. Yes sir. Because litigations attend to be pursued in court. How many cases are pending? Text dispute cases. I'm not... The courts. What is the volume? What is the worth of them? How much money is involved? I'm not a person. It is to resolve those. Right? Almost five lakh cases are pending. The money involved is more than nine lakhs. Nine lakh crores. Right? Now to resolve this and to get some money out of those disputes. These are unending disputes. They go right up to the Supreme Court. So government has brought out this scheme. It's kind of an amnesty scheme. Now tell me what are the amnesty part of it? Amnesty part is that they pay the principal amount. Interest won't be asked to be paid. Up to which date no interest or penalty will be charged? Sir, I'm not aware of this. You are not right. I've just read that this isn't... You must read. These are the current issues. You must know. 31st of March this year is the final date. Penalty will be waived as well as interest. The scheme will remain open till I think June end. But in that period, after 31st of March till June, people will have to pay, affected party will have to pay something extra. But not the whole penalty system. Why is this scheme being introduced? Sir, just the litigations because they tend to... Government must get this money. Government must get this money plus the amount. Amount is huge and economy is already struggling. If we get that amount, then it will be helpful. You've done mathematics. You've got some award in Olympiad or something. Yes, that was in class too. Now, maths, is there a Nobel Prize in maths? I haven't heard of a Nobel Prize in maths. Any other? Is there just four Nobel Prizes? Five. So maths is not... There is no Nobel Prize for maths as such. Is there any prize which gives a person international recognition? So yes, there are but I don't think there is much. What is the significance of PIL in the constitution? Public interest litigation. And what is the article on which this PIL is based? Article in the constitution. Sir, PIL, public interest litigation is basically a mechanism by which any public spirited citizen can file a petition on behalf of the people who cannot do it themselves. And article, I think, is huge. Yes, article. I'm not sure, sir. Can you name one or two important cases, very landmark cases, which have been decided by the Supreme Court in the interest of public welfare through the mechanism of PIL? I don't know. But constitution, you have not read any of the paper? Sir, I have read but specifically the cases which were decided by PIL. What is the significance of preamble of the constitution? What is its significance? Sir, it explains the philosophy of the constitution and the vision of the founding fathers of the constitution. So, name some keywords in it? Can I read the people of India having solemnly resorted to constitution into a sovereign socialist secular republic and to ensure liberty of thought expression. Liberty of thought expression? Believe, faith, worship. Is that what it says? Yes, it is. Now, there is something called social equality. Yes, sir. Does the constitution guarantee social equality? Yes, sir. Name some specific articles of the constitution which promote social equality? Social equality. Sir, there is article 15 which prohibits discrimination on any court. Good, good, 15. Sir, there is actually... Only name the articles. If I want to ask you more questions, I'll ask. 15 is one excellent, what else? Sir, article 39. Oh, you've gone to 39? 39 is Directive Principles. But yes, it's an article. Directive Principles of... Something which is enforcement. I think it's an article... So, right to equality, I think all articles 14, 15, 16, 17, they all... for social equality as well. All right. 15 to 17, you said, any other article? Sir, these are the ones I... You don't know? There are two amendments in the constitution which devolve political power, political and administrative power to the local bodies. Yes, sir. Which are those? So, 73rd and 74th. 73rd and 74th, very correct. Do they also provide some benefits to shulkas and women? Yes, sir, because one-third of the seats have to be reserved for women. And for SES and STs, seats in proportion to the population. All right. You are a aspirant for foreign service also, second choice. Yes, sir. Right? What is the size of Indian diaspora around the world? Approximate idea, no idea. What is the importance of diaspora for us? The nature of our diaspora in other countries helps us to achieve the objectives of a foreign policy. For example, in USA, the Indian diaspora is very influential. And I've read somewhere that it did play a significant role in negotiating civil... Are they influencing our foreign policy in... I'm saying that influential... In what way? Tell me one area where foreign policy has been controlled by these diasporas in America's foreign policy. So Mr. Trump had one of his... I think he was... There was this person I don't remember the name, but he was with Mr. Trump and in the initial stages when Mr. Trump assumed the presidency, he played a favourable role in... Who was he and in what field did he play a role? In the initial talks between the two countries after Mr. Trump assumed presidency. Check your facts. But there are other benefits which diaspora bring to India. So remittances? Remittances. How much remittances we got in 2018-19? Sir, I remember this much that Indian remittances... The remittances that we get are the highest according to the World Bank, but I'm not aware of the exact figure. You're not aware. $76 billion. How much have we got this year from April to December because after December we have no figures? Sir, I'm not aware of the figures. You don't read newspapers? I do, but... This report came yesterday only. $41 billion. If you don't read newspapers then you're missing a lot of information. I do read newspaper, but this wasn't perhaps in the newspaper that I read. Alright, final question. There's a saying called history repeats itself. Can you give any strong example, not obscure example, strong example, which supports this argument? History repeats itself. Number one, do you agree with this? Maxim or not? Your history student? Do you agree with this Maxim? History repeats itself. Sir, my take is that we should repeat history only if it is readable. You're not answering that. If you are a student of history from history, you would say it. Are there any episodes or incidents which support this argument or don't support this argument? That's what your answer should be, not what you're saying. Listen to a question carefully. Alright, we close the interview, I'll give you a little feedback. Is this your first attempt? So this is my first interview or overall third attempt? Third attempt. How do you think you performed? So I think I need to brush up on my parts. We have asked you questions largely based on your budget. And we feel that is how your interview will be. Some of the areas are like you're from Haryana. The problems of Haryana, the progress it has made in some sector like automobiles, and also IT, etc. GDP growth is very high. So all those things you should work on. And second is history. Next is your subjects like management, etc. We have asked you questions. Fourth is that IFS is also your choice. These are the fourth broad area. There will be Olympics, Maths and all. There are two medals. Maths, Fields Medal is the most field that is able. These are, Fields Medal is considered the Nobel Prize. So check upon this. And also who have received these in the last 3-4 years. Foreign service is your important area. We have asked you questions. In this we have also asked you about Kalatana, Kulboush, etc. These are all very important questions. Then what is happening around India? Pakistan, CPEC, China's moves around our periphery. What is happening in Maldives or Napa, etc. Since you are aspiring for foreign service, these are the things where questions can arise. We have 3-4 initiatives in foreign policy. Act East, Neighborhood First, right? Yes. So who is the most important foreign dignitary who is listening here? He was president was perhaps... You are only speculating. If you are ready to tell? No sir, no. Yes, you are speculating. 20 seconds to 24 seconds. Then you also have to know why he is coming. What are the important issues on the table which we will discuss. Don't forget to like, share and subscribe to our channel. And press the bell icon to never miss an update.


President Alexander Lukashenko has said that Belarus was ready to prevent any internal insurrection that the neighbouring Poland may be planning to support. Lukashenko made the remarks while commenting on a Polish general statement that Poland should prepare for an impending uprising in Belarus and aid the military operation in this country. We are keeping an eye on these people that this confused Polish general was talking about. They say they are putting together regiments or battalions, at least they proclaim that they are doing this but in fact we know where they are. We know them by name. We are ready. Let them come. Lukashenko said on Thursday. A few months ago I said that they were preparing an uprising, a revolution, whatever. We have known for a long time that they are up to something, perhaps preparing another revolt, he added.


Hello and welcome to today's episode of the International Daily Roundup by People's Dispatch where you bring you some of the top stories from around the world. Let's take a look at the headlines. Hundreds injured as protests continue in Sudan, Canadian court allows private prosecution of indigenous activists, delivery workers go on strike in South Korea, and indigenous groups resist dam project in the Philippines. In our first story, at least four people were shot and killed by Sudanese forces on December 30th, the day marked the 11th March of millions against the October military coup. The Central Committee of Sudanese doctors stated that around 200 protesters were also injured. This included approximately 40 injuries with live bullets. With the number of casualties expected to rise, the current death toll in Sudan's protests stands at 52. Communication services including the internet had been shut down ahead of Thursday's protests. Security forces also closed off bridges including those linking Omdurman and Bahri to Karthung. Heavy use of tear gas and stun grenades were reported as tens of thousands of people took to the streets. Protesters could be heard chanting as much as we sacrifice and die, we won't be ruled by the boot. Hunta forces also attacked protesters as they reached within two kilometers of the presidential palace. Sudan's revolutionary forces including the Sudanese Professionals Association are leading the resistance. They have refused in negotiations with the military and have demanded complete civilian rule. Thursday's march of millions followed just days after a similar protest on December 25th. At least 235 demonstrators were injured including those hit by live bullets and gas canisters. Sudanese doctors have also reported repeated attacks on hospitals, ambulances and medical staff. Meanwhile, powers of arrest, detention and seizure have also been reinstated to the intelligence services. In our next story, the Canadian National Railway has been allowed to privately prosecute anti-pipeline activists. The disturbing ruling has been issued by the Supreme Court of British Columbia. The case is related to a rail blockade carried out by Hitsa and heritory chiefs in activists in February 2020. The action was part of the countrywide protests and blockades in solidarity with the Wetsuit and heritory chiefs. The clans were fighting to remove the coastal gas link pipeline project from their unseeded territory. However, CGL obtained a Supreme Court injunction sparking unrest. Twelve people including Hitsa and chiefs were arrested from a blockade at a CN rail line in New Hazelton on February 24th. The company has been able to obtain an injunction against the blockade two weeks before. In the proceedings that followed, CN Rail tried to win the right to pursue criminal contempt charges. These are punishable by fines and even jail time. However, in April 2021, British Columbia's public prosecution service declined to press charges. They also raised concerns about a private company being allowed to conduct a criminal prosecution. Despite this, the British Columbia Supreme Court ruled in favor of CN Rail this month. Judge Ward Branch stated that the prosecution service was not the sole arbitrary of public interest. The criminal contempt proceedings would be allowed against three out of the 12 arrested people. Kai Nagata from the Dogwood Initiative has also pointed to the branch's previous employment at Fraskin. The law firm currently represents coastal gas link. Next, we go to South Korea where delivery workers employed by CG Logistics went on strike this week. This is the fourth such action undertaken in the country in 2021. Unionized workers staged a walkout on December 29th after the strike was approved by a majority of 93.6%. CJ Logistics workers have been mobilizing for better pay and working conditions. The company holds around 50% of the market share in South Korea's delivery sector. Outrage grew following reports that at least 16 workers had died due to overwork in 2020. Conditions became worse during the pandemic as South Korea witnessed a 21% increase in the number of delivery parcels. Workers reported being forced to work 6 days a week and up to 14 hours a day. They were also forced to make same-day deliveries, failing which they incurred late fees. Logistics companies and unions had signed a deal in January 2021 to prevent overwork. This included hiring more workers and limiting the work week to 60 hours, among other steps. However, as per reports, the January agreement's implementation has been pushed to January 2022. Workers at CG Logistics have also stated that the company hiked its shipping charges by 171. However, just over 51 won or 30% of this increase was allocated to improve working conditions. And for our final story, we go to the Apaya Province in the Philippines. The Kabugawa ancestral domain is passed with third resolution against the National Commission on Indigenous Peoples. The text has declared NCIP official Joffrey Calderon as a persona non grata on the Isnak tribal lands. The move is part of the years-long resistance to the proposed Genet II dam to be built over the Apaya River. The project involves a state-run Pan-Pacific Renewable Power Philippines Corporation, or PPR-PPC. The late-last resolution was passed after the affected residents were barred from attending a consensus-building activity. This is part of the mandatory, free, prior and informed consent process for projects on Indigenous lands. The NCIP also deployed the National Police, who used threats and violence to prevent people from participating. On December 18, the community passed a resolution declaring non-consent to the dam. It stated that the agency and the NCIP did not engage with the people in good faith and have disregarded the Kabugawa ancestral domain's customs. Now, redesigned with a 225 megawatt capacity, the Genet II dam is among the several projects along the Apaya Abulag River. It is expected to cause massive encroachment on Indigenous lands, including those of the Isnak people. Eight administrative divisions in Kabugawa are expected to be submerged. This will directly affect over 17,000 people. And that's all we have time for today. We'll bring you more such news next year. Until then, keep watching People's Dispatch.


Phonemic awareness is the ability to hear sounds and sound patterns within a spoken word. Students need to learn how sounds make words before they can learn to read and spell. Children naturally listen to the sounds around them. This includes the sounds of spoken words. The teacher helps children listen to these sounds carefully and learn to hear the patterns within words. Over time, they learn how different sounds work together to make the words we say. The students also discover that by taking a word and adding, changing, or taking away a sound, a new word can be created. For example, if we take the word ox and add the sound b at the beginning, we create the word box. If we take the word bag and change the middle sound to a, we have bug. And if we take the word belt and take away the ending sound t, we have bell. These skills are all about sounds, so they are taught with speaking and listening only, not with written words or letters. Learning about letters and printed words comes later. To learn phonemic awareness, students need a lot of practice, both in the classroom and at home.


So I'm doing the research for today's lesson. I learned something really amazing. I'm still like shocked about this. Did you know that a water droplet is a lens? I was like, woohoo, how cool is that? So as you can probably tell, today's lesson is going to be about, it's going to be about lenses. So in this image, if you look in the background, you can see that there is a bridge back here. You can see me kind of doing a tracing of the shape in the background. That bridge is out of focus because it's kind of far off in the distance and we're zoomed in. Our eyes are focusing in on, or the camera and this photograph is focusing in on these water droplets. They're sitting on a window, looking in the direction of the bridge. Now if you look inside of each of those water droplets, you can see this bridge. There it is. Little spans. You can see I just copied that. You can see it's flipping it. So you can see that it's also upside down, which is something that lenses do. You can see the cable stretching out between it. Maybe if we were to look at the top of this water droplet, so if we were to look from maybe like up here and slice the water droplet in half, it would look something like this. So you'd have your bridge on the other side over here and the light is then falling from that bridge. It's then being sent out in lots and lots and lots of different directions. So from each little point, it's going in every direction. But the lens is looking at it from one particular point. The lens, the light is coming off of it, hitting the lens and then traveling through the lens. And in this case, we've got our eye or our sensor down here and the light is then going down onto whatever it is that it's being focused onto. So the really amazing thing about lenses is that they're all over the place. And you don't maybe think about it, but this is actually a lens. An eye is a lens. It's one of the most advanced and amazing lenses that there are out there. And we'll talk a little bit more about that a little bit later. Now this is what a lens of a camera looks like on the inside. This is a diagram of a lens. And so we're going to go through and maybe label this. I'm going to delete that. That was just telling us a little bit of information about the lens. I'm also going to delete this because it's just telling us where the focus, focusing elements of this lens are. Right here you can see that one, there's lots and lots of little pieces. And these are actually made out of glass. So they're going to be sort of maybe we'll color them blue because everyone likes to color glass blue. And each piece of glass is called an element. Now you put these pieces of glass together in a group and they are called a group. Pretty simple, right? So you've got your elements and your groups. And in this lens you have a whole bunch of pieces.  You have another group here, another group here, one here and then even just sort of the single one in the back here which is still I think maybe possibly technically called a group. To put them all together you need one big piece of metal or lots and lots and lots of pieces of metal or plastic or whatever the lens is made out of. So we're going to make here what's called the barrel. So here we've got our barrel and connecting to the barrel are lots and lots of little pieces of metal that are holding all of these pieces of glass in place or sometimes lenses are also made out of plastic so they don't necessarily have to be a piece of glass. And so here we can have maybe divide our lens into a few pieces. Our barrel is connected right here for example. Some often see that on a lens. There will be many sort of different outside pieces. And this lens maybe has a mount right here that connects it to the camera. So here's your camera. We've already talked about cameras in this course so you kind of we're not going to explain too much about that. Back here is maybe your sensor. And the light is then coming through the lens. Boop, flying in this way all the way through all of this and then landing here on the sensor in the back. So how does all what are all these other pieces in here? There's quite a lot to building a lens. So we go through and diagram some of that. Now we've already got our barrel here. We've talked about that. Now this lens I remember if you remember I erased a line that was on the bottom here and it was said it was internal focusing which basically means that each of these little pieces of glass or each of these groups is on sort of a roller. So we'll put like a little roller right here. And they are sliding back and forth to focus. And those are all connected with some complicated screws and pulleys and stuff like that. And they're connected to a ring that's on the whole outside of the lens so we'll make it right here. And you as the user you're holding the camera with your one hand so one hand is holding the camera right here. That's not a very good hand. Sorry about that. And you are then turning this ring on the outside of the barrel so it's sort of maybe like a rubber gummy piece of something and you're turning it in one way or the other. And that is then sliding these pieces of glass back and forth and letting you focus the image that you're seeing on the other side. So maybe you have a tree that you want to focus on. It's not going to be yellow. It's going to be brown. We'll put our tree right here. So you're focusing it on that tree and it's a certain distance from you. So to set that you need to turn your focus ring, zoom in or not zoom but focus in on that tree. And then it'll appear focused here in your sensor and in the eye that is looking through the prism that is looking at the tree. And you'll see your tree will then be focused onto your sensor or onto your film. We don't want to discriminate against people who use film. So there we've got our, let's see, that is called our focus ring. And inside of the camera at another spot is another important part of the lens which is also often connected to a ring and this is called the aperture. And connected to the aperture is also another ring. It might be like right here. Have some numbers on it often. And this will be called the aperture. I'm just going to put it A period aperture ring. So the A stands for A period, stands for aperture like you see down here. And the aperture is basically just a little, it's like a, kind of like the shutters on a window. It opens and closes and we'll talk about aperture in more in-depth in future lessons and we've already kind of discussed it in a very broad sense already in this course. The aperture just closes and opens. It's like a big barn door allowing more or less light into this lens.  And the only thing that I really want to tell you about before we move on to the next slide is about the lens coating. Now on the front end of this lens is going to be sort of a layer of, it's going to kind of maybe look purple or red or blue. It depends on the type of lens you have, the type of coating it has. But there will be some sort of synthetic material that's put on the front that's called the lens coating. And lens coating is important because it allows you to shoot into light sources without getting what's called ghosting, which just means that the light source inside of the lens will make all kinds of weird, the light will bounce around and make it kind of look funny on the other end when it's actually being looked at as a picture. So a lens coating is important to helping keep that light focused and making sure that when it hits the lens it just goes straight in and it doesn't bounce around and make all kinds of, make a big mess on your photo. So lens coating is very important. It's important because you want to protect it. You don't want to get it scratched. And you also don't want to change it by using maybe some sort of lens cleaning fluid that might affect it in some way. So just as a piece of information that's really important for all photographers, protect your lens coatings. Protect your lens on both ends. You're actually going to be able to have, you're going to have lens coating on both sides. And you're going to want to protect the glass and the coating on both ends of your lenses. We're going to do a really quick little literacy lesson here about how to read about the lens without even having to open the manual. So you can see here on this lens it's actually detracted, it's put inside of the, it's pulled itself inside of the camera, which most compact cameras do. You can see here that the lens is made by Canon. You can see that it is a zoom lens. And we'll talk about what that is here in a second. And you can see that it is a three-time zoom. And all this means is that it takes the widest version of your photo. So say you are looking through the viewfinder and you see this scene right here and you've got a person in the room who you want to maybe zoom in on. If you go from the widest setting, the widest setting that your lens will do, and you zoom in on that person say right here, the difference between this widest image and the smallest image is a zoom factor of three. And that actually, you can actually, if you do the math right here, this 3.8 and the 17.4, that is reflected in these numbers down here. Now 4.8 is actually the real size of the zoom lens. And there's something called 35 millimeter equivalent. So if you're reading a lot of times in the literature, you'll see a 35 millimeter equivalent marked. And that just has to do with the size if it was on a 35 millimeter camera. This camera has a tiny little sensor that's a lot smaller than 35 millimeters. So it has a smaller lens. But what they're saying when you see a 35 millimeter equivalent, they're saying if the lens was bigger, if the sensor was a 35 millimeter sensor like this yellow would be, then this number would be a lot bigger. It would stay instead of 5, it might say 18 or something like that. So that's just an important piece of information. Now over here, you can see 1 to 2.8 dash and then 4.9. And this right here is the widest possible aperture at these different settings. So at 5.8, your aperture, your maximum aperture is 2.8. And at 17.4, your maximum aperture is 4.9. The widest aperture is very important. It's very important actually. And having a widest aperture of 2.8 for a camera like this is actually not too bad. 2.8 just means that it's a very wide aperture. If it was 5.6, it would be smaller. And thereby a slower lens. So you don't necessarily want a slower lens.  So at 2.8 is letting in more light than a 5.6. So you can see here when you're zooming in, when you're zoomed in at three times, zoom, that you're getting less light than you are when you're shooting at the widest angle. And we'll talk about why that's important in the exposure lesson, but it's just important to know the smaller these numbers, the better. Now here you can see two different types of lenses. You can see here this is a zoom lens. And this is what's called a fixed or a prime lens. Now a prime lens is just 50 millimeters. So it just has one length that it can go. And this one right here is actually if you look closely, I think it says 28, yep, so this is a 28 to 200 millimeter lens. So this means that this lens can zoom in, so it can take a really wide shot, and then you could zoom in and take a really small shot. But this lens over here can only take a wide shot. It can only take, well, what's called a normal shot, because 50 millimeters is what's called normal. And it can't zoom. There's no zoom factor on it. The only way that you can zoom in on this lens is by walking closer to the subject. Now how about we take a look at what that would look like. So we've got our zoom lens. Let's say we have that 28 to 200 millimeter lens. When you've got it on your camera, so here's your camera, here's your lens, and you're looking out over a room. Let's say you've got a really long table right here and some people are sitting at it. You're taking a picture of them. You've got your lens on, you're looking to the viewfinder, and a little ways away, you're pretty far away from them. Maybe you're, let's say, you're two meters or something like that. So you're looking at them, and you've got your lens on at 28 millimeters. It's going to look something like this. You're going to see, you're going to see have a pretty wide range of view from left to right, and you're going to be able to see quite a lot, also up and down. This is it, the zoom goes in both directions. Now say you zoom in from 28 millimeters to 50. Fifty millimeters might look something like this.  So your 50 millimeter is going to be a little bit more narrow, and this is actually pretty exaggerated, probably be a little bit wider than that. Now say you wanted to zoom in on just one person, it was maybe sitting in the very center of the table, and maybe you would move up to your 200. So here we've got that. It would be your range, this would be maybe your 200 millimeter. So here you can, when you're looking at a 200 millimeter image, all of this right here has been cut off. You can't see any of that. All you're seeing is this one person right here. This is your picture about. And in this case over here maybe your picture would actually look something like this, and in this other picture you're obviously looking at a much, much wider image, something like that. And all of this is just basically called focal length, so the length of your lens. All right, so that was your lesson about lenses. In the next video we're going to talk about what the difference is between a good lens and a bad lens.


All right, thank you. So I guess this is my last chance to try to get my point across. So before getting buried in calculations again, let me try to explain what's the general idea of obstruction bundled gluing. So suppose we want to glue a bunch of things together. Usually, we'll talk about holomorphic curves, although my example's a far worse theory. So suppose I have a bunch of holomorphic curves to UI to want to glue them together. Now, so there's some moduli space M. So this moduli space M consists of configurations of the UI together with some gluing parameters. So in general, these holomorphic curves might themselves live in some moduli spaces. So for any collection of holomorphic curves in their respective moduli spaces, and then some gluing parameters, saying like how much we translate things, et cetera, we can try to glue. Now, the best situation is where all of these linearized operators, so DI is the linearized equation for UI, the best situation is when these operators are surjective. And then there's no obstruction to gluing, and you can just use the usual construction. Everything's fine. The second best situation is where these operators are maybe not surjective, but their co-kernels have constant dimension. So as you move around in the moduli space, the dimension of the co-kernel of this operator doesn't jump so that I have an actual vector bundle over the space M of configurations, where the fiber over a point in this vector bundle is the sum of the co-kernels of all of the operators for all of the pieces. And this is when we have a well-defined obstruction bundle. The worst case is where these co-kernels dimensions may jump. Then you're in Kerinishi land or polyfold land. So that's when you really need to do something fancy. So these obstruction bundle cases sort of in between the classical case or everything works fine, and the general case where you need really high technology to solve the problems. So now let's consider this obstruction bundle case. So then you can try to glue, and you write down the gluing equations, and there's an obstruction. So you can glue up to elements of the co-kernels of all of these operators. And so you can then define an obstruction section S. So S inputs a gluing configuration and gluing parameters and outputs the obstruction to gluing. And whenever S is equal to 0, you can glue. And in general, you can prove that this construction gives you a bijection from the 0 set of S to the set of possible gluings. And then you want to understand what are the zeros of S. And the section S is defined in some strange, indirect way from the gluing equations. So you want to approximate S by something easier to understand. So S0 is this linearized section, which is obtained basically from taking the leading order terms in S and throwing everything else away. And then you argue, so if you're trying to do some count of the number of zeros of S, then in general, this moduli space M is going to have some boundary. So the gluing parameters can only be in some certain range. And you argue that on the boundary of this moduli space, the terms in S0 are much bigger than all of the other terms in S. And so if you linearly deform S to S0, zeros never cross the boundary. So any kind of count of zeros that you can make will be the same for S and S0. So then you just have to understand the zeros of S0. And then the idea of S0, I think I didn't explain this so well last time, so let me try again. So for each of these holomorphic curves or whatever that you try in the glu, there's a component of S0 in the co-colonel of that operator. And the contribution of S0 for one of these curves has contributions for all of the things that are adjacent to it, for all of the things that are being glued to it. So if I look at this U5, I want to say what is the U5 component of this section? So for each of these curves, U1, U2, and U7, there is a contribution which comes from the asymptotics of the end of that curve that's getting glued to U5.  So basically, you look at the asymptotic expansion of this curve, and you can write a kind of Fourier expansion of it in terms of eigenvalues of the linearized operator, and you look at the leading order term of that. Excuse me, eigenvalues of the asymptotic operator. And you look at the leading order term, it's going to have some exponential decay. So this contribution is going to look like e to the minus r lambda, where lambda is the eigenvalue, and r is the amount that you translate this curve. So that's the basic idea. And then if all goes well, you can then actually count the zeros of S0 to figure out how many gluings there are. So are there any questions about this general picture before I plunge into my next example? Yeah. So when we do the gluing, for example, U7 part dominates in that end, and we do the gluing construction using S0 corresponding to that whatever highest eigenvalue it is. And then we do it for U2, U1, but I'm confused about how we do the gluing in between these, like, in the interior of U5. Like, how do we glue the part coming from U2 and the part coming from U7? How does that happen automatically? I'm confused about that. How does what glue to what? So we glue near the connection with U7. So up here? Yeah. And then we glue near the connection with U2. But those two things should come together to. Well, these don't really interact. I mean, these are sort of far away from each other. But the levels of the gluing, I guess, I mean, there should still be a U5 giving those two gluings, no? So we're sort of doing the whole gluing at once. So we're sort of pre-gluing. So I guess in this example, I've got three levels. So I translate these guys up by some large amount. They can move by different amounts up. I translate this one down by some amount. And actually, these guys can also translate up and down with respect to each other. And I pre-glue using some Cotto functions. And then I deform using sections of the tangent bundle over each of these curves. And then I solve this whole system of equations for all these statuses to be 0, or at least in the co-kernel of the operators. So somehow, in the glute homomorphic curve, in the interior of U5, nothing seems to change. Oh, change the whole thing. But how does that a little bit change happen? I'm missing that point. I mean, there's a section psi 5, which describes how much I'm moving U5 in the gluing.  I mean, there's a Cotto function beta 5 for U5, which is supported on U5. And it dies on the ends of U5. And then there's a function psi 5, which tells me how much I'm moving U5. So I am moving the whole thing. And OK, so the abstraction, OK, I understand now. There is no equation for psi 5 which contributes to this abstraction gluing procedure. That part is already solved. That part is 0 on the nodes. And then for the other, like this psi 2 part, then I get some condition. Something should vanish in the of this kernel of the i. Some number, that number should be 0. But for the other one, number is already 0. Yeah, I don't really quite understand. OK, all right. If you're happy, I'm happy. OK, other questions? If you could ask another question. How many equations are you going to write down? Seven. So what's so funny? So philosophically, say, if you look at the top floor, you look at maps, model, or common r, actually, that is on each level. In principle, the object is, if you look at asomorphisms, it is a map model of the image shifted by r in r cross v. Well, it depends what you're trying to do. I mean, you could shift the different components. But this are different asomorphism classes of objects. OK, OK. But that would, of course, in the modernized space be a different point. So from my perspective, I think basically you have a blue asomorphism class. It's essentially a blue asomorphism class. And then you have two blue parameters in this case, which is sort of, if you look at the average value, at some points distributed here, the average value here is how they go up. I guess it depends on exactly what problem you're trying to solve, whether you want to glue things. Well, exactly what kind of equivalence class do you want to glue? So am I right in conclusion, in this picture, there are two gluing parameters? Well, for Helmut's point of view, yes. For my point of view, there would be more, because there would be one, two, three, four, five, six gluing parameters. But he can't sort of moving the thing on a level against each other as the same, where I don't. So then I have more objects there. That's where the parameters then come. Maybe, I think the definition of this, if you multiply space, also counts those as different things. OK. All right, that's fine. So let's just not find. From that point, if you're trying to solve that problem, then there are just two gluing parameters. Yes? I think the more or less space would be the same. So I guess the tangential question is, again, you're assuming the dimensions of all the co-carnals are the same, is that right? So then what about, I guess you also need that dimension of the co-carnal on the interior of the moduli space is also the same? Well, I'm not sure what that means. I mean, each of these pieces lives in some moduli space, which is not breaking or anything. So this U5 lives in some moduli space of unbroken things. Right, so presumably that's a piece of the boundary of some larger moduli space. Yeah, so this whole configuration is part of the boundary of some moduli space. And we're trying to understand how many ways can we glue this boundary or corner configuration to something in the interior of that moduli space. All right, so let's do an example. So I'm going to talk about a non-degenerate contact three manifold. So we have the ray vector field R. We have J, a generic almost complex structure on R cross Y satisfying the usual conditions, which I wrote down last time. We want to glue holomorphic curves in R cross Y. So let me talk a little bit about branched covers of trivial cylinders. So suppose gamma is an elliptic ray border, elliptic simple ray border. So what does elliptic mean? Well, here's my orbit gamma. So it's a periodic orbit of the vector field R. And you can look at the normal bundle to this ray of orbit, which is actually sort of identified with the contact plane field restricted to this ray of orbit. And the derivative of this flow defines a map from this normal plane at a given point in gamma to itself. So you just look at the derivative of the flow like this. Defines a linear map. So linearized return map from the contact plane at, say, gamma of 0. This has a symplectic form on a defined by d lambda. The linearized return map preserves this symplectic form. So it's a 2 by 2 symplectic matrix. Non-degenerate means it does not have 1 in its spectrum. And elliptic means that this map is conjugate to a rotation by some angle 2 pi theta. So theta is sort of like a rotation number of this orbit. And the non-degeneracy of the orbit and all of its multiple covers tells us that theta is irrational.  So this is a holomorphic cylinder because of the usual conditions on J. Then I want to talk about branched covers of this trivial cylinder. It's lost. Where's this took? So you go down more. By the time I figure out really, by the time I master these chalkboards, my course will be over. All right, so suppose I have a branched cover. So this is a degree D branched cover of the cylinder where sigma, let's say, has genus G. It's connected in genus G. And let's say it has positive ends of multiplicity a1 through ak. So in other words, so here's R cross gamma. And I have this surface that's covering it. So each end of the surface is going to give me a covering of the sort of end of the cylinder of some multiplicity. So the positive ends of multiplicity is ai. The negative ends have multiplicity b1 through bl. And the sum of the ai's and the sum of the bj's must equal D, because it's the whole thing is degree D. So this has a complex structure on it, pulled back from the complex structure on the cylinder. This is a perfectly good holomorphic curve. It's a nice example of failure of transversality. So Chris Wendell did the sort of a closed version of this for closed curves in his talks. And this is a sort of relative version of that. So why does the transversality fail? What's the Fredholm index of this curve? So the index of this map u0, well, in general, for the index of any curve in a R cross contact three manifold is minus the Euler characteristic of the domain plus twice some relative first turn class. So this notation is, so tau is a trivialization of the contact plane over all of the ray orbits. And this c tau is the relative first turn class of the contact plane field restricted to this curve u. Then there's a sum of conlead-sander terms. So you have the sum of the positive ends of the conlead-sander index of the ray of orbit with respect to the trivialization tau minus the sum over the negative ends of the conlead-sander index of the negative ends. The same formula works in any number of dimensions, except the minus chi gets replaced by n minus 3 times chi. Now, in this particular case, I can just use the same trivialization for everything coming from some given trivialization of gamma. And then the trivialization is xi trivialized along with the gamma. So tau is a trivialization of xi over the ray orbits. So the index of u0 is minus Euler characteristic of sigma. And this first turn class goes away if you use the same trivialization of gamma. So then I have plus the conlead-sander stuff. So what that comes out to be, so it's 2g minus 2 plus k plus l. So that's minus the Euler characteristic of the curve. And then, so if I fix some trivialization, then the linearized return map for this ray orbit gamma is rotation by 2 pi theta. And the conlead-sander index is a, so if I have the conlead-sander index of gamma itself is 2 times the floor of theta plus 1. And then if I'm looking at an ai-fold multiple cover, then that has the linearized return map for that as a rotation by ai times theta. So I get 2 times the floor of ai theta plus 1. So that's the sum of the conlead-sander index terms. For the positive ends and then for the negative ends, I have minus sum from j equals 1 to l of 2 times the floor of bj theta plus 1. And for convenience, I'm going to write this little differently. I'll write this as 2 times the ceiling of bj theta minus 1. Could you say again why the relative turn-class transition? Because the trivialization, if I use the same trivialization coming from a given trivialization of gamma, then that trivialization extends over the entire curve. And so the first turn-class is 0. So the first, relative first turn-class is sort of the obstruction to extending the boundary trivialization over the whole curve. Is it a branch point? Yeah, yeah, it's still OK. But it's because you have a cylinder, right? Otherwise, you add some ding-a-ding or a cylinder. Yeah, I mean, everything's just mapping to this one ray of orbit. I trivialized the contact point field over it. OK, so now I can, oh, I did that. Sorry, that's not the way I wanted to do it. Excuse me. Let's make this, let's keep this the floor. I actually wanted to change this to the ceiling. Because now I can cancel some stuff out. So I can cancel this plus k with this minus 1. I can cancel this plus l with this plus 1 over here. So this whole thing comes out to be 2 times g minus 1 plus sum over i of ceiling of ai theta minus sum over j of floor of bj theta. And then I always get confused by which way it goes. But the sum of the ceilings is greater than or equal to the ceiling of the sum. So this thing is greater than or equal to the ceiling of d theta. And this thing is less than or equal to the floor of d theta. So this whole thing is greater than or equal to 2 times g minus 1 plus ceiling of d theta minus floor of d theta. And because d theta is not an integer, the ceiling minus the floor is equal to 1. So this is equal to 2g. So what does that tell us? So it tells us that the index of this multiple cover is greater than or equal to 0. That's a nice fact. It's not true in higher dimensions, unfortunately. But in three dimensions, this branch cover trivial cylinder always has index greater than or equal to 0. And it can be equal to 0. So the index of u0 is equal to 0 if and only if the genus is equal to 0. And the sum over i of ceiling of ai theta is actually equal to ceiling of d theta. And the sum over j of floor of bj theta is equal to the floor of d theta. So this is sometimes true. So in some cases, the index is 0. So little combinatorics just for comic relief here. Because we need some jokes in between all the analysis. OK. Great. So why did I tell you this? So what's the situation we can have? As we could have a Fredholm index 1 curve here.  So this u plus could have negative ends at gamma of multiplicities a1 through ak. Then I could have some curve u minus of also Fredholm index 1, which has positive ends at covers of gamma of multiplicities bj. And then in between them, I could have this curve u0, which sometimes has index 0. So suppose we're in a situation where this thing actually has index 0. So then in principle, maybe I could glue this configuration. And I want to know how many gluings there are. So the reason why I looked at this problem, or Cliff Tobbs and I worked on this problem in our paper, was to prove that the differential in embedded contact tomology satisfies d squared equals 0. So for that application, you actually need to glue stuff like this. Although in our paper, we actually did something much more general. So we glued more or less anything of this kind, even much more generally than what's needed for embedded contact tomology. We also did orientations very carefully, if you ever want to read about orientations. Very carefully. Is it? Yeah, question? Does it have to be a cover of a trivial cylinder? Maybe that goes more like different pieces that go to different parts of gamma. I guess that it wouldn't. Well, you could certainly do this kind of thing more generally for branch covers is something which is not a trivial cylinder. We didn't do that, although that's of interest when you're trying to look at cowardice maps in embedded contact tomology. So Chris Garry gives thinking about that. It's also an example of the Bowen-Honda paper. That's true, yes. The Bowen-Honda paper looks at covers of cylinders, non-trivial cylinders. They're not branched, but they're non-trivial, which come up in Shane Hometopes for invariance of cylindrical contact tomology. It's a very nice example of obstruction bundles that they do. Why do you think the same thing with more general?  We just did dimension 3, yeah. But the kinds of configurations that we glued were more general than what you need for your CH. I think one advantage of dimension 3 here is there's sort of low dimensional phenomena related to automatic countisality that will help you prove that the obstruction bubble is well defined. Yeah, that's coming in a second. Right, so in this situation, a situation like this, we have an obstruction bundle. So I say I want to try to glue this thing. So what is this M? Well, it's basically a modular space of branched covers. So the reason why this is an example of failure of transversality is because the dimension, so the modular space of all such curves U0 is a manifold of branched covers of the cylinder. And the dimension is 2 times the number of branch points. And the number of branch points is k plus l minus 2. So whenever this thing is not a cylinder, there are some branch points. So it's a modular space of positive dimension, manifestly. But its index might be 0. So then transversality just doesn't work.  OK, so now in this problem, we could sort of fix the gluing parameters so we could fix how much we're going to translate the stuff up. And then we have a modular space of branched covers. And then over each point in this modular space, we have the co-kernel of the operator D0 for the curve U0.  So we'll assume that the operators D plus and minus are surjective, which you can do if at least these things are somewhere injective. You can assume that for generic J. However, D0 definitely has a co-kernel. And as Chris just pointed out, the dimension of the co-kernel never varies. So the lemma, while I have all these combinatorics up here, one more lemma. I'm just certainly doing this badly. It's not the right way to do this. Yeah, in general, you could look at non-disconnected branch covers, too, on how to use the board. Well, they should have sent us an instruction manual. What do you see in the restroom?  But you just have to be like that for the boards. Then you'll have 25 steps. OK. So the lemma is that the kernel of D0 is always equal to 0. So this is Elkris Wendell's talk. He talked about unbranched covers of closed surfaces, improving that those had kernel equal to 0. And here I'm talking about branched covers of trivial cylinders. So it's a slightly different thing. So this is just a simple calculation. And it's related to automatic transversality things. Right. In fact, I forgot to tell you what D0 is. So D0 is not what you think it is. I'm very sorry. I'm getting scattered here. So first of all, this is a sort of, I want a normal deformation operator. So this operator, D0n, was a go-to. Or you could take sort of L21 sections of U0 pullback of the normal bundle to the trivial cylinder. So these are basically sections ascribing how to move my curve U0 in the direction normal to the trivial cylinder. So you could think of this as deformations that don't move the branch points, roughly speaking. And then this maps to L2 sections of, well, 0, 1 forms on sigma with values in this bundle. So we're measuring deformations that don't move the branch points. So the index, right? So this thing, the dimension of m is 2 times the number of branch points. And the index of this operator, D0n, is minus 2b. So the curve U0 itself is index 0. So the sort of usual operator for that curve would have index 0. But because I'm sort of restricting the domain of this operator by not allowing the branch points to move, that decreases the index by the number of constraints I put on, which is 2 times the number of branch points. So that's why the index of this operator is minus 2b. And my lemma is that the kernel is 0 so that this co-kernel has dimension equal to 2b again. So I have a rank 2b bundle over a 2b dimensional base. And I can expect to get some finite count of 0s, which would be the number of ways to glue. And what's the proof of this? Well, so you let psi be a non-zero element of the kernel, suppose such a thing exists. You want to count how many 0s does it have. So by this sort of Coralman similarity principle, these have isolated 0s, which look like the 0s of a holomorphic function. So the algebraic count of 0s is greater than or equal to 0 with equality if and only if it's non-vanishing. Now, it will be non-vanishing sort of on the ends. So at each end, it has a kind of winding number around 0. So as you go around one of these circles in the ends, you can count the winding number of this section around 0. Then the number of 0s in the interior is going to be the difference between the winding numbers and the ends. So it's going to be the sum over i, sum from i equals 1 to k, of the winding number with respect to this trivialization tau of psi around the i-th positive end minus sum from j equals 1 to l of the winding number of respect to tau of psi over the j's negative end. And then you can put bounds on these winding numbers. So basically, the asymptotic behavior of psi is described in terms of asymptotic eigenvalue of an asymptotic operator. I certainly don't have time to explain that in detail. But you find that the winding number is less than or equal to the floor of ai theta, while one of these winding numbers is greater than or equal to the ceiling of bj theta. So then this whole thing is less than or equal to, you're right, thank you. So this whole thing is less than or equal to the floor of d theta minus the ceiling of d theta, which is minus 1. So I just show that 0's, less than or equal to minus 1. That can't happen. Well, whoever's reading this is unhappy, but this nonsense. All right, so we have a well-defined instruction bundle. This didn't actually disuse the index to be for 0 assumption? No, the kernel's always 0. I actually suspicious as to whether this really is a low-dimensional phenomenon. It seems like it should be related to the fact that the operator for the trivial cylinder is always invertible, which is very general. All right, let's talk about that later and see if it generalizes. By the way, how much time is left? 20? Really? Well, 17. 17 minutes? OK. Wow, OK. Time flies. OK, well anyway, all right, great. I don't know what I'm saying. All right, so we have an instruction bundle. So this instruction bundle is good. So we're happy over here. All right, so now we can do obstruction bundle gluing. So what's the picture? Is that thing as happy or sad as the question? Well, it makes us happy that this method is applicable. And lucky you, you didn't have to do it. So I think people greeted my papers with tobs with a mixture of horror and shod and froida. OK, so let me do the simplest non-trivial example. So simplest non-trivial example. So this is where k equals 1 and a1 equals 2 and l equals 2 and b1 equals b2 equals 1. So my curve u0 just looks like this, just a pair of pants. The top has degree 2. The bottom has degree 1. This is mapping to r cross gamma. And let's assume, we should assume, that theta is between 0 and 1 half. And this is what you need for the index. Well, up to shifting it by an integer, this is what you need for the index to be 0. If theta is between 1 half and 1, the index will be 2. OK. And there's how many branch points are there going to be? There's going to be one branch point. So it's going to be somewhere over here, mapping to somewhere over there. So the modular space of branch covers, we can identify this with a cylinder. Basically, you just need to keep track of where this branch point is. It's actually, the modular space is actually a double cover of the cylinder, because you need to keep track of asymptotic markers. Well, let's not go into that. Now, when you're actually gluing, you're not looking at all branch covers, because I've only shifted u plus and u minus down by some finite amount. So there's only sort of a finite range in which this branch point can be. So this really is actually something, some interval, say minus r, r cross s1. Then we try to glue these things. So this is going to be an instruction section. So can you say, actually, why you trumpeted r at that point? Because I shift u plus up by some finite amount. I shift u minus down by some finite amount. And there's only this finite interval in which I can put this branch cover. So the branch point can't be way up there, because then I'm far away from gamma. So when you're writing this, it's really kind of annoying to sort of sort this out and figure out where exactly the branch point can be, and so on. Then there's another simplifying thing you can do, which is you can arrange that the linearized rate flow around gamma is j linear. In other words, you can choose your j in such a way that the rate flow from one point in gamma to another point in gamma is actually a complex linear map. When you do that, this causes the normal operator d0n to be actually a complex linear operator, and not just a real linear operator. It simplifies the calculations a little bit. So the co-kernel, well, it's a two-dimensional real vector space or one-dimensional complex vector space. So now I have a complex line bundle over m. So this m is an interval process 1. I have a complex line bundle over it. But this is more interesting, because now in my Morse theory example, the bundle was trivial, because you had the same co-kernel over every point. But now these curves use zero in varying, so they have different co-kernels. So it's a more interesting bundle. Then I have some instruction section s, and I have no idea what it is, but I have an approximate section s0. So I fix the gluing parameters.  So the gluing parameters, like some constant times this are. OK. So they have an s0. So s0 has contributions from the negative end of u plus and the two positive ends of u minus. And you can actually sort of figure out what these are. So I'll spare you the whole story, but I'll show you what it comes out to. So you get that s0, well, let's give these things names. So this is, let's call this s and t. So s, oh, not this again. I have some of your s0 and s0 sections. No, I have. OK. Just make a little R. Let's make this stigma entail. When I was taking undergraduate quantum mechanics, they would write equations on the board like this. It looked like this. p, p equals p, p. The letter p would mean three different things. So there's three different things in this equation all represented by the letter p. I sort of couldn't. I found this very difficult to deal with. But then on the exam, when I had to write my answers, I would just sort of write the same letter everywhere. And then I got an A. Let's put a hat on this. How about that? Is it your trivialization fix so you can just use tau? Or are you going to change it? Like dynamically scoped variables or something. I forgot this. All right. Anyway, just s0 hat. So s0 hat of s t, what's it going to be?  Or lambda is an eigenvalue, some operator. So basically, this e to the lambda s means that when the branch point is higher up, you're sort of closer to u plus, so you're seeing more of a contribution. And this i t reflects the fact that as the branch point moves around, well, so you have to compare these co-current elements for different elements in the moduli space. But to do that, you basically need to understand the winding numbers of these co-current elements as you go around the end. So the fact that this term is here represents the fact that the co-current element has winding number 1 as you go around the end. And then there's going to be some, from the negative end, there's going to be some contribution, let's call this, I don't know, lambda plus, the form e to the minus lambda minus s. And these, on the negative end, the co-current elements have winding number 0, so there's actually no t term up here. So I don't really have time to explain this in full detail, but you get an expression like this. So this is actually quite simple. The general case where you have lots of branch points, it becomes very challenging to calculate the number of zeros of the section. You'll search the tricks. But this case is simple enough that you can see directly what's going on. So what's going on? So if s is large, then this term is much bigger than this term. So in the complex plane, when s is large, as t goes around s1, the section's going like this. So let me label this circle here. If s is small, then this term is very small. And then this term is big, but this is, as t goes around the circle, it's a constant. So if s is small, it's some little circle like this. And then on that finite cylinder, you're interpolating between these two circles. So one of these circles has winding number 1 around 0, and the other one has winding number 0. So the number of zeros is equal to 1. There's one way to glue, up to science. In the general case, we calculated the number of ways to glue something like this. We found kind of remarkably that the number of ways to glue is equal to 1, and only if you have exactly the kind of data that comes up in embedded contact tomology. Otherwise, you get some number other than 1. Isn't that crazy? All right. So in my remaining three minutes, let me just close with a question, which we can think about next week. But next week, how am I going to tell us the definition of s of t? So s of t is supposed to count index 1 things, but not just index 1 curves, but any sort of index 1 building. So you could have an index 1 building, which looks like this. So there's some nice, let's say, we can make it embedded, if you like, embedded index equals 1 piece. Perfectly good curve. This is the kind of thing we want to count. And then maybe attached to it are some branch covers of trivial cylinders. So they can be attached above it. So these have some branch points in them or below it. They could also, in sort of worst cases, this thing could have some additional ends. The covers are the same orbit, and there could be some further branch covers of trivial cylinders on the side. So you have some building where 1 piece is an embedded index 1 curve, and then there are a bunch of other pieces either above it, below it, or on the side, which are index 0 branch covers of trivial cylinders. And something like this should make some contribution to the s of t differential. And the question is, how much? So my question is, what is the contribution to the s of t differential? So I have some conjectures about this, but it's not really well-defined question until we've seen the definition of the s of t differential. I think it might, depending on, I mean, so you have solutions on the boundary, on corners, and you have an algorithm kicking certain things in, systematic way, kicking certain things in and out. And there are some conventions. So depending on the conventions, I mean, they lead to the same algebraic theory. So it's conceivable that, actually, depending on the convention, the answer to the question would be different. Yeah, that's exactly what I expect. So I expect this to depend on some choices. So just to draw the picture for the simple example that I did. So we did this example where you glue a pair of pants in between two index 1 things, like this. So this is index 1. This is index 0. This is index 1. We found that the number of ways to glue this equals 1. Now, this building consisting of the upper piece plus the middle piece makes some contribution to the s of t differential. I don't know what. Let's call it c plus. And this lower piece, obtained by gluing the lower piece to the middle, makes some other contribution. Let's call it c minus. And this forces us, modulo. I guess, in general, you need some combinatorial factors in the s of t differential. But ignoring that, the fact that there's one way to glue basically forces that c plus plus c minus equals 1 in order to get d squared equals 0. And I would expect that you could choose one of these to be whatever you want. And then the other one is forced by this equation. So I have some more general conjectures along these lines. Can you just say it again? Because the number of gluings is 1. So there's an end of the index 2 moduli space converging to this thing. So it has to be somehow counted by d squared. So either this thing, it could be counted as c plus times this thing, which is contributing 1. I'm ignoring the signs here. Or as this thing, c minus times 1. So the total contribution to the differential squared is c plus plus c minus. That has to be 1. It'd be nice if I could say that you could make the choices such that whenever there's some junk on the top, it counts 0, for example. Although I'm a little confused with what to do when there's stuff on the side. I don't really understand that. I was actually thinking about this question 10 years ago. And I think I gave a completely incomprehensible talk about this 10 years ago. But I think maybe now we have the language in place where we can actually answer it. All right, thanks. What if I saw on this, in the needle part, some further picture, in the embedded part, do you assume like a standard ECH assumption? No multiple powers of the type of voltage? No, I'm not assuming that. Doesn't have to be one of the things counted by ECH. Although that's, of course, an interesting case. But those are the assumptions that need to be answered if I'm correct. Right, well, this is a different question. This is not a question about how many gluings. It's a question about how do you want to count this? So SFT says there are some numbers, maybe depending on some choices, that have the properties to give you an invariant. And silly people like me say, what are these numbers exactly in these examples? So if you had a slightly more complicated thing where you had, say, three times winding, just one end at the top and three times winding, but then you have three separate disconnected things, order one at the bottom. And I seem to remember you had this partition. I mean, it's a one-wave to do that, or do you have to partition it into two things? And I seem to remember these trajectory. I think in that case, the number of gluings was also one. But it's, in general, it's a kind of complicated combinatorial formula, which I don't remember off the top of my head. It's in the beginning of part one of the two-part paper with top. So there's some complicated combinatorial formula for the number of gluings. Right, but when you split things, when you sort of stretch along the torus and you have a small orbit, you have something sort of that the ECH thing that you get is sort of one multi-covered thing at the top, the generic one, and then lots of single ends at the bottom. Right, yeah, so these partition conditions would say that. Yeah, so in this picture, if you're actually looking at, if your U plus and U minus are counted by ECH, then the numbers A, I, and B, J are determined by D and the angle theta. So if the angle theta is positive and very small, then at the top, as you said, you have just one end. And at the bottom, you have D ends, well, with multiplicity. And it's just splitting into one thing in the middle. Or would it split into lots of things in the middle? Just one thing. So I don't really think of splitting the middle into levels. We don't really look at it that way. Yeah, so Chris Garry will answer all of your technical questions. And he also, if you don't have technical questions, he has a couple of sort of other examples of this phenomenon prepared to tell you about. Guys, it was brought to my attention that the next speaker wants to have plenty of time for a leisurely coffee after lunch before we reconvene at 2.15 in the stink microwave.


Christianization Christianization or Christianization is the conversion of individuals to Christianity or the conversion of entire groups at once. Various strategies and techniques were employed in Christianization campaigns from late antiquity and throughout the Middle Ages. Often the conversion of the ruler was followed by the compulsory baptism of his subjects. Some were evangelization by monks or priests, organic growth within and already partly Christianized society, or by campaigns against paganism such as the conversion of pagan temples into Christian churches or the condemnation of pagan gods and practices. A strategy for Christianization was Interpretation Christianity, the practice of converting native pagan practices and culture, pagan religious imagery, pagan sites and the pagan calendar to Christian uses. Due to the Christian efforts at proselytism evangelism based on the Great Commission.S