Okay, I think everything set up welcome everyone. I'm here with little Robin. We're gonna do vocabulary workshop Six I think it's six I Get all my vocabulary from IELTS IELTS tests We're not really studying IELTS. We're just looking at vocabulary that IELTS tests recommends You know Okay, I'm gonna put a little little Robin down. Thanks for your help. He made the PPT's today. So if there's any mistake Blame him. Okay. Welcome guys. Let's see who's here. We have a lot of channel members and Thank you very much to the channel members if you're a channel member Thank you so much. I know I don't give enough time to the channel members, but you I Really appreciate everyone who supports the channel and if you don't please help out Helps me with tea and coffee and lets me know that you care Anyway, Lolly Lolly was here first today who and Layla sleepwalkers here Trudis Welcome guys. Do we have any men? Are there any men today? Sarah Carrie Michelle Michelle and Carlos You guys are men. I'm at men. Okay. We got some men today. Thank you That's good to see oh nicks and nicks Welcome nicks and nicks. We're starting now welcome I'm Robin.  Maybe homeless man? The last the last time I got a haircut was May 2020 I think May 2020 was the last time I got a haircut. So my hair has been growing almost a year Getting long. Yeah, I'll get it cut eventually. Maybe May 2021 Carlos calls me sir Robin. I'm not sir Robin. I'm just Robin Alex some testosterone. Yeah, good. Good Alex could be a female name like Rob. My name is Robin. Rob Robin is More common as a female name And Alex could be male or female. So it's hard for me to guess sometimes Yeah, all right Kadesa Zaki. Hello Hello. Welcome. All right. We're doing a Doing vocabulary workshop today. So I got 10 vocabulary words And I think Students are here Everyone's here, you know, I don't want to start exactly at 1030 because a few people come late that's okay And I guess we'll get into it slide one boom Vocabulary workshop number six if you did not watch vocabulary workshop one two three four five They're not connected They're each each one is separate and different and sometimes I might repeat a vocabulary I don't know because I don't remember the vocabulary. I taught. I've been teaching this series over a year so Once every two months, maybe Hey, Fanny.  I Don't have coronavirus, but I still have some problem Nixon next day what time? Schedules right above me day and time right above me You look different with your beard. I think you Most videos I've made I've made with a beard Seven seconds delay some people will have 20 seconds delay some people have one or two minutes delay depends on your country and your Wi-Fi and I don't see win Wins not here win is from Myanmar and Myanmar is Also known as Burma's going through a very difficult time. So if there's any students from Myanmar You got a very tough future Fighting for your democracy. So good luck To you guys, I'm always watching the news and checking about Myanmar You know, it's ridiculous in 2021 that there's a military is taking over a country Ridiculous, so I hope the people in Myanmar get their country back Lopez St. Val. Hello If any I'm not in your group again how to get in your group just follow the links under this video or any video Okay, I had my this is a lemon tea. I had my lemon tea a Little Robin made all the PPTs. I think we're ready to go here. So let's get into it boom This is vocabulary one And let's let's look up at the sentence as so it says Although he is famous He is very Blank now we got to choose our vocabulary We have to choose the best answer So we have one popular two approachable three rich for easy giving Five you could see five So to make things quick you could just put the number of which vocabulary you think it is One two three four five you could just punch that To see What your guess is and you got to think about this although he's famous. He is very Love lemon tea. Layla is more concerned about the tea. All right.  Yeah. Yeah. Okay. I got your popular So one two four So this is this I Think half of you got it correct, but half of you got it wrong. So let's take a look The keyword is although so remember Although is similar to a bot Just we're using it in different grammar grammar form or we were different structuring the sentence differently So although he is famous. He is very Now I have the word famous Now anyone who is famous Anyone who's famous is popular so Be if the sentence was because he is faint because he is famous. He is very popular That makes sense, but although he is famous because People all famous people are popular aren't they? Unless they're infamous or notorious which is kind of bad Famous, but we're not I'm not going to talk about that popular like, you know, Tom Cruise is popular He's famous and popular So usually famous and popular go together, but this is although he is famous He is very so Let's let's reconstruct this sentence using but So he is famous but So remember all those Brother is but if we use but that means we got to use something Unexpected or the opposite So although he is famous. He is popular popular and famous go together. They're not opulence opposites and Rich can't work. Some of you said for easy giving There's no expression called easy. There's no expression easy giving. There's easy going But this is not easy going. This is easy easy giving This is easy giving which is not an English expression number five is Quite that's not quiet.  That's quite. So it's definitely not number one Not number three not number four not number five and as half of you got that it is approachable So although he is famous. He is very approachable now approachable. That's an adjective and It means friendly and easy to talk to easy to understand So when we think of famous people we think they are not easy to talk to right? I cannot call up Tom Cruz. Hey buddy. What's going on? He's not approachable It's you can't talk to him But in this sentence Although he is famous. So although he is famous. We expect famous people to be unapproachable He is very approachable. So he is famous but Approachable okay, and I think I am an approachable English teacher or youtuber A lot of youtubers are unapproachable. You can never talk to them But I think it's very easy for Any of my students on this channel or the other channel if you want to talk to me today You can talk to me today. It's very easy. I'm very approachable very approachable Yeah So all of you who put five I'm very disappointed because that's not quiet.  Let's see what you guys are talking about Because and although are not the same word But and although are the same Layla I thought easy going. Yeah, I put easy giving again. I wanted to confuse you And I wonder if little Robin is approachable since he's famous. Well little Robin His social media is private. So he's unapproachable. He does not make friends and Right, you don't like you don't like talking to strangers or students. So he just talks to me He's very unapproachable if you want to send him a message you got to go through me. He will not talk to you and Gertrude is Tom Cruise is popular, but he is not accessible. Yeah, that's another good expression Not accessible, but that's very formal work more common would be he's not approachable Alyssa oh Alyssa's here. Okay, and start them all sure. Hello.  Hello. Welcome back. I forget the year your Zaynen All right, let's go into number two. Oh Monday, Krispy's here Love Kumar a few people jumped in Okay, we're going to the number two here number two. All right Number two says sorry the flight is booked all seats are One unavailable two available three occupation for busy five filled in and This one should be a little easier. I got 10 words today. Some of them are easy. So I'm a little difficult But maybe Lollie Lollie gets them all Lollie Lollie Lollie makes some mistakes Maybe once every six months Marine chimisee. Yes, you're approachable. Thank you. So what do you got one five unavailable one I Agree to say Robin is approachable. Thank you, Michelle Alyssa's one a Love Kumar surely five Love Kumar surely five must be filled in Lollie Lollie. I'm watching you. I love it when Lollie Lollie makes mistakes Okay, John Lollie Lollie always types too fast. Yes She has good internet she has no delay Sarah I got disconnected. It was too bad. All right. Let's take a look. Sorry the flight is booked all seats are I Start the monster puts one or five now filled in Let's talk about five filled in makes sense. But actually We might use filled up to mean the seats are full or all the seats filled up We don't usually use for filled in to mean people are sitting in the seat the seat is filled up or Usually forms we fill in we put in the answers So fill in kind of sounds right and in some specific situations that I cannot think of It might work But here it is not the best word. It is not the best word and the best word is number one unavailable and You have to understand in the sentence sorry, the flight is booked Now booked Means full no more seats. It's booked booked full All seats unavailable. There's no so unavailable the opposite of Available you can't get by or find something it's unavailable. So all the seats are unavailable The flight is booked Yeah, filled in usually feel you might filled up would work better than filled in So yeah, I added I added a few words to confuse people and let's see most people put Occupied Yeah, you can use occupied Little strange to a little all seats are occupied that can work But I didn't have occupied here number three says occupation because I was hoping some people confused occupied in occupation The flight is complete that can't that can't work The flight is complete the flight is full the flight is booked or booked up We can also use a phrasal verb booked up all the seats are Unavailable no seats available. Okay, let's move on to number three This is a political one Hey, may I see you? Layla you always want us to make mistakes. Yes. I want to trick you. I want you to make a mistake I don't like perfect people If you're perfect, you're now you're not I can't teach you anything. Oh Lopez st. Val what does valable stand for I? Put available there is no word available.  Number two. Valable is not a word. It's just to trick you So I don't want to confuse it. That is not a word should be available All right going back to Love Kumar surely for You guys sure you're picking for are you sure? Or everyone's just copying lolly lolly because they have confidence in her Well, let's take a look me and Mars New government so last last few days Last week last I don't I don't remember which day maybe last week Thursday Friday. I don't remember it. Sorry They had the coup so the military took over the government suddenly The democratic government they took over which is Terrible terrible thing and the new governments, you know, because they took over they want to control information so they're Sometimes blocking internet blocking Facebook Excuse me blocking YouTube stuff like that And I got through some words here number one author that's definitely not author number two is sweet. They're very sweet I don't think they're very sweet number three is very politician politician is a noun doesn't work and Number five authority. No, it's very authority. No, it has to be number four Authoritarian and that's a big word guys authority authoritarian authoritarian Layla you should just guess the meanings what you think it is you don't have to get perfect. I Really don't want you guys going through the dictionary check you get coming back and then putting the perfect answer I'd rather you just take a guess When you take a guess then I know how everyone's thinking So I consider checking the dictionary cheating, right? Let's get a little Robin's opinion Yeah, he definitely thinks going to the dictionary is cheating. All right, me and not me and Mars new government is very authoritarian and what is authoritarian? It's an adjective Believes that people should obey authority authority the people in power authority and rules Even when these are Unfair or even when this means the loss of personal freedom So governments that have unfair rules They expect you to obey the rules. You cannot protest the rules These are authoritarian Governments and that's gonna be like me and my right now Maybe Thailand we can throw in there.  Can we say it's dominant? The dominant has a little bit different meaning Dominant doesn't necessarily mean Nate is not negative authoritarian is negative. It's a negative word if anything is authoritarian. It's it's You know, you could have you could have authoritarian parents. That means they're strict parents So it's not just governments you could have authoritarian parents authoritarian boss So it's huge as negative dominant is used for other situations and dominant is not a negative word Tyrant yeah a tyrant and a dictator their government is authoritarian Robin be careful. You don't need to be in trouble. Yeah, it's you know, these days You're not allowed to give opinions anymore. So if I give my opinion about something YouTube might Cut my video. All right, let's go on to number four So mana treyore. Hello. Welcome. Welcome Kertrude's we have a democratic government good to hear here in Korea, they had Long history of authoritarian governments, but democracy won And it's too bad Myanmar, Thailand. They really tried for democracy, but they're struggling Going through that. All right, number four we blank interviews every two months We blank interviews every two months one example to conduct three section four grade five eight and the answers are coming in Oh Lolli-Lolli says three Three conduct two is conduct. I'm confused Lolli Lolli sleepwalker to conduct carry grade two to Layla for Love Kumar surely conduct Conduct to to All right, we have two and four Carry, okay, all people here said two so I should change my answer Carrie is having doubt in her own answer and is just going with the majority That's a strategy on a test cheating All right, let's take a look So we need a verb We need a verb here example example is not a verb Conduct is a verb section is not a verb grade is a verb and Eat is a verb. So we need a verb to four and five can work as possible We're gonna get ready to eat We don't eat interviews. So that's a silly word if you said five Nobody said five. Thank you. That's out. So we're down to two which you guys figured out It's either number two or number four and most of you are saying number two And I can see yeah, most of you and the correct answer is number two. So conduct So when you conduct something you you know conduct has remember all these words have various meanings not just one meaning and conduct conduct is a Few meanings but the meaning we're looking for here is using it as a verb conduct And it's to do something. It's a formal word To organize or do a particular Activity so I used it for conduct an interview interview is a formal activity, right? So let's do an interview. Mmm. Let's conduct an interview very formal of doing Okay What else do we conduct? We conduct a poll or a survey Politicians conduct surveys they do surveys Hey, we in we talked about me and Mars government All right, let's go sleepwalk eat the interviews is funny. Thank you for Appreciating my jokes number five She has blank levels of sugar in her blood One delicious to abnormal three hesitate for quick five highly She has blank levels of sugar in her blood You conduct our business. No usually we just say conduct business. I need to conduct business. I Need to conduct a meeting I'm meeting. I think there should be a preposition before every in the sentences Go No No, no, no, no, no, I don't know where you got that love Kumar In every no, no, that that that is not a rule. All right. Let's take a look again She has blank levels of sugar in her blood One delicious to abnormal three hesitate for quick five highly The answers are coming in I see two and five two five two five May May are you a little bit behind some of you are still answering conducted? There might be a delay two five two five Well, I'm glad nobody chose number one Layla, what did what did you say Layla? You didn't you didn't make it. Oh, you said five I'm very happy when I trick with my students. That's my joy in life Confusing my students.  So that is possible. Maybe for a vampire, but on an IELTS test No, we're not going to use delicious levels of blood and Number five highly No, we have to use just the adjectives High She has high high levels of Sugar that can work not highly levels Okay, so highly doesn't work, but high would work, but I didn't write high I wrote highly that doesn't work Hesitate and quick they don't work. So yeah, the correct answer would be Abnormal abnormal. What is abnormal? You can think of normal levels normal levels of sugar in the blood That's good and abnormal which is not normal So the opposite of normal Adjective Different from what is the definition here says different from what is usual or expected Especially in a way that worries somebody or is harmful were not wanted So when we say she has abnormal levels of sugar in her blood This is bad this is a bad thing We got to worry it might be bad it might be helpful something like that Not normal Like Alyssa's abnormal Well, this is personality is highly Abnormal delicious sure Robin is a tricky teacher. Yes Yes, I am so You know if I'm losing hair. I'm not I am losing hair But I'm not if I'm losing a lot of hair in the shower. I might be oh, this is abnormal hair loss Not normal abnormal All right, let's go to number six You guys are putting in your answers and I'll read it out for Alyssa number six, we could go to the Thai restaurant or Blank we could try that new Italian place So we could go to the Thai restaurant or Blank we could try that new Italian place number one Alternatively number two secondarily number three accept number four in doubt and number five optionally so we could go to the Thai restaurant or Alternatively secondarily except in doubt optionally we could try that new Italian place So Let's see what you guys are doing All right sleepwalker one alternative carry is five one alternatively Alternatively for What's for how you deleted it? Love Kumar surely alternatively secondarily Wayne secondarily Alyssa one Five optionally Well all of these words are very similar family, but I'm tricky. This is very tricky guys. This is probably The hardest question yet Monday crispy this one kind of hard. That's beautiful words. I'd love to hear from my students like gets you thinking Alex alternatively And lately I can't imagine the meanings for any of them We shall alternatively Johnny Philip. Hello, buddy. All right. Let's let's take a look We could go to the Thai restaurant or We could try that new Italian place now the correct answer is Number one alternative So let's take a look at alternatively Used to introduce a suggestion that is second choice. All right alternatively Alternatively is a good word to say a second choice or possibility. All right So let's go back So we could go to the Thai restaurant or alternatively alternatively a second choice We could try that new Italian place so secondarily That's Native speaker would know right away. It's not secondarily. Okay secondarily sounds like it could work But we don't use secondarily As a word. I don't think it's secondarily a word. It must be Yes yeah, secondarily is Is a is used when something is less important now in this case These things are equal importance. I wish I could have more time to teach secondarily, but it doesn't work Except is wrong in doubt is wrong now optionally. That's interesting Optionally sounds like it'll work but it doesn't work the Very quickly a native speaker would know that it's alternatively Which means another choice another equal choice is we could try that new Italian place. So yeah guys, it would be alternatively Let's see. I think If you don't know like this is a good word to use guys, we'll practice this a little later alternatively Mira and Sarah. There's a little bit of delay for you guys so I think you're Hopefully you're talking about number six Samira one alternatively. Yes, Sarah secondarily. No, all right, so Yeah, alternatively another choice. That's a good word to know we'll practice that late again later Let's move on number seven pollution levels have blank increased these days Pollution levels have blank increased these days number one considerably number two exactly number three specifically number four calmly and number five eventually So pollution levels have considerably increased exactly increased specifically increased calmly increased eventually increased Carry it's not even easy to pronounce. We will practice pronunciation later Samira hush man. Thank you for coming. Oh Gertrude's That's a good answering system seven one Everyone look at Gertrude's Put vocabulary seven seven one. That's great. That would let me know if there's a delay on some of you people so Put the question number and then put your answer So this is vocabulary seven so seven one seven two seven three whatever your answer if you got a delay Some of you have a delay guys. Some of you are Like I said five second delay. Some of you are one minute delay So it's a little difficult sometimes So Gertrude is thanks for the put in that answer. Anyway, what do you guys put? Lolli-Lolli says considerably sleepwalkers specifically win specifically specifically Samira Considerably can Alex considerably Gertrude is considerably considerably Alyssa considerably Layla maybe one Star demolisher one Michelle one love Kumar surely considerably Zuma not rare. I'm not sure if you got a delay, but you put five eventually nicks nicks one marine Chimesi five eventually me Considerably start a monster. Thank you seven one considerably and Sarah Tara hurry Taherian Hope I pronounced that correctly number one considerably and marine Chimesi you have a delay here too. Yeah, and everyone has a delay of something All right. What's the answer here? Well, the answer is Considerably number one. So most of you got that right and Considerably means much a lot. That's it considerably, I have a Considerly considerably amount of hair just means a lot. So it's a big word Considerably five syllables. That's a very big word difficult to say, but it just means a lot a lot So pollution levels pollution levels increased a lot that's all means considerably considerably Exactly increased doesn't make sense specifically increased I've specifically increased to Whatever parts per million if you had a specific answer it might work calmly pollution levels don't calmly increase and Eventually what kind of mean in the future? It says these days so we can't use eventually eventually means Pollution level I could say pollution levels will eventually increase over time So in the future we could use eventually so eventually means a time in the future not these days So considerably just means a lot so if you guys knew considerably this was very easy And if you didn't know considerably you now have a great word To write in an essay that means a lot Considerably we'll worry about pronunciation at the end Layla tricky word. I have no experience with this vocabulary. Where did you get them? These are IELTS vocabulary So these are good words to improve your speaking use improve your speaking and writing and they're all very popular They're all very common every native speaker knows these words and so should you Eventually grace you got a little delay eventually doesn't work Eventually means a time in the future. This says these days All right, let's go on to number eight Number eight says the computer generates a random blank of numbers Number one option number two digits number three sequence number four basic Number five category the computer generates a random option of numbers a random digits of numbers a Random sequence of numbers a random basic of numbers a random category of numbers Which is the best answer. All right. Thank you. Lollie. Lollie Alex's sequence Monday, Krispy. He says digits Lollie Lollie says sequence. Alyssa's the sequence. Layla says option Coutreuse may sequence grace Is still on seven one Got a little bit delay. What was seven one considerably? Grace's right? love Kumar option grace Changes it to seven five. No, don't change it eventually Best answer is Layla's answer Yes, Layla Rain Chimazee three is that eight three Nick's eight three sequence sequence. All right, a lot of you are put in sequence Well, let's take a look the computer generates a random option of numbers Random that's a little strange. What is an option of numbers? It's a little awkward The computer generates a random digits of numbers digits means digits has two meanings digits can mean fingers and digits can mean numbers So a random numbers of numbers that doesn't make sense a number two is wrong because digits means numbers now sequence That is the correct answer Basic of numbers doesn't make sense basic is used more as an adjective We need a noun here and a category of numbers while numbers is already a category so The computer generates a random sequence our word to know is sequence of numbers sequences a noun a Set of events actions or numbers which have a particular order at which lead to a particular result so a sequence is similar to order and We have sequence adverbs in English sequence adverbs are like first next after that Finally a sequence an order. Okay So the computer generates now a lot of you what is generates? That's a big word makes it makes it creates Random no a random order of numbers a random sequence of numbers Okay, thanks. Wow. I'm correct. It's me. It's easier to do it Grace am I in delay? Yeah, you're on a little bit of delay but if you put Whatever Answer you know guys if you some some people pause the video if you pause the video you're going to delay the video So never pause the video and you got to keep watching live, but anyway, Grace you just put your Eight one eight two and I'll get it starting to molesters at eight two or eight three. Well That was a three Sequence Okay, let's move on. There's only two more. Let's move on to number nine Number nine the first blank for the new leader is to focus on the economy So the first number one task to List three options for purpose five Transfer So the first task for the new leader the first list for the new leader the first options for the new leader the first purpose For the new leader the first transfer for the new leader is to focus on the economy Michelle sequence is the most difficult to find until now for me sequence. Good word to know Michelle may practice a few sentences Desk desk body is here You're late. You have to do late students do 10 push-ups So send me a video in the WhatsApp group of you doing 10 push-ups. I love Kumar surely task 9 5 transfer 9 1 9 2 so we got tasks 9 1 9 2 is a list 9 5 transfer Zumana trari 8 4 let's go back to 8 8 Was 3 8 3 sequence All right, so my day. I was sleeping. I still it's no excuse a teacher doesn't care about excuses All right. So number 9 the first For the new leader is to focus on the economy. Let's see Let's see what Laila put Laila said 9 2 Womp womp Sorry Laila the first list The focus on the economy is not a list And It's not 2. It's not 3. It's not for the first purpose Purpose I know it sounds okay the The leader has a purpose But we we don't usually there's only one purpose We don't say the first purpose the second purpose you can say that but it sounds a little awkward. You would say the first Reason second reason But these are not Purpose really or reason all right and 5 is wrong. So Alyssa you might have misheard 5 is definitely wrong. The answer is task So a task is like a job. You got to do a job So the first job for the new leader now a task is like a smaller job So the definition it's a noun a piece of work that somebody has to do especially a hard or unpleasant one You know these are tasks you could also say chores Something like that. So the first task the first job for the new leader is to focus on the economy So that's his job first job first task so that was 91 and a lot of you put four First purpose sounds awkward if we got rid of first we said the purpose the purpose For the new leader. Yeah, the first purpose is just awkward. Sorry guys Layla I wanted to choose number one, but actually I put number two sure Layla Grace everybody talking about question 9. I'm still in question 8 or anybody grace Love come on, but after purpose the preposition will be of yet purpose I'm gonna say Generally purpose of is correct But there might be some situation where there's purpose for I can't think of any but purpose. Yeah, I love come on. That's That sounds right purpose of grace 9-1 task you got it right grace All right, let's go on to number 10 I've just finished an eight-hour blank at the factory task Occupation career job shift I've just finished an eight-hour Task at the factory. I've just finished an eight-hour occupation at the factory I just finished an eight-hour career at the factory I just finished an eight-hour job at the factory and I just finished an eight-hour shift at the factory one, two, three, four, five. This is the last one, guys. This is your last chance to guess. Arabic guy, number four, it could be true. I think you're talking about number nine. Purpose, no. Like commarser, we usually say purpose of. All right, what are you guys saying? 10, five, shift, shift, shift, shift. And lolly lolly is going with job. So you guys sure about shift? You wanna change it to job, like lolly lolly? Layla says maybe four in 10, four, job. Love, Kumar, task. We just had task.  And number nine, Michelle says shift. Marine, Shemazi, task. Arabic guy, shift. Wadi says shift. Wow. Let's get to the answer. Shift, so lolly lolly says job, but it is shift. But job is not exactly wrong, but we'll talk about that. So let's go back. I've just finished an eight hour shift at the factory. Now we have to know what shift work is. Shift work is when like a factory might be open 24 hours. And in 24 hours, you can have three shifts. Person works eight hours, another person works eight hours, another person works eight hours. That's shift work, shift work. Also hospitals, doctors and nurses, they have shift work. So they might have morning shift, afternoon shift, night shift, something like that, shift work. So this person says I've just finished an eight hour shift at the factory. Now for those of you who chose task, eight hours, usually if something is eight hours, we don't call that a task. So just the leader of a country can have a big task, but as normal people, a task is usually small, like one hour or 10 minutes, something like that, a small little task. Checking email in the morning is a task, but eight hours, that is not a task. That's too big for a task. I just finished an eight hour occupation. Occupation means job, but this doesn't work here. Occupation career means job, but this doesn't work here. Occupation career, that's like teacher, lawyer, doctor, and that just does not work here. Now, Lolly Lolly went with job. I've just finished an eight hour job at the factory. Now, Lolly Lolly, that can work. You can, anyone who's put job, actually that could work, but we are looking for the best answer and a native speaker will recognize right away that if we said eight hours, that is a shift, a shift at work, an eight hour shift. So the time working, so shift, usually shift, a period of time worked by a group of workers who start work as another group finishes. Like there's usually, there could be three shifts at a hospital or a factory, or there's other jobs with shift work. And Lolly Lolly, no, I'm wrong. You're not wrong, Lolly Lolly, but your word is correct. Just the best word would be shift. A native speaker would definitely put shift, eight hour shift, eight hour block of working. All right, let's practice. Here's your homework, but I wanna practice on some pronunciation. So I'm gonna go through these words. They're very big words. And let's practice a little bit of pronunciation. Number one, approachable.  Approachable, everyone. Approachable, approachable, approachable, approachable, four syllables, approachable. Number two, unavailable. So we have the base word available and the opposite, unavailable. Approachable, unavailable. Number three, a little bit bigger, a little more difficult, authoritarian. Authoritarian, authoritarian. So Myanmar's new government is authoritarian, authoritarian, authoritarian. Approachable, unavailable, authoritarian. Four, conduct. Conduct, conduct. Number five, normal, opposite, abnormal. Abnormal, abnormal. Abnormal. Number six, alternatively. Alternatively, alternatively. So say that out loud, guys, alternatively. Alternatively, alternatively, alternatively. Alternatively, number seven, considerably. Considerably, considerably. Number eight, sequence, sequence, sequence. Number nine, task. And number 10, shift, shift, shift, all right. Which word was the most difficult? Tell me. Love, Kumar, the new leader is approachable. Who is approachable? Sleepwalker, I like vocabulary workshop. If you like it, please press like. If you like the video, hit like. So, Mahdi, I used to work for a company that had three shifts. Make sure you got the F or the S, three shifts. What does it mean? What are you talking about? Maybe, Mahdi, you missed the beginning. I'm not reteaching. I'm not teaching all these again, just for you. You can watch the video anytime you want. So, Layla says, number six is alternatively. All right, let's, let's practice. So, let's see. Let's go to Google Docs here. These are from last week. Let's remove that. Let's put our list here. If you guys have any questions, I'll answer your questions about the list. Let's just put our list here. All right. So, when you're giving suggestions to people, so let's give a suggestion. Someone asked me, how can I improve my English? Get that question 100 times every day. You guys cannot imagine how often I get this question. So, someone asked that question. How can I improve my question? I could say, you must read every day or I want to give a second option. Alternatively, go to the US. All right, so I want to give a second option here. Alternatively, so another option that works. Any other question? Authoritarian, so yeah, okay, so that was alternatively. Authoritarian, dictators, dictators are very authoritarian. So remember, authoritarian, they make rules. Those rules might be unfair. People might be getting hurt, but they make the rules and they expect you to follow those rules. You cannot protest those rules. So, dictators or you could even say, my parents are very authoritarian. Parents make rules and the kids must follow the rules even if they're unfair. And teachers are very, well, depends on the country. Korea not so much anymore. Star demolisher, that cute girl over there is approachable. So, star demolisher needs some really practical use for this new vocabulary. Yeah, you could tell your friend that cute old girl over there is approachable. And you could also, you could say it looks approachable. Looks like I could just go up and talk to her. I don't know yet. It looks like it. But if you do know, yeah, that cute girl over there is approachable. Good luck. Anyway, I don't like vocabulary. The work trap is so hard to get the meanings even though I want to learn something new. That's why we're doing it now. Alyssa, authoritative versus authoritarian. I think it's just gonna go, I mean, I'm just checking dictionary. So Alyssa, authoritative, the definition of authoritative is showing that you expect people to obey and respect you. So we usually describe my teacher had a authoritative voice. She sounds like she wants people to. So it's describing kind of the action of the person. I guess you could describe the government, but authoritative is better way to describe people of power in this case. So think of authoritarian as a more powerful very unfair and authoritative is just powerful. So let's make it simple. Authoritarian, powerful and unfair. Authoritive sounds powerful. Okay, I hope that helps. You win. The military here is very authoritarian, yep. And when I use that example for vocabulary number three here, Myanmar's new government is very authoritarian, number four. So I use that example. I know you guys are struggling there, so good luck fighting for your democracy. Can a boss be authoritarian? Yes, a boss can be authoritarian. Can I say even though instead of all though? Yeah, they're the same family. I can't think of every situation right now. I even did a video on that, but basically it's the same. It says even though is a little stronger. So a little more emphasis we use even though, but they basically mean the same thing. All right, let's take a look at Leila's sentences here. I want to ask Robin about something, but I am afraid he doesn't reply. I am afraid he doesn't reply to me. You can ask him. He is, you can ask, okay. You've got to change this to won't. So in the future, I'm afraid he won't. He will not reply to me. And your friend says, you can ask him. He is approachable. This is correct. This is correct. He is approachable. So it's easy to approach him, easy to talk to him. Love Kumar, what is the meaning of over there? We're not talking about other vocabulary. Sorry, love Kumar. All right, I think, oh, Michelle has a sentence here. My first task is to write a sentence with the new words. This is five-star example sentence. This is perfect, Michelle. So my first task is to write a sentence with the new words. I'm gonna say your only task. You only have that task, write a sentence with the new words. But that's a good example. My first task is to write a sentence with the new words. Any other questions about this vocabulary? Now is the time. Oh, sequence. I didn't talk about sequence. Sorry, Leila. So if we have one, two, three, four, five, this is a sequence. If we have five, 10, 15, 20, this is a sequence. If we have 19, 10, 19, 11, 19, 12, this is a sequence. If we have a coronavirus detected in China, spread to the world, we're running out of room here. See if I can move this up. Vaccine is coming soon. All right, so we have, this is a sequence of events. This is the first event. This is the second event, and this is the third event. It's a sequence. It happens. So think of sequence as just order, the order of something. One, two, three, four, five, five, 10, 15, these are 19, 10, 19, 12. This is the order. First is this, next is this, next is this. That's the sequence. Zumbana, one and seven are difficult to pronounce. Let's try it again. Approachable, approachable, approachable. And considerably, considerably, considerably. And love Kumar, the cars are standing in a sequence. Now, this is a confusing sentence. Love Kumar, cars don't stand in a sequence. So, I'm gonna change your verb here. Arriving, so there's an order. The cars are arriving. Alyssa has a sentence. Oh, I got a spelling here. Arriving. Alyssa wrote, people are obliged. Okay, they must oblige to, I'm gonna make it oblige to nothing wrong with that, but I'm gonna just say must to make the sentence simpler. People must wear face masks. Alternatively, they can use a biohazard suit. I'll change that to suit. So yeah, so people must wear face masks, but alternatively, another option, another thing you can do is use the whole biohazard suit. Those are the suits you get in, they cover your entire body. So it's your choice. You could wear the face mask or the whole biohazard suit. Number five, abnormal, ab, abnormal. Just, I'm gonna put a little break right there. Think of it like that, abnormal. But don't say abnormal like two words, abnormal, abnormal, abnormal. All right, starred demolisher has an interesting. The amount of hours in his shift are considerably abnormal. Great, this is another five star example because he's, let's fix your spelling. He's using shift, he's using considerably, he's using abnormal. All right, so maybe let's say he has a 15 hour shift. So if he has a 15 hour shift, the amount of hours in his shift are considerably abnormal. Yeah, maybe a 20, doctors, sometimes doctors have 20 hour shifts, right? That's considerably abnormal. My brother, my sister, and I are the sequence of my parents. Well, now it's the sequence of birth. So sequence of birth. Now, in this case, you're talking about your own family and sequence is a little bit formal. Okay, sequence is a little bit formal. So if you're just talking about your own family, that is the sequence, but probably we're just gonna go with order. You could say sequence, but again, it's too formal. If you're just talking about your own family, the order of birth. Wean is biohazard suit, PPE suit. I don't know what a PPE suit. Let me check. Yes, looks like that. I'm just looking, this is PPE. No, I think the biohazard suit. So the PPE suit, what would that be? Personal protective equipment. So this PPE suit, they don't really cover the face, but a biohazard, it covers the face with the mask. You guys can see that. So I think the biohazard suit is 100%, nothing gets in like that. That's definitely a biohazard suit, but the PPE looks a little more practical. Like it is technically possible. I think just the biohazard is stronger. So I think probably PPE suit is what people are wearing these days. They're not really wearing some places they might have biohazard suit. When I went for my coronavirus test a few days ago, they just had this PPE suit. Let's take a look at Leila's. The restaurant is only available for a few people. So after 10 p.m., you couldn't go because all seats are unavailable. Yep, that works now. Gertrude's, in my last shift, I had to assist a lot of people with COVID-19. Yep, that works. A lot of people. So a considerable amount of people. Sequence of reports, that can work. All right, guys, I'm gonna finish this live stream now. Thanks for watching and I hope you learn some new vocabulary and of course you can do your homework. You can write the sentences. Come back in two days. Don't do it now. Come back tomorrow or the day after. In the description of this video, in the comment section of this video you can write out some other sentences to practice using these words. So you can remember them. All right, I hope I helped you guys today and I'll see you next week and be sure to check out teacher Sherry's video this week. So I'll leave it in a minute. Any last final words or goodbyes, now is your chance. Tell me that you're still here. Who's still here? Stardomolisher, thank you. And Sleepwalker, Lolly Lolly, thank you. Wynn, thank you. Alyssa, thank you. So only those five people are still here, thank you. Sarah, okay. Okay, sorry about your connection, Sarah. Grace, the dresses here are considerably cheap. I'll just fix up Grace's sentence. She has a little delay, so let's make sure we got her sentence. Dresses here are, let's put a B verb there, are considerably cheap. Works perfectly. Petrudas, thank you. Lolly Lolly, thank you. All right, oh Grace is saying bye now. All right, thanks a lot guys and see you next week. Take care. Thanks, love, Khmer.


And this is by Peter Forrester, speaking about decomposition of measures in random matrix theory applied to number theory and integral geometry. I've clicked my microphone on quite a wild echo here. Why is that? Try over here. Still got an echo, but take two. Right, take two. Yeah, the echo is gone and I'll resume. Ten years ago, I was fortunate, well, I was going to start by thanking the organisers, particularly Alice and Ivan and in the background, Alexi. Ten years ago, I was lucky enough to be in the state of Utah for a meeting organised by the random matrix community. I think Dino had something to do with that meeting. One of the participants was Ballant Varag. Ballant was talking about his newly introduced Brownian motion model relating to the beta ensembles, and he wanted to explore beta-dependent quantities. And he asked a particular question of Alexi, and Alexi passed the question on, or Ballant, over to me. So perhaps say something about this question. So this question goes back to the times of Meteor and Dyson. Well, it's actually a theorem from their earlier papers, where they consider the circular ensemble, the beta equals one, with two end points. Circular orthogonal ensemble. So with the two end points, the eigenvalues on the circle. Now I didn't keep count there if I got an even number. Well, we'll find out. If one integrates out every second eigenvalue, we get left with a new point process. And the theorem of Meteor and Dyson is that every second eigenvalue, the operation they introduce there is alt, is actually equal to the PDF defining as an eigenvalue PDF, the circular symplectic ensemble. So that's going between beta equals one and beta equals four. I was very interested in these sorts of interrelations. And what Ballant asked was, let's suppose we take, now that we've got particular interest in general beta, let's suppose we don't restrict to just the classical values here, one and four. We take another beta, and the simplest example would be, can we find a value of beta such that if we integrate over two eigenvalues in a row and observe the third, integrate two in a row, observe the third, integrate two in a row, observe the third. Is there a interrelationship between a certain circular beta ensemble? We have to nominate what beta is here. We'll have three n eigenvalues. And the answer is, in fact, there is. And the interrelation goes between beta and four on beta. Well, this is when beta, in this setting, what's it going to be? So I've got to get my values correct here. So we're going to have beta, the usual interrelation is between two on beta and so beta equals one. It's going to go, let's excuse this, slightly impromptu, but I won't continue. But to say that we can find a beta value that for this relationship to actually hold, we can find a beta value that if we integrate out three eigenvalues in succession, we can find another, there we would have a four n to n, et cetera, et cetera. So this is something that actually came out of discussions in a conference like this. And I think that's one of the great advantages of having people together and hearing about different perspectives. So at a technical level, it's also of interest how can one make a contribution of this sort? Well, I have my own technical tools. My technical tools were working with Solberg type integrals. So I knew how to actually perform integrations over these sort of domains and I was very popular in studying models relating to share processes, McDonald's processes, et cetera, where we have interlacings and I knew of such integrations and I knew that perhaps one could extend this with values of beta although it wasn't obvious at the time how to break away from structures that are special to beta equals one, a determinational structure that appears there how to break away from that, but it was possible to get extended beta. So here, this would be actually thinking of it, beta equals six. And on this side, well, I think we go down to a third but I'll double check that afterwards, but it's certainly a theorem. It's a theorem that holds for finite internet. It's something that came out of the discussions here 10 years ago. Hopefully, as Percy Deaconis said, I will try and pose some questions where I'm at in this present line of study where I'm a little bit stuck on some points and maybe I can get some advice that will help me out. So let me talk about the material for today. The material for today is a little bit ambitious. Less is better, so I perhaps won't continue on to the very last dot point but I will give an indication of why I'm interested in that very last dot point. So let me get underway. Problems in number theory and integral geometry are not necessarily mainstream. It's random matrix conferences. It is a particular line that we could follow if we were to take an historical perspective and ask where do certain studies within random matrix theory show themselves? And if one was to actually start off with the work of Hurwitz, one would actually be giving a talk of this sort. So I did start off with the work of Hurwitz. I found a collaborator in that pursuit, Percy Deaconis over there and wanted to explore further that particular lineage and this is what comes out of such a study. So I will talk about three particular problems. Concrete problems. So the first one will actually be counting matrices with integer entries and determine it equals a one. So they form a group, the simplest example, 2 by 2 case, SL2Z. How many matrices are there with a bounded norm when that norm gets large? That's an asymptotic counting question and it turns out that question can be answered through notions of random matrix theory. The second question I'll inquire about is to do with a random lattice. A random lattice is defined by taking a basis chosen with uniform measure and I'll explain what uniform measure means. So take this again, just take the 2 by 2 case. We're just considering two vectors, linearly independent. We're going to form now the integer span of those two vectors. That will give us a lattice. We've defined the lattice as the integer span of our two given vectors. Within that particular lattice there'll be a smallest vector and a second smallest linearly independent vector. And the question we'd like to ask is what is the actual statistical properties of that smallest vector and that second smallest vector? And that is again a question that can be answered through notions in random matrix theory. The third problem that I'd like to discuss is one of statistical properties of the convex hull. So Santa, sitting over there, the ginger has studied convex hulls relating to Brownian motions. What I'd like to talk about is convex hulls, for example, as the concrete example take three points in a disc at uniform, uniformly, form a triangle and ask about the statistical properties of that triangle. How does one actually perform calculations like that? That would be my last topic if I had time. So let's proceed. This is the first motivating question, asymptotic counting problem. As I say, very concrete. If we just think in the 2 by 2 case, we're asking ourselves how many matrices are there that have some bounded norm? And notions in random matrix theory are relevant because of an asymptotic counting formula due to those particular authors, Duke Rudnick Sane, 1993, where they say that asymptotic counting formula is related to a ratio, one on the volume of the fundamental domain. The fundamental domain is an arithmetic quantity that's actually known. So in n equals 2, it's z to function at 2 pi squared on 6. The remaining question is a random matrix problem question. How do we actually integrate a volume where we're choosing invariant measure? I have to define what the invariant measure means. On the space of matrices in the n equals 2 case, SL2R. So that's the question. We'll be answering it as we go along. The second problem I made mention of is this random lattice problem. Random lattice problem is very much in the lineage of invariant measures as we'll see. So here's a picture of what was going on. v1 and v2 are our original basis vectors that we're spanning with an integer span. The question we would like to know is what is given v1 and v2 are actually chosen uniformly at random in a sense that we'll be made clear in a moment. What is the distribution of the shortest basis vectors, u1 and u2? Very concrete question. You can see it. It's a question in geometry of numbers. What about general dimensions? So n equals 2 hasn't necessarily got that much prestige in our random matrix community. The large n limits are usually preferred, but still. n equals 2 is very good for drawing pictures. Here's n equals 2 again. Here is the third and final problem. This problem is not a number theory problem per se. It's a problem in integral geometry. Or the geometrical probability theory. So we've got a disk here. We're choosing three points uniformly at random. We're forming a triangle. We're asking about the statistical distribution of the area of that triangle. So here I've denoted that area by delta, or the region by delta, the volume of delta in general, the area. We could ask about moments, but let's be happy to compute the average moment. Now, I saw this remarkable formula and really got interested in this topic because of this remarkable formula. I wanted to see, understand why we could get the binomial coefficients with n squares in there. This is a result of Kingman. And I just sat down trying to actually compute this integral in the case n equals 2. So we can try that. We've got how much time I've got left. Let's see if we can do that in the next three quarters of an hour. It's an elementary integral in printable, but the method of computation is not elementary at all. Kingman's method. In fact, I read about this first in an old American math monthly article. And the question was posed. It wasn't answered except sort of a footnote to say that, I think the person's name is Clee, he was thanking Kingman for a note saying that he knows how to do this, but he kept us all in suspense. He didn't explain how that might be done. So if I get time, I'll say a little bit about how some of our random matrix theory gives rise to an understanding of a computation of that sort. So this is moments of the volume of a convex hull point. Okay, so the very beginnings that I would like to talk about is one of the beginnings of random matrix theory. We can say there's a number of different beginnings. There is this beginnings of Hurwitz that's not necessarily that high profile, the beginning in mathematical statistics due to Wishart, a beginning in theoretical physics due to Wigner at least. There are those beginnings. So the beginning due to Hurwitz goes right back to the days where invariant theory was of interest. Hurwitz following on from some famous work of Hilbert on a problem of finiteness problem, a particular notion relating to invariance of higher-order polynomials. Hurwitz got to ask a similar question in the continuous setting and he was led to the notion of what Percy introduced earlier as the Ha measure, the invariant left and right measure with respect to the group action. And he did a number of calculations that are just lasting interest, I think in random matrix theory, I'd like to make that point as we go. So what's the notion here? The notion is the definition of this left and right invariant measure introduced, as I say, by Hurwitz some time ago. Hurwitz actually gave a specification of that in terms of the volume element. So if we're taking orthogonal matrices, the Hurwitz specification is to take this matrix R, orthogonal matrix, R transpose DR. The meaning of DR, I'll give an example over the page. How can one see that that's actually invariant? What do we mean? Well, we mean that central equation, but let's look here for a left invariant. So we take a fixed R naught fixed orthogonal matrix, multiply on the left. We want to see that we actually want to see it at all. We want to see that we get a... that our nominated volume form that we're claiming is invariant under left and right action, group action, is actually unchanged. It's a very simple calculation using the fact that R naught is a member of the orthogonal group that that particular volume element's unchanged. So the correct combination R transpose DR, that's fundamental. And we can form a calculus out of that. And here is an example. So we'll take a parameterization. This is what Hurwitz did as well. One could give a lecture, and Percy certainly over the years, give a number of lectures relating to the use of Euler angles and generalizations for the permutation matrices. It's just quite remarkable that it all goes through for permutation matrices structurally. I'm just taking the 2 by 2 case. So we're taking a standard parameterization of a rotation matrix and using the formalism to do a calculation of what that actual invariant measure is in these variables. So one finds that R transpose DR is an anti-symmetric matrix. The meaning of the volume element is that we take the product of the independent differentials there. Well, there's only d theta that is independent. So our actual volume element is d theta for a 2 by 2 orthogonal matrix. If we went to the 2 by 2 unitary matrix, we'd actually have four variables. If we normalize the first and second column to have a positive real first entry, we just go down to our two variables and we can do the same calculation. We can form U dagger to U and then take the product of the independent differentials. It's slightly trickier here because we do have something on the diagonal. That's something on the diagonal is repeated. We have another DL for on the off diagonal. In fact, what we do in practice is multiply together the independent differentials, the real and imaginary part on the off diagonal only just as we did it in the orthogonal case. So this is the 2 by 2 setting. Very concrete final expression here in terms of the Euler angles in both examples. One has to know how to parameterize a general orthogonal matrix for general N and unitary matrix for general N in terms of Euler angles to proceed. The first person to do that was actually Euler himself in the N equals 3 case. Excuse this technical glitch. And apparently in that paper, Euler sketched how to do this for general N. Herwitz repeated Euler's parameterization in the real case and in the case of ON for UN he gave a generalization was then able to give explicit parameterizations of these volume forms for general N and his application computed the volumes themselves. So I remember for our application for these asymptotic counting formulas we want to compute volumes and here is the very first volume calculation so this is way back in 1897. Well, perhaps the next major development along this line was while in his book 1939 on the classical groups he introduced this notion of a class function, a function unchanged by conjugation by orthogonal and unitary matrices and for that purpose he was interested in decomposing these invariant measures not in terms of Euler angles but in terms of eigenvalues and eigenvectors. So in his book he showed how to do such calculations technically they're not necessarily that difficult and one gets this well-known decomposition to measure that somewhat underlie what I was sketching out over there that we have these product differences to some power. We have a factorization of this eigenvalue part and the eigenvector part so the V's here is our matrix of eigenvectors so we see that invariant measure introduced by Hurwitz V-daggerty V appear in this particular calculation this is where the V's are have indeed the first entry normalized to be positive so we've actually got two lots of unitary matrices going on here where U-daggerty U on the left and V-daggerty V on the right that's Vile's calculation. In the 1940s Siegel got interested in the... well, had already been interested in the geometry of numbers he wanted to introduce this notion of a random lattice so to introduce in the notion of a random lattice he wanted the vectors to be unchanged by all elements of GLNR he wanted his measure to be invariant so let's go back to the... I don't even need to... two dimensional... can we go to the one dimensional case and ask ourselves what measure is unchanged by multiplicative group action so we want to replace X by alpha times X and have our measure unchanged the answer to that would be dx divided by X the matrix generalization just isolated here by Siegel very simple is just dividing out by the absolute value of the determinant of the power of N an example like I just said was N equals 1 in the study Geometry of Numbers these lattice that Siegel was interested in forming at random he normalized things so that a volume, a unit volume of cell was equal to 1 we know that the determinant of the matrix if we're thinking of the columns as defining the vectors gives us the volume of a cell parallel piped what one has to then do is go to a... I like to think of it as a distribution on this particular invariant measure that has a delta function when the determinant is equal to 1 now since the determinant is equal to 1 the factor that we saw in the denominator in that first displayed equation is no longer present and it's replaced by this Siegel distribution Siegel did not actually use this same formalism but the calculation, in a calculation sense it's actually equivalent so Siegel had some interest and I'll mention later in the talk he calculated or gave a mean value theorem using this formalism it'll come up later continuing studies of the Geometry of Numbers this guy Jack who's well known to us studying Jack McDonnell polynomials etc and one of the collaborators of a specialist at the time in the Geometry of Numbers this guy Macbeth they partnered together to calculate for a different purpose but they did a calculation which now could be used and plugged into that formula of Duke S.R. because it is the actual volume where we make a truncation according to a certain norm so a norm is chosen to be measured according to the largest singular value so that's the operator norm they show us how to compute the volume so we know how to compute the volume by integrating out over the orthogonal matrices that's her which result what remains is how do we actually do this computation over the singular values and we have to perform this particular integral after many pages they were able to perform that integral their purposes were very technical they wanted to show an equivalence of a particular limiting process under Riemann integration as opposed to Lebesgue integration they had very technical reasons for that purpose there's a different application in mind here but before that if you look back at their paper it goes on many pages and then if I flashed over it I didn't say it at all their final result is actually we start with an indimensional integral their final result is a single dimensional integral so that's the sort of my sort of stuff playing around with those integrations turns out integrals are more techniques that are about to revise here sort of come back into vogue there's an activity in integral aspects of random matrix theory for random matrix products and one of the technical tools there is using this Mellon transform so how can we introduce the Mellon transform here in a way that allows us with just in a few lines to compute this integral go from this indimensional integral down to one dimensional integral what we can do is introduce a dummy variable T into the problem before we can strain our determinant to equal one we can now constrain our determinant to equal this free parameter T and now we'll take a transform with respect to T a Mellon transform and the Mellon transform undoes the delta function for us that's fine and we know how to take an inverse Mellon transform so if it happens that by undoing the delta function we've got a multi-dimensional integral that's well structured then we've made progress and indeed the R dependence in this multi-dimensional integral depends on S and R the R dependence factorizes out we're left with something that's very familiar and occurred many times in random matrix theory it's a particular example of the Solberg integral the Solberg integral in turn is evaluated in terms of products of gamma functions so we've evaluated that multi-dimensional integral we still get left with one integral because we have to take the inverse Mellon transform so at the end of the calculation we get a single integral and that's good value the application we have in mind is a large R the large R form of that and the large R form of that is actually easy to to compute and we've gotten ourselves if I have the next yes it's down the bottom there we've gotten ourselves an actual prediction or corollary of knowledge of this or an application better still of the knowledge of this particular volume for the invariant measure of matrices in SL and R to see get a little bit better understanding of this result of Duke Rudnick and Sarnac one has to know a little bit about the geometry of let's again take the n equals 2 case very very well known famous picture up there the geometry relating to the quotient space SL and R SL and Z when n is 2 is hyperbolic geometry in the upper half plane where each of these particular regions is identical under this Mobius transformation so the the mappings of SL2Z themselves basically each of these cells is can be thought of as one of these particular matrices so all we have to do is count up the number of cells within some region but they're all equivalent so we're basically just work out the total area and divide by the area of a single cell so that's the way of thinking of their result so it's very natural result n equals 2 it's not even too easy to find an elementary derivation of this the final expression is that the number of matrices goes like 6 times R squared so remember R is the bound on the norm and turns out you can look at different norms and in the original paper of Duke et al they consider the Frobenius norm the here this is the operator norm you still get the same answer in n equals 2 only you get 6 6 times R squared that's the end of the first problem the second problem why we might be interested in exploring invariant measure following the line age of Hurwitz is this smallest shortest lattice problem so let's get under way with that I've stated the problem a few times you can think of as a minimization problem the if we specialise to n equals 2 some sort of remarkable analogies with the diagram I put before relating to the upper half plane appear so we need to get an understanding of criterion for our 2 vectors to be the shortest vectors that are basis vectors the key one is this formula in the middle there 2 times the absolute value of u dot v less than or equal to the absolute value of u squared the other one is just saying that the is the second shortest vector and u is the shorter one there's an ordering in our 2 vectors if we align this shortest vector u with the considering these vectors in R2 we align that shortest vector along the x-axis we look at those 2 inequalities those 2 inequalities actually give us exactly the same region as we saw for the fundamental domain of hyperbolic geometry in fact it's not too hard to go from those inequalities and perhaps it'll become clearer when we go to a particular coordinate system and see that we get precisely the inequalities that define the fundamental domain the absolute value of this z has to be bigger than 1 and the absolute value of the real part has to be worrying about the boundary here equal to a half this second inequality here is actually precisely 2 times u dot v less than or equal to absolute value of u squared provided we parameterise u and v in a way that's going to come up on the next slide so let's have a look at the next slide how is our random matrix very relevant to this problem well it's relevant because we have a natural what did I say in that previous diagram we wanted to rotate the shortest vector to align along the x-axis naturally the q-r-d composition does that for us this further we're taking out two basis vectors m1,1,m2,1 is one of them that's the first column, second column we're rotating by a 2 by 2 member of SO2 this is basically the Gram-Schmidt what we get left with a triangular matrix these are though the correct coordinates to use for the problem I just described the vector we just aligned along the x-axis is actually r1,1,0 the second shortest lattice vector, a basis vector is r1,2 and actually it's 1 on r1,1 because we require the determinant to be equal to 1, we've said we're normalising things to determine trigger to 1 it's um we can generalise that so I've been just descriptive purposes talking about the n equals 2 case but this works for general n at least this part of the calculation works for general n what doesn't work for general n is an easy way to describe these inequalities here we've only described the shortest lattice basis vectors for n equals 2 and it's these particular inequalities which I say are strictly equivalent inequalities that we see in the upper half hyperbolic model geometry upper half plane model of hyperbolic geometry what might one be interested in here, one might be interested in the distribution of the shortest lattice vector because where our original basis vectors are at random our shortest lattice vector will be described by probability density function and that calculation is relatively elementary we're just integrating over r12 it's the coordinate we don't observe, we observe the coordinate r11 no big deal, we have a particular domain to integrate over and something we're fairly familiar with in random matrix theory, we get some linear repulsion we see linear repulsions of course in this circular orthogonal ensemble, beta equals 1 very characteristic of beta equals 1 they're chosen with uniform measures so that's a very important point so the original vectors are chosen at random with respect to the invariant measure for SLNR we have to then normalise things so that we have less than or equal to r so that's the notion of Siegel we want to choose, what does it mean to have a random lattice it means that our basis vectors are unchanged by all group actions of SLNR and that is the invariant measure so there's a unique variant measure it's not normalisable, we had to cut off the absolute value although in the case I considered the largest singular value it doesn't, that cutoff does not show itself in the final calculation you don't require a cutoff as it turns out that's a point I haven't tried to attempt to discuss here but there is no need to have the cutoff in the final calculation that's in the conceptual underpinnings of the calculation we get this linear repulsion in the sense, it's not really repulsion because it's just a distribution of the shortest lattice vector but it's linear so that's sort of something that we've seen in random measure theory for other reasons let's say what's this number four-thirds to the power of one-quarter that comes about for the geometry of a triangle lattice that's an extreme the best packing is the triangle lattice case the second shortest lattice vector doesn't start till s equals one what's the s equals one significance that's a square lattice so that's when the shortest and the longest and the second the other basis vectors are actually equal to each other now to perhaps better illustrate that the setup of this one can do a numerical experiment so we want to choose our basis vectors from the sample from the invariant measure where we put some bound on the operator norm of the matrices we know how to do that we go back to our singular value decomposition to do that so we know how to sample uniformly at random from harm measure Percy gave us some some details talk and it's n equals two one can explicitly sample from the distribution specifying the singular values if n is not equal to two one can do a Monte Carlo calculation so we can easily sample uniformly at random with this particular bound what we need then next to do is use other knowledge of how one computes the shortest lattice vectors and that's classical algorithm due to Lagrange and Gauss in another setting in setting of quadratic forms and it's a very easy it's very elementary the idea behind it you take you take your original vectors order them so that you is length less than v and try and create this red vector you cannot actually create this red vector the orthogonal projection or the orthogonal component of the orthogonal projection because in our lattice we are only allowed to use integer linear combinations so what you do is form the closest integer this is exactly the same as minimizing v minus m times u the norm of v minus m times u what value of m gives the smallest length it's precisely this projected very familiar projection quantity from this geometrical procedure of very familiar from Graham Schmidt just repeat that algorithm that's the way it goes it's very closely related to the greatest common denominator divisor algorithm actually you can see that but I won't attempt and then you can compare the theoretical predictions against the numerics the first two shortest lattice vector the second shortest basis vector and this third graph here is a prediction for the cosine of the angle between the two vectors so that all works very well for n equals 2 okay I'll now just briefly say well no one more thing as I sort of said n equals 2 perhaps not very prestigious and one actually do calculations for general n which I've now changed to d here you can say a little bit about the general n case because one has this tool the Seagull's mean value theorem in this geometry of numbers so he says that if one wants to average a function over all lattice points provided that the lattices themselves are chosen uniformly at random then it's just the same thing as averaging the function now if one takes the function to be the indicator function of a ball of radius r you can make some predictions relating to the shortest lattice vector so I run through a little bit of a calculation here the end of the day it gives a prediction for what the small s behavior should be of the shortest lattice vector very simple calculation and it gives a prediction that this particular coefficient relates to the Riemann Zeta function because of this interesting calculation how do you integrate nought to one where you've got one on s you're taking the integer part this one on s just comes about as a number of s here is to be thought of the length of the shortest lattice vector r divided by s is the number of multiples of that shortest lattice vector you can have up to the value r and then we're multiplying that by the distribution of the shortest lattice vector so that's the content c is unknown here but since we know the values of the left hand side that's just the volume of the unit ball we can work out what c has to be and if one superimposes that prediction for the small s behavior over an exact numerical calculation numerically exact sampling and then lattice reduction that's an interesting point in itself lattice reduction is a non-trivial task for d equals 2, 3 and 4 it can be done in polynomial time exactly beyond that one only has available approximate approximate what's it called LLL algorithm a very famous algorithm but it's very approximate in the sense it's exponentially far away it can be exponentially far away from the shortest lattice vector as the dimension increases but up to up to four dimensions there is a polynomial time natural extension of the Lagrange-Gauss algorithm if one does superimpose that prediction here involving the Riemann Zeta function evaluated at three one finds very good agreement up to around about 0.3 or something even though this is supposed to now just be the very first term in an expansion still still exhibited in the numerical data and the prediction is that that's the case for general d in the limit d goes to infinity this problem in some sense becomes less interesting because you get a Poisson process for all the different lengths of the basis vectors there is some sort of decoupling okay my final point which I won't go over in too great detail but of interest from the random matrix perspective because it's now making use of a different decomposition we started off with a decomposition the Euler decomposition of the orthogonal matrices we then made use of singular value decompositions we made use of the QR decomposition and his final subtopic the decomposition that is relevant is actually the polar decomposition so the polar decomposition is closely related to the singular value decomposition and I have a little bit of discussion there what we want to do is decompose our general rectangular matrix in terms of a what's our Q it's going to be a unitreme a matrix but times is that correct there I got to get this right n by n and v itself n by n so it's not actually so u times v transpose n by n I think that's correct multiplied by a symmetric matrix so that's the difference it's a little manipulation of the singular value decomposition we know how to change variables we know what the Jacobian is for this particular decomposition what's the where is this sort of particularly heading why we're interested in this particular subtle difference from what we saw before with the QR decomposition we're interested in this because we have a different problem in mine the different problem is the one from integral geometry of giving some statistical information about the distribution of the volume of this simplex so it's this subtle, subtle difference okay so let me continue on I got as I say interested in this problem because I wanted to know how one could actually compute that formula of Kingman and I was doing reading a lot of the literature and I just came across this particular how is it now it's doing it on its own who knows excuse that so how to okay where are we yeah we're up to that slide next slide here I came across this particular paper in one of the local journals let's say the bulletin in the Australian Math Society um um what this paper did was relate the polar decomposition which um we might come across in our random matrix theory to a key a key decomposition of measure in the integral geometry so I just like to um lead towards that particular decomposition of measure that particular decomposition of measure that Kingman at least implicitly rediscovered and made use of in his um his calculation so from the polar decomposition we can write for ourselves an integration formula now it's clear to me we're stumbling over before this space of matrices used is actually in this stifle manifold so it's not actually unitary matrices we have n by n so we have each of the individual columns are mutually orthogonal but we don't have as many columns as there are rows so um that's this stifle manifold so it's subtly different we have um our positive definite matrices so the polar decompositions um second part of the polar decomposition is symmetric matrices so we have here a positive definite I should say matrices I didn't emphasize that before on the previous slide I should have the positive definite matrices all eigenvalues are positive so um that's our starting point one of Mogadish's Mogadish's observations is that if you use this formula twice you get this sort of very subtle point sort of a magic trick of a person if you use this twice you get a formula that um that is of interest in random matrix theory because it goes between um a distribution on say a Geneva matrices rectangular Geneva matrices n by small n by big n to a distribution on n by n matrices and this is something that in this interest of having the random matrix products has been quite useful so it's a very subtle change but if you examine what that formula is saying integration formula over rectangular matrices integration formula over square matrices is saying how they're related they're related by this Jacobian factor so that was already a bit of a um interesting point that one can read into the workings of in that paper one can uh these these sigma here is just a notation for the surface areas that come in integrating in this dipole manifold the uh ingredient that goes beyond what one would do entirely from a random matrix perspective perhaps is this next step here where one introduces the uh grasmanian this grasmanian the concept here is that the space of subspaces where each subspace is to be thought of as coming with a preferred basis and it's a conceptual um challenge of sorts to get a head around but if we get a head around that particular way of thinking we can we can introduce an invariant measure on this space of grasmanians one can with minor manipulation of the previous formula deduce this decomposition of measure this decomposition of measure is close to what is used implicitly at least in Kingman's work so it's not really too different to what we've seen before we've just written into some different notation we've certainly got the determinant of the matrix so I'm thinking here of all the columns as defining vectors that's how the people in the integral geometry think we have a factor that's our determinant we have this new invariant measure relating to this grasmanians whereas before we had an invariant measure either on a file or manifold or on an orthogonal matrices so it's not too different there's yet one other little step that one has to do the formula is actually due to these two authors, Blaski Petts Ganskson and there's an affine version and one wants to make use of this affine version with some a few subtleties but it's very similar we've got this preferred basis and there is an extra coordinate that appears this coordinate that I've denoted by R and that's in a direction orthogonal to the column space of these preferred coordinates B and that's really if one reads Kingman's paper the essential idea that he introduces this orthogonal direction so if you try to do that integration I mentioned at the very beginning what is the average area of a triangle in a disk I don't think it can be done elementarily without this insight that Kingman was able to give to the problem how can we see that this is useful for computing volumes of simplices well we have in our determinant now not the determinant of the original matrix that has columns V we have this combination of a fixed vector subtracting off each of these and that's exactly the formula for the volume of a triangle for example you have to what's one of the formulas for the volume of a triangle if we have our three points denoted V1 V2 V3 in the plane then the formula for the area relates to the determinant of the difference say V3 minus V1 as a column and V2 minus V1 as a column and the absolute value of that determinant so this is exactly what we want and that's exactly what this particular decomposition and measure that appeared in the classical literature that was sort of rediscovered by by Kingman at least implicitly shows us the interesting development that I try to emphasize there is that this really can be viewed from a random matrix perspective from this polar decomposition alright that's about all I want to say there is quite a few ongoing aspects of this one ongoing aspect is in a random matrix theory I have particularly in this talk really been talking about the beta equals one case the real entries but we could ask about the invariant measure on SLNC or we could ask about the invariant measure on SLNH where H are the caternions once we've gotten that we're sort of starting off being able to perhaps answer some counting questions asymptotic counting questions of the sort posed by Duke et al we need a quotient space if we take SLNC the natural quotient space perhaps is the Gaussian integers so we're now going to do lattice reduction where our integers are Gaussian integers well actually do we only have to work with Gaussian integers when we're in the two dimensional case we could perhaps work with other families of integers like Eisenstein integers for example turns out it would seem from the calculations doing to date that any integers that have a generalized integers that have the Euclidean algorithm works for that we actually can do division and get a smaller remainder that seems to be really very much the mechanism we did calculations that relate to the distribution in the two dimensional case can we do those calculations as I say for SL2C well yes we can do those calculations and I have a student working on that and we can compute for example what's interesting even to compute the fundamental domain the volume of the fundamental domain is fairly challenging there it involves the Catalan constant for example and as well as the Riemann Zeta function that's done if one moves on to the Caternians a bit more difficult because these lift-shift integers do not form a Euclidean domain you have to go to something called the Hurwitz integers makes it a little bit technically more difficult what I haven't said that is also very interesting about this random lattice's problem is that this is all very close to related to continued fractions firstly the problem in that I've been talking about in this talk two dimensional real case is closely related to continued fractions in the complex plane if one looks at this a problem that's already made a fair bit of progress on SL2C the continued fractions are now themselves Caternians actually go up to the Caternians if you look at the complex case and there's some quite a bit of geometry relating to all of this that I believe is a topic that's fairly rich it's also quite fascinating that the line I've been taking is begun by Hurwitz in the 1890s Hurwitz is the person who proved that the only normed real division algebras are the real numbers the complex numbers and the Caternians if we require them to be also associative so that's something that we have a Dyson 3 file way that's very important to us in random matrix theory one of Hurwitz's other interests even before his main result in the 1897 paper was complex continued fractions so there is three interests of Hurwitz that well over 100 years later are quite prominent in following this particular line of random matrix theory any questions yeah that's very much the exactly right Percy so what Miles noticed was that underpinning Kingman's calculation was this particular decomposition of measure due to Blasky Pinkaskin and then he was able to extend the calculation by computing all of the moments very much that's exactly right well then that's yeah so once you know all the moments you can try and compute distribution in some cases that was actually done and it involves it's a little bit like the problem in random matrix theory of asking about what is the limiting distribution for finite N of a determinant it's very much in that class of problems so the determinant case so if we took our Gaussian sort of problem GUE problem and interpretation of what we're doing there is computing a volume of a parallel piper that's pinned at the origin what I showed that Kingman did he didn't have his simplex pinned at all it was moving around that's an interpretation of the average or moment of the determinant then you can ask the question for finite N what is the distribution from knowledge of the all the moments then you can write that in terms of my AG function and that's how things all go around because my AG functions is what another line in random matrix theory of present interest of integrable properties of products of random matrices my AG function is very prominent so perhaps that was another reason why I was looking in this direction interested in that sort of side of things


The reason why you should not follow your passion. A few days ago, I was invited to speak to a group of mostly young people. After my session, one man from the audience asked me, I've always heard about following one's passion, what do you think? I then told him about my personal business. As I time of making this video, I have three businesses. I loved one so much that I can do it for free. So you can call that my passion, right? The second one has a part of it that makes me high, while the third one is nothing but a boring job for me.   What do you think about the cliche, follow your passion? In this video, I'll explain the reason why the advice to follow your passion is not always a good thing. If you're new here, by the way, consider subscribing so you won't miss other interesting videos like this. Now let's get into it. You see, there's a complex formula that makes people truly successful in business and in their career. That formula is your passion plus the market willingness to pay for the product of your passion, minus easy entrance for most people. To understand this, let's imagine a building or a car. It's one, but with many compartments that have to be put together before the house or a car could work. Now, there's a problem. If the engine of the car is ready but the body isn't available, we don't have a car yet. If the building is constructed but there's no roofing, we can't have a house yet. If you're passionate about something but the market isn't willing to pay for it, you don't have anything that can give you money yet. That's why the follow your passion advice is not always the best thing to do. For example, I have a friend who is passionate about writing poems and novels but unfortunately, he lives in a country where most people hate reading, hence no one is willing to pay for his passion. In this case, would you still advise him to follow his passion? Now, if I've been able to convince you that follow your passion isn't necessarily a good advice, what should you do first? 1. Follow money first Maybe this is a bad advice but you need to keep your body and soul together. If my friend should continue writing poems that no one wants to read, hunger will bend his waist. Maybe if you're still young, your priority shouldn't be to be happy.  2. Keep your passion alive That the market isn't willing to pay for your passion today doesn't mean that they won't pay tomorrow. Human preferences change all the time. In fact, it may be your fault that you cannot market the product of your passion. It may be that you don't know certain things now that you know in the nearest future. Such things may be the open door for you to start making money from your passion. Keep your passion alive. Keep practicing and making plans about what you're passionate about while you walk on things you hate because of money. 3.  While I think that you can't always wait till you are able to make money from your passion because you need money to remain alive, I also think that you should never get used to a job that makes your life miserable. As you're working on what you hate just because you have a family and financial responsibilities, always remind yourself that you are in the prison for a purpose. Don't get used to the prison. Instead, keep working and making plans on your passion by the side. Many people say this is not easy and I agree but many ungodly people usually have side chicks after they are married, you know, so that's not easy too. Manage your side chick 4.  I do the first one for money. I do one for a little love and I do one for the real passion. At the end of the day, my goal is to get rid of the one I do for money. Maybe I'll sell it or just throw it away but before then, I'm going to use the money this ugly thing is giving me to build the business I truly love and that has been what I've been busy working on. My advice to the employees I need to speak specifically to the employees because this is very common among them. According to Gullop 4, more than 85% of employees hate their jobs, even those who have the so-called good jobs. But even though these people hate their jobs, which is fine since I too hate one of my businesses, they simply get used to it. They tell themselves that it's normal or they think that the next promotion will give them more money and make them happier. That's the wrong way of thinking. It's like me saying, though I hate this business but if I can make more money, I'll be happier running it. That's simply not going to happen. It doesn't matter how much money you get from a business or a job you hate.  Instead of thinking that the next pay rise will make you happier, why don't you use the checks you're getting from this job or business to start financing what you will truly love? Now, these are things I think you can do. 1. Buy no luxury for now The categories of people who ought not to buy luxuries are the ones who buy luxuries most and there's a psychological reason for that. You see, when you're doing what you hate, you won't be fulfilled. When you don't feel fulfilled, you'll think that latest iPhone or wristwatches will make you happy. Unfortunately, that's the last thing you ought to think about. When you're doing something you hate, you should never buy luxuries because buying luxurious things means that you will never have enough to save and that obviously means that you will forever remain in the prison of a business or a job that you hate. Be disciplined. Leave on very little things. Don't buy a car if you don't seriously need one and even if you need one, buy small and used ones. You obviously don't need the iPhones and the $10,000 wristwatches when you are working on things you hate. Who told you that you need the Rolex watch? 2. Make friends with people who are living your dreamed life I have a friend who was an employee many years after I have been an entrepreneur but one year after we leave near each other, he decided to quit his job. I'm not suggesting that you should quit your job though. I'm only sharing with you how making friends with whom you want to be like can inspire you to make some really big sacrifices. If you are in a business or a job you hate, don't hang around people in that same environment. Make friends with people who are already doing something similar to what you want to do. When you see the way these people live and how fulfilled they are, you'll get inspiration to make sacrifices like saving a significant part of your earnings, reading some good books instead of partying and doing all manner of things. 3. Start small Almost everything in the world can start in a very little way but because little is ugly, most people don't like starting little. I started my entrepreneurial journey in one of the worst ways you can ever imagine. It was little, ugly and dirty but I have to keep that fight for eight years before I got a big breakthrough.  Now let me conclude this video this way. You see, the advice follow your passion is not always a good advice. It's like someone advising you be happy always. It sounds good but it's not realistic because before you can really appreciate joy, you need to know sadness. Most times you won't be able to follow your passion whether because the markets don't yet appreciate your passion because you don't know how to market your passion or because it may take a lot of time to make money from your passion. While you're waiting for the good, you may need to hang around the ugly. My only advice is don't lose sight of the good.  Thank you very much for watching our videos. I would like to give you another interesting video for you to enjoy next but before then our team will be very happy. If you can like this video and share it with your friends on social media. If you are new here don't forget to subscribe so you won't miss other interesting videos like this. Now look at your screen to see two other videos we handpicked for you to enjoy next. We love you.


And I'll give the floor to the observer of the International Institute for Democracy and Electoral Assistance. Thank you, Madam Vice President. Excellencies, International IDEA considers the high-level multi-stakeholder summits of the future in 2023, proposing our common agenda, as an opportunity for reframing multilateralism and the relevance of the UN system in the face of the mandate challenges that have been addressed during the series of thematic consultations. The Secretary-General reports proposes a set of measures aimed at making multilateralism or networked inclusive and effective. Firstly, we underscore that this pandemic has laid bare the dramatic shortcomings of current multilateral rules to deal with health emergencies and global crisis. All multilateral institutions were put to a most stern test and have been found wanting. The damage has been of such scale and could be much worse the next time around. They're refusing to make a deliberate attempt to revamp those rules and institutions before the next calamity would be nothing short of suicidal. Secondly, while acknowledging the threats to the right to life trump all other rights, it is incumbent that we try to protect as widely as we can the human rights principles and norms that all states have signed up to. The sustainable development goals have been conceived and couched in the language of human rights. So upholding the protection as a foundational principle of global responses to future global crisis is not just possible, but essential. Thirdly, we should acknowledge a crucial point. Enhancing international cooperation requires that there is such a thing as a rules based international order. And that is precisely what is at stake now in Ukraine. A world in which might is right is a world in which global cooperation to solve global challenges is all but impossible. Fourthly, we welcome the emphasis placed by our common agenda on promoting the UN's role as a source of reliable data and evidence. As a think-and-do tank, specialized in producing policy-relevant, non-prescriptive and comparative analysis on political and institutional reforms and processes around the world, we, an international idea, attach great importance to this aspect. We should recognize the potential of combining official and non-official data in monitoring progress on implementation of agreed agendas, including the variables that are oftentimes less measured despite the relevance to assess the human rights dimensions of transformational shifts. In this perspective, the Scientific Advisory Board envisaged in paragraph 117 that it could support the Secretary General in strengthening the capacity of the UN, not only to better link knowledge centers across the system, but also to enhance partnerships with the wide network of research institutions and think tanks engaged in this endeavor. International Idea reaffirms its commitment to collaborating with the UN for the promotion of such partnerships with stakeholders from all regions of the world to help member states base future choices and analysis on solid and relevant data. I thank you. I thank the observer of the International Institute for Democracy and Electrical Assistance.