Daily Tech News show is made possible by its listeners. Thanks to all of you, including Paley Glendale, Dr. X17 and Dustin Campbell. Coming up on DTNS, Nika Monford talks to us about some things it takes to make an autonomous vehicle, plus how CEO and inventor Jessica Matthews powers the world and why AI and radiologists together are better at identifying cancer than either one is alone. This is the Daily Tech News for Monday, July 11th, 2022 in Los Angeles. And from Studio Redwood, I'm Sarah Lane. We have got some excellent stuff on the docket today, as you just heard. So let's start with a few tech things you should know. The Guardian obtained a trove of over 124,000 internal documents from Uber during the tenure of CEO Travis Kalanek dating from 2013 through 2017. These detail lobbying efforts, growth strategies and include texts and emails from executives. The Guardian shared access to these documents through the International Consortium of Investigative Journalists with media outlets like LeMond, The Washington Post and the BBC publishing further investigative pieces. Uber said it made mistakes and missteps, but the company has been transformed since 2017 under its current CEO, Dara Khazver Shahi. Uber's former head of European lobbying, Mark McGann, came forward as the whistleblower, saying, quote, It is my duty to speak up and help governments and parliamentarians right some fundamental wrongs. Morally, I had no choice in the matter. Hmm, okay. FOSS patents reports that Ericsson won an injunction against Apple in Columbia on a patent dispute and has begun trying to enforce a sales and import ban on 5G iPhones and iPads in Columbia. The current patent dispute isn't about whether Ericsson's 5G patents are valid. It's about whether they are charging a reasonable royalty for them. In response, Apple brought an emergency motion to a court in the Eastern District of Texas asking Ericsson to quote, indemnify Apple from any fines, fees, penalties and costs that it incurs as a result of the injunction. In a letter to Twitter's legal counsel, disclosed in a securities and exchange commission filing, Elon Musk's attorney, Mike Ringler, said Musk would not proceed with an acquisition of Twitter, citing materially inaccurate representations and that Twitter failed or refused to provide relevant business information regarding spam accounts. Twitter board chair Brett Taylor says the company remains committed to closing the deal at the agreed price. Bloomberg sources say that Twitter hired the firm, Wachel Lipton, Rosen and Katz, who's founder, Michael Lipton, is credited with inventing the poison pill defense against hostile takeovers. You might recall, we talked about the fact that that is what Twitter has been doing as of late as this. But they tried to anyway, yeah, yeah. Yeah, exactly. Twitter plans to sue Musk as early as this week for backing out of the deal in the Delaware Chancellor Lee Court. Musk waved due diligence when he agreed to buy Twitter, but is pursuing the idea of material adverse effect, which requires him to prove that Twitter's actual business differs dramatically from what he agreed to buy. I don't think the story is going to end any time soon. Somebody wants a cheaper price. That's what it feels like to me. Consumer intelligence research partners reports that Amazon had roughly 172 million US Prime members on June 30th, which is about the same as six months earlier. That's no growth, if that's true. This comes after Amazon added 60 million US Prime members in 2020 and in 2021. Back in February, Amazon raised the annual price of Prime from 20 bucks to or raised it by 20 bucks to 139 bucks. Meta open sourced a tool called Sphere designed to provide a knowledge base for AI and other complex platforms using information from the open web. Sphere was trained on 134 million public web pages. Wikipedia will use Sphere to automatically scan entries to identify how strong a citation is supported. It's unclear if use of Sphere is part of a paid deal between the two organizations. Some people think that is probably the case, but no, neither company has said such. Yeah, if it works well, seems like it might be a decent use of good deal. Yeah. And we need more good uses of AI. Yeah, we do. So we talk a lot about the best uses of AI on the show. Obviously, if you listen, you know that to be true. Many folks are worried about giving it control too much AI control can be a little scary. What's it? Well, you know what's gonna happen, especially given that it's only as good as its data and data is often biased. And we've had lots of stories about AI being used as a diagnostic assistant, not making decisions necessarily that humans would make but helping medical personnel identify them. But as we say, often on the show, it's always better to find out if things actually work than just implement policy based on what sounds like it makes sense at the time before knowing exactly how it's going to work. That's why we're pleased to see a study in the medical journal, the Lancet this month called combining the strengths of radiologists and AI for breast cancer screening a retrospective analysis. So Tom, let's talk a little bit about what this report says. Yeah, the authors of the study used software from a German startup called VARA. VARA is in use in around a quarter of Germany's breast cancer screening centers. It's also in use in Mexico and Greece. They worked with radiologists on the study as well from Essen University Hospital in Germany and Memorial Sloan Kettering Cancer Center in New York. VARA has been trained on 367,000 mammograms. The radiologist notes and assessments of those mammograms and the eventual patient results. Did they end up having cancer? Did they survive where they treated etc. VARA learned to classify confident normal, not confident and confident cancer. So they had three categories. One was pretty sure there's cancer there. One was pretty sure there's not. The other was not enough to tell. Yeah, so the study focused on that but also focused on how well the AI did identifying cancer, but there's a little bit more to it, right? Yeah, they compared two approaches in one, VARA alone analyzed the mammograms and they assessed how well it did there. In the other, radiologist reviewed scans without VARA's help. So they reviewed them on their own and then if the radiologist cleared a scan that VARA had said was concerning, VARA would send an alert to the radiologist to review it a second time. So you got three numbers out of those two approaches. You got how VARA did alone. You got how the radiologist did alone doing anything about what VARA would say and then how well they did once the team was combined. And you should know that this was a study, so nobody's actual like prognosis rested on this. This was to see how well the system worked. The radiologist had better accuracy than VARA alone. So the humans outperformed VARA in this case, but the combined team, when VARA said take a second look just to be sure, the combined team was 3.6% better than the radiologist alone and reduced the number of false alarms where they thought there was something there and turned out there wasn't. In addition, VARA classified 63% of scans as normal where it was like confident that it was normal, which sped up the process of review. Humans still looked at them, but they got pre-filled reports, meaning if the radiologist agreed they could just press enter and move on. So that helped speed things up. That can be especially useful in regions that don't have enough radiologists. India, for example, has one radiologist for every 100,000 people, so you need things that can allow those radiologists to review more scans. The next step here, of course, is to judge how well VARA performs over time in actual clinics, not just in the study.  We think you're good. I wasn't. And that is no fault of anybody's. These things happen. So, you know, personally, I'm like, yeah, A.I., 100% do it. I don't know why this would ever be a bad thing, even if it was a false positive, even if it was that, just to have some sort of machine learning help to say, maybe take another look at this. I don't think my outcome would have been different personally, but I think a lot of other peoples might be. And I think from this study, they really got two things right. And one of those things is that it's the notion that A.I. along with human intervention is the best course of how we think about A.I. I think we have this idealized view that A.I. is here, machine, go out into the world and do all these things for us so that we don't have to do it. I think we need to shift our focus into thinking more along the lines of what this study proved is that machines along with humans makes for the most ideal outcome, because again, you are still teaching a machine to do human things, but it still does lack that human aspect. The second thing that I thought that was important, that they got right, was the data. It all really starts with the data. If you have a bad data set, if you're manipulating the data to get the outcome that you want, it's not going to give you anything you can actually use in the long term. Looking at the notes from the Lancet website, what really stood out to me is that for the data that they used to train it and validate it, the test data set didn't include any of those people. So it's not one of those things where they can sort of kind of tweak the data to get the outcome that they want. It's more along the lines of this is what we started with. This is what we're training on. This is what we're validating on. Let's inject this new set of data that's not linked in any way to what we use to validate and to train. Let's test it against this new set. And I think those are the two biggest things that they got right in the study. Yeah, I'm excited about this study because I have suspected that what we're going to use is to help us because there are so many things that are you just need a more intuitive sense of knowledge to understand them, especially if you're a professional like a radiologist that are hard to manipulate but humans get tired. Humans make mistakes and even the most highly trained, most professional human will make a mistake and so having a backstop like that seemed to me like the best but this study says, yeah, in this particular case, we found that to be true. We still need to do more work but it points to the direction of don't look at AI as a replacement for humans. Look at AI as a useful tool that in certain situations like this can be helpful. Absolutely. A companion, so to speak. Yeah, yeah. Well, folks, it's time again for, you know, Nico Bonford's on the show. We're going to do tech and while black, shining the light on a technology leader you might not have heard of, but probably should have. Nika, who are we highlighting this episode? So this week we're highlighting someone who comes from a little bit of an untraditional background. We're highlighting Jessica Matthews. She is an inventor, a CEO and a founder and she is the current founder and CEO of Uncharted Power, which is a global company that specializes in building and operating renewable power infrastructure and she created this company at the age of 22. But what I think is probably even more impressive is that her first invention is called Socket and it's an energy generating soccer ball and she did this at 19. Jessica is both a dual citizen of the United States and of Nigeria and as we know, soccer is one of the biggest sports, if not the biggest sports in the world and to be able to use something as, I won't say common, but as easily accessible as a soccer ball to be able to generate energy. And we know in some parts of the world, energy is very hard to come by. It can be very scarce. It can be hard to keep on a consistent basis and that's one of the things that she said was that her goal was to inspire and to find ways, she says, quote, I asked myself, how do I design solutions that inspire them to try? So it's one of those things where you want to, her goal was to find ways to not only help her native country, which she, again, like I said, has a dual citizenship of, but how can we do this for the whole world? And one of the visions of uncharted power is to aid cities in becoming smart cities and we all know climate change and things that we're experiencing. If we can find ways, which this company seems to be doing, to find a way to have an alternative power source, a renewable power source that helps all nations across the world. And for this company, she has backers. I think Maddie Johnson recently joined the board of directors. She has backing from large companies like Disney. So it's one of those things where this is a huge, you know, step forward in how we power the country, this country and nations all around. In addition to her young age of creating this, her offering, her CSA offering for her company, initially when it was established back in 2016, it broke the record for funding by a black female founder. So to me that says, hey, this is a very important aspect that we are definitely trying to try to fulfill and meet. And she has the backing to continue to do that. How many times have you looked at a bunch of kids playing something like soccer and think, man, if I could harness that energy? And Jessica Matthews was like, I think I can harness that energy. I can't do that. That was sort of my question is like, I mean, is this something where, you know, if kids are playing soccer in the backyard type thing, it could power the house? I think so. That's what it is. Yeah, the idea behind it is to be able to generate energy that can be actually used by people playing something that most children play. And that was just the opening move for Jessica Matthews. That was it. Yeah, that's crazy. Yeah, it's pretty wild. And like I said, she comes from, I say an untraditional tech background because when we think of scientists and technology, we think of folks in maybe computer science or chemists or environmentalists, those type of individuals. But she has her undergraduate degrees from Harvard in psychology and economics and an MBA from Harvard Business School. So she has a little bit of an untraditional tech background. But when you think of the human aspects of something as trying to find ways to generate renewable power, you need the aspects of how people think, how people act. And also, where's the money coming from? How are we going to get this done? And also, in addition to inventing the socket, which was her first invention, she has over 12 patents and patents pending. So she didn't stop there. She's still going and there's more to come. Some of the honors and accolades, which are tremendous, I kind of narrowed it down. Harvard Foundation Science of the Year in 2012. Fortune's 10 most powerful women entrepreneurs. Fortune's one of the most promising women entrepreneurs. She's actually ambassador for entrepreneurship for the country of Nigeria. So it's one of those things where it's something that's continuing to build and she's continuing to build because she's still relatively young. So it was definitely something I think definitely needs to be highlighted, especially where we are in what we're seeing with our global economies and our global infrastructures. So shouts out definitely to Jessica Matthews. You can find information about the company at Uncharted.city and you can find her over on Twitter at Gesso Matt. Yeah, go save kudos to Gesso Matt on Twitter. Good stuff. Thank you, Nika. And it's part of DTNS's mission to sort of help us understand the world by meeting different people and finding out about different people that are doing different things. Next week is special guest week. New guests we haven't had on the show before, including Jack Rissider from Dark Net Diaries, Will Smith from Tested.com, Quinn Nelson from Snazzy Labs, Joel Telling, the 3D printing nerd. Don't miss it and most importantly, tell your friends, hey, new folks, interesting folks coming on to Daily Tech News Show next week, starting on July 18th, DailyTechNewsShow.com. Well, much like Hope reports about Apple's car project, Spring Eternal, folks. Recently, we have gotten updates on the project, mostly when there are staffing changes. Executive Left is the Apple car not happening anymore and nobody really knows. But the information published a wider ranging report covering the last eight years of the so-called Project Titan. Yes, the report claimed Apple's Senior Vice President of Software Engineering Craig Federighi is skeptical of the project, at least according to those sources. CEO Tim Cook, quote, rarely visits its offices and has not yet committed to producing a vehicle. If you haven't followed the frequent management changes, retired senior hardware engineer Bob Mansfield ran the project from 2016 to 2018, followed by former Tesla's VP Doug Field. He just left in September for Ford. The project is now overseen by Kevin Lynch, who was the person who led Apple Watch Development. Yeah, and as far as capabilities, the Project Titan team showed executives a prototype self-driving vehicle that was able to travel through a known 40-mile route in Montana successfully. Sounds good. However, in real-world testing, prototypes have shown lane issues, hitting curbs. In one instance, a human safety driver had to intervene when the car didn't stop for a jogger, but only slightly adjusted its path forward. So, you know, it's not perfect. Apple concluded that the safety driver hadn't intervened. If the safety driver hadn't intervened, the car would likely have struck the person. Not good. Want to get that stuff ironed out before you start selling cars to people. Yeah, and honestly, that kind of thing is not super unusual where the safety driver intervenes. That's why the safety driver's there, but it does indicate that Apple's not as far down the road as some of the others. That's what Apple's reportedly doing. Nika, you're working on these type of systems every day, so can you shed a little light into what goes into navigating routes, so to speak, in this kind of testing? So, it is quite difficult, and one of the things is this is a long-haul effort. This is something you can kind of just turn out over a couple of years. You have to have a genuine commitment to, you know, getting into this and following it through. As you did say, Tom, I do work in this every day. I work on the features engineering team for General Motors Ultra Cruise, which was announced, I believe, October of last year. And what the system does is it's not fully autonomous, it's semi-autonomous, but it does offer hands-free driving, and it does, it is able to navigate, you know, different types of avenues. And one of the things is when you're doing this type of work, you have to make sure you take into consideration the terrain that you're on. Are you going to be traveling on highways? Are you going to be traveling on residential areas? Are you going to be traveling on streets, interstates? How does this work? And you think about interchanges. If you're going around a steep curve to get on and off an exit, those types of things have to be taken into account. And it's not just something that is easily done, because you have, when you think about it, if you're having a vehicle drive for your hands-free, hands off, it has to know things like, where's the dotted line? Is it a straight line? Can I cross? Can I not cross? Is this, am I going on to the median? You know, those type of things, what's the speed limit? Those different types of features have to all come into play, and it has to work seamlessly, because you are dealing with humans. And one of the things that I have encountered is, you have to be able to make sure that you're looking at just not the basic things, being on a straight road. You have to look at different edge cases as well, because again, as we mentioned earlier with the story regarding breast cancer, this is still a machine, and it still needs human intervention. You can't just be completely hands off. Well, I guess in some vehicles you can, but particularly on what I work on, it's not intended to be, you know, completely just checked out. It has safety protocols in place to make sure that the driver is still engaged, should the need arise where you have to kind of take over. But it's one of those things where you have a route. I think the article mentioned that Apple had a 40 mile route. We as well have a specific number mile route that we test on. We've tested on it, you know, in the prototype, and we've tested on it in more real-world scenarios. And those things are imperative when you're thinking about human life, when you're thinking about how you engage with other vehicles, how you engage with someone who may be crossing the street. Those things are of the utmost importance. And what this says to me as a relation to the Apple Car is that this is not something you kind of just jump in for the fun of it because it's novel. It's something you truly have to plan out. I think it was just announced in October of last year for Ultra Cruise, but it was worked on significantly before and is still a significant amount of work being done before it's ready to, you know, be out in the wild. And there are different aspects that you definitely have to take into account because, again, human interaction and machines, they need to work cohesively together. Yeah, because I feel like there's a few levels you're describing there. One is like there's a test track. We know exactly what it looks like. We're pretty much, you know, except for maybe a bird or something, we're pretty much can control what it's going to encounter. Then there's what you're dealing with when you develop, which is, well, we can handle most situations, but we've always got the human because it's not an fully autonomous system. So that makes it a little easier because as long as you keep the human in the loop, you can fall back to that. But that just shows how much more complicated it is to make a fully autonomous car that can handle every single thing without a human involved, which it sounds like from this story from the information is what Apple is targeting.  And like GM started with supercruise. That was the first foray into this semi-autonomous driving. I think it was like a level two. And so this next iteration of ultra cruise kind of kicks it up a notch because now we are on, you know, all types of roads and we are expected to be able to cover, I think over 2 million miles of road. And this includes all roads. This includes interstates. It includes neighborhoods. It includes, you know, side streets. And the other thing you have to take into account as well is where are you sourcing your data from? Are you going to buy Google's map data or are you going to create your own map data that adds in a whole nother level to it? If you're going to use some existing map data, sure, that helps. But that has to be pretty expensive, I would presume. So if you're talking about generating your own map data that adds a whole nother level, how are you going to map the whole country to be able to find every kind of nook and cranny road? So it's one of those things where it's a long haul and, you know, the fact that Apple announced this so early, it was exciting because, I mean, it sounds cool, but the thing is how ready are you for this? Did you really think how convoluted this could possibly get? Did they announce it though? I think that's the thing that I always try to remember is they didn't announce anything. This has always just been like, well, we're here and they're doing a thing. So, you know, like Apple's been spending a lot of time and money on it, but it's been mostly rumors about what they're doing. But don't you think some of this stuff is probably leaked by some people like strategically? Maybe, maybe. I don't know. It might be. It's a good point. Yeah. But yeah. So it's a lot to take into consideration when doing something or taking something like this on. Yeah. Well, Nika, thank you so much for sharing your hands out like literally hands on information in this. That's super invaluable. Indeed. Indeed. Speaking of cars. If you ever looked at a car and said, oh, you know, it's a car painted black, but it could be blacker. The team over at Pit 1 Customs in Japan has just the thing for you. I was going to do a Staphon voice for this, but it will just keep it pushing. It painted a portion 911 and co-pros Musso black acrylic paint, which absorbs acclaimed 99.6% of inbound lights. The result is kind of a shapeless shadow of a portion 911 because the paint makes it hard to make out contours of the vehicle. Some people might like that. It also costs about $130 per liter. So it's not going to be the cheapest paint job out there, but boy, does it look nice. The video will be in our show notes. Now, there aren't many ways to get a darker car to indulge your nihilistic dreams if you have them. Vantablack coatings, some people might be familiar with that, made from randomly aligned carbon nanotubes can absorb up to 99.965% of visible light, but can't be supplied to private individuals unless your name is Anish Kapoor, because Anish Kapoor holds the exclusive license for this particular artistic work. If you follow the art world at all, you might know many other artists are not pleased about anybody having exclusive rights to a particular color, even in the name of art. Little context here. Another artist named Stuart Semple developed a new pigment. He deemed the pinkest pink. It's very hot. Back in 2016, he posted it for sale with this legal ease. And I quote, by adding this product to your cart, you confirm that you are not Anish Kapoor. You are in no way affiliated with Anish Kapoor. You are not purchasing this item on behalf of Anish Kapoor or an associate of Anish Kapoor. To the best of your knowledge, information and belief, this paint will not make its way into the hands of Anish Kapoor. Yeah. The fact that he tied it up and won't let anybody else license it is what got everybody upset at Anish Kapoor. Yeah. And that, even that Porsche 911 with the GoPro Muzo Black sounds hot. I don't, I need my car to not absorb sunlight right now in the middle of July.  I never had one before. When you live in the country, terrible idea. I mean, it's, you know, I come back from the car wash 30 minutes later. I'm like, dirty car. So yeah, whatever. But it looks pretty. It does. Do you want that? My thing goes, my thought automatically goes to when you're driving at night, like, do you have to have like extra light? Do they have like put in like another light? Yeah. It sounds like it's so dark. The running lights become extremely important in that situation. Yeah. All right. Let's check out the mail bag before we get out of here. Yeah, let's do it. Yeah. And then one comes in from Tyson who wanted to share. We're extremely excited to have a star link option for ships. Because Tyson says, my job is to build a cloud on a ship where students travel around for a semester and take college courses on the ship. Like semester at sea, I guess kind of our internet was about 35 K to install about 30 K per month for 10 megabit down and four megabit up connection sharing that with 500 to 300 students and staff painfully slow. Having star link would enable us to lower cost and use some cloud services that are currently extremely painful to build local versions of Tyson says, hopefully this helps understand just how expensive current maritime solutions are and why even at that price, star link is a game changer on ships. Thank you, Tyson. That yeah, that's amazing. Patrick and I were suspecting that it would feel cheaper to those who had to pay and Tyson confirmed that we really appreciate that. Absolutely. Yeah. Thanks. Thanks for the on the ground report, Tyson. And if you have any reports on anything we talk about, anything we might talk about in the future, feedback at dailytechnewshow.com is where to send an email because we really love your feedback. We also love having Nika Montford on the show with us. Always a delight Nika. Let folks know where they can keep up with what you do. I am pretty much on Twitter most of the time at tech savvy diva. You can also follow me over on snobby westcast. That is podcast that I host with Terrence Gaines, where we talk about a whole bunch of Apple steps because we are Apple snobs. You can definitely find information about that. We are at snobby westcast on all the social media outlets and our website is snobbywestcast.com. It's a good show. Subscribe early and often, everybody. Also, we wanted to thank some brand new bosses we got over the weekend. Alex, Yaneve, and Eric all just started backing us on Patreon. Thank you, Alex. Thank you, Yaneve. And thank you, Eric. One for each day. Indeed. Yeah. Saturday, Sunday, and Monday. Thanks for that. Perfect. Just need one for tomorrow. It's perfect. Yes. Just keep it rolling, everybody. Just a reminder, there's a longer version of the show called Good Day Internet available at patreon.com slash DTNS, which we roll into right after we wrap up DTNS. But just a reminder, we do the show live. We're live Monday through Friday, 4 p.m. Eastern, 2,800 UTC. Find out more at dailytechnewshow.com slash live, and we'll be back to it all again tomorrow with our old co-host, Ayaz Akhtar, join us. Don't miss it. Talk to you soon. This show is part of the Frog Pants Network. Get more at frogpants.com. Fireman Club hopes you have enjoyed this program.


Having heard some of our remarks earlier, I think it's now clear to just about everyone that the nonprofit sector alone is not enough to reach the sustainable development goals. This is a massive undertaking is going to require all hands on deck. Governments NGOs individuals and most definitely the private sector. We are thrilled that so many companies both in the United States and globally are joining in this effort, including the one represented by our keynote speaker today. Based in the United States, Xylem water solutions as a global provider of water technology, serving the commercial agricultural residential industrial and public utility markets. The company operates in more than 150 countries worldwide and employs more than 17,000 people. Xylem has committed to donate 1% of its profits and employee time to water related causes, including technology solutions to reducing pollution and reusing wastewater. Our keynote speaker today is Austin Alexander silence vice president responsible for sustainability and social impact. In that role, Austin oversees the company's efforts to increase diversity, equity and inclusion in the workplace and Xylem's Xylems corporate social value program watermark. Since she joined Xylem in 2013, Austin has worked in customer service, sales engineering and investor relations throughout her tenure she has been involved in volunteering and leading events through watermark. Xylems corporate social corporate citizenship program that provides water resources and communities around the world and education about water issues. Before joining Xylem, Austin held what sounds like one of the most fun jobs outdoors in the United States. She was a range technician for the US Forest Service, working out of Big Timber, Montana. She studied, studied engineering management at Gonzaga University and earned an MBA at Wake Forest. We are delighted to have her as our keynote speaker today. Please join me in welcoming Austin Alexander. Thank you so much for the introduction and good morning, good afternoon, good evening to everyone. That was incredibly powerful and a different way to introduce a conference so I really appreciate as means creativity that really grounded me ahead of this morning. Again, thank you for the opportunity to join today, I have been learning about this program the impact engineering and the impact the engineering for change work the last week or so and just completely impressed about the incredible work that this program is doing. And something I hope we can continue the conversation on it's absolutely wonderful. As, as I said my name is Austin Alexander and Vice President sustainability and social impact at Xylem. I'll share more about Xylem in a minute. But in short, we are a global water technology company. I'm currently working remote from Montana which I was where I was born and raised and had the pleasure of moving back to in 2020. Today I want to talk briefly about what, what I think engineers and the role engineers play and how corporations and companies in partnership with nonprofits can really work to achieve the SDGs. I'm also as mentioned it engineering management degree but a civil engineer underneath that so please don't hold that against me. I have a few slides to help with the first few minutes of this session. There we go. And if you can move to the next side please. I'm not very great at talking about myself so I decided to do it with some pictures. I'm from an entire family of engineers, my mom, my dad, my two brothers are all various engineers. So it may be easy to think that I was that I stumbled or perhaps was directed into becoming an engineer. And yes, there was some pressure to smooth that direction. And the real reason I was leaned into engineering was in high school. And it always been stem inclined, particularly a really like math and I was pretty good at it. I also found I had a deep love for toilets. And it was early in high school that I realized the, the immense impact that safe sanitation has on elevating individuals and communities out of poverty with profound health impacts. And so as I move into my education journey. I was considering how I could take this newfound passion and interest around waste and something many people do not want to build a career around. And it was something my mother, who's also an engineer she's pictured below in the blue hard hat with me. I continued to buzz in the end into my ear that engineers saved the world. So fast forward a few years later I'm walking across the graduation stage with my newly minted engineering degree, EIT certification, and I took a job at Xylem. And it's been an incredibly rewarding journey so far. Both my background as an engineer but also working at a company that's global in nature and crosses so many different geographies has opened a lot of doors for me, both in opportunities for my career and expanding into different areas, but also in meeting people around the world and traveling to different geographies and being able to stay rooted in my deepest passion, wastewater. So if you can move to the next slide let me touch quickly on who is Xylem. We are a global publicly traded company roughly 16,000 employees, we operate 150 countries and our mission is to solve water. But why water, and then I'll sum it up across three challenges that we think about scarcity resiliency and affordability. As you can see on this slide, three out of 10 people globally do not have access to safe drinking water. And by 2050 about 40% of the world's population will face absolute water scarcity, unless they're a major intervention interventions. And this issue of scarcity here I feel it in Western United States, as well as many of you around the world in different ways that we're experiencing a significant drought over the past several years. And this is being exasperated by climate change. Additionally exasperated by climate change is the resiliency of our existing infrastructure by 2050 over 200 million people could be displaced by desertification sea level rise and extreme weather events. And all of this has a price to pay roughly $40 billion of clean water is lost annually we call that non revenue water. The water that where infrastructure is broken pipes leak, and we're losing the most valuable resource we have, and the resource we spent so much energy to obtain and treat is lost the long way. And if you're here in the West like myself infrastructure has been a hot topic and the cost to make the improvements we need across our water infrastructure in particular is immense. There's also a couple other stats not on this slide I'd like you to consider that water and wastewater utilities globally account for just over 2% of the global greenhouse gas emissions. So that's about the same as the shipping industry and water use and management across all sectors accounts for 10% a tenth of the total global greenhouse gas emissions in the world. So not only do we face the challenge to continue to provide improved and expanded access to sanitation and clean water. By doing so we need to do so in a way that has a reduced carbon footprint, so we cannot exasperate those challenges, I laid out head just now. So to give you a flavor where we play, you can move to the next slide. Perfect. Just to give you a flavor for xylem we are the company that provides the equipment the software and the services to treat and distribute clean drinking water. Collect and treat wastewater assist the users of water like industrial companies and commercial buildings to be more efficient and to protect and monitor the water bodies around the world. Roughly, half of our revenue comes from public utilities to give you a flavor. So what I want to talk about today is what is under the surface of that set of solutions under the surface of what we think about the technology, the equipment, the hardware that we as engineers are so familiar with working with. And so for that I'm going to turn to the sustainable development goals. And it sounds like from the previous comments. Let me move to the next slide please. This audience is very familiar up one more I'm sorry. Thank you. This audience is very familiar with these 17 goals that the world aims to achieve. And if you are not I highly encourage you to go to the UN SDGs website to learn more. So while all of these goals are important at xylem we're focusing on six in particular that we align not only our sustainability strategy, but our business strategy as well. That includes the good health and well being gender equality, industrial industry innovation and infrastructure, sustainable cities and communities climate change. And the SDG that I spent much of my time considering access to clean water and safe sanitation. So how are we going to get there on these goals I mentioned earlier in the session we have a steep curve to reach in particular SDG six but many of the SDGs. And if we continue at the rate of expanding services that we are today we will not reach our goal. By doing so if we continue to use the same technology and processes we use today, we will expand the greenhouse gas emission impact of our sector even worse. As I just laid out those stats on greenhouse gases they're expected to double across the water sector by 2040. If we continue to expand services in the same way that we do today. If we do so, using the existing technology and processes we use. It's going to cost a whole lot of money. Typically impacting and raising the rates to those that cannot afford services today. So how can we do this in a way that expand services is good for the environment provides a social benefits of clean water and sanitation. And is it a reasonable price that we can reasonably make happen over the next few years. And where my call to the engineering community comes in. And I have three areas where I think we can solve these challenges as engineers. The first is innovation. The second is accountability. And the third is a depth of consideration across equality and social considerations. So first innovation something this group is probably very familiar with. In my role I have the great pleasure of meeting with hundreds of startups nonprofits, individual designers coming up with the next greatest invention or innovation to solve water. And these are really important it is the only way we'll be able to advance to get to solving the SDG six and the other SDGs. But I'm also the most comfortable with that, because of the rate of innovation that we see. We have mentioned the technology in our hands today. And we, if we don't, it's in line to be achieved quite quickly. So I'm less worried about innovation because I know that will be an important aspect to solving the SDGs, but that's well underway. One example of that is how quickly we're able to produce vaccines and vaccines and respond to a global pandemic using technology that's been developed the past 10 to 20 years. And I think the same thing will be rapid innovation across the water and sanitation sectors over the next years. And the second area is perhaps a little less clear. And it's what my boss, who's our chief sustainability officer told me when she hired me into the sustainability role that she wanted an engineer running sustainability, because the most important thing that we can have as a global company based on sustainability as part of our strategy is that we are accountable, incredible in our sustainability claims. You're probably familiar with the term greenwashing happens a lot. Some claims of really fabulous sustainability goals or impact or technology that will save the world. And while I don't want to dismiss the excitement around sustainability, we should be as engineers holding ourselves extremely credible or accountable excuse me, that the claims we are making about the innovations we are putting out there are accountable and impacted the impact is assessed in a really engineering and thoughtful way. And then later, can you be the engineer that is purely chugging out technology which many of you I know are not, but you must also be credible in the in the way that we're accounting for the impact, and holding our organizations accountable as well. Finally, the third is, we need engineers that can think about the breadth of challenges across our collective society in all dimensions, not only the technology technological advances, but also the environmental and social impacts that we need to balance. And we need to incorporate all three of those into our design and application of technology. So, for example, on climate change like I mentioned, using if you go to visit a wastewater treatment plant in the US, you're up, even the new new plants that are being built at a rapid rate in areas like China and India. Roughly, they look the same. You're really and they probably look the same as they did in the 1990s and the 1980s. And that's because it works. We also have to think about the environmental impacts and the associated greenhouse gases with each new implication of that treatment plan. And we also have to think about the communities in which we're putting in new technology. We can't only use the technology we've always used, we must also innovate in a way that is thoughtful of the entire community. And we also need to bring the community into the design process. We'll wrap here in just a minute, but a book that I recently read over the last year, which I really like, it's called Waste by Catherine Pullman Flowers. And it's a book guarding the issues of septic tank access, sanitation access here in the US in what we call the Black Belt or Southern US. And we as engineers can have a tendency to design a system without thinking about the full impact and cost to the community we're serving. And in order for us to truly achieve the sustainable development, we must consider the implications of the end user and bring them into the design process as we are designing new innovations and designing the applications. Thinking about the full circularity of the system we're applying the environmental impact, the social impact, the technology technological advances and associating community communicating credible and accountable data to hold ourselves accountable as we apply those new advances. But who is better prepared to do this and balance this and think about the different inputs than you engineers. So my call to you is you're already here, you already care about this you're thinking about this continue educate the next generation of engineers and inspire those being educated today to take the same thoughtful approach to the technological advances we need. So again, thank you for the opportunity to join. I'd love to hear from you find me on LinkedIn or Twitter or elsewhere of shoot me an email I'd love to hear from you and thank you ASME and the program for the opportunity to join you. Thank you so much, Austin. Those reflections are really powerful and we are so grateful that you have this commitment to redefining social and environmental value creation and focusing on circularity, as you noted. So in this form, engineers need to be holding themselves and their employers accountable and make those claims credible by applying engineering tools and mindsets. We are so honored Austin to have you as one of our programmatic optimists in our corner.


I'm Sophia and I was born and raised in a Christian family, but ever since I was little, Christianity was just a religion to me and I did whatever my parents did or what they told me to do and whatever the church was doing around me. But a little bit ago it hit me that Christianity was not just a religion, it was a relationship with Jesus. And now I'm announcing that I made a decision to follow Jesus for the rest of my life. Sophia, do you believe that Jesus Christ is the Son of God? Yes. You put your trust to follow Him all the days of your life and you commit your life to follow Him. Yes. And based on that decision, we are going to baptize you today in the name of the Father, in the name of the Son, and in the name of the Holy Spirit.


So this lecture is part of an online mathematics course on group theory and will be about the symmetric and alternating groups. So we will start just by recalling what these are. So the symmetric group SN is all permutations of N points. And we may as well call these points 1, 2, up to N. And its order is N factorial because it can take 1 to any one of these N points. And the permutation can then take 2 to any of the N minus 1 points that aren't in the image of 1 and so on.  It's got a subgroup AN called the alternating group, which is index 2 and order N factorial over 2, at least if N is not equal to if N is greater than 1. And AN is the group of all permutations fixing the following polynomial. You just take the product over nought less than i, less than j, less than n, of xi minus xj. So for instance, for N equals 3, this looks like x1 minus x3, x2 minus x3, x1 minus x2. And you can see if you permute these numbers 1, 2, and 3, you will permute these factors, except you may change the sign of some of these factors. So any permutation of the symmetric group maps this polynomial either to this polynomial or minus this polynomial. And we actually get a homomorphism from SN to the group with two elements plus or minus 1, depending on whether it multiplies this polynomial by 1 or minus 1. And AN is just the kernel of this homomorphism. So we get a little exact sequence of groups like this. And this map is onto, if N is at least 2, so AN has half the order of SN. And we notice that AN is normal in SN because it's the kernel of a homomorphism. So we'll now just quickly recall what the symmetric groups look like for small values. So the symmetric group S1 is kind of uninteresting. It's just a trivial group, as is A1. S2 has ordered 2, so it's just isomorphic to the cyclic group of order 2, and A2 is again trivial. S3 is isomorphic to the dihedral group of order 6, consisting of all symmetries of a triangle. And the alternating group A3 is, of course, just the rotations of a triangle, which is just a cyclic group of order 3. So far, nothing very much is going on. S4 is a bit unusual. It contains the alternating group A4, but the alternating group A4 has an extra normal subgroup, z over 2z times z over 2z. So this is normal. You remember this group here is generated by the permutations 1, 2, 3, 4, 1, 4, 2, 3, and 1, 3, 2, 4. And this is the only case in which an alternating group has a normal subgroup other than the trivial subgroup and the whole group. So the alternating group is A4. It's also rotations of a tetrahedron. You can think of these four points as being the four corners of a tetrahedron. The group S4 has an extra normal subgroup. It's got this normal subgroup z2 times z2. And again, it's the only symmetric group that has a normal subgroup other than the symmetric group and the alternating group and the trivial group. And it's equal to rotations of a cube or an octahedron. So you can see this. If we've got a cube, a cube actually has four diagonals. And any rotation of a cube acts on these four diagonals. And by fiddling around with a cube, you can convince yourself that the rotations of a cube give you exactly all permutations of these four diagonals. S4 is also equal to all symmetries of the tetrahedron where you allow reflections as well as rotations. And then we get to S5, which only contains A5 and contains one. And A5 is rotations of an icosahedron, as we will see later when we talk about it in more detail. A5 is an example of a simple group. So this is no normal subgroups other than one and itself. So groups with no normal subgroups other than one of themselves are, of course, called simple groups. So some simple groups are the cyclic groups of prime order. And A5 is the smallest simple group that isn't cyclic of prime order. We'll probably say more about those later. Symmetric groups. Beyond this, we get S6 and A6. S6 is unusual because it has some extra automorphisms, which aren't at all obvious. So any group has automorphisms called inner automorphisms given by conjugacy by an element. So if you match x2, gx, g to the minus 1, this is an automorphism of the group. And S6 is rather unusual in that it's got some extra automorphisms, not of this form, as we will see a bit later. Maybe. So next, we want to look at conjugacy classes of Sn and An. And the conjugacy classes of Sn are quite easy to describe. In fact, the symmetric groups are one of the very few groups where the conjugacy classes are easy to describe. So let's look at a typical element of Sn. So it might do something like this. So I suppose it takes, let me number these, 1, 2, 3, 4, 5, 6. So it might take the element 1 to 3, 3 to 5, 5 to 6, 6 back to 1. And then it might take 2 to 4 and 4 to 2. And you see any permutation of these six points can be split up into cycles. So we write this permutation in shorthand notation like this. We write 1, 3, 5, 6. This just means the cycle taking 1 to 3, 3 to 5, 5 to 6, 6 to 1. And then we've got another cycle 2, 4. So any element of Sn is a product of disjoint cycles. So a cycle is just where you cyclically permute the elements as if they were sort of around the edges of a circle or something. Also, if A is this element here, if we conjugate A by some element g, all that's doing is it's renaming these elements according to g. So this has the same cycle shape. So it will be a cycle of four elements and a cycle of two elements. So we see that a conjugacy class gives us a cycle shape, where a cycle shape is something of the form 1 to the n1, 2 to the n2, 3 to the n3, and so on. Meaning we've got n1 cycles of length 1, n2 cycles of length 2, and so on. So this would be a cycle shape, 1 to the 0, 2 to the 1, 3 to the 0, 4 to the 1. And of course, you usually miss out 1 to the 0 and 3 to the 0 out of laziness. Conversely, any two elements of the same cycle shape are conjugate. And this is sort of obvious if you remember that saying two things are conjugate just means you're relabeling the things you're permuting. So let's do an example. Suppose A is equal to 1, 2, 3, 4, 5, 6, 7. And B is equal to 6, 4, 3, 2, 1, 7, 5. And the problem is find G. So G, A, G to the minus 1 equals B. So we want to actually, I mean, we can sort of see informally that A and B are conjugate because all we have to do is to find G relabeling the elements to get from there to there. And you do it like this. All we have to do is do this permutation. So this is the permutation G in blue. So in other words, G takes 1 to 6, then it takes 6 to 7, then it takes 7 to 5, then it takes 5 to 1, and it takes 2 to 4, and it takes 4 to 2, and it takes 3 to itself. And now you can check that G to the minus 1, A, G equals B. So if I start with this element 6, I apply G to the minus 1, I go up to here, then I apply A, I get to 2, then I apply G, and I get to 4. You see, I've met 6 to 4, which is what B is. So this sort of makes it obvious that if two elements have the same cycle shape than they're conjugate and gives you a constructive way of finding an element mapping one to the other. Of course, G is not unique because I could have written B as say 4, 3, 6, 1, 2, 5, 7. And then I would get a different element conjugating A to B. So for example, we can now work out the conjugacy classes of any symmetric group. So the conjugacy classes of Sn just correspond to cycle shapes of length n. In other words, we want 1 to the n1, 2 to the n2, k to the nk, with n1 plus 2n2 plus 3n3 plus knk equals n. And if you think about it, this is the number. It's just the number of partitions of n because this is just the number of ways of writing n as a number of smaller integers. For example, let's find the conjugacy classes of S4.  And this gives us the conjugacy classes either a 4 cycle, a 3 cycle. So this might be A, B, C, D, or A, B, C, or it might be 2, 2 cycles that might look like A, B, C, D, or it might look like A, B, or it might just be the identity element, where I'm not bothered to write the cycles of length 1 because they're just trivial and uninteresting. The next question is, how many elements are there in each conjugacy class? Well, again, this is fairly easy to work out. The size of the conjugacy class is equal to the size of an orbit under G because the conjugacy class is just acted on by G. So you can think of it as being an orbit of G acting on itself by conjugation. And the size of an orbit is just G over the size of the subgroup fixing a point. And the subgroup fixing a point of this conjugacy class is just G over the subgroup commuting with one permutation. So we've got to figure out what the subgroup fixing a permutation is. So let's write down a permutation and figure out what its subgroup is. So here's a permutation. We might fix a few points. And then we might have a few points that are mapped to each other. Let me do these all in the same direction. Then we might have some triples. We might have some cycles of length 3 and some cycles of length 4 and so on. So here's a typical permutation. So this particular one is cycle shape 1 cubed 2 squared, 3 to the 1, 4 squared. And now let's try and find permutations that commute with this. Well, first of all, we can permute any of these three points. So we get a subgroup S3 of order 3 factorial permuting these. Then this cycle here commutes with this permutation. So we also get a 2 cycle like this. This gives us a group of order 2. And we've got another group of order 2 here because this cycle of order 2 commutes with everything. But then we can permute these two. We can act on these two with a symmetric group S2. And this gives us another factor of 2 factorial. And here we've got a cycle of order 3 commuting with everything. So that gives us a factor of 3. And then we can commute this 1 element, which gives us a symmetric group S1 of order 1 factorial. And similarly, here we get a factor of 4 from this. So I guess that should be going that way around. And a factor of 4 from this. And we can permute these with a symmetric group S2. So we get an element of order 2 factorial. So we see that this element here commutes with a group of order 1 to the 3 times 3 factorial times 2 squared times 2 factorial times 3 to the 1 times 1 factorial times 4 squared times 2 factorial. And you can check that these, in fact, give you all things commuting with this particular cycle. So the order of the centralizer of a permutation of shape 1 to the n1, 2 to the n2, 3 to the n3, and so on, is just 1 to the n1 times n1 factorial, 2 to the n2 times n2 factorial, and so on. So the size of the conjugacy class is equal to n factorial divided by 1 to the n1 times n1 factorial, 2 to the n2 times n2 factorial, and so on. So this gives us an easy way to work out the size of every conjugacy class. So let's do this for S4. Just work out the size of the conjugacy classes. So the permutations have cycle-shaped 4, 3, 1, 2 squared, 1 squared, that doesn't sound right, 2 squared, 2, 1 squared, and 1 to the 4. And then the order of the centralizer is 4, or 3, or 2 squared times 2, or 2 times 1 squared times 2, or 4 factorial, which is 24. So the size of the class is just 24 divided by this number here, which is 6, or 8, or 3, or 6, or 1. And now, as a reality check, we should check that these numbers here add up to 24, which is the size of the group. So we have, indeed, found all the conjugacy classes and their sizes. Conjugacy classes in the alternating group are similar, but slightly trickier, or similar. But classes of Sn sometimes split into two classes of An. Let's look at A3 contained in S3. So the conjugacy classes of S3, there's 1, there's 1, 2, 3, and 1, 3, 2, and there's 1, 2, 2, 3, 3, 1. So these are the three conjugacy classes. Now these elements aren't in An, so we can forget about them. But in A3 has three conjugacy classes. There's 1, 1, 2, 3, and 1, 3, 2. So this class of S, the symmetric group, has split into two different conjugacy classes of the alternating group. And the problem is the element that conjugates this class into this class is one of these elements, which isn't actually in the alternating group. Similarly, if we look at A4 contained in S4, we find that there's one conjugacy class of elements of cycle shape like that. But if we think of A4 as a group of rotations of a tetrahedron, it's got two conjugacy class of elements of order 3 because we can point a vertex towards us and either rotate clockwise or anticlockwise. So this again has two classes of order 3. This only has one class order 3. So you might think, well, maybe classes of elements of order 3 are always going to split into two classes, but that doesn't hold because if we look at A5 contained in S5, this contains one class of elements of order 3. But A5 also contains one class of elements of order 3. Well, how come? Well, you might say, why are these two in the same conjugacy class? Well, we can actually conjugate this element into this element by using the permutation 2, 3. Well, that's no good because 2, 3 isn't in the alternating group, but now I can multiply it by the element 4, 5. And this is equal to g times 1, 2, 3 times g to the minus 1. So these two elements are conjugated in A5 because you've now got these two spare elements, 4 and 5, that you can use to make this odd, this element not in the symmetric group into an element of the symmetric group. So conjugacy classes of alternating groups are a bit tricky. The conjugacy classes of the symmetric group sometimes split into two classes and sometimes they don't. And there's a rule for telling you when it doesn't, when it doesn't, but I can never remember what this rule is because I won't bother giving it. Next, we want some generators for the group SN. So generators mean we want to find some particularly simple elements such that any element of SN can be written as a product of them. And the generators are going to be these transpositions 1, 2, 2, 3, 3, 4, 4, 5, and so on, up to n minus 1n. These are called transpositions. So transposition is a permutation that just exchanges two elements and nothing else. And we want to show that every permutation is a product of these. And to do this, we use the notorious bubble sort. So the bubble sort algorithm is the following problem. Suppose you've got a collection of objects which are not in order and you want to put them in order. For instance, suppose I've got several pens and I want to put them in order of putting the lightest colors to the left. So I want yellow, orange, red, and blue. And what I do is I keep switching the order of two of them so I can switch orange and red and then I can switch these two and then I can switch these two and then I can switch these two and then I can switch these two and I finally got them in order. So I can get them all in order just by switching two neighboring elements. The bubble sort, this is called bubble sort and it's kind of notorious for being a really awful sorting algorithm. It's much, much, much slower than the best sorting algorithms. But I mean, writing bubble sort used to be given as an exercise and introductory programming classes. And the problem with this is that bubble sort ended up being used as a sorting algorithm in actual computers which made them all rather slow in a way fortunately bubble sort has been more or less eradicated from computing but it's still very useful in the theory of the symmetric group. So you see bubble sort is really just using these particular transpositions to change the order of some object. In other words, give you a permutation. So I think rather than give a formal proof of these generate S and I'll just give an example which should hopefully make it obvious. So suppose you want to change one, two, three, the permutation one. Suppose you want to have the permutation that takes one, two, three, four, five to three, four, two, five, one. Then I can sort of do a sort of bubble sort. So I take one, two, three, four, five and now I'm going to switch one and two and now I'm going to switch one and three to get two, three, one, four, five. So I'm bringing one up to the last position two, three, one, four, one. Now I switch these two and then I switch one to here, two, three, four, five, one. And now you see by switching adjacent ones I brought the element one to this position here. And now nearly everything is in the right position. Now I've got to get, I want to move three up a bit. So I can, so I want to move three down to here. So I can switch two and three and now I want to switch the order of two and four. So I get three, four, two, five, one. And it's kind of obvious that by doing this you can change this to any other permutation. And now if you look at this, what has happened is this is the transposition one, two. This is the transposition two, three. And that's a bit confusing. So what this means is if I do one, two. So if I do the transposition one, two, followed by two, three it means it's a function that takes one to position two and then this function takes two to three. So it's rather confusing because this is acting on positions not on the actual element. So this isn't the transposition one, three. It's the transposition two, three. And then I do three, four and four, five and then one, two and two, three. So this permutation is equal to one, two, two, three, three, four, four, five, one, two, two, three. And it's sort of fairly obvious that you can get any permutation of any symmetric group by doing something similar. What we want to do now is discuss relations between these transpositions. And I think I'll leave that for next lecture since this has gone on for about half an hour. So next lecture we will be doing the Cox sort of TOD algorithm.


You don't want to get what the action of L is, what do I do? Well, I find what the action of t is inside the coefficient function. Then I multiply this action of t by z to the pi n plus 1, and I integrate dz to the value of pi. If I isolate the part of the correlation function, then the A is like one of the z to the pi n plus 2. And there's a long way to do that using consciousness. Is this clear? Remember we said that the action of L might depend, you know, is given by this kind of path integral, this kind of integral, but changes, you know, depends on whether you can't just go through operator's actions. Yes, so we're in the state, right? We take a slice of constant time. A is some operator defined in some time. And if I wanted to insert L at that time, what do I do? I use the definition. I insert t everywhere on the circle, and do the following contour integral on correlation functions of t. That would give me correlation functions of n. Yeah.  Why is that? Why is that? Yeah, because correlation functions in a path integral are always time limit. So if you put something before or something else, you get one order. If you put something after something else, you get second order. And unless these two operators come in, you should get a different state. So we say that in terms of this picture. You see, my torche is there. Provided this t here, the function in question, is completely analytic. It does kind of work. What's on your main set? Is this clear? It doesn't sound good. That's it.  So suppose I come from some operator in the set. I wanted to put L at some time. So I do this. What makes the path integral that it needs a correlation function of t with all these other operators? Thought it was a function of the insertion point of z. And then you'd be written in this way. By the answer, the correlation function is analytic in z. In z for which it's analytic, I get freely formed along the door. It won't change my answer. That's what you said. For any analytic function, prenatally form your conflict in regions of analytics. But you can't go through regions of non-analytistic and reach to singularities with this. And we know that in general, t with other operators inserted will have singularities.  Because that will be a multiplication of singularities that you're not taking without. And that is the mathematical analog of the physical statement that operators, the insertion of time order operators, depend, you know, putting something just before or just after, is different unless the operators communicate. Take this intuation and use it as an activation device. So that will be a function of the head. To the rest of you. And it should also be provided. Commutate that. Okay, so let's actually use this as a nice calculation. You know, in order to calculate something. What we have to do is to take the definition we had of the LNs, which defines our operators on our open space. And I want to compute the commutation relations between these operators. So what I want to do is to compute what I get LN minus LNl. It is of various conformal transformations. You see that, right? We had a formula for the generators of conformal transformations last time. Okay, it was, the current was multiply epsilon of z with t of z. That was current. And if you wanted to charge, then you have to take this thing and integrate it over space, which is over the surface. Okay, so now, you remember we parameterized conformal transformations instead of taking arbitrary epsilon z and it empowers them. So it's like some power times t of z. Okay, integrate it over the surface.  We do this integral over the surface if k is only when k is equal to N plus 1. Okay, so you get actually LN plus 1. So these LN plus 1 are the generators of conformal transformations corresponding to the conformal transformation in our k. Some link between k and N. These objects, because we've got a huge symmetry in our theory. Symmetry of that conformal transformation. And we want to see what symmetric group the commutation relation of the symmetry algebra generate. So what we're doing is conceptual equivalent to computing the combination relations of the generators in a direct relation. Let's go on. This is what we want to compute. Well, let's see. Let's write down the expressions for LN and LN. The first is to ask, how do you compute LN times LN inside a path integral? Well, remember, inside a path integral, anything that is inserted at an earlier time goes to the left. Things inserted at a later time go to the left. The left times, in z-lay, is smaller in this. So if I want to compute LN times LN, I should do the following. Remember, LN was equal to integral d z z to the pi n plus 1 t of z divided by 2 by n. LN was equal to, let's take w, w to the power n plus 1 by 2 pi i d of z. If I want to compute LN times LN, what I do is insertion with two t's, 1 and w and 1 and z to two contouring tables where the outer contour integral is z and inner contour integral. Let's make a final statement. The insertion of LN times LN is equal to the product of these two things. So d w, d z by 2 pi i, 2 pi i, z to pi n plus 1, w to pi n plus 1, t of z on this contour. Right? On this one. d w, d z to pi i, the whole thing squared. And then the same thing, z to pi n plus 1, w to pi n plus 1, t of z and t of w. But I move inside. Doesn't matter. Because unless these two t's approach each other, the correlation becomes fanatic. The question is there of what contour do you choose? What contour do you choose? Unless, you know, you pass through a point of insertion of t, it doesn't matter. There is inside and outside whatever v d shape you take, it doesn't matter. In-choose circles, but it doesn't matter.  Once you've chosen that, you may deform it however you want. So let's deform it in a way that is of a boundary choice. Let's see. w and there are two integral values. I can do the w integral and the z. Let me first do the w and let it first do the z. So let me freeze w to a particular value. So let me freeze w to a particular value here. I imagine that the w integral is always at a particular contour and the z thing goes outside inside. Let me freeze w to a particular value here. First, we can form a contour. We can form this contour to here. I've got to go to w. Minus this. I'm just doing the integral. w is w. And the z integral is just in this. I have to do dw divided by 2 pi i. Let's use the pole because I've got a model of 2 pi i there. So it's just a residue of the pole. z to the pi plus 1. w to the pi plus 1. t to the z. t to the w. So it's a residue of the pole in value of the z. Because I've got so nicely replaced that the top case is no pressure. So let's move on. I'll be with the values. So all I can do is compute the residue of the pole. But frankly, who people supposed to have it at? I don't know. But where does the... I'm so sorry. Can we just have it now? Why not? We have to wind this up. Yeah, yeah, yeah. Okay, let's wind. Let me leave you as an exercise for you to compute this residue. I'll ask somebody in terms of the pole. Okay, so we stop for a bit. The reason for that string is technically complicated.