I'll check the chat here. Okay. Okay. Awesome. So let's get started. Let's just wait one more minute because I think we have a few other people joining but as we're getting started, I'll say, we had, first of all, apologies if you were roused from slumbers just a little while ago. We were talking about the history of time zones and time changes, daylight savings time as a standardizing technology that still to this day occasionally goes, goes awry. So we were, I don't know if you're even aware. We were all joined the call an hour ago but apparently the time shift here and daylight savings time mixed up the appointment schedules. I don't know on someone's end. So we had an interesting conversation about that and I'll confess, I felt badly about any effort that Francis was making from your office who has been nothing but professional, supportive and fantastic. I hope she didn't rouse you from slumbers. Okay. Yeah, I woke up I think half an hour before, checked that an item from my calendar moved downward like one hour and I really didn't think much of it. So it didn't affect me personally but I saw that it affect all of you. So yeah, I guess this is your late gratification or something, what was it? All right, we've got the link there in the chat to slide out. So why don't we go ahead and get started. So Minister Tang, thank you so much for making some time to join this class at Stanford University on Ethics and Policy of Technological Change. We're in the last week of the quarter here. So the students have gone through nearly the entirety of the course and there's a team that teaches the class. I am a philosopher, I bring an ethical lens. There's a public policy or social scientist, my colleague, Jeremy Weinstein and then a computer scientist, my colleague, Meron Sahami. So three different perspectives brought to bear in the course. As a philosopher, I wanna start off in introducing you by saying that I am probably more oriented towards a kind of Socratic orientation, both in the class and in the discussions we have with guests. But today with you, I have to confess or gush in a way. You serve for me as a kind of modern day hero or at least a role model of sorts in that you have taken the extraordinary technical talents that you've acquired and deployed them for civic rather than money making purposes directly found ways to use technology in the way that Stanford so distinctively and proudly trains on people for ways that are apart from the entire system of Silicon Valley here, which is important and has produced incredible things. But too rarely in my judgment these days people acquire those skills and use them for explicitly civic purposes. So to introduce minister Tang, a minister without portfolio in the government of Taiwan, worked for a number of years as I understand it, never here physically located in Silicon Valley for Apple, but you've done all kinds of things in technology, including working within some of the companies that everyone here, in fact everyone across the world are familiar with. And you've been in office now in Taiwan for, let me see if I got this right for about five years. Okay, super. All right, with that I wanna begin in the spirit of my introduction by talking about a phrase which I gather you've used about yourself, that you consider yourself a kind of civic hacker. Yes, sir. I have in mind here thinking particularly about the students in the class as a way of suggesting, I've seen you describe democracy as a technology itself, a technology. A type of technology, yes. A type of technology, right, a form of social organization for decision-making, equal voice, civic hacking as a way to support the efforts, perhaps of civic spaces as well as democratic institutions. The names that everyone at Stanford are familiar with are the founders and the dropouts who go on to found the large companies. And another hero of mine is Aaron Swartz, a kind of leader of the open software movement that I'm sure you're aware of. But too infrequently people at Stanford look to people like you doing civic work. So I wanted to ask just if you could give a capsule summary about the path from acquiring the technical skills that you have to deciding to devote yourself with those skills to civic purposes rather than to disruption within the marketplace. Certainly. So I think the journey for me began truly when I was 14 years old, that was 1995. And I discovered this new thing, then new thing called the World Web, and told my teachers that my textbooks were out of date because knowledge was being created. And I'm part of this knowledge creating process. Well, I'm really just offering free spell checking services to people who pre-print publishers archive.org or XIV, you know, the Cornell thing that was actually, wow, it's still around. So a very large enterprise by now, it is spawned like bio archive and many other archives after that, right? So yeah, I was offering free spell checking services not just writing to the pre-print authors with the suggestions to fix typos or something. And they over the internet, they didn't know I was just 14 years old. And so they considered me their peers and we were able to have like real conversations over email and so on.   And the head of the principal really liked the confrontational style, actually. And she, after looking at email printouts, considered for a minute and said, okay, from tomorrow you don't have to go to Moscow anymore. And I'm like, what about compulsory education? And she's like, I'll cover for you, which means that she faked the records. And so, so I as speaking like autonomous learning right then and started a few companies and each one after another exploring this idea of how people come to a swift trust over the internet and how the internet can very quickly connect people and also the dangers of some parts of social interaction that makes people lose trust with each other as easily, actually much more easily than in face-to-face settings online. And so social interaction design was definitely my main line of work. And so that brought me to the Silicon Valley and building more pro-social, rather the anti-social networks and also brought me to the Sunflower Occupy where we occupied the parliament for three weeks with half a million people in the industry and many more are willing to deliberate a trade deal, but using listening skills, facilitated conversation, open space technology, technology, which is a social technology. Now, while communication and this digital counterpart we did arrive on very coherent set of demands which was then agreed in 2014 by the head of the parliament and ever after that we were hired then as a reverse mentors to the cabinet then in end of 2014. And after I guess a couple of years of apprenticeship the civil service just considered this to be a good idea overall. And so I'm hired a full time promoter for an intern to a full ministry at large and then starting to work with mimetic producers in each and every ministry. We've got hundreds of people now participation officers that lives with cute companion animals and translates like communication material like physical distancing when you're outdoor keep two of those shivas away or wear a mask. And if you're indoor keep three of them away.  But communicated in a way that five years old and 17 years old alike would very much willingly share. And so it has a more higher R value that I don't know respect elderly kind of messages and so on. So that brought me to this pro-social messaging parts which is more like a trainer trainer network so that each and every ministry and agency now gains this idea to listen as skill using the cute dogs and also occasionally cats. Yeah, looking at the what you just showed us on your iPad or whatever device that was that's an iPad. Reminds me, I've also from seeing some of your presentations online in the past, I formed the impression that you believe that either some form of humor or play of humbleness, a kind of humorous orientation is important to the delivery of technological services. And in particular in government, I think very few people here in the United States associate playfulness or humor with public services. Can you say a little bit about how that plays a role in thinking through your own delivery of public services in Taiwan? Yeah, I saw the slide of the question is literally a humor over a rumor which is some poetry right there. So anyway, the humor over a rumor that a fast, fair, fun idea of communication basically says so in Taiwan, we still remember the martial law. We've had the longest martial law in any country in the planet. So we don't want to go back there. We don't want to go back to the white terror, right? So people could not even fathom the idea of a otherwise would be reasonable like administrative takedown. For example, in Germany, they consider it reasonable to tell the largest internet platforms to do a takedown if his speech is involved, right? In Thailand is about royalty and the image of the royalty. So each country or each jurisdiction has some ideas like for these particular part of speech, it's not enough just to address the damage later. It's actually necessary to do censorship and takedown for these very specific things. In the US used to be about pattern entry, secret and copyright violations, but I see that that part has been relaxed a little bit. Anyway, so the point is that it's jurisdiction has its things that's worth taking out, right? Taking down and in Taiwan, none. So because of our very strong aversion to the martial law era regimes of authoritarianism, of the guard against authoritarianism, anytime anyone says something, that's a takedown like I don't know, Psyhub, right? And then even considering a administrative takedown is a non-starter, which is why Psyhub lived heavily as a DTW domain for quite a while before other administrative reasons took them down, not our government's work. And so because of that, when we see a pretty damaging viral disinformation that attack the election integrity of accusing a like bad count, a bad telling or a public health related disinformation that sells like a purpose to sell masks but actually sells computer virus instead if you do try to purchase and so on. These are like downright criminal, but even for that we can't even fathom a kind of censorship or takedown that other regimes would perfectly reasonably consider. So we have to investigate what's the, both the reason of the higher than one basic reproduction value, the basic R value of those disinformation, because if it doesn't spread it doesn't really cause damage, right? And we discovered that it's really about outrage. It's really about a emotion of something injustice is happening and that people would want to highlight it and while they get into this mood of outrage they don't bother to fact check. And so that gets very easily connected to the idea of discrimination or the idea of revenge not the idea, the affect, the feeling of such and has social feelings. And when people are in those states there is no media competence to speak of. And so then we start to ask what's the vaccine? What's the antidote of the mind? And it turns out it's humor. If you have left about something then it channels the energy in the outrage into care creation, into creativity, into a sense of playfulness. This is like a transitional space on the internet that makes people's imagination into a more pro-social dimension. And so we now have this really teams of comedians that's just rose out whenever we detect there's a rising trending misinformation. At any given time there may be just three because the total mental bandwidth is limited in a society they also compete with one another like virus in it. And so we roll out those very fun vaccines and then that gets even more viral than the angry disinformation, the outrage based disinformation. And when people left about there's no going back. So because this is the one way street in people's minds. So that allows us to fight disinformation without a takedown much like we fight, I don't know the pandemic was no long. Yeah. Well, so let me, can I just verify? So the government Taiwan has comedians on the payroll for this purpose? Yes, definitely. Very well paid, very well paid. Amazing, okay. Well, I wanna use that as a bridge now to ask you some questions like maybe this is going to be your answer, fight with humor. But we've covered in class the issues of free speech online, content moderation strategies, how the First Amendment interacts in the United States setting with such questions. First Amendment of course makes it difficult to carry out here what you described in Germany in certain respects, although other democratic societies have lots of different ways of thinking about this. So we've lived through here in the US just recently the takedown of Donald Trump from Twitter and Facebook. If you were either employed by the US government or employed by Twitter or Facebook, what would you have done on January 6th and January 7th about Donald Trump's tweets or posts? And independent of that particular case, I'd love to hear like guidance or advice from your perspective to people in Silicon Valley with respect to content moderation. Right. So we had in January, 2020 our presidential election almost exactly the same conspiracy theory that was spread and although it's addressed differently and so it has different results and I think it really is worth sharing. So for example, right before the election around the end of 2019, one of the most trending piece of this information and I quote is, well, the words are probably in a legible, so I'll just read quote, Hong Kong thugs compensation exposed, killing a police owns these teenagers up to 20 million. So, and with a rather scary looking young protestors image. And that's because that was shaping to be the deciding factor in our presidential election January, 2020. So this is a Reuters photo that photo actually is genuine but the original caption only said there's teenage protestors seen during the march end of story, right? So this alternate reality caption, this something talk about this 13 year old said both view iPhones and recruiting his brothers. It's entirely made up. It's intentional misinformation which means it's disinformation but we didn't take anything down. We used as I mentioned the regime called notice and public notice which is like the Canadian copyright enforcement system notice and notice, except it's on public notice. So first the people who use the line which is like WhatsApp, Antoine encrypted chat channels they flag the early warnings by a long pressing a message and this message and flagging as spun. And then when they do that, just like when you're flagging your inbox as spun it sends signals to spun house and in our case to international fact-checking network or the co-fax project from the Gev Zero community so that people see which ideas are trending. And then we made this deal with all the leading social media starting with the Taiwanese domestic pro-social social media called the PTT which is a national Taiwanese university student PTT project that's been running for 25 years with no advertisers or shareholders is still remains a national university subsidized by governments, PTT project open source co-governed audits. And so they don't have shareholder values they're very happy to do notice and public notice and with the social norm firmly in place we then negotiate with Facebook and the like so that they also do notice and public notice for example, the transparency around community nations and advertisement during election and so on. And so this is no different when the fact-checkers discovered that this alternate reality caption was indeed first seen in Zhongyang, Zhenhua, Weichang and see in the Weibo accounts of the communist Chinese central law and political unit their state propaganda unit then everyone who shared this on any social media platforms immediately shares it but with a mandatory frame that says this is proudly sponsored by the communist party maybe not proudly sponsored by the state organ of the communist party and probably also violating Reuters copyright. And so because of that anyone who shared this becomes a vector of sharing the clarification and sharing the vector of the frame also so people learn about it they learn about how to be more media competent not just media literates because when you have this notice and public notice you can also generate your own narrative based on that so you later on when you see for example on also January 2020 literally the day of the election there's another rumor that says the CIA make two invisible inks so no matter which person you vote this CIA special ink make sure Dr. Tsai's ballots always appears during the counting and whatever candidate you vote those ink will disappear basically it's on the democratic process itself. And we solve that very simply by inviting and we've been doing that for close to a decade now well to invite the YouTubers of all different parties to go into the counting process arbitrarily close to the counting stations and then they can film the entire process and each major party have these custom developed apps where they did the counting during the official counting so they have a unofficial count but from each and every major parties even during the counting process and we use paper only counting paper only ballots which means that it's very much verifiable if there are invisible inks there's no way that is escaping the YouTubers eyes and so that the point here is that if people don't believe the opposition party's YouTubers most of them don't they do believe their own party's YouTubers and when those YouTubers report the same counts people don't buy into this invisible ink story anyway so this is radical transparency and participatory accountability inclusive accountability anyone can be a auditor in the sense of that that's so we see this rumor goes up and then once the YouTubers numbers start to agree and people get their count from their favorite party members then this goes down again so I would say that whatever happened in the US was more of a symptom of these systems not in place and that's my take. All right, super interesting so let me expand from that and ask one last question before we open it up and take some questions from Slido Mark Andreessen wrote a widely noted I guess it was a blog post about a year ago or just a year ago basically saying it's time to build and at least what I inferred he meant by that was that it's time for entrepreneurs to come to the fore again and build companies and creative ways that get beyond the great stagnation we wanted flying cars and we got 140 characters instead a kind of exhortation to the software geniuses at Stanford and beyond to create the new generation of companies if you were able to give a similar exhortation to the talent at Stanford University and beyond about what they should be building or what they should be doing if not building I'm curious what software do we need to support rather than subvert democratic institutions what types of career pathways do we need for people at Stanford and beyond so that social systems and individuals thrive rather than being threatened by the current or kind of landscape of big tech Well I'm just free associating here because I am seeing your face in the middle and a circle of students and each name with a mute button mute icon next to them and so my free association when hearing your question is that maybe you know unmute yourself and unmute others like literally give voice to people and I think that this idea of democracy as a type of technology is really about increasing the bit rate because in traditional way of thinking about democracy each of us get three bits every four years per person of upload and that's it, call voting and in other times we're muted or we're literally muted and there's really no way for us to study agenda I mean we may answer a online poll or a SMS poll or a survey call from be part of focus group or whatever but we're anonymous there we're just aggregated data to the democratic process and the commercial world and the social sector have all developed ways of co-creation I'm thinking about crowdfunding crowd sourcing and many like now staples already infrastructures of the whole creative process to get your customers to be not just customers but essentially your partners in co-creation and so people would pivot their ideas very quickly I think that idea came from Stanford actually pivot their idea very quickly when their stakeholders discover a better way of doing things but the government's because of the limited bandwidth of the democratic process couldn't do so until four years or two years down the line right so my invitation really is to think of democracy as a bit-rate constrained process and work on ways to empower the voiceless just as in Taiwan you are a national participation platform already more than one quarter of the 10 million or so visitors per year in the country with 23 million a lot of people the initiatives more than a quarter are from people who are not even 18 years old who are still in the basic education schools and so on and because of that they don't get this idea that I need to mute myself until I'm 18 years old and therefore matter to the civic process anyone who's like six years old, 12 years old they always perfectly good citizens initiatives especially around sustainability because they are on the business end of climate change and when we empower these people more they will probably not strike and go to street on Fridays but they effect real change all right let's move over to Slido Ryan we're a small enough group here that I'm just going to invite you to unmute yourself yes thank you that's the call to action oh okay sure hi Audrey thank you so much yeah so I was just reading some things I also saw a podcast or a part of an episode of a TEDx technology podcast that you recorded once along sort of the like information campaign in Taiwan but I was I don't know this question is more for you personally like where did your I guess like transparency idea for government come from like I guess maybe what were you reading or who you were listening or talking to and then also aside from just government how could how might this help companies such as like before Facebook especially when it comes to user data yeah thank you very good question personally I participated in internet governance as I mentioned when I was just a 14 or 15 years old across the internet nobody know I was just 14 right and and so I work on technologies such as the ATEM technology who's a successor of the RSS or alternative to the RSS system to connect the the blocks together and so on so in that case I'm working on what Aaron Schwartz among others have created and bring it into a more real time publication subscription and so on and I discovered that's really in the bloke's fear or in the internet engineering task force which is where we're doing those standard making nobody can force anyone to do anything that the internet society doesn't have a army or a navy's for the matter so all we have is transparency is rough consensus and running code and making the process transparent so people if they are negatively affected can get that addressed immediately and when nobody feel that this will hurt them or we can at least all live with it then we literally hum right either in the same space or on the mailing list we literally hum our consensus a rough consensus into action and then we go go and develop our own coach and so this to me seems to sign that if the transparency is inclusive that is to say anyone who could get negatively affected and has an email address could join the decision-making process that and if nobody can force anyone to do anything then by definition by default it works toward a common good it avoids the tragedy of commons and is more likely to avoid a tragedy of horizons because the people who represent the future generations interest because it's not a voting process they could also get their idea more nuanced eclectic ideas into the mix and into the rough consensus setting so I would say that I learned it personally from internet governance now for corporations I think that also helps a lot because we're now in an era of stakeholder capitalism we are seeing now in the some parts of the world like Taiwan and the party are in the especially young people but more and more consumers are not satisfied of the companies just doing their like minimally efforts to avoid polluting the environment or polluting the society nowadays in Taiwan people consider the company that do nothing to be proud of the problem and they demand to see where they are in the you know solution to for example reducing carbon footprint among other things also reducing conspiracy theory footprint in that case of the more anti-social social media so because of that any company that is not transparent with the customers probably face more a higher chance of a social boycott or social sanction and the only way to turn the co-creation energy of your customers with your leadership team and so one is to be basically the same as the open government principle to be inclusively transparent and also invite participatory accountability all right there are a bunch of other questions that are anonymous we're going to come to those in a minute but Bagarath I've got a question from you actually a couple of them but why don't you just unmute if you're willing and and and ask your question to the minister sure hi minister Tang it's so great to be with you I had a question on just like what are some lessons you can share with us about successfully integrating technology with government and what are like some of the challenges you have faced and how how did you overcome them because here there's tons of regulation and sometimes bureaucracy can be slow so I was speaking with the former chief data scientist of the U.S. DJ Patil and he mentioned how it can be challenging to navigate through all of that so I was just wondering what do you think helped you become successful yeah the career public service can be slow but it also could be really fast and the tempo the iteration in agile speech that the length of the sprint or the scrum is is where it's mattering so personally I hold my office hours to social innovators in Taiwan every Wednesday so what I'm doing is essentially improving the the kind of bandwidth from social meditators anywhere in Taiwan with the career public service by bringing out fresh ideas every Wednesday which means that after this video conference I'm off to the social innovation lab which is my office and the office is itself co-created with the leading social innovators in Taiwan and beyond so this is my my office like literally and soccer field here is by people with Down syndrome who turns out not are only good in bakery they're very famous bakers but also in Taiwan and but also very good at creating art so they just draw whatever they see and it's like the the lens of Van Gogh's eyes and people automatically become more creative when they step into this place with pretty good music pretty good food and and things like that and I do bring in this ideas but I don't command anyone in my office in the space to to realize it for people who are in the public additional innovation space my office if you check out digital ministry at TW you see the people there are secondments from pretty much half of the ministries so more than 12 ministries have sent secondments or dispatches to my office and they're all career public service but they don't report to me I don't order them to do anything they don't give me orders all I ask is that we work out loud we use a Kanban board started physically but now the office has like 20 staff and 30 interns and you know many the network of comedians and preservation offices more than 100 people in total we can't use physical Kanbans anymore so we would send it online but and nevertheless the spirit is that we just have fun and share food every week on a weekly basis incorporate those social innovation ideas into visible things and for the career public service station in my office they know that first of all I absorb all the risk if things go wrong if we had to pivot it's all order spot so say they are happy about proposing genuinely new idea sometime anonymously through Slido actually and also it saves time for most of the bureaucracy they're wary about implementing something that saves their time but actually waste people time other the citizens during the process if they're not familiar with it or some other departments which will absorb the time cost or the like the retraining cost and so on so they're very conservative in the innovation that they deploy but because pretty much all the people facing bureaucrats are already in my office so whenever they hear something from a video conference call with a remote area and so on they don't say oh I have to check with the Ministry of Interior the Ministry of Interior dispatches sitting right next to them I have to check with Ministry of Health and Welfare because they're sitting right next to them and so because of that they were able to in a facilitated conversation gets to the rough consensus about okay this seems that nobody is losing from it and because every other Ministry's dispatch even if they're not related to it they can also play outside customer because we're all citizens ourselves right so if people get above consensus then it's likely to save everyone's time and for ideas that reduce everyone's risk and save everyone's time the career public service is very fast in implementing it at that they're like oh what way don't we get it done yesterday and so that's then improve the mutual trust because it improves the feedback cycle the speed from people calling into the office or the hotline saying hey I don't know I invented a new method of using traditional rice cooker to cure the virus but doesn't destroy the musk and that gets amplified immediately we make a creative film of me cooking the musks dry steaming without adding water or a young boy calling saying you're rationing our musk and all I get was those pink medical musk or the boy in my class have navy blue surgical great musk and I don't want to wear pink to school well then we talk to the persuasion officer the person who lived with that cute dog and they made a suggestion and lo and behold at the next day on 2 p.m. all the medical officers in the central epidemic command center regardless of junta or pink and with pink medical musk and all and it became really sensational all the all the leading brands then just changed their social media profiles pink and I think our ministry of health even said that pink panther was the child of a hero and so the young boy become the most hip boy in the class for only he has the color that heroes wear and I guess the heroes hero wear so on and so because of this it massively increased the mutual trust and then we get more fresh ideas from the civil servants as anonymously on slide sometimes and also from decision makers all right so you're painting for us this kind of wondrous picture of radical inclusivity and participatory governance and openness and transparency a playfulness and humor written into it comedians on on the staff just as important as other civil servants a couple of the anonymous questions on slide out the next on on my screen ask questions again to put you know just as you call democracy a form of a technology and we're looking at the bit rate of participation well you know American democracy is is at a much greater scale than Taiwanese democracy we're 12 or 15 X the size in terms of the population scale in in many respects introduces new problems in into the mix and I wonder if you think the kinds of things you've accomplished in Taiwan would translate more or less smoothly or cleanly into an American context to the scale or is there something different about the scale of American democracy take the question that's asked about trust in particular as a as a different kind of challenge that you would need to rethink if you were you know in these in these quarters rather than where you are in Taiwan well the population of of say New New Jersey is one third of Taiwan and California is just I think slightly more people right we have 23 24 million California's was 39 million and so we're on comparable scales is what I'm saying because to work across different time zones is is difficult as like proven today and and people who live in different time zones there's less chance of real-time interaction right so I think for this cross-time zone conversation the ideas of maybe something like immersive virtual augmented reality is probably needed if you want to people to feel this kind of empathy in real time across time zones but within the same time zone is always much easier and I think that the two foundational requirements one is broadband as human rights universal broadband in Taiwan even if you're on the tip of Taiwan almost four thousand meters high the Jade Mountains or the Saviyah or pen of the known you know indigenous languages all these interpretations could get a full live streaming when people go to the top of Taiwan because they know they have to make this per second both ways for just 16 euros per month unlimited data and they know that if they don't it's my fault they can actually call me so this is a solemn guarantee that we use you know helicopters spectrum auction methods whatever to to make sure that everyone has universal broadband access because otherwise when we introduce these mechanisms we're essentially leaving people behind we're excluding them from the democracy where you're muting people who were previously unmuted and as unthinkable and so that's the first thing and the second as I alluded to is the idea of digital competence we don't teach media literacy anymore we teach media competence the same for data and digital competence meaning that the young people are not just a consumer which they had to remain consumer until they're 18 years old but rather they're already producers of narratives they fact check the presidential debates and forums they make their own environmental sensing machines the PM 2.5 sensing machines in particular called Airbox where it's pretty standard actually in the primary school curriculum so each and every dot you see here is some primary schooler and their teachers contributing to a distributed ledger that measures the sensing the environments in real time and really to me there's no other way to teach about data stewardship about bias about the idea of a feedback cycle of stakeholder of data and so on data pipeline and so on without people actually participating in the production and the narrative making of data so that's our primary school education so with data and media competence and with Providence Human Rights I don't I think this is remarkably scale free if people end well with overlapping time zones then we can get overlapping consensus uh-huh fantastic I will confess that that sounds almost too good to be true for the sporadic orientation in me but I mean well we're in the future right like full 12 hours that's true that's true and it's distributed unevenly as the as the Institute for the Future here in Palo Alto well done we've got some more questions on Slido but you know with folks here there is a hand raising function so if you if you have a question for Minister Tang don't be shy about raising your hand right now let's go over back to Slido in the meantime and you know the issues of radical transparency of openness seem also to bump up against conventional concerns about data privacy we've covered those in the class a good deal I wonder if you can say how you how you square that circle how on the one hand with with media competence that you're just describing radical forms of transparency in inclusive ways do you nevertheless find ways to protect data and ownership even for parts of people's digital lives that they don't necessarily want to be open and transparent to others certainly so when we're in a public square whatever we contribute to the commons are voluntary and in the social innovation lab which I just show you a minute ago people come to my office out with the understanding that everything what I said to them will be on public record however if they relate something about a power imbalance in their work or they saw something wrong and they tell an anecdote about their friends who wish to remain anonymous and so on I fully respect their wishes so while it's true that we publish the entire transcript to the the internet I think the say it function is the I'm just pasting a link here you can see that after I become digital minister I've held 1600 meetings with 6000 people in 300 000 speeches nevertheless if the people wish to remain anonymous and or if they wish to just censor themselves whatever they said in extreme cases transcript become a monologue of just me talking and people can take out any part of it that pertains to say privacy or trade secrets or whatever else so this link in particular is a conversation with the senior leadership team of Microsoft with all the usual suspects and they want to protect their privacy I guess because they're a privacy conscious company or something and so they decide to to censor everything that I said from the senior leadership team but you still have this one side transcript from my side so you can still like sit out whatever they have asked but in no way compromise this review is zero bits of their information because I give my visitors 10 days to just scrub their transcript so if they notice some part of my speech that reviews their trade security or private details they can tell me and I'll always reframe that so that it becomes non-identifying let me just push one more dimension on that so I understand the the principle there about when people show up in a public or in an interaction with you in your capacity as minister these are strong expectations about transparency for citizens to inspect what you do and to be able to participate in ways with equal information flows but you know it's often sometimes said that for COVID responses for example the expectations of either freedom or privacy here about employers or companies or worse the government surveilling us for the sake of public health spills over into other forms of surveillance Edward Snowden style and in that respect you know undermines our liberty and compromises our privacy even when there is a public warrant for it so so what I want to push on is just to get you to say something more specifically about if you see trade-offs or tensions it does Taiwan or in your head do you simply wait the interests of the public over and more heavily than the interests of individuals to determine for themselves what they want to share and what they don't whether it's for the COVID response or for the purposes you were describing or do you think that it's not a it's not just a weighing of the of the values or a different balance but but you there's a you know an understanding about public and private that's different from what I've described mm-hmm yeah I don't think it's a trade-off because if you frame it as a trade-off there's there's no winner it's essentially a knowing situation I think Ben Franklin says something about that right deserving leader liberty or safety and so what the Taiwan model shows essentially is that basically liberty is a form of like self-expression and privacy is respected when everyone who voluntarily participate in a social relationship or transaction gets probably the same or overlapping norms and just act within those norms and so just one concrete example of during COVID we did what we call participatory self-surveillance that doesn't require the state or the capitalist to be the awesome I right and so in Taiwan we require people to register to the business they visited their contact so that when a local outbreak happens we know who to contact with the contact tracing now we didn't communicate that very well I guess because back in last April there is a professional hostess in the hostess bar part of a nine life district that gets diagnosed but on the first day she didn't want to review the contact numbers for the fear that her patrons will be re-identified which would violate the privacy norm in those parts of the nine life district and this is a classic trade-off situation and Taiwan is not the only jurisdiction facing that pretty much the same week Japan is facing the same thing and later on Korea faces the same thing but our response is very different instead of threatening to put people in jail or finding the business or ordering them to close and essentially driven them underground as some other jurisdiction did the US did during prohibition and fostered an unaccountable underground economy and therefore have a lose-lose situation we instead invited basically publicly challenged without imposing prison or fine saying that you know all we need is a way to contact people so we don't really care who they are and also we don't want their contact numbers the business owners can retain those so they brainstorm and in just a couple weeks adjust it to basically have scratch pads so anyone visiting the night clubs and or whatever have to maybe get a pre-pays and called maybe a throw away proton mail or whatever email address and they file that in make sure they could be contacted but that's between the business owner and them and nothing more and they keep those shards of papers like literally 28 days or four weeks and after no outbreak for those four weeks they shred whatever that was 28 so it's like a one-time path but it's not really a one-time path but you can think of that way and so because of that what we're seeing is that it preserved the privacy norm there because the patrons understand this will not get aggregated by the state or the local government it will not get into you know the surveillance capitalism world which will be really bad for the people who will be frequent there and so because of that they become part of the solution not part of the problem and then the nightclubs and the drinking pubs and whatever gradually reopened after just a few short weeks after they invented this way of what we call a real contact system this is just one anecdote but there is a philosophy behind it this is basically saying privacy is about norms norms are good for co-creation and if we involve the people who participate it's including host and hostesses in a hostess boss then they do the right thing but if you give no trust you'll get no trust yeah yeah I mean I myself have never been to Taiwan but hearing you describe it now puts it even higher atop my wish list to experience because what you describe sounds so fantastical from an American perspective with not merely are there issues of lack of compliance with contact tracing but merely showing up in public with a mask as of course you know being reports was a technology here has often served a polarized society rather than to create norms of trust and co-creation and reaching a point where playful settings of co-creation is the norm in public space among citizens sounds a happy distant future to this socratic I invite you to the future and check it out yourself I paste this goad card link you can apply it overseas and get a free pass to the Taiwan society even during the COVID more than 2,000 people visited this way that's more wondrous every minute we speak Mr. Minister Tang all right we've got just under 10 minutes left I want to make sure that other folks get a chance to interact personally with minister Tang if they wish so please raise your hand if you've got a question I've got many more but I've been the mouthpiece for the most part so far please get into the conversation I'm going to leave a pregnant pause here for first people to get in I'm just making some instant coffee all right there's a hand Thena go ahead hello thank you so much for talking to us it's a real pleasure I was just wondering what you think or any examples that you can think of that is the worst possible way to deal with misinformation that you've seen here in the US yeah a really good question it's a seminar topic okay so what I think is feasible and what we've seen in the more polarized democracies is to begin with things that are broadly speaking people could relate to for example in Bowling Green Kentucky they run a public consultation using Polis the same software we used to cross those rough consensus on the lack of UberX or Airbnb qualities so they make sure that only the rough consensus the resonating ideas get print get visibility rather than on the more anti-social corner of social media which to me are like the nine life district you know with additive drinks and private balancers and things like that I mean Facebook and so on and trying to you to convert that into a town square well no right tough luck so anyway so the point is that if you do have digital public infrastructure like Polis and as evidence by people in Bowling Green Kentucky who actually did a consultation with no preset topic so basically agenda setting instead of getting people to comment on particularly agenda and then you suspect you have an agenda you just begin saying what about a civic town hall that's really unconstrained you just do do whatever and then people actually come to trust one another more because they could see no matter whether they identify as Democrats or Republican or whatever they all have the same priorities in the Bowling Green Civic Assembly homepage you can see that everybody cares about the art in STEM education so people see that the STEM need to become steam and it's bipartisan actually it's non-partisan it's trans-partisan meaning that everyone feels very strongly it need to be done yesterday but people didn't know that they actually agreed on those things before they all care about universal broadband access more diversity more choices in telecom operators in broadband suppliers again that is a non-partisan trans-partisan issue and so on so once people see the picture from Polish and this is the actual Bowling Green picture and this is I think one of the most important picture that people can see during a democratic process so this picture looks like this and then this reads divisive statements so that you know polarized ideological statements this is a Bowling Green conversation and these are the rough consensus like things that people probably agree on and then sorted by the consensus you see the country statements that people do agree on like steam education like the diversifying of broadband access traffic flows shaping and like it need to be improved especially on the Scottsville road okay anyway so I'm just reading that report so then people get into the feeling that we are a polity after all this public rough consensus get reflected to the people's minds and this powerful picture just like humor triumph over rumor this overrides people's self-identity of thinking half a population is somehow known people right and so just getting them into this coherent blended coherent volition moment for a few times and then people develop their own agency to work across the party or ideological differences so that's this like cheap and cheerful like no cost way of starting a public communication just talking about the traffic flow on Scottsville road super James go ahead thank you so much thank you so much minister Audrey oh yes my question is how do you balance between automation and job security in Taiwan right so in in Taiwan I call AI assistive intelligence just like I I call collective intelligence CI so they give us ACI I guess assisted collective intelligence and by by saying assistive this means that we hold the same expectations as we will hold any human assistant right that the basic ideas of alignment like this glass actually showing me things more clearly rather than more blurry or pop up advertisements goes without saying and accountability goes without saying when you frame it in an assistive way and so by framing this as ACI what what do our assistants do well like Francis I guess she helps making this communication possible enable human to human communication connecting me to the people who otherwise I wouldn't have the bandwidth to to communicate and so basically a good human assistant actually Francis a professional diplomat from foreign service so I'm just kind of using this as an example and not really saying that foreign service is some sort of AI but anyway what I'm trying to say is that just like that the foreign service does not take over Ministry of Interior or really any other essential function of a state but rather enable collaboration between previously less trusting parties then this makes collective intelligence more possible so if we frame AI this way this is strictly speaking improving the capability of people to co-create together which needs a lot of amplification because we all have this wet wear hot coded Dumbass number that we don't function very well with more than 150 strong links right so some sort of assistant intelligence is needed anyway video conferencing is actually one part of it right so so if we develop AI in our national agenda in the AI ethics and so on strictly toward divisions of a assistive collective intelligence then the ideas of human replacement just just go away because that's not what we're interested in that's not why we spend taxpayer money to we don't design incentive structures for corporations working on those things that take away job security and so on All right well we're a minute shy of 6 p.m. on Pacific time and and I guess 9 a.m. at your time in Taiwan on behalf of the entire group I want to express our gratitude to you at an early hour of the morning for making time for for for us and for our questions as I said at the start you're an inspiration to me and and I hope to to many others on this call and in our class as well I thank you so much Minister Tang it's been a delight and and so illuminating thank you live long and prosper bye


So we're having Tim and Dr. Chitra's second talk on the inverse Scala problem. So please go ahead. Oh, thanks, Bjorn. All right, so just to remind you where we were, was it only yesterday? We proved that, reviewed Hilbert's Revisibility Theorem and proved that as San is a Galois group of Q. And remember the proof was that he start by a field where you're just formally joined in determinants, just an independent transcendental variables. You let S and act on this field and then you compute the invariance. And because in this case, the invariant field just happens to be purely transcendental over Q, it automatically gives you a family of SN extensions over some rational function fields such as this. And then by Hilbert's Revisibility we can get a San as a Galois group of Q as well. Now, there's an obvious question, is that can you do it for other groups rather than SN? And so let me start with this, it's called a NERTHES method or NERTHES program because she basically asked this question and initiated quite a lot of work and invariant theory to look into this. So suppose now G is not a symmetric group but just an arbitrary group. Find it, of course, like all groups here. So it's a subgroup of some SD. In other words, G acts again as permutations on D things. So let me again take D independent transcendentals, D determinants and let G act on them by permutation. And now this is not a full SD action but nevertheless it does. And you can do the same thing. You can compute the, you can start with this field capital K which is Q a join alpha one alpha D, rational function field in D variables, G acts on it linearly preserving Q and you can ask yourself what's the field of invariance and in particular if this field a purely transcendental extension of Q. And then if the answer is yes, then G is a, we've just realized G is a Galois group over this field and by Hilbert's visibility, we also realize it over Q then exactly as we did for SN. In fact, what you did, what you do need is a little bit less. I said, let G be a transitive subgroup of SD so it acts on D letters. Actually you need a little bit less. You need a rational representation of G. So it's enough for G to act on Q vector space and then we can do this whole yoga but it doesn't really matter. So what NASA did is that she approved for all subgroups of S4. This is indeed the case and this field that you get here is purely transcendental in particular. You can prove like this that all the subgroups are realizable as Galois groups over Q. So this is a very encouraging beginning but it turned out that in general, these fields, this rational function fields, so these fields of invariance, they can be quite complicated and the answer in general is no, they are not in general purely transcendental extensions of Q. So the first example was by Swan in 1969 and then there was a for cyclical group of 47 and then there was quite a detailed analysis for a billion groups by another Henry Glenstra who proved in particular that C8 is the smallest example of a cyclical group for which the answer is no and this field here is not purely transcendental. And I should also say that as far as I know, this is also not really a practical way to realize Galois group over Q. This fields are quite complicated, they can be complicated to compute and to decide whether they're purely transcendental extensions of Q or not. But anyway, this sort of, this was on the, I guess still is an approach to the inverse Galois problem and just one remark on it, you can reformulate it in an algebraic variety terms. So what is this field capital K that we started with? The field Q join alpha one, alpha n. So this is the field of rational functions on an affine space A n over Q. And when we compute its invariance, that's the same as today the quotient of A n by the action of G. So again, G acts linearly on this vector space in our simplest case, by just permuting the coordinates. So this quotient here is some n dimensional variety, which is to be affine. And notice question is simply equivalent to the fact to the question, whether this quotient to be is a rational variety. So we call rational variety is the one which is bi-rational to P n or A n. And that's a question about its field of rational functions. So it's a question, whether it's field of rational functions is purely transcendental of Q. Now, of course, you can say, well, asking whether it's rational variety, even if the answer is no, well, that's a little bit too much to ask, is it? You don't need it to be a rational variety. For example, even if it has at least a rational point, at least let's say that this map here, you have a natural projection map from A n to A n over G and assume it's underemified over that point, then the fiber of that point gives you a G extension of Q. And therefore you just realize G is a Galois group of a Q. So if you want to solve the inverse Galois problem, really you don't just need this to be a rational variety. That's between overkill. You just need to be able to prove that it has or it has enough rational points. Or for example, if it has rational curves or rational varieties of high dimension, then you can realize this group G over some function fields, possibly with less variables than M. But in any case, you've solved the inverse Galois problem over Q of t as well. So this is just nice way to think about it. This is just a geometric interpretation of what Hilbert has done for SN and what you could try to do for other groups. Okay, so as far as I know, this approach kind of doesn't get you that far in the inverse Galois problem. So let me talk a little bit about the approaches that do and focus at least this lecture, mostly on soluble groups. So let me start by one very easy observation that if you realize group G, then automatically you've realized every quotient to G. So in other words, let's say it was little field K, maybe Q, if you try to realize Galois groups over Q or Q of t or Q t, Q t 1 t n. And suppose you found an extension of little K, capital K, which has Galois group G, then and Q is a quotient of G. So suppose you have a normal subgroup and in G, such as quotient is Q. Then of course, just from this tower and from Galois theory, you see that this quotient is, it is also, if you find a Galois extension, it's Galois group Q. See, there was a question in the chat. It's okay, I just, let's go ahead and take care of it. So in this story, when you take a quotient of f i in space, maybe you get a finite map. So the quotient variety is always n dimensional, but it's not necessarily a n or a rational. All right, so again, if you've realized the group, you can realize a quotient of it. And by our definition of regular, if you realize G regulate, let's say over Q t 1 t n, so no subfields are constant. Or if you want, there are no algebraic elements in capital K over Q, then of course, every such subfield is going to be regular as well. So every quotient group you've realized as well. So in other words, it's completely elementary that if you've realized the group, you've realized a quotient notation here. As I know, not a little unfortunate, but I don't know a good notation for quotient groups. So this Q over Q is really a quotient group of the rationals. I try to distinguish between what we're writing. And similarly for the inverse Galois problem over Q of t. And by the way, if you're playing with this package of computing Galois groups, it has a function which is called a subfield, which does let you compute subfields of families and therefore construct quotient groups like this explicitly. And now a natural question is, what about the converse? So can we always embed, can we always go the other way and embed a Q extension, where Q is a quotient of G into a G extension. So in other words, suppose you have a group, it's a quotient of a larger group. You've realized a smaller group as a Galois group. You want to realize a larger group as a Galois group is that always possible? Because if yes, there is a large class of groups, soluble groups on your photon groups, which you can build step by step with very simple steps, just simply groups. And then so for instance, every two extension of Q, you can build by repeated quadratic extensions. So then you can ask yourself, can I realize a soluble group? For example, Q over Q of t by this kind of repeated construction. And for that, you need to be able to solve this question. Well, this question goes under the name extension problem or actually embedding problem. I think maybe I should have said this is the one which normally use. And the answer turned out to be no. So in general, it is not possible to extend just an arbitrary Q extension to a G extension. So let me just look at one example of that and to see that there are local obstructions for doing that. So for this example, I am essentially talking about an embedding problem when you try to embed a quadratic field into a C4 extension. So let's suppose first that you've constructed an extension of Galois group C4. So let K over Q have Galois group cyclic of order four. So in other words, so this is C4 here and it has a unique intermediate field which I'll call L. So the top field is K, the bottom field is Q and there's a quadratic extension in between which is called L. And let us look what sort of restrictions this imposes on a field L to be contained in a C4 extension and you can do it by considering places above infinity. So Q has an infinite place and you can see how it splits in L and how it splits in K. So how do you do that? You have to look at the, what's called decomposition group, the infinity which is a subgroup of C4. So again, C4 is my Galois group. I have a unique place infinity here. I take an infinite place above it in K and I look at the set of elements in the group which leave that place in there yet. Now this decomposition group because I'm dealing in the infinite place, it's a Galois group R over R trivial or C over R of order two. So this is the group of order one or two which is contained in C4. So it's really two possibilities. It could be trivial or it could be C2 contained in C4. And now if you decode what it means. So in this case, there are four places above infinity in K because places above infinity in K are in one-to-one correspondence with left cosets of the Galois group by the decomposition group. So there's four of them in this case. And in the second case, when the infinity C2, there are two places above infinity in K because these are cosets of C4 by C2. And if you want to compute places in an intermediate field, there is a general result which is for which you don't need this group to cyclic or Galois or anything. If you know the decomposition group and you want to know how your prime decomposes not in the top field, but in any intermediate field, then the only thing you need to know, you need to know what is a subgroup that cuts out this intermediate field. So in this case, C2 in C4. And you have to look at double cosets of that group and the decomposition group. So in this case, the places above infinity in L are double cosets C2 backslash C4 forward slash D infinity. And if you compute what it means in these two cases, you see that, well, this is either C1 or C2 and you see immediately that there is a one, sorry, there are always two of these places above infinity. So in other words, which has proved that independently of what this decomposition group is, there's always two places above infinity in L. So the two possibilities again are that there can be four places above infinity in K or two, but in any case in the quadratic field in between, there's always two infinite places. And that means that this field is real quadratic. So K doesn't have to be real, it can be real or complex, but L has to be real quadratic field. So what we just proved is that if you have a C4 extension, then the unique quadratic inside it is a real quadratic. And in other words, what it means is that if you try to solve an embedding problem, you start with the quadratic extension and you say, well, I have a surjection from C4, which is this Galois group to C2 to this Galois group and I have a quadratic extension, can I embed it into a C4 extension? And the answer is no. So for example, QI or Q join, well, any negative square root so any dimension of the quadratic field cannot be embedded into a C4 extension of Q. So this embedding problem is quite, it's not trivial, already for simplest groups you can think of such as C4 and C2. And it's really Witt who studied this in great detail for at least small groups and he has complete results in particular for this case that I'm talking about C4 to C2 and the Quaternion group created, he was also interested in. So let me state his results and I outsourced the first theorem as an exercise, Q1 on the whole page. There's mini course. So with theorem is the very pretty statement. It says suppose K is a field of characteristic not two so little K and we're interested again in a question whether you can embed a quadratic extension over K into a C4 extension. So let's take a quadratic extension, non-trivial, K join square root of E and then it can be embedded into a C4 extension of little K if and only if this number E, which is in K is a sum of two squares. It's a very nice, very pretty result. So if you do it over Q, then for example, five is a sum of two rational squares. It's two squared plus one squared. And therefore, you know that Q square root of five, you can embed into a C4 extension. And in fact, there are many such C4 extensions and one of them is Q join Z divide. So this cyclotomic field here has got a lot of that much for that and it contains Q of five. But on the other hand, if you take QI or Q would three or any other rational number, which is not a sum of two squares, then such quadratic extensions cannot be embedded into a C4 extension. And you see therefore that if you try to, well, do what Schultz-Reinhardt and Schubert-H have done to prove, for example, that every soluble group is realized over Q by some sort of inductive proof, which is essentially what they did, then you do run into this an obstruction problem because on various steps in your construction, if you are not careful, these extensions will not even exist. Just see if there are any questions which are unanswered, no, not yet, okay. All right, so and another result, which is also very nice and there have been generalizations of it by Sarah and others by looking at these quadratic forms, which play a role here for other groups, but this is the first result like this proved by Witt in 1936 for Q8. It's the same sort of problem, but for the Quaternion group Q8. So if you try to construct the Quaternion group Q8, it kind of goes like this. Q8 has center plus minus one. So this is a group which has eight elements, plus minus one, plus minus i, plus minus j, plus minus k. It has center plus minus one. And if you divide by the center, you get C2 cross C2, generated by classes of i and j in the quotient. So the way you can construct Q8 extension, you can start with a C2 cross C2 extension. That's very easy, right? It's just by quadratic, take arbitrary numbers A and B in your field, as long as they're independent quadratically and you do get a big quadratic extension, then, well, this is a C2 cross C2 extension. And then you ask yourself a question, when can I extend to a Quaternion extension? Is it possible? And if not, what's the necessary and sufficient conditions? And this sufficient necessary and sufficient conditions were found by Witt. He proves that a big quadratic extension like this can be embedded in the Quaternion extension. So in other words, there is a quadratic extension on top, which makes the whole thing Quaternion, if and only if the following two quadratic forms are equivalent over, sorry, it's supposed to be a little k over the ground field. So this is just a standard form, x squared plus y squared plus z squared. And this form has coefficients A, B and A times B. So A times B defines a third quadratic, which you don't see here inside this big Quaternion extension, k a joint square root of A. So again, this is a very neat, necessary and sufficient condition and you can make it completely explicit in a sense that if you have a witness to the fact that these two quadratic forms are equivalent, then you can actually construct this Quaternion extension explicitly. So the second part of the theorem says that if P is a three by three matrix, which let's say was determined one over A, B, normalize it like this, such that it transforms this quadratic form into that quadratic form. So P transpose times diagonal matrix with A, B and B on the diagonal is the identity matrix. Then you can classify all Quaternion extensions containing a little k, A, B, root A, root B. There's essentially only one line of them. So you can a joint square root of this expression here, where this P11, P22 and P33 are the diagonal entries in this matrix capital P and root A root B and root A, B are in your big quadratic. So this gives you a Quaternion extension and every other Quaternion extension which contains this big quadratic is really the same except before taking a square root you multiply it by an element of a little k. I see that my little k became capital K and fortunately somewhere here, but it's the same people. So this again is a very nice result. But again, you see that this is, well, it's a non-trivial condition. For example, you see if you stare at this that if you want this quadratic forms to be equivalent they represent the same numbers and the numbers represented by the form on the right are sums of three squares. So anything represented by this form, for example, A, if you plug in y equals zero and z equals zero, x equals one, you see that A is represented by this form and similarly B and AB, they're all sums of three squares. So in other words, if you take a quadratic extension K and join a root A and you ask yourself can I embed it into a Quaternion extension? And then the answer is no unless A is a sum of three squares and that's more or less if and only if in fact there's a very few exceptions to that. But again, it means for example that if you work over the rationals, then well, sums of three squares are positive. So things like QI, for example, again, you can not embed it to a Quaternion extension of Q. How hard is it to find such a matrix P? Yeah, that's a good question. And as far as I know, there are computer algebra systems like Magma which have nowadays equivalence of quadratic forms over general number fields. But from what I understand, it's quite a difficult problem to do it. And in sort of, I don't really know very much about it, but it seems not very easy over a general number field as far as I know. For the second question, do we have a result where the extension of degree to the K can be embedded in a Quaternion to the K plus one extension? I don't know actually. So this is, this is a type of as I mentioned, this is a type of problem which has been considered quite a lot because in the example that you ask or the example that I do here, this is what's called a central extension. So the little C2 which you want to put on top is contained in the center of the group. And this is exactly the type of extensions which have been considered a lot by Sarah, Master and others, because they looked especially at double covers of alternating groups, so-called AM twiddle, which are like this where you construct AM and then you want to extend it to an extra quadratic which is a central quadratic extension. And so in that case, there is a result which essentially involves a trace form of the field in question, which kind of explains where this form comes from and which quadratic forms you need to be equivalent. So I think there is a result like this. I don't know if anyone has formulated explicitly for Quaternion extensions and well, how practical it is, I guess, depends how efficient your algorithms are for proving that quadratic forms are equivalent or rather giving a matrix which does that. Okay, so right. So as I just mentioned, especially for double covers of alternating groups, people have studied this a lot. And in particular, it has been proved that this double covers of AM, so the smallest interesting one is a group SL25, can be realized over Q of T using these methods. But as I would like to mention here, these proofs are not sufficiently explicit in that they do not give a family. So as far as I know, for these groups, even again, for SL25, when MS5, it's a double cover of N5, there are no families known over Q of T and I stated this as a research problem P1 to find such thing. And again, as I mentioned before, Schultz-Reichhardt and Schubert, they do construct soluble groups. For example, the soluble groups over Q inductively and they need to deal with this problem. It's essentially sort of by ramifying a lot of primes along the way and occasionally changing groups in order to prove that these instructions do not exist in their context. All right, so what I would like to do next is to talk about cyclic groups because as you've seen from this extension problem business, even for a cyclic group of order four, which is not like a very large group, it's not entirely clear how to construct C4 extensions of, let's say Q of T because, well, if you do it very naively by starting with a quadratic and trying to extend it, this is not entirely trivial. So the question is how we do it for general cyclic groups and generally what are the, and after that I'll talk a little bit about one case where this abstraction problem does not exist that gives a large class of groups that we can construct called semi-abelian groups. All right, so let's look at cyclic groups next and construct a regular CN extensions of Q of T for arbitrary M. So this is the problem I'd like to talk about to start with Q of T. How do we construct a family over Q of T of regular CN extensions for arbitrary M? Now on the one hand, you can say, well, from algebraic number theory, we know everything about CN extensions because this is just part of class field theory. We know everything about abelian extensions of Q and every CN extension is contained in some cyclotomic field, Q zeta M. We know the Galois groups. We know what these fields look like. So, you know, how hard can it be? But if you think about it, if you try to do this, you find that class field theory is a little bit kind of orthogonal to this families over Q of T business. And while, for instance, you can just construct lots of C4 extensions in the class field theory, take, for example, the prime one, what four, five, 13 and so on. Then in Q zeta M, there will be a C4 extension. It's very difficult then to put them into a family over Q of T. So, and there is rather easy solution to this, how to construct this CN extension, which I think has been rediscovered a few times. And I think first goes back to Smith and then Dan Sar and Schnapps have refined it a little bit and it uses Kuhner theory. So, let me explain how it works. And this is the one which is also implemented in the package. So, suppose you've already constructed an extension, let's say over Q, with Galois group CN, C4, if you wish. Then, no, no, how do we understand CN extensions without class fields theory? Well, there's something that's called Kuhner theory, which is in some sense much more explicit and much more amenable to what we are trying to do. For example, every quadratic extension of Q is just obtained by joining square root. And similarly, every CN extension can be obtained by joining nth root of something, but it's only true over fields which contain nth root of Q. So, in other words, if you start with a CN extension, so Q, so degree n extension Galois with Galois group CN, then it's not true that it's obtained by joining nth root of rational number. That's completely untrue, because if you join nth root of rational number, you don't even get a Galois extension. You need nth root of unity to make it Galois. But if you join nth root of unity, then this becomes true. So if you join zeta n, then this extension here, K zeta n or Q zeta n, which of course usually will also be a CN extension unless you've been very unlucky and you've chosen zeta n to overlap with K. So this will be again normally CN extension. And now by Kuhner theory, this is obtained by joining nth root. Of course, this nth root is not of an element Q, it's of an element in Q zeta n. So this K zeta n can be obtained by joining nth root of something to Q zeta n and this is something I'm going to call G. So G is going to be some element of Q zeta n cross. So in other words, every extension that we care about, every CN extension of Q, it's contained in this extension here where you take Q zeta n and then you join nth root of everything or if you want nth root of G for varying G. So let's look at this fields a little bit. Now, if you look at this field here, it's a Galois closure, it's actually reasonably large and this is what it looks like. It has the Galois group over Q, which is what's called the wreath product CN by Z mod n Z cross. So where does this come from? So first of all, this Z mod n Z cross, this is just a Galois group of Q zeta n over Q. So if you recall, the Galois group of Q zeta n over Q is Z mod n Z cross and this isomorphism is completely explicit. If you take an element J in this group here, so it's just an integer core prime to n, that there's a unique automorphism in Q zeta n over Q which sends a fixed primitive n through the unity to its J's power. And this gives an isomorphism like a completely canonical isomorphism between these two groups. Yes, so that's the wreath product CN with Z mod n Z cross. I think that's what I, I don't know my sign of wreath product maybe is not very well visible, but this is what it is. So generally, so this is the group here, Z mod n Z cross. So what I'm doing here, I'm taking a Galois Z mod n Z cross extension by joining n through the unity and then I take a CN extension of top on top. Now normally when you do this, so when you take an extension with Galois group G and then on top of it an extension with Galois group H, the Galois group you get for the full extension over Q is this wreath product of G by H. So the wreath and which is quite a large group. So it's, it's the way it's defined, you take this number. So that's what ends at cross. So all the five function of M copies of CN. So you take CN to the power five of N and then you let that one ends at cross act on it by permuting the, by permuting the copies in a regular action. And the reason you get the group which is so large if you think about it is the following. So when you are joined to Q z to N, N through to G you get a CN extension. But of course this group here is that what ends at cross it has a lot of elements. So there are a lot of automorphisms of Q z to N over Q and if you apply such an automorphism to G so to the element of which I take the N through then I get a different element in Q z to N and taking N through to it gives you a completely different degree of extension. So if you want like this to be small so this whole gala group to be small then this field here has somehow had to be invariant under the action of z mod N z cross and then generally it won't be. There's a completely independent CN extensions which you get by joining N through to G and through to sigma two of G and sigma N minus one of G where the sigma J's I denote this automorphism which sends zeta N to zeta N to G. So there's lots of CN extensions of top on top and if you compute the whole thing make it gala by taking the gala closure you get this very large group here. But nevertheless it turns out that if you play with this and cleverly choose a combination of these sigma and J of G's then inside this group you can construct a CN quotient. So here is a theorem which I think goes back to Smith in some form and as I said to then search and snaps in more generality but essentially it's a very explicit results which says that here is the CN extension for you. So this is how it goes. So in the notation as above let G be an arbitrary element of Q zeta N and let's identify the gala group of Q zeta N over Q with z mod N z cross as above calling the elements sigma J where sigma J sends zeta to zeta to the J. Then you just define the following polynomial. So it looks a bit large but essentially what happens here is that you take this N through of this different sigma J of G's. So again G's an arbitrary element in here you hit it by sigma by various elements in the gala group here and then you take N throughs you pick a choice of an N through an arbitrary choice for each of them. They don't have to be compatible in any way with some complex number or number in this field if you want where you compute it, it's just a complex number. And then you take these guys here and raise them to some powers throw in some root of unity for a good measure and consider these expressions as roots of a polynomial of degree N and then it turns out that this polynomial has the right Gala invariance properties. So this polynomial f sub G it depends on this auxiliary parameter G that you've chosen q zeta N is a polynomial with rational coefficients. And moreover from the computational perspective it has integer coefficients if you start with G which is integral so which is an maximal order zeta N. And of course polynomials with integer coefficients are very nice to compute over the complex numbers. You just do a complex computation. You take these guys here compute them as complex numbers to some precision multiply everything together round to the nearest integer as long as you control your precision you'll find these polynomials and very, very quickly. And for most choices of G again from the risky density point if you want this polynomial will be irreducible and it's Gala group is going to be C N over q. And moreover, if you think about it this construction again every C N extension of q is contained in one of these fields. So in fact, you can prove that in some under some assumptions I think the assumption is that N is called prime to five M or something like that then you do get an arbitrary C N extension of q like that. So let me just give you an example for N equals three and four where these cyclotomic fields q, z, or q they're very small they're just quadratic extensions of q, q, z, as three so q root minus three if you want or qi respectively. So you can just do it in full generality this whole construction. So you take G to be an arbitrary element in this field q, z to N. So A plus B z to N if you want where A and B just you can think of them as independent transcendentals. So it's an arbitrary element of q, z to N and then you plug them through the formula above it's quite easy to do by hand we don't even need a computer algebra system for this and then you find two polynomials which I wrote here for N equals three and N equals four f sub G of x which is this cubic and f sub G of x which is this quartic and these are families which turn out to be generic so they cover every single C three and C four extension of q families over q, a, b those two parameters a and b and this is how you construct C three and C four extensions and this works beautifully for a general N. So if you take G to be just make it one parameter now A plus z to N and then this is the choice that I implemented with the packages family of cycling group of N. So for any N if N becomes large then it's a wait for a little bit longer because these polynomials grow quite quickly and they have large degree in A but it will give you such thing hopefully eventually and these families are not very useful for large N because the so their degree in A so in other words the degree in these variables here is quite large for large N is N times five N and as far as I know for even for cyclic groups it is not known what are sort of the best possible families for cyclic groups in other words for lowest degree in A. I think it's a very interesting question and I sort of made it as problem p five on the homepage but in the case of this air construction and it does allow you to construct regular families over q of t and sorry and so at least for cyclic groups we know that they can be realized as gamma groups. Now to answer a question in the chat is it well understood which g work and why some fail? Well it's kind of the usual thing if you take g which is by accident happens to be an N power in q zeta N and it's not entirely clear how do you explicitly formulate this in terms of coefficients then of course it fails and you get a trivial extension and similarly if it has some sort of bad in various property so you just take it with zero I don't know you could get this polynomial to be inseparable but it's generally a little bit messy so I think the best thing you can really say is that most choices of g will work I don't think you can do much more than that really. Okay so now that we've constructed cyclic groups and then I didn't mention it before but if you can construct regular families for two groups it's very easy to construct regular family for the product of two groups just basically you're multiplying them together then once you know how to construct cyclic groups you can construct a billion groups and the next very important case which kind of brings you close to soluble groups is this one which is called split extensions by a billion groups so this is the one I'd like to talk about in detail. So recall we've already observed that there are these embedding problems or extension problems which are in general not soluble so you cannot for example if you have a group G and a quotient Q you cannot always extend a family or just field if you want with Galois group Q to embed it to a field with Galois group G and the two examples where it's not possible is C4 surjecting onto C2 and Q8 surjecting onto C2 cross C2 where you can't always lift C2 extension to a C4 extension and similarly here. So the problem here is that these extensions are what's called non-split extensions and for split extensions with a billion kernel it turns out that you can always solve this embedding problem. So let me say a few words about this. So suppose A and Q are groups. A for the moment is arbitrary but in the moment I'll take it to be a billion which is why I call it A. So just maybe to answer the question when I say that most G work I really mean that's the risky sense that if you go back and for example here you think of A and B are two independent transcendental so there is a whole sort of FI playing worth of choices then there will be as a risky close subset of those EMBs which do not work or at least the risky open which will work even though I think what I said is also true. So it's a risky dense and not a risky open, that's the same. Okay, so let me look at the split extensions with a billion kernel again. So A is going to be a group in a moment it's going to be a billion for the moment arbitrary and Q is also going to be a group. So this will be a normal subgroup and this will be a quotient. First of all, just recall that an extension of Q by A it's an exact sequence of groups like this. One goes to A goes to G goes to Q goes to one. This just encodes the fact if you want that A is a normal subgroup of G and Q is a quotient. So in other words, whenever you have a group which is a normal subgroup in a quotient group it's an extension of Q by A. So it's an exact sequence like this. And this extension is called split if this exact sequence splits. So in other words, if there's a group homomorphism going back from Q to G, let's say phi such that if you compose by taking phi and then pi again you get the identity map, it's the usual definition. So if you think about it, the image of Q under phi will be a subgroup of G, which does not meet A and which projects subjectively onto the quotient. And such subgroup is called the complement to A and G. So it generates G together with A and it meets A trivially. But it's not necessarily a normal subgroup. Otherwise this will be a direct product. So if such a thing exists then you call it a split extension and if it doesn't you call it a non-split extension and group theories very often use notation like this with a colon or a dot, the right G equals A colon Q if G is a split extension of Q by A and A dot Q if not. Now, every finite group can be built from finite simple groups repeatedly deductively if you want to use any extensions. So if you start just with simple groups and then you allow extensions of simple groups by simple groups, extensions of what you've got by simple groups and the other way around and so on and eventually you'll get to all groups. And therefore this notion of extension is very important group theory because it allows all kinds of inductive constructions as long as whatever it is that you're trying to do works for simple groups and it works for extensions. And in particular soluble groups can be always built from cyclic groups or even cyclic groups prime order like this. And now that we know from the inverse Galois theory point if you everything about cyclic groups we know that they can be constructed regularly if you want to get from there to soluble groups it's really trying to understand extensions. So now suppose from now on A is a billion then what happens is that, well we have an action of G on A by conjugation, right? When you have a normal subgroup A of G then conjugation preserves A that's precisely definition of a normal subgroup. So every element G acts on A by conjugation but if A is a billion then A itself acts trivially on itself by conjugation and therefore this action descends to the quotient. So it's really Q that acts on A. So the action of conjugation of G to odd A has A in the kernel. So we get a map alpha, well let me call it alpha from Q to automorphisms of A by taking an element of Q, lifting it to G and this action does not depend on this lift and then acting by conjugation on A, okay? All right, so now we have three pieces of data. Whenever you have an extension it has a normal subgroup A it has a quotient Q and it has an action which is also canonically associated to this picture of Q by A. And it turns out that in the case of split extensions which I should have said here or I should correct my notes. So in the case of a split extension this data A, Q and this homomorphism here from Q to odd A, they determine G uniquely. So you can construct G from this data from A, Q and such homomorphism and you get a unique split extension of Q by A which has this action here. So in other words, if you want to understand split extensions that's very easy. You just need to decide what your A is, what your Q is and you pick a homomorphism from Q to odd A and for any such homomorphism there's a unique split extension like this. So there's a formula which I wrote down here how to construct this split extension which is called a semi-direct product. In this case, it looks almost like a direct product except not quite in the first component because it uses this alpha somewhere along the way from Q by of Q from Q, A and alpha. Okay, so split extensions we understand completely and if you want to understand all extensions of Q by A with a given action alpha then they're classified by some homology group. So there's a homology group H2 of Q, A. So this is the group homology where you have a group acting on a Nabilian group and the action is given by that alpha there and it's this homology group which classifies all extension up to an obvious notion of an isomorphism of extensions where the zero element of this group corresponds to the semi-direct product to a split extension. Okay, so maybe let me just end with an example here. So let us look at extensions of C2 by C4. So in other words, I'm looking at groups G which fit into an exact sequence like this. They have C4 up in the kernel and C2 as a quotient. So G divided by C4 equals C2. So in this case, what you need to know the quotient is C2, the normal subgroup is C4. So what could be the possible actions here? What are the actions of Q1A? In other words, what are the possible group homomorphisms from Q to odd A? Now odd C4 is just Z cross, so it's just plus minus one. So there are only two automorphisms, it's identity on C4 and there's inversion on C4 which sends X to X inverse. So there are two possible actions, alpha one, let's say which sends C2 to odd C4 by letting it generate X identity. It's a trivial action and alpha inverse, alpha minus one which sends C2 to automorphism of C4 by sending a generator to the inversion X goes to X inverse. And now in both cases, it turns out that this homology group which I alluded to, it's cyclic of order two. There is a state homology theory for finite groups which says that if this group here Q cyclic which it is in this case, then you can compute this homology groups very explicitly by taking in this case just invariance of A under the Q action and dividing by the image of the endomorphism of A which is where you take the sum of all elements in Q and then you let it act on A. Now if you do this computation in this case for both this homomorphisms, you find that the quotient is Z mod two Z. So there are two extensions for each alpha one and alpha minus one. So for each of the choice of this action of Q and A there are two extensions like this, the trivial one, the semi-direct product and the non-trivial one which is non-split. So let me just list them here. So in case of alpha one, the trivial action the split extension of C2 by C4 is C4 cross C2. So this is a mableon group. This is a normal subgroup. This is the quotient and the quotient acts trivially by conjugation and the non-split extension, the unique one is cyclic group of order eight. Again, this is a normal subgroup in here, quotient is C2, the action is trivial but this is a non-split extension. There is no way to lift this to a homomorphism into here such as the composition is going to be identity. And similarly, in the case of alpha inverse if you want to construct extensions of C4 by C2 by C4 you find that there are two, one split extension which is by Hidl group of order eight D4 and one, which is non-split extension quaternary group of order eight. In both cases there is a cyclic subgroup of order four in the case of D4 these are rotations. In this case, the reflection gives you a splitting of this exact sequence. And in this case, you think for example subgroup change by I, but there is no splitting. So in this case, this is a non-split extension. And I will start next time by explaining what it has to do with solving the embedding problem. Okay, I think it's been 15 minutes so let me just stop here. Okay, thank you. So let's please join me in thanking Tim for this lecture.


So I'm very pleased to be here at the Pacific Central Station on behalf of Prime Minister Stephen Harper to announce an extension of the border clearance services to Amtrak's second daily train between Seattle and Vancouver. Obviously a very good day for British Columbia and the United States. The announcement of the second Amtrak train being permanent obviously means a great deal to both of our economies. You know, thousands upon thousands of people travel this train each and every day and it began as a trial of the second train during just prior to the Olympics in 2010. The response was overwhelming so we worked very collaboratively on both sides of the border to ensure that we could maintain this service that actually as I said earlier benefits both British Columbia and Canada as well as the United States. On behalf of the Washington State Department of Transportation and Amtrak, I wish to thank everyone who made possible the continuation of our second Amtrak Cascades that's servicing Vancouver, British Columbia. The extension of these services will ensure that Canada and in particular British Columbia continues to be a destination of choice for tourists and for business travelers alike.


I think dynamic front is an excellent opportunity not only just for me but your artillery community as a whole to work in a multinational environment develop our interoperability so there's definitely communications gap in a language barrier but I think I've been pretty fortunate in that the liaison officers at the Italian Spain and Great Britain provided a very amicable I've met some really good guys over there it's just a really good working fire and very friendly and we wanted for nothing all the support that we could have actually been given. Our mission today is to work with our allies and our peers here in NATO and in GERP and do a live fire firing reduce-range practice rackets along with our British counterpart. The MROS is an effective weapon system versatility being able to fire in all weather conditions where others can't. The family munitions ranging from unguided to guided are range and the letality of the munitions. Exercise Saber Guardian is a mission that we're doing partnered with our Bulgaria allies. We're going to go out on the C-17 aircraft download move to a certain point execute our fire mission and return within the same day. This exercise is going to help us to demonstrate our capability to rapidly deploy and then integrate with our multinational partners across Europe. U.S. Army soldiers assigned to ARI's battery 177 field artillery battalions were rapidly deployed from Graffenburg, Germany for a dynamic force employment exercise during African Line 2021. Brigade North is conducting a large combined arms live fire exercise. They're having multiple battalions move down through the area while sinking fires from our MROS to their K-9 cannon systems, their mortars, as well as having F-16s flying overhead and dropping live ordnance. Witness today that was a tactical exercise live firing with the artillery, U.S. artillery fully integrated in the Norwegian artillery battalion and together this is a joint combined event. So it proves that we are fully integrated so it was really good to see.