So the latest Trab for this week at Bungie has not long dropped and it beholds some interesting news that is for sure. How's it going guys? My name's DP Jen. I'd like to thank you for stopping by and checking out my video. If you do enjoy it leaving a like really helps out and if you are new around here and enjoy daily Destiny videos be sure to subscribe. So the Trab has not long dropped and it does behold some great interesting information to talk about so let's get into it. So the Trab starts off with basically what I talked about earlier in a video if you missed it basically the Shadowkeep DLC has been delayed until October 1st basically a couple more weeks. This is what they say as we've been getting closer and closer to serving up Shadowkeep a new light has become increasingly clear towards the out releases for this fool would benefit from a bit more time in the oven. Basically they're saying that there isn't quite ready to push out yet so they would need a couple more weeks to get things perfect which is fair enough. Now we do see a couple new images of Shadowkeep and some of the things here look at the armour there people look at that hunter mask look at that look at all he looks in look at a wallop mask I don't know what's going on there it's got like a mask on top of his mask and the Titan as usual just looks like a spaceman but looking absolutely great the theme of the weapons is awesome too. Cannot wait to dive in so yes the new Shadowkeep releases now October 1st. It then goes to talk about the sources of heroes right now sources of heroes rages under European areas on his live guardians are dishing out the mayhem with their elemental buffs. Today marks the third and final day of the rotation if you haven't gotten a chance to experience them just yet here's your briefing and they're going to showcase the praise the sun which is for solar unlimited power which is for the arc and gaze into the void which is for void. We're only on day three of the event and we're already seeing legendary armor sets and the moments of triumphs title we'll be watching to see who among you earns all three armor sets and unlocks the converted void streak sparrow. They then state ending on a high note with two more weeks of our season of opulence calendar we're taking an opportunity to make place for your truly opulent moments of triumphs is being extended for those hunting their pride ship sparrow and t-shirt multiple iron banners will happen during the final month and a few valet infamy boosts to help guardians reach some milestones before the end of the season here's a quick calendar for what we have planned and because you can see that on screen now we're getting iron banners on the 3rd of September we're also getting iron banners on the 17th of September we've also the date for the update 2.5.2 2 in the destiny update 2.5.2 you'll find a quick quality of life update to address some trending feedback from players specifically concerning rewards here's a preview moments of triumphs has been extended to the 17th of September gambit prime and reckoning weapon reward rates will be highly increased that's great they will also have a bad luck protection to guarantee a drop after a set amount of runs pretty cool reckoning difficulty will be tuned to be more welcoming as you hunt for rewards great tech schematics will no longer have a daily look at and will give better chances toward weapons that players do not own manting top and when the gold gl3 pinnacle quest will become more accessible specific quest subjects is will be tuned based on player feedback there will be multiple weeks of increased valet infamy to look forward to in september iron banner will be accompanied by increased valet in each event as the season comes to a close menagerie chess will become more rewarding as calluses fit on the week of destiny update 2.5.2 2 will be hosting a community challenge of sorts stay tuned for more details note while moments of triumphs has been extended sources of heroes will end on august 27th if you wish to complete all moments of triumph be sure to earn your legendary sources set and master at one piece before the dead line we'll have more specifics on these notes as we get closer but we wanted to give you a few things to look forward to as we embark on the final sprint toward our four release on october 1st now the interesting thing for me here is the menagerie chess and how it states they will become more rewarding as calluses fit now what does this mean now i put out a tweet the other day talking about how well how perfect the ea z is the more progress you do the more boss you destroy the more bosses you kill at start the more chess you get a chance of hunting down i mean that's like the perfect system which would perfectly fit the menagerie now in my opinion the menagerie was a great great pve edition probably the best we've had in destiny all together and the chess at the end they could farm it over and over until the five minute time it ran down was great it was an amazing addition one that many thought bunch had put into place implemented into place on purpose and when they took it away i'm not gonna lie to you guys i've done a menagerie once since i took it away i was doing that thing every single day so fixing the loot system here fixing the rewards fixing the chess in my opinion would be a great great addition so as soon as we find out more about that people i will have you covered right here on my channel but yeah guys if you want to read through the whole travel i have covered basically the important part you will find it linked within the video description but tell me what you think down below in the comment section what they're going to be doing to the menagerie to fix this thing to make the rewards more appealing to make people want to play a game because that thing is dry as heck well yeah guys and i know we have come to the end of the video if you guys enjoyed living like really does help out and if you are new around here and enjoy daily destiny videos be sure to subscribe also people stay tuned for the specific stay drop uncertain changes with the <IP_ADDRESS> update but guys hopefully you enjoyed the video and hopefully i will see you on that next one


Live from Las Vegas, Nevada, it's theCUBE, covering Accelerate 2017, brought to you by Fortinet. Now, here are your hosts, Lisa Martin and Peter Burris. Hi, welcome back to theCUBE. We are SiliconANGLE's flagship program where we go out to the events and extract the signal to the noise bringing it directly to you. Today we are in beautiful Las Vegas with Fortinet. It's their Accelerate 2017 event. I'm your host, Lisa Martin, joined by my co-host, Peter Burris, and we're very excited to be joined by a technology alliance partner, Nozomi Networks, Edgar Capdeville, you are the CEO. And welcome to theCUBE. Thank you, happy to be here. So a couple of great things that Nozomi announced just a couple of months ago. One was they just secured fantastic 7.5 million in series A funding, and the second thing they announced was you as the new CEO. So congratulations on your new post. Thank you very much, thank you. So Nozomi is focused on the industrial control systems technology industry. What was it about this particular opportunity that attracted you to want to lead Nozomi? Yeah, great question. Two things mainly. One is the team. The two founders are truly rockstars. They have a great background in cybersecurity and how do you apply artificial intelligence to industrial cybersecurity? And two was I had been working with the founders for a little bit, and I saw with my own eyes how the customers adopted the technology, how easy it was to deploy in an industrial setting, which tends to have a lot of friction. Not a lot of equipment gets into those networks. And the ease of proof of concepts, soy with my own eyes and the frictionless interactions was made me join. So Nozomi was started in 2013. You're already monitoring over 50,000 industrial installations. Some of the themes that we've talked about at the event today so far with Fortinet's senior leaders is the evolution of security where they're positioning really at this third generation of that. As we're seeing that and we're seeing that in order for businesses to digitalize successfully, they have to have trust in that data. What is Nozomi seeing in terms of your industrial customers? What are some of the biggest concerns that they have regarding security? And how are you working with Fortinet to help mitigate or limit damage from cyber attacks? A lot of our customers in our space are going through what's called IT-OT convergence. OT networks have traditionally been serial, point to point, run over two-step pair copper, and they've recently adopted Ethernet. When you adopt Ethernet, you have a gravitational force, which is to connect. So these OT networks used to be air-gapped, segregated, and now they're being converged with IT technology under sometimes IT operation, and therefore they start suffering the traditional IT attacks. Those traditional IT attacks are particularly harmful when it comes to industrial critical infrastructure, and they require a special technology that understands those protocols to be able to detect anomalies and whitelist or blacklist certain activities. Give us an example of an IoT network. So what is, you say critical infrastructure. Give us some examples. What are you talking about? IoT is a very broad term. The focus is very specifically on industrial IoT. Industrial IoT could be a network that controls a refining, so the refining process in a refinery. It could be electrical distribution, any form of electrical generation, oil and gas, upstream or downstream, manufacturing, everything that moves in manufacturing is controlled by an industrial control networks, pharma in the same sub-segment, if you will, some transportation. We're based in San Francisco, so our BART system is controlled with industrial control systems. So we're talking about, as you say, critical infrastructure. We're talking about things that we're getting control of some element of that critical infrastructure, especially in process manufacturing businesses, can have enormously harmful effects on not only business, but an entire community. The disruption that it can cause is tremendous, from lights out in the city to harm to people in a transportation case, oil and gas case, environmental damage, leakage, the damage can be tremendous. And that's basically one of the huge differences between IT and OT. In IT, if your network blinks, your email may be two seconds late, my print job may need to be resent. In OT, you may not be able to turn off that valve or stop this process from happening or receive an alarm in time. Right, so like, I live in Palo Alto, not too far from me, is some of the big refineries up in Richmond, California, not too long ago, they had an OT outage and it led to nearly a billion dollars over the damage to that plant and the local environment. Correct. This is real serious stuff. With a product like Nozomi, you can detect anomalies. Anomalies come in three flavors. One could be equipment, damage malfunction. The other one could be human error, which is very, very common. And the other one could be cyber. Any one of those could be an anomaly. And if it tries to throw the process into a critical state, we would detect that. And that's where, yeah. Talking about cyber, from a cyber attack perspective, what is it about industrial control systems that makes them such a target? Yeah. It is that they had been used to be isolated networks. Just like I said, IT and OT converges, are taking networks that used to be, serial security was not really a concern. In industrial control networks, you don't have identity, you don't have authentication. You're just starting to have encryption. Basically, if you drop a command in the network, that command will get executed. So it's about the vulnerability of- Vulnerability, maybe it's an easy target. And then from a proliferation perspective, we've mentioned kind of the evolution of security, but the evolution of cyber attacks, the threat surface is increasing. What is the potential? Give us some examples, some real world examples, of the proliferation that a cyber attack and an high industrial control system can have on our retail, a bank, energy company. The industry was put in the map in 2010 with Stuxnet. Stuxnet was the first attack. Everybody talked about Stuxnet for a while, and it was very hard to create a market out of that because it was done really by a nation state and it was done like once. Since then, 2010, you know, till 2013 and now till today, attacks have increased in frequency dramatically and in use cases. Not only are nation states attacking each other, like in the case now of the Ukraine, but now you have traditional security use cases, your malicious insider, your compromise insiders, doing industrial cyber attacks. In 2015, the Department of Homeland Security reported 29795 industrial cyber attacks in our nation's critical infrastructure, and those are not mandated, they don't have a reporting mandate, so those are voluntary reporting. So that number could be two or three times as big. If you think about it, from 2010, we've gone from once a year to 2015 once per day. So it's happening, it's happening all the time, and it's increasing in not only frequency, but in sophistication.  Correct. So you're talking about potentially thousands of efforts, and you're trying with Fortinet and others to bring the technology as well as sort of best practices and thought leadership for how to mitigate those problems. That's right. For Fortinet, we have a very comprehensive solution. We basically combine Fortinet's sophistication and robustness from a cyber security platform with Nozomi's industrial knowledge. Really, we provide anomaly detection, we detect, like I said, any sort of anomaly when it comes to error, cyber, or malfunction, and we feed it to Fortinet. Fortinet can be our enforcement arm, if you will, to stop, quarantine, block cyber attacks. So Nozomi's building models based on your expertise of how industrial IoT works, and you're deploying those models with clients, but integrating them back into the Fortinet sandbox and other types of places. So when problems are identified, it immediately gets published, communicated to Fortinet, and then all Fortinet customers get visibility into some of those problems. We connect with Fortinet in two ways. One is we have FortiSim, so we alert everybody. We become part of the information, security information environment, but we also use Nozomi 40 gates to block, to become active in the network. Our product is 100% passive. We have to be passive to be friendly, deployed in industrial networks, but for the level of attack or the level of risk, it's very high. You can actually configure Fortinet to receive a command from Fortinet and from Nozomi and actually block or quarantine a particular contaminated node or something like that. Does that make sense? Oh, totally, it makes 100%. Because as you said, so you let Fortinet do the active work of actually saying yes or no, something can or cannot happen based on the output of your models. That's right, yep. So when you think about IoT or industrial IoT, there's an enormous amount of investment being made of turning all this analog feeds into digital signals that then can be modeled. Tell us a little bit about how your customers are altering their perspective on what analog information needs to be captured so that your models can get smarter and smarter and better and better at predicting and anticipating and stopping problems. When it comes to industrial models, you need to pretty much capture all the data. So we size the deployment of our product based on the number of nodes or PLCs that exist in a industrial network. We have designed our product to scale so the more information or the more number of nodes, the better our models are going to be and our product will scale to build those models. But capturing all the data is required, not only capturing but parsing all the data and extracting the insights and the correlations between all the data is a requirement for us to have the accuracy in anomaly detection that we have. What is the customer looking at? In terms of going along that, that seems like an arduous task, a journey. What does, you don't have to give us a customer name, but what does that journey look like working together with Nizomi and Fortegard to facilitate that transformation from analog to digital? All the information is critical. That transformation is happening already. A lot of these industrial networks are already working on top of Ethernet, standard DCP IP. The way the journey works for us is we provide, as soon as we show up, an immediate amount of visibility. These networks don't have the same tool sets from a visibility and asset management perspective that IT networks have. So the first value at is visibility. We capture an incredible amount of information and the first and best way to deploy it initially is with, let me look at my network, understand how many PLCs do I have, how many, how the segmentation should be properly done. And then, during all this time, our model building is happening or we're learning about the physical process and about the network. After we've done with the learning, our system determines that now it's ready to enforce or detect anomalies and we become, at that point, active in anomaly detection. At that point, the customer may connect us with Fortinet and we may be able to enforce quarantine activities or blocking activities if the problem requires it. Is there any one particular use case that sticks out in your mind is a considerable attack that Nozomi has helped to stop? We obviously can name any one in particular, but when it comes to defending yourself against cyber criminals, we have defended companies against malicious insiders. Sometimes an employee didn't like how something may have happened with them or with somebody else and that person leaves the company, but nobody removed their industrial credentials and they decide to do something harmful and it's very hard. Industrial malicious insider activity is extremely hard to pinpoint, extremely hard to troubleshoot. Industrial issues, in general, are very hard to troubleshoot. So one of the things that Nozomi has a lot of value with is allowing troubleshooting from the keyboard without eliminating trucks and Excel sheets. You quickly can pinpoint a problem and stop the bad things before they happen. One more quick question for you. With the announcements that Fortinet has made today regarding, you mentioned some of the products, what are you looking forward to most in 2017 in terms of being able to take it to the next level with your customers and help themselves? Listen, the solution works amazingly well. We have to tell more people about it. I think the critical infrastructure has not had the attention in prior years and I think this year is going to be a year where ICS security is going to be, and Fortinet, of course, is very aware of this, is going to be a lot more relevant for a lot more people. It's the number of attacks and the attack surface, the vulnerability, it's all playing so that this year is going to be a big year. Yeah, I think we were talking before we started that the U.S. Department of Homeland Security has just identified the U.S. election system as a critical infrastructure. So maybe it's going to take more visible things that have global implications to really help move this forward. I think the one point I would make when it comes to government, government has been great. If you make an analogy, this is an analogy that I have in the top of my head. If you look at cars and the automotive industry, seat belts and airbags have saved a lot of lives. We don't have that in industrial cybersecurity and we need the government to tell us what are the seat belts and what are the minimum set of requirements that our electrical infrastructure should be able to sustain. And that way it makes the job easier for a lot of us because nobody can tell you today how much security to invest and what's the mix of security solutions that you should have. And therefore, in places where you don't have a lot of investment, you don't have none and you become very vulnerable.  And that makes it a minimum bar for proper operation if you will. But the proper, the way it typically works is government's going to turn to folks like yourself to help advise and deliver visibility into what should be the appropriate statements about regulation and what needs to be in place. So it's going to be interesting because you and companies like you will in fact be able to generate much of the data that will lead to hopefully less ambiguous regulations. Yes, that's right. That's right. I agree 100%. Well, what's an exciting prospect. Edgar Keptaville, thank you so much, CEO of Nizami Network. It's been a pleasure to have you on the program today. On behalf of my co-host, Peter Burris, Peter, thank you. We thank you for watching theCUBE, but stick around, we've got some more up, so stay tuned.


We often say we evaluate a function or a formula and solve for a given variable. But the only real difference is that a function or formula is already solved for one of the variables. And in general, given any formula, you should be prepared to solve for any of the quantities it uses. So, for example, our simple interest formula has four quantities. The amount a k t at some time t, the principal k, the interest rate s, and the time t. We can use the formula to calculate an amount and solve for any of the other quantities. So, for example, suppose a friend borrows $20 and pays you back $25 a week later. What kind of person are you that charges a friend interest? Wait, wait, moral and ethical questions are not part of mathematics, so let's just consider the mathematical question about the interest rate per day. Now, since we're looking for the interest rate per day, our time t must be measured in days. And so we have the amount borrowed, the principal is 20, and the amount of time is 7. And since they're paying us back $25 after t equals 7 days, then a 27 is equal to 25. Now, equals means replaceable, so we know the value of a 27 k and t, so we'll replace those. And to solve an equation, the thing to keep in mind is the last thing you do is the first thing you undo. So, over here on the right, we're multiplying by 20. And since the right-hand side is a product, we'll begin by dividing both sides by 20. Then we'll subtract 1 and then divide by 7 to get s, our interest rate per day, which we'll convert into a percentage by multiplying by 100 and then rounding to 3.6%. And at this point, it's important to consider rounding. Now, in physics, engineering or chemistry, a rounding error might cause minor consequences like a bridge collapsing or people dying, but in finance, a rounding error can cause people to lose money, so it's important to worry about rounding. And so here's an important idea. Never use a rounded value in any computation. And in particular, only round the number you report. In other words, once you've rounded, don't ever use that number again in anything. Now, the best approach here is to do all your computations at the same time, but that can be a little bit cumbersome. The second best approach is to use the capability of your computing device to store intermediate calculations. And most calculators and spreadsheets are designed to do this, so take advantage of your technology. So if we wanted to solve 25 equals 20 times 1 plus 7 s, we'd go through the same steps. We divide by 20. We subtract 1 and divide the whole thing by 7. And here we've recorded what our computation should be, but we don't actually do them until we calculate at the end of the problem this single expression. Now, if you plan to take something like the actuarial exam, a certain amount of technology is allowable, and it's important for you to learn to use the available and allowable technology. And in particular, I'm not going to talk about that because of two reasons. First of all, some of you already know how to use it, and second, that technology changes constantly, so you will lead to learn how to use technology on your own at some point. And the best place to learn technology on your own is in the classroom where, one, you can determine whether or not you've used it correctly by comparing your answer with the right answer, and two, the worst thing that will happen if you don't get it right is you'll get a bad grade. In contrast, once you're out in the real world, if you don't learn how to use technology correctly, your company will find someone who can. In any case, we want to evaluate this expression to find the interest rate, so we can use a calculator, and most modern scientific calculators will actually allow you to input something that looks exactly like standard math notation, for example this, and so we find our interest rate 0.03571. So, another quantity in our simple interest formula that we could solve for is the amount of time. So I suppose you invest $1,000 at 7% simple interest annually, how long before the investment is worth $2,000? From our simple interest formula, akt equals k times 1 plus st, we have the principal amount $1,000, we have the interest rate 7%, that's 0.07, and we want a 1,000t to be 2,000. We want to know when our investment will be worth $2,000, and we don't know t, so we can write down our formula and solve. And again, keeping all of our calculations until the end, we find t is approximately 14.29 years. And just this day I got an email from a Russian nobleman who says he's willing to pay 45% simple interest annually. How much should we invest to make $100,000 in two years? So, from our simple interest formula, we have our interest rate 0.45, our time 2, and after two years we want the amount to be $100,000, with k the principal amount as our unknown. So we can write and solve. And so this means you should invest probably nothing because something that sounds too good to be true probably is. But it's your money.


There have been many versions of capitalism, and the current iteration we're in is generally sort of the Friedman capitalism, which is the 1970s Friedman doctrine, and it already sort of acknowledged that as externalities were a thing, but the explicit instruction was that businessmen should not worry about externalities just about profits. Well, that didn't make the negative externalities go away, but we know that negative externalities are a thing, and we should be doing something about them, and I think there's this growing recognition of that. So generally, when we talk about impact capitalism, it's not like a fully flushed doctrine. I'm not pretending that, and any of us, I don't think we're pretending, we're like have it figured out at like an economics sort of noble price level, but we're saying that the organizing system seems to be good. Generally, we're like talking about the same principles of like supply, demand, human motivation, being driven by greed and often self-interest. We're just trying to say that we can use those mechanisms and create those micro-experiments and create incentive structures that will use exactly the mechanisms of capitalism to drive the type of positive change we want, and it sounds, and I think the carbon credits market are like one, and there's more examples, but that's an example where like that seems to be working, that like we have created something where it sort of still feeds into the very typical capitalist engine, but it generates positive externalities, and there's more examples like that. So that's kind of the idea for today. Other speakers will explore their experiments and their approaches. Most people are early, this is a long-term vision, yeah, it's possible to change the system and that's why we're here. So, Kueh Mai, I'm an entrepreneur with some successes and some failures on my account. I'm a founding partner of Venture Fund called 50 Years. It's been, this is kind of how I've been portrayed, but I'm also a burner and a very physical active person, and that's kind of how I prefer to think of myself, and I think I'm much more interested in sort of the, that sort of dynamic approach to life, and I think there's a lot of creative energy that comes from that. So, why 50 Years? We'll just run through that, we don't have that much time. So, 50 Years is named after Winston Churchill's essay from 1931, it was called 50 Years Hence, and you don't have to read the essay, although you should because it's amazing, but generally the TLDR is we can extrapolate from today into the future. Scientists and technologies will have an outside impact on how the world looks. He thought that in 1931 already, sort of beyond even politicians, and then the second part of the essay is pretty much talking about the sort of responsibility that comes with power. And so, our experiment is sort of simple and then I'll go to kind of the broader vision of it. So, it's a precedency VC firm started in 2016 by two or I come down to an alumni entrepreneur myself and said, say right there, we started with a tiny fund of four, I mean for a venture fund it's tiny, a little over four million, essentially emptying our bank accounts and asking all the friends from tech to give us the money. And then the other funds have been bigger and you've grown the team and we've been at it for a few years right now. So, our theory of change has been that we want to back impactful early stage founders. And then the question is why? So, you can focus on many interventions when you're trying to drive change but obviously comes from cognitive bias of being startup founders but startup tend to be incredible vessels for change. They align incentives very well, there's equity sharing in the team, it's sort of the process of getting money is relatively streamlined right now in the startup ecosystem and generally startup has speed.  So, one of the first lessons we learned two or three years into what we're doing is that deep tech is really where we can have more impact and specifically translation which is the process of spinning out from academia and helping more companies in deep tech companies get established is like how we can actually drive more of the type of innovation we want. So, some of the slides are long so you can pause if you're looking at online or it's going to be uploaded so don't worry too much about reading them but generally speaking, these are the reasons why deep tech is where we should be looking. If we knew, if there were easy solutions the general assumptions like if there were easy solutions to the big problems we all see and get frustrated by and suffer from those easy solutions would have been deployed. So, generally you have to develop something that hasn't existed before, you have to commercialize it, governments are bad at innovation, there can be better at distribution, tech entrepreneurship in deep tech is where a lot of positive change can come from. It's also extremely impactful to help those deep tech companies sort of spun out of academia and that's a messy process and we have been spending a lot of time on that. So, we also learn money is not enough as you mean, the money is sort of abundant in the early stage ecosystems like support is extremely important and the type of support we provide actually has been extremely hopefully meaningful and we see that there's so much more we can do. So, why support is important because it's much harder to commercialize if you think of like novel therapeutics, if you think of a new communication, new satellite communication stack, if you think of clean chemicals or like things that can sequester carbon from the atmosphere, like it's not just sort of building what I've done before B2B SaaS tool and like you know kind of it's the level of complexity, it's much harder in the level of coordination and the funding stack you need to get it's much harder. So, those people just need, those founders need more support. And yeah, so we, early stage investors and like us and many other in this ecosystem can be extremely helpful. One thing we notice is extremely important to identify the people who are mission aligned in the first place, because it's very easy to pivot when things get hard and things get hard almost always. So, generally speaking, when you think of like the Venn diagram of what we do, it's the big problems, a billion revenue potential, deep tech. So essentially, you know, the Mr. Burns here is something that we use in this thing we call Mr. Burns test. It's like, you know, for now the types of companies we back, we wanna make sure they sort of can, they pass this test, which is, you know, with a greedy sort of prototypical sort of capitalist of Mr. Burns driven only by his self interest, pick, you know, a company or a service because it's, you know, better, cheaper and more convenient to him. So like, we ultimately believe, like there's some kick starting and helping for those types of projects and companies in the beginning, but like they will never scale and have like the type of global impact that like we are all looking for if they don't have sort of a sustainable engine of sort of revenue or some sort of commercial flywheel. So yeah, in the current phase, we look a lot like a normal VC and we seem to know what we're doing. I think I've just like five or six or seven years in, like I think we have a good grip on this and it's been a learning curve. Oh, but the longer term vision, which is really the exciting one and that's what drives us is to like, how do we take the learnings from what we're doing and like how do we scale it to move humanity to a better economic system and like here, you know, this is hard. Everyone here is trying to think about that. We don't have all the answers. We have some learnings and some experiments that we're designing. So the four phases of 50 years, there's going to be a lot of text again, pause on YouTube and don't try to maybe jump through it. The first phase, which it's sort of behind us is the product market fit. Can we do this? Can we help? Is there a need for that? And then what can we learn from this essentially? So, and this is how we learn with deep tech and have more impact. This is why we decided with Fun2 and Fun3 to refocus on deep tech. This is why we realized like, oh, there should be even more companies spinning out. Why aren't there more companies spinning out? Oh, there's like things we can do and we'll talk about in a second. And then supporting founders can be extremely impactful. So then phase two is scaling operations and then focusing on deep tech. And we don't have to go through all the details, but we've learned that like this is the part when like we like to really, you know, support the scale of what we're trying to do and we need to scale our operations and like there's been a lot of focus on that in the last two years, which I'll talk about in a second. And then there's a ton of the sort of big levers which sort of maybe sound simple is as much sort of access to fundraising, operational support, also storytelling, events and helping with communication. And then people, one of the biggest learnings as well is people get distracted with easy money. And in terms of how capitalism transitions to this more sort of positive impact phase is that we need to actually kind of recreate incentive designs. So like if we don't find a way, you know, think carbon credits market, if we don't find some of those like loops and mechanisms, then I think it will be hard to have impact at scale. And then so phase three, which we're sort of like in phase two, looking into phase three is to develop scalable tools to identify and measure and support impact both for the early stage and like maybe looking also for the slightly later stage, specifically tech entrepreneurship, that's what we're going to focus on. And then help create markets and incentive structures. And this is where we're talking with protocol apps and this is how we're participating in those discussions because there's like things that can be done. And then obviously for everyone in attending, I'm sure funding the comments like phase four for many organizations will be to kind of merge this approach with master branch of the global financial system. Okay, so some experiments that I will just run through because I have just a few minutes and you'll have to just bear with me. We run events, connecting investor scientists and repeat entrepreneurs to see if they can like start new companies and not in just gaming or marketing automation, but on things that are really important. Some areas of events that we run, we run a translation podcast where we like will help scientists talk about their papers and their work and the potential for translation. We try to like help sort of uncover some of the potential for translation there. We run programs for PhDs to help them understand the world of investing. And we also run a community of 50 50, which is the community of North America's top 50 researchers to like with this hypothesis that like community is often what's missing and also sort of getting them on the entrepreneurial path and that's been fairly successful in terms of like company creation actually entrepreneurs spinning out. We're working on a spin out playbook which essentially we're saying, it's so easy to start a Delaware company. It's so easy to like sign a safe and get some money in, why combinators are safe as a financial fundraising standard. Why isn't there a standard for spinning out? And like why do people have to like negotiate and they don't know the terms. It's like asymmetry of negotiations. A lot of this could be massively impactful, very hard because there's many different systems at university systems and tech transfer offices but you know, this is something we're very excited about for scaling this vision and for helping much more innovation, important innovation to happen. We had a failed experiment of a physical lab space in LA which we're going to repeat probably because it was mostly failed.  How can we ecosystem for how can we enable more change? So some of the results we backed, you know, 44 PhD founders and we have 22 PhD CEOs. These are like just from our founder community. This is like kind of the caliber of the people we're talking about. Some of them are really incredible. We have a Nobel Prize winning co-founder. So we've backed some of these amazing companies. These are some numbers over here of, you know, fund right now we backed over nine companies. The next company we'll end up partnering with will be the hundred companies over 200 founders. We have multiple deep tech unicorns in the sort of clean chemical space and satellite connectivity space and in like many other areas. There's reputation flables which we're already experiencing when it's easier for us to attract the type of people interested in. We have amazing investors who are actually really, you know, 44 founders of billion dollar tech companies have backed us.  So it's, you know, it's been working in many ways. Protocol Apps team asked me to like talk about why, which public goods this approach is good or bad for and actually like just going through like the types of public goods. I think whether directly or indirectly it can be pretty much good for anything just because pretty much anything that the governments do has to come from somewhere. Governments do not do tech development. So like when we kind of acknowledge that they have to buy it from somewhere and pay it for it and including the, even like democracy itself, which relies on, you know, the independent press and sort of ad-free press. When you think of sub-stack, this is really their vision. It can be used for anything. So yeah, we have some unsolved problems. For example, we're still trying to uncover how to commercialize even more scientific research and make it easier for people to spin out. You know, as I mentioned that already, that we have the Delaware C Corp standard of starting a company where it's pretty much like, you don't have to get lawyers involved to get your incorporation documents done. And like, yes, everybody thinks, yeah, of course we don't, but like at some point that was not the case and that was like much more painful and now we can use like Stripe Atlas and some of those tools. And then the innovation of, first the VCA sort of funding documents and then the, why Combinator is safe. Created a standard for like, what a protocol for fundraising is. And like, you know, right now it's not uncommon that, you know, we talk to a founder, we have a handshake, we confirm it over email, the same as a safe, we wire the money within an hour, right? How much easier is that?  And then it just like, they don't have to run the like cycles of like, who do I talk to? Who's the lawyers? And then like negotiating, like all that BS that like nobody really wants, that does not exist for tech transfers. Like if we could have that, you know, this is one of the big things that like, if any of you have been thinking of and of any of you watching that talk have been thinking of like, this is, we should talk about that because we're trying to, yeah, we really want to solve that. You know, the other challenges is definitely like, in terms of liquid returns, it take a long time. So it's, you know, it's like harder to have a massive, massive operations team, which again drives operational efficiency and like productization of some of what we're doing, but like definitely that's like, you know, if we're talking about like, what are some of the unsolved approaches to this, that's definitely the things that's on our mind. And yeah, it's, we would love to see, yeah, the, you know, we have seen and we have inspired other firms to do this. And this is really our goal. Like if you want to start a venture fund, start doing exactly what you're doing, please do, we can help you. There needs to be so many more sort of local firms that can focus on different areas of tech, deep tech and specializing and sort of having like this ethos and hopefully helping solve important problems through that or at least creating the pipeline of innovation that then could be distributed to solve some of the problems. And yeah, we're excited about the, some of the talks today about impact certificates because this definitely, you know, we see some of the, we see already that there is going to be, you know, for all the climate tech companies that we've backed and we've backed quite a few. Like they will be able to use, right now they're in like the technology validation assessment stage. Some of those companies are generating revenue, but it takes time to get certified. So, but hopefully that can be brought into the present, but in other areas of sort of positive impact, there isn't a way to essentially get additional revenue or credibility. Yeah, so I think the impact certificate work could be extremely impactful. So anyone working in impact certificate work, I mean, I know some of you, some of you, but if also talk to us like how this could be, how this could be used in an early stage ecosystem. Yeah, and I think this is kind of all I have. I'm very, I'm sorry, I sprinted through it, but this is kind of the time I have. Thank you so much.


Thank you all for, okay, thank you all for inviting me back here to speak. So, yeah, so let me just get started. All right, so I'm talking we're talking about integer solutions to polynomial equations. So, well a big open question of the field. So I'm going to start very general and we'll specialize later is well given a degree D. Is there an algorithm to determine all the integer solutions to all the polynomial equations of degree D. And I'm sorry, sorry, not degree D, D isn't, I shouldn't have called D, D is the number of variables. So all all polynomial equations and D variables. All right, and so, well, I guess you all know that if you have a if you have a polynomial in one variable, you can find all the integer solutions. You know, rationing test, etc. And for each of the degrees from two to 10, it's unknown. So that's why there are nine question marks here. And then, and then for D, at least 11, it falls from the work on Hilbert's 10th problem by a bunch of people that there is no algorithm. So, so my talk today is going to focus on the case of two variable polynomials. And so, these, okay, so this is, this is about integral solutions to a polynomial f of x, y equals zero. And then this problem, finding the integer solutions such an equation that that's equivalent to the problem of finding integral points on on smooth affine curves. I mean, this curve doesn't have to be smooth, but you can always normalize the curve to get a smooth curve and then you can do a reduction. The problems, you can you can figure out the difference between the integral points of one and the other. And, and also the other going the other direction if you know how to, if you, if, if you have an affine curve, then you can always you can always map it to a curve and a to and so so yeah so these are so that that's these, these are equivalent. So from now on we're going to talk in more geometric language so talk about integral points on smooth affine curves. Okay, so before going into the, the, the number theory, let me just say a little bit about the geometry of affine curves. So the complex numbers, if you start with a smooth affine curve, then you can complete it to get a smooth projective curve, and all my curves are going to be geometrically integral as well. So, and so, and then, yeah, so then your, your affine curve will be the nice it'll be this nice curve smooth projective geometrically integral curve with some number of points removed so some some number of punctures. And the curve will have, I'll call it, it's genius G, and I'll call the number of punctures a little R. And then in terms of those parameters you can, you can say you can, you can give a formula for the other characteristic of this topological space. And it's the other characteristic of X minus the number of punctures. Okay, and I'm working my way up to stating Ziegels theorem, reminding you what Ziegels theorem says, and it's and it applies to hyperbolic curves, and where hyperbolic means that the other characteristic is negative. And well because there's such a simple formula for the other characters you can say exactly when this happens, and it happens most of the time. If you're going to be able to zero, then you need are to be at least three for this thing to be negative. And, and yeah, and then G equals one you are at least one and for G at least to any R will work, although we're actually we're always assuming that are as positive anyway because we're not talking about projective curves we always, I'm always talking about affine curves today. Okay, so, yeah, so these are, so these are the curve that we're going to talk about where the other characteristic is negative. All right, so now let me state the Ziegels theorem. And so here's the setup. So start with the number field K, find set of places, and all the R community ones. And then this always the ring of S integers. And so it's the elements of the number field whose valuation is now negative for all the places outside s. And, yeah, so, yeah, and then, and then you is going to be, well, it's as in the previous slide except now everything is defined over K. So you is, you is the smooth affine curve over K. And then, well to talk about integral points, technically speaking, we should choose an integral model of you. So it's it's an O scheme. Yeah, so that that, yeah, who's generic fiber is you. And then Ziegels theorem set tells you that if you if the curve is hyperbolic after yeah, which you you test by base changing from K to the complex numbers. If it's yeah then the set of integral points is fine. All right, so yeah so here's what you're just one quick example. So if you take one example is if you take P one, and you should in you delete three points. And then so that you can also think about that, that curve as being the curve given by the x plus y equals one in GM cross GM, because I mean if you if you write if you if you imagine drawing this curve so there's you know this diagonal line. And then, but you remove the axes, the two axes because we're looking at GM cross GM.  And then your name is a point square x plus y equals one crosses the x and y axes. And then you also have the point infinity removed because everything's affine here. And so if you take that punctured curve it just projects onto the x if you project it on the exponent you just see you get P one minus three points. Okay, so, so that's my curve and then the other characteristics you can calculate it's two minus three. And so that's, well, I guess you know that we're all number three is here we know that two minus three is negative. So, so. Yeah, so Z goes theorem that applies, and it tells you that the set of integral points on this curve is funny. And concretely what the set of integral points are, what the set of point integral points is, it's the set of solutions to this equation where, but but it's in GM cross GM so that means that X and Y are have to be units in O. And so they have to be units that have to one. So yeah so this is sometimes called the S unit equation. And so this is saying that the S unit equation has finally many solutions. Okay, so that's, yeah, so that's that's an example. Now the big problem, as you all I'm sure many of you know that with the big problem with Z goes theorem is that it's ineffective meaning that at least even in principle, it doesn't give you a bound for the, for the solution so and so and so it doesn't give you an algorithm to actually find the all the solutions, it just it just a, yeah, just a theoretical finding statement. Okay, so, so because of that baker developed a method to that that that could make that could try to in order to, in order to make this make this theorem effective in some cases. So, let me stay here's just one version of baker's theorem which I mostly took from the paper by you. So, and okay so the baker's original theorem. So okay this theorem involves a choice of absolute value on a number field. And I, and Baker originally I think, I mean they're probably people here know that the history better than I do but I think Baker originally did it for Archimedean absolute value and then later as generalized to arbitrary. Also, non Archimedean petic absolute values. So this baker's theorem says this is this theorem on linear forms and logarithms. So, and this general version you choose a number field, you choose an absolute value on K. And you choose you fix some n elements of the, okay, oh, I guess they sorry they should be in verbal, not zero verbal elements. And then you fix some real number positive number epsilon. And then you look at all the, the exponent tuples tuples of integers, such that this power, this power product is very close to one. And very close means it should be sort of exponentially close relative to the size of the exponents. Yeah. So, I mean, so, yeah, but and so this is if you take. So this thing this is close to one and if you take logarithms of that that means that be one times log alpha one plus data dot plus bn log alpha one should be very close to zero. So, so this is tell this theorem is telling you that that it's rare for that linear form in logarithms to be very small. So it's with finally many exceptions it's giving you a lower bound for how close a linear form of logarithms can be can be to zero. And the great thing about this theorem is that it's effective so not only is this set of not only are these is this fine set of exceptions, the finite but you can actually bound the size of the of the x forms. And so from this. So there's a few once you know this theorem then there's a short argument that I guess that that that I guess was due to Baker is that you can use this to compute. You can do integral points on certain curves. I mean, doesn't it doesn't. I mean, it's only for certain curves so you implies in particular for the s unit equation. And there's some other cases too, but yeah, but it's but the the what I'm when I'm talking about today is, is, is asking. Well, what what curves doesn't really apply to how how how if you if you combine this met this met this theorem with other like other methods in geometry what what can you actually get out of this method. All right so. So yeah, okay so here I'm just copying the, the corollary things that you could be this, and Yuri Bulu on the generalization of bakers. Well he generalized the occupation of Baker's method. He used the theorem combined it with some, some arguments in geometry to prove to expand the class of curves to which Baker's method could apply. And so he proved that if you have a, if you have a curve affine curve you that it now not just it's not just necessarily a sub variety of GM cross GM but it's something that has a map to GM cross GM. It's not degenerate in the sense that it's not a, it's not a subgroup of this, so it's not something like why squared equals x cubed, which inside GM question. It's not a coset of a subgroup. And it's not a, it's not it's not or it's not contained in the coset. Well, I guess. Yeah, so it's not a constant map, for example, so. So if you have if you have such a curve that some he's some some interesting curve and GM cross GM or mapping to an interesting curve and GM cross GM. Then, then you can use Baker's method to compute the integral points. Okay, so then, well, I mean so be lose once you know what's what's now that be lose prove this, the big question is, if somebody hands you an affine curve can you figure out whether it has a non degenerate to GM question. I mean, maybe is it possible that all curves over all number, all affine curves or number fields have such a morphism. And, well, well, that's not quite true.  What you need are two independent invertible functions on you because because those which will those will serve as the coordinate functions of the map from you to GM. In other words, you have the two coordinate functions on GM question and you can pull them back to functions on you just by composing with this non degenerate morphism.  Another way of saying that is if you take the group of invertible functions on on on you, and you quotient out by the scalar functions, then, well that that finally generate that finally Jerry group should have rank, at least two. Okay, and, and to say this, I mean if you think about the divisors of these two independent functions to say that you have those two independent functions the same is saying the F2 principle divisors. I mean those two principle divisors I mean they're not these these function not supposed to have zeros or polls on you. So that means that all the zeros and polls are going to be in the complement in this in the set of punctures are. And so what you so this condition is saying is the same is saying the of two independent principle divisors. So Z independent principle divisors on the, on the nice on the protective curve acts that are supported on that fine set of points are. And another way of saying that is. Well, if you have the divisor degree zero divisor on X and you want to know is it a principle divisor. That's the same as asking whether those that divisor adds up to zero in the card group or in the, yeah, or even or if you can think of as the Jacobian if you want. And so, having this condition this is equivalent then to having two independent relations between the punctures. And where the relations are supposed to happen inside the card group inside. Yeah. Okay, so, so it's a it's a sort of subtle condition it's some kind of arithmetic condition you need to have these points. Have, have, have and have and have these relations. So, I mean, there are many curves which you will not have such relations. So I mean a simple example is if you only have one puncture, you only have one puncture if R is just a single point, then they're not going to be any relations at all between that one point. So, so yeah so you need at least, yeah so you need to you need more than one point in order to apply this method directly. All right so. Now, say that. So okay so okay so wait a second sorry I think I. Yeah, oh, sorry okay yeah so um. So okay so if this theorem does not apply if you can't find those two independent rational functions to begin to the on your affine curve, then what you can do is you can you can well there are various things you can do you can try to enlarge the circle or enlarge the set of places s and now just make your ring of s integers bigger. And so if even if you can find the finally many integral points over that larger ring of s integers then you can also do it over the ground ring. But a more interesting new thing you can do also is to is to use this the what you might call the method of scent, which is where you were, you replace you by a finite tall cover. And the reason that that's, that's an allowable operation is that it's a, it comes out of descent theory that every, that every integral point on you comes from an integral point on you prime. But I mean maybe not not an integral point over the same ring of s integers, but over some larger ring of s integers and it may be a larger number of fields. So that means if you can find all the integral points on this you prime over this bigger ring, then you have, you have an upper bound for the set of integral points on on the original you. And so you can just look, look at which you can just look at that look at those points one by one and figure out which ones are up and figure out which ones are actually your points on you. Okay, so the upshot of this is that you're always free in, in, in solving this problem of an interval points, you're always free to replace your original smooth affine curve by an unranked I cover finite tall cover. So, yeah, so here's, I mean, I mean this this this reduction I think it's been known for a long time fact I'm not even sure where it first originated I mean I guess Chevrolet and they I know they did something like this but they're I think they're maybe, maybe they were considering mostly projective curse but it's a, and you can be in that case it turns out to be a statement about rational points but anyway, and might even be older than that, maybe, maybe somebody can tell me what the history is. Anyway, in this context, in, in a for applying for combining this with Baker's method. This was already done in a paper by Baker and coats to give an algorithm for finding the integral points on elliptic curve. And when I say integral points and elliptic curve I really mean on a punctured elliptic curve, technically speaking so. So, you're really looking at the affine curve that's given by an elliptic curve equation like y squared equals x people say x was be. And so, so then, and that's the, that's so that affine curve is the you. And, and the, the big Baker's method or be lose generalization it doesn't as I said it doesn't apply directly to you because it's only one puncture and there are no relations between just that one point. So what you can do is you can replace it by a fine at all cover that's given by the multiplication by two map. So, so the multiplication by two map is an at all the finite at all map from E to E. And if you just restrict it to the part that lies above the punctured curve, then you get a map from. Well, you get a map from E minus the pre image of which and the pre image of always is the four. two torsion points. And so this, so this this is now a finite at all cover this and the event and so now now you instead of finding the integral points here, you can, you can try to find an integral points on this you prime. And what has been gained by passing to this finite at all cover is that now you have a curve with more punctures. So now they're for punctures, and, and there actually are relations between these points in the in the Jacobian, namely, Well, the, the difference between any two of these points to torsion points is is something who's that's a two torsion in the Jacobian.  So, and it is explicitly if you're working in terms of a fire stress equation of E, then, and if you call the x coordinates of the two torsion points, you want to E3, then you have these two rational functions, for example, are independent. And they have no zeros or poles on this so the zeros and all of these are supported on the punctures. And so that means you can apply, you can apply, you can apply Baker's method to this curve you prime and actually and get a bound on the size of the integral points on this curve. Yeah, so okay so one feature about this example is that I mean this was not just a finite at all cover of you it actually came from a finite at all cover of the projected curve of the nice current E. And so, yeah, so, anyway, so. All right, so. So okay so now the question, I mean, which is implicit and be lose work is, is whether this always works, can you can you always find some finite at all cover that works like this, given any starting with any smooth integral affine curve you have. I mean, and over, I'll assume genius at least two from now on because by the by the Baker quotes theorem on the previous slide the genius one curves are taken care of already. So the interesting case they're genius of these two. And well, they're always over some number of fields I'm going to say it's over a bar. And now it's a, it's sort of a purely geometric question. Is it true that every such affine curve you has a finite at all cover that has a non degenerate morphism to GM GM cross GM, in other words, has a finite at all cover to which be lose theorem can apply. The answer is yes, that means then then the big problem has been solved so that then you can determine the integral points on any curve over any, over any curve over any smooth affine curve over any number field. And okay so that's, so that's the maybe that's the most. Yeah, that's that's the most fundamental question we want to answer. Like, we could ask for even more. We could ask for for it was the same question except I want to find a talk cover to come from a finite talk cover of the productive curves or the nice curves, as it did in the in the previous, and in the Baker quotes method where the, I actually started with a finite cover of the E, and then I just restricted it to the, to the punctured curve. Okay, so these are the two questions that I want to, I want to study in the rest of my talk. All right, so. Well, let's see. So, okay, so here's. So okay so here's just a cop I just copies in here the main question or this is the second question and previous slide. And now let me tell you what I can actually prove so I can't quite answer this question. But I can, I can, so I can, even though it's a purely geometric question. I can I need, I can prove, I can prove a negative answer to this. If I replace q bar by an uncountable algebraic close feel like complex numbers. I'm saying that there is an as an affine curve genius lease to over the complex numbers, such that no matter what find a tall cover, you try to try to take of it. So, of where the atoll cover is coming from a productive atoll atoll cover. Yeah, all none of those fine atoll covers will have these two independent rational functions on it, you won't be able to find those two things. Yes, that's so okay so this is maybe, well yeah one of the one of the main theorems, and actually I prove a stronger version of this. And not only does this fail for some curve genius really great and equal to actually fails for every curve of genius at least two. If you let me put if you let me choose which puncture where I put the puncture. So, yeah, so for every, for every curve there exists a point such that it fails if I puncture that puncture it there. Yeah, so, so what this means in terms of algorithms for integral points is that, I mean if there is a positive solution if it is possible to use this to be loose theorem over for curves over q bar. And it means that, well, it, there's not going to be a general a generic method that works no matter where for all the punctured curves. It'll have to be a, if it works at all, it'll have to be a separate fine h I'll cover, depending in an arithmetic way on where the puncture is. So you'd have to, it'll, I mean it has to be something like it what happens in belly's theorem, where there's, if you have a. Belly's theorem is the way where you have if you have a curve over over over q bar you can represent it as a ramified cover of P1 ramified only above 01 infinity, but it's not, it's not a single construction that works for a given genius. The, the, the construction of the morphism sort of depends on the, the actual, well it depends very much on being over q bar and it's this and you have to, the degree of the map depends on the height of the, of the, of the, of the, of the curve in that case. So it would have to be something similar here so, I mean you can try so, I mean, various people have done various constructions where for certain, certain special curves you can actually find, you can find these non-generomorphisms but there's not going to be a general method for a given genius, the way there was for a genius one. Okay, so that's so that's that that's what. So that's so actually, we'll actually prove this. And actually will prove some, I mean for this one, this is saying that if you have a current genius release to then you can puncture it so that you cannot find these two independent rational functions. In other words, you cannot find two independent group. You cannot find two independent relations among the pre-images of your puncture in, in, in any, in any etel cover coming from nice curves. And, but you can, but here's here I can prove something even better, namely, that you can't even find one, it not only can you not find two independent relations among the pre-images. There's going to be some puncture, such that you cannot even find one relation between the pre-images of that point. Yeah, so, so given, yeah, so, okay, so you can, well you can read what it says, it says that given a, this is a nice curve of genius release to you can find this, you can find some point in it. So if you punch, if you, if you puncture there and you look at its pre-image, then for any finite etel cover, the pre-images will always be independent in the Picard group of y. So yeah, so you, so it means that you cannot even get the rank of the, of the group of invertible rational functions, module scales. You can't even get it to be at least one, let alone the greater than equal to that you would need for Beelow's theorem. Okay, so, all right, so I want to explain how, how, how I prove this, this theorem. And it comes from another theorem, which maybe, well, maybe this is my main theorem. I can't decide which is my main theorem, but let's maybe maybe it's this one. So this one's, this one actually works not over, not just over the complex numbers, but it works for any algebraic closed field of character zero. So in particular, it would work for Q bar as well. And it's actually a little easier to prove over the complex numbers and that would suffice for the previous one, but for that. Okay, so what does this say? So, so given a given a given algebraic closed field, so think about Q bar for example. And now suppose you have any morphism of nice curves of genius lease to it doesn't even have to be fine at all anymore. Just, just some subjective map of curves. Then you can find a point. So if you take X in here, such as you take the pre images and in this one cover, why, then there, those pre images are independent in the Picard, the part of the card. So no relation in their image copy. So, okay, so, so I'll explain the proof of this a little later but for now let me just say that you I mean in fact what happens is that for each if you, given the why going to X. You can ask. Okay, I mean this says there exists a point but in fact, almost a very general point little X will work. And in other words, the conclusion will fail only for countably many little X. And so, so, and so that means that if you're working over the complex numbers. Well, then you then, well then, then you can use you can use this, not you not just for one why but you can. You can use countably many X coming from each of the countably many sign until covers of wide X you can apply this kind of many many times, and you'll still have only countably many bad choices for little X. So, so that's why this theorem applies this theorem that lets you choose one point over the complex numbers that works for all the financial covers such that for any financial cover, the pre images are independent. Okay, so. Yeah, so I mean I would, it would be nice if we could prove this theorem also over Q bar, but you would have to use something more. I mean this argument as it stands doesn't work because you can't. I mean accountable union of accountable sets could be those of kind of that could be all the Q bar points so there might be nothing left to choose. If you're working over accountable feel like you bar. So if you so maybe, maybe there would be a way to prove to prove this theorem over Q bar but it, but for that, I mean, I would, I guess you'd have to know something about the height of the of the bad points in theorem B. If you could prove some uniform bound for the heights of these bad points for the, for the heights of the points X for which this fails, then then you would get a theorem like this. Okay. So, alright, so, so, so now what I want to do. In the rest of the time I want to explain the proof. So the thing that remains is to prove this theorem B. And so I want to sketch the proof of this. Okay, so, so here's the, okay here I just copied the statement, and then this in a smaller text here of theorem B. Yeah, so much so let me just remind you what it says we're over working our field such as two bar, we have a morphism of nice curves, maybe ramified. And we want to find a point, little X here, such that it's pre images in why are independent in the car. So, the first thing you can do is, you can take this ramified cover and you can replace why by its Galois closure, which will be a higher cover of X. And now it's a Galois cover and. But if you if you can, yeah, but if you prove it for that higher one. I mean, then you will, you will also prove it for the original why because any. I mean, if you because any relation that you have between the pre images and why will give you a relation between the pre images and the Gallo cover as well. So yeah, so you can reduce to the case where the cover is Galois and let G be the Gallo group. The other thing I want to assume is that I have an I can embed why in a Jacobian and in a natural way, and a way that respects the action of G. And so one way to do that is, I mean, the normal way the normal way of embedding a curve and it's Jacobian is to choose a base point and use the oval Jacobi map from why into its Jacobian, but, but that base point might not be fixed by the group G. And so it might not give rise to an echo variant embedding of Y and J. So what you can do instead of choosing a base point is you can use instead of that a base point which is a point of. The canonical divisor as you're sort of as a substitute for the base point. And so we're talking about a curve genius release to so the canonical divisor case is a positive degree. And you can use this sort of sort of modified apple Jacobi map for you, you send why to, well you can't send it to like what I mean normally you do why little y goes to little y minus the base point. It might work because this is not degree one so you multiply a little y by the degree, but 2g minus two. Yeah. Okay, so anyway this is a map from y to j and it's it's a morphism and it's, it's, it's so canonical that it's, it's, it's going to be echo branch for any group acting on why. One, one thing that it might not be is injective, because it's because of this multiplication that's happening inside the Jacobian but. But I mean, often it is injective and so just for some simplicity just so I don't have to say extra words in this talk. Let me, let me suppose that actually is injected. All right, and then what I need is to use this lemma. And that says, okay, so now okay so I have this curve inside this abelian variety now this curve winding around inside j. And I want to say that there exists a point in why that on the curve that outside every algebraic subgroup of j. Well, other than J itself. In other words, I'm intersecting why with all the algebraic subgroups I'm saying that there there is an outside of that intersection there I can still find a point of why. And this is, I mean they're constantly many such such subgroups. H. And by the way I'm also considering disconnected algebraic subgroups so for example, this is saying that the why has to be in particular has to avoid all the torsion points for example. And all the torsion cosets of algebraic of abelian sub varieties of J. Yeah, so. Well, okay, so here's here's a sketch of proof of this lemma. So, yeah, you can. I mean, so it's, I mean, this would be obvious if you're over the or be easier if I were over the complex numbers because then I could just, I could just say that well all the, all these h's are of lower dimension. I mean, if each each H intersects wine finally many points and then take the Union and then the set of bad why it's just accountable set and so over the complex numbers it would be easy to find such a why, but over to bar you have to do a little bit more work. Well, well, one way you can do it is to do a P use a pediatric method, which is similar to the method that I used in a paper with Davesh Malik for proving for proving something else. But anyway, so you can this soft case something like it could be something like you bar, but in any case, you can first restrict to the case where a case finally find transcendence degree because you can just. So all the input data is defined over some finally jaren field just look at the field journey by all the coefficients of all the equations defining why and J. And, and so, yeah, and so you can. So and you can take the algebraic closure of that finally jaren field. And so you can reduce the case for that's your new K. So if you have such a K it's an algebraic closed field of fine transcendence degree then you, you can embed that into QP bar because QP bar is a uncountable transcendence degree over Q bar over Q. So in this way. Yeah, so now we can work over to P bar and what and what comes out of this, these arguments in my paper with Malik is that this. So if you look at all the algebraic subgroups and you look at how they intersect why and you take the union of all those, all those points to all the points and why that lie on some algebraic subgroup. Then that is, I mean, not only does it not cover all the points but it's actually not even dense anywhere in any pediatric open subset and any non empty open sets. It's nowhere dense in this set. And the K points like the QP, like the Q bar points would be periodically dense in here. And so that means you can always find a K point that's outside all these this big in this outside all these intersections. All right, so that's that's the proof of this lemma so so you can forget the proof now and just just remember the statement that there's a point outside all the other breaks up. All right, so now let me continue with the proof of the R&B. So let me just summarize where we are again. So we're we have a morphism of curves over something like a Q bar, and we're trying to find a point here whose pre images are independent of the precarity of Y. So, and we already reduced to the case where the cover is Gawa and let me give a name to them. The morphism that's called F. And we also were assuming that Y embeds into its Jacobian we have this G equivariate embedding. And the lemma told us that we can find a point of why that's outside all the algebraic subgroups of J. Okay, so fix such a Y that's outside all those subgroups, and my pump my X. I'm going to take the pre images of that's just going to be the image of Y under under this map. And so, well, then the pre image of that little X. Well, it's going to have one of the pre images is going to be why, and all the other, and it's a Gallo cover so all the other pre images are just going to be the other things in the of that of that Y. So that's what I'm this so here this G why is the G orbit of why. So those are the pre images. Okay, so I want to I want to be able to say that these pre images are independent in the in the in the card group of why. Now if, if so, I will do a proof by contradiction suppose suppose that suppose that the pre images were actually dependent is the dependent. So that means that there's some z linear combination of of these Gallo conduits, or not, yeah, or the G or there, there's some there's some z, z and z linear combination of these points in this orbit that gives that gives zero on the card group. So, another way of saying that is that there's an element of the group ring, some s, some non zero element, such that s times y is zero in the card group or in the Jacobian. So, so, yeah, so this, so this is why I want to get a contradiction of those. Yeah, so, so if you have this if you if s if you have this group of this group ring element s and s y is equal to zero that that just saying saying that why isn't the kernel of s, if I think of s as an as an endomorphism of the Jacobian. So which is just a z linear combination of the endomorphism is given by the elements of the group G. All right, so why belongs to kernel of s, but I mean the kernel of s that's that's an algebraic subgroup of J, and why it was chosen so that it was not did not belong to any of the algebraic subgroups of J, except for of course the whole J itself. So what this means is that the current the only way this can be true is if the kernel of s is actually the whole, the whole Jacobian. So, yeah, so so the kernel of s is the whole Jacobian that that means that s kitten s kills the entire Jacobian. All right, so now we have this group non zero group ring element that kills the, as an endomorphism, it kills the entire Jacobian. And then by functoriality that that same as will have to kill the lead algebra of J, where and so I'm thinking here that the lead algebra of J is it's I mean it's a vector space. It's a K vector space but it's also has a G action. So it's a rep this lead algebra as a representation of G, and it's killed by this by this group ring element. Okay, so that so we're there. On the other hand, it turns out that for when you when you're in this situation where you have a gala cover of curves of at least of Jansely's to. And you look at the lead algebra of the Jacobian of the curve upstairs. Then it turns out that, and you think of that lead algebra as a representation of the Gallup group. It turns out that all the all the irreducible representations of G occur in that lead algebra.   I mean, if s kills a regular representation of G, that that that's possible only if s is zero. And so so get the s equals zero but that that that's a contradiction since we were assuming that we have a non zero s that was actually giving a relation. So, yeah, so that's the end of the proof this proves that there cannot be any dependent Z dependency between these pre images of that. This x that was chosen as the image of this why. This is the end of the proofs of the theorems. So, yeah, so I guess I have a little bit of time left so maybe I'll just end with with one slide saying, where, where did you go from here. So I mean, so far I mean this is not the, I mean, I mean it's not necessarily the death now for Baker's method, but it just means you're going to have to work harder than if you if you if you if you think that this method of bakers and bakers bakers method together with finite covers if you if you still think that this is going to apply to all smooth out fine curves, you're going to have to work harder. So, I would like so I so, and so on my talk I was mostly talking about question to all my theorems about question to remember the death, the different theme question one and question to is that in question to, we were looking at the finite tell covers of the affine curve that came from finite tell covers of the projective curves. It would be nice if we could prove the analog for for finite tell covers of just the affine curve itself. So, I mean, in other words, though that means we'd be looking at covers of the affine of the. That's the same as looking at covers of the nice curve that are allowed to ramify above the punctures. So they're more of those and there's still some hope that you could have some generic construction involving those covers to that to which you could and you could use be lose theorem for those. Yeah, so it would be nice to be able to study those those finite tell covers of this punctured curve, not just acts. Yeah, by the way, it's enough to study the case of, I mean you're trying to get a, it's enough to study just one puncture, because the problem just gets easier if you put more punctures so the main cases have a curve punctured one point. So anyway, so yeah so, so we'd like to see if we could do some, some method. There's something similar to the way to prove this theorem be for two but but to apply it to the spine to call covers. But I mean, the algebraic geometry, I mean, I, the long a risk for CS theorem is not going to be enough because it's now we're not just looking at one. Yeah, as we move X around in the puncture around an X, the, the cut the financial coverage, it's not just one curve anymore, the financial covers will actually deform, because as as the Lex moves the ramification locus of the cover is going to be the branch of the cover is moving into the curve, the financial cover is deforming as well and so maybe there's some way to use some deformation theory or something but well, I'm trying to work that out so I don't know if it will be able to do that. Okay, so that's one open that's one possible further direction for this. Another thing is what I mentioned already earlier that maybe it's possible to upgrade the some of these theorems that I proved over the complex numbers to theorems over Q bar. Yeah, so, for example, I mean going back to this llama here, maybe is this known already I mean I think there are people in here who know this better than I do. Is it known that, and if I have a, if I have this curve and then in a Chicago in here. Is it known that if I intersect it with all the algebraic subgroups, then all those intersection points have bounded height is that is that a theorem. Okay, maybe, okay, maybe somebody can tell me after after I finish. Okay, so that if that's true, then that would give well that would be a uniform this and then but I ideally you want to do this. Not just for one, why over X but I mean eventually in approving the, the theorem a double primary you want to take a union of this so you'd want to have a height bound that was uniform over all the financial covers live acts to in order to get the, the negative answer for for two, for two itself and instead of two over negative answer to over the complex numbers. So, um, yeah okay and if you well and if, well, if I mean if you end up proving negative answer for for one, well that sort of kills off that make me it means you cannot use, and then you make, you make your review sad because you can't make his method will not apply to all smooth affine curse. So then it's time to invent some totally new approach for it to find the interval points and curse that goes beyond because theorem and financial covers I mean I don't know what that would be but I guess that's the way it is. Oh, okay, so. Alright. Oh, I see. I see your ability to be lose here so maybe. Okay, well, I mean I haven't made you completely sad yet so but yeah so alright so I'll end here so thank you all for your attention.


Alright, so we know that Wonderland posted a teaser today and we all thought the teaser was just for an announcement. Well it turns out, thanks to Coaster Circuits on Twitter, that we were all wrong. This is packed with some information and we don't know if there's some red herrings, but there is a blurred out picture of the layout that was hidden in this teaser and that is going to come up next. But I froze it at this point because it honestly looks like to us, Giga is carved in this spot and we are not fully sure, so I'm posting it here to have you guys help me. Do you see a GI on the left side of the three and a GA on the right side of the three? Right on that lit up section, if you see the white highlight underneath the three, it almost looks like Giga is like carved in there slightly. We also noticed if you watched the teaser at the beginning and keep watching it and freezing it, it looks like it spells out ZZZZZ. Now what's interesting is, in this next freeze frame, which you guys are going to enjoy and I can finally reveal the secret element that we've been hiding, I'm just waiting for it to pop up. They definitely gave away a pretty big kind of section of the layout for this small teaser that is just 19 to 18 days away from announcement. So here's the freeze frame and at first you're kind of like, wait, what am I looking at? Well let me tell you, you're looking at a 0G roll or a 0G stall, we're not sure yet. It also looks like it may confirm that there's an MCBR after the exit on the tunnel. Again, we're not fully sure yet because it is blurred, but this definitely looks like an inversion and it looks like it is over vortex. Now a few things to note is, it looks like it's suggesting that this inversion is extremely tall and this is a 0G stall or roll, whatever the 2 it ends up being and it's directly over vortex and you can tell by the kind of like viewpoint of it, it's extremely high. Now the only thing is, the mountain is missing in the background out of this freeze frame and we're not sure if that's, you know, the wrong angle that we're looking at this or if there's, it's just a rendering for the ride and off-ride POV for the announcement and it's not showing it. We're not fully sure yet, but it's definitely something that we're not fully aware of quite yet, but you're definitely looking at an inversion here. And then if you look in the bottom right section, again, we just saw Ziz carved in the teaser. You can see Val Ravens crown. Val Ravens crown is definitely in the bottom right section. If you look very closely, it's not small. You're looking at that big brown blur at the bottom right. You can see the crown points. That's definitely Val Ravens crown and maybe that's suggesting that Val Ravens crown is being handed over to Ziz. We know that Ziz, if this does end up being Ziz, is the king of all birds. Now we're not suggesting this is going to end up being Ziz, but I think this is getting pretty like hard to ignore. We think there's going to be a lot of Canadian theming in it and they'll definitely go down the minoring route for this coaster, but incorporate Ziz somehow. We're not fully sure. What do you guys think? What do you guys think of these teasers? I hope you enjoy it because, wow, I wasn't expecting this at all. I mean, we're what? Almost 18 days to announcement. Anyways, I want to hear from you guys down below. What do you think of this? Do you think that I was right about the giga carved in the three or do you think I'm horribly wrong and I just need to like go back into bed and sleep some more? I don't know. Anyways, give the video a like, subscribe if you haven't. And guys, please share this video. I'm trying to get to 6,000 subscribers before the announcement. We're doing a big live stream of the announcement and all that, so I would love to get to 6,000 subscribers and we're so close before the announcement. So 18 days to get to 6K, help me get there, share this video and thanks so much for watching. Have a good one guys, bye.


Sunday Igbo has joined a host of Nigerians to condemn the terror attack on St. Francis Catholic Church over understate that left scores of people dead, reacting to the incident Igbo in a viral audio released parade to God that we don't experience such a game.