Welcome, everyone. I think everyone probably came to see Josh, but I'm going to kick things off. My name is Colin Stephenson. I'm with Pivotal. I lead our partner architect team. And so when talking to Josh about the open service broker API, which hopefully you'll hear to learn more about, we thought about, is this really necessary today? We work with partners. We talk with ISV vendors, with system integrators, our various cloud provider partners within the Cloud Foundry ecosystem. And the open service broker API, there's some questions about it. Is this overkill? Is this something that we really need to be doing today? And if we look at the history of service-oriented architecture now moving into microservices and how we're provisioning these and interacting with these, my proposition is that this may not be necessary. And so Josh is here to prove me wrong. I'm a huge fan of the open service broker API. I think it is the most important thing to happen in computing sense object-oriented programming and probably more significant. This is basically like cats and dogs living together in perfect harmony. It's like rainbows and kittens falling from the sky. So the best thing ever. And paradoxically, it's essentially like hell has frozen over and we're all going ice skating. How many of you have heard of the open service broker API? OK, good. So picture this. You can take an Oracle database product run by Microsoft and connected to from on top of Google's cloud. So if you had predicted that to me a few years ago, I would have told you like that will never happen.  You can take a Microsoft database, run it in Linux, and deploy it on top of Oracle cloud. You can take Google's database and connect it to applications running in vSphere. Like you can mix and match service brokers in ways that their authors never intended. In fact, that I don't think anyone ever intended. And so this is exciting. This is beautiful. This is freedom. I don't know. I don't know about that. So if we, yeah, you're reading down to the bottom already. If we look at the term service, and we're talking about service brokers, right? If the term in Latin servidium goes back to talking about slavery, are we just talking about slavery brokerage to these different technologies and these services? I'm a little concerned here. We looked at all these big logos from these large vendors. Are they just honing us in into some new slavery of API service provisioning? I don't know. I'm a little concerned here. Let me cast aside the shackles for you for a second. Let's talk about great moments in technology. And I mentioned object-oriented programming, but I'm going to go back further than that. So if you think about the invention of programming as a discipline, I don't mean Turing, and I don't mean the invention of the computer. I mean Steve Wozniak. Hear me out. With the invention of the Apple computer, it was the first time you could be a software engineer without understanding hardware. It was the first time, and it was actually my first computer was an Apple, you could not know how the hardware worked. You didn't have to understand logic circuits. You didn't have to understand the physical memory or the CPU or what a logic gate was. You just had to understand programming. That was a really powerful abstraction. It took software development and computer programming from the domain of thousands of professionals to millions and tens of millions of enthusiasts. It actually opened up an entire field. And that we did that by creating the separation of concerns. We think about highways. If you want to go really, really fast on a highway like the Autobahn, you have separate lanes. Traffic goes in one direction on one lane and another direction on another lane. This is a very powerful concept. You keep these things separate, they don't crash into each other. There are other great examples of this. How many of you are programmers or have written code? Okay, how many of you have ever used MVC? Another great abstraction written by a man whose name I can't even pronounce. Which is because I'm American, I'm sorry. Xerox PARC 1979, right? The separation of, oh, I don't know, logic that relates to the front of the program, from logic that relates to the back end of the program, the separation of data and logic. A super powerful concept. So when we separate concerns like this, we end up accelerating both teams. If we get the abstraction right. Let's look at another one. Let's look at networking. The entire software-defined networking movement over the last 10 years has been powered by the separation of the data plane from the control plane. Packets move in the data plane, deciding the rules about where those packets should be routed and which ones should be allowed and which ones should be transformed happens in the control plane. We can keep the data plane really, really fast by separating the control plane from it. That's actually what the open service broker API does. It's the control plane for data services. So we're applying that same magic of the separation of concerns to these shackles, the relationship between state and stateless, the heart of the cloud native movement is separating these two things. But we still had the control plane logic stuck in the data plane. So I mean, this should be enough convincing for anyone, right? Even you. But I'll give you one more concept here. The word broker comes from broaching a wine keg. So the original brokers were wine merchants, which already, I love the concept, right? But even more importantly is tenancy. If you have a single wine keg, it's yours. It's one person. It's a single tenant model. As soon as you have a broker, you can share that wine keg with your friends one bottle at a time. This is really important. As soon as you have a service broker, you're making it easy to provision multiple copies of a service with different credentials. You're creating a new tenancy abstraction. And in fact, it gives you a way to take advantage of the divisibility of your underlying resource. Wine is infinitely divisible, but people prefer to have their own wine glasses. So far, you've convinced me that I'm ready for one of the beers from the SAP booth out there. Day drinking is the hallmark of all great programmers. Yes, exactly. That's not actually true. So I get the excitement, we're getting there. But then I look at some models, right? You look at Apple, maybe some other technologies out there where they're able to successfully go from generation to generation. They can even mix up the interface and the protocol that you're using. And people will still buy it, right? It's perfectly fine. You get some complaints, right?  But for the most part, that mark is still there. And even though the interface changes with something open like the Service Booker API, we can, it's not fundamentally changing how our application actually interacts with the service. So you're making something maybe incrementally better in terms of how that service is provisioned. But the application still has to use that service as libraries. We're not completely getting the loose coupling that maybe we desire. So I'm not really sure that proprietary, in some cases still okay, that we can actually optimize the interface for my service, right? My customers are happy. I do care about them. But I don't know if I have to go and now add yet another layer of abstraction there. So I don't know if that's, not quite there. You're talking about this as if it's gonna be a lot of work. And it would be a lot of work if this wasn't an open standard. And it would be a lot of work if this wasn't open source. But the fact is, we fight these risks, these sort of potential lock-ins of saying, okay, well I've made it really, really easy to go use all these commercial vendors. Have I actually just created a new kind of slavery? Am I tying myself into new economic slavery? By saying, no, we're gonna make it just as easy to leave, right? So the notion of rough consensus and working code, this is not a standards effort in the tradition of the W3C. This is not a, hey, we spent 10 years thinking carefully about this problem without ever writing code. And now we're gonna subject you all to that standard. We've tried that before in the brokerage space. That would be UDDI. Or 800 different efforts at soap marketplaces.  But the difference of the life of using JSON versus using XML is huge. And this is the same thing with Service Brokers. The difference between the Open Service Broker API and just a different way to get access to a data service is kind of like the difference between XML and JSON. But I've sort of realized I'm not gonna be able to convince you without showing you this, right? So maybe what I should do is demo really quick. Not how an Open Service Broker works, because I know you're gonna get a really cool deep dive tomorrow when Alex Lay is gonna walk you through Service Brokers and how they can be bound to applications both in Cloud Foundry or in Kubernetes. But since I'm challenging you to build Service Brokers, I wanna show you how to write one. And I think I can demo writing a Service Broker in under a minute. So use an angle there, not a curly. Oh. That's not JSON. All right, scary live demo. Live demos are always a bad idea. Just so, for those of you who've ever done this before, let me start with some code. And I'll try and make it big enough. I actually wrote the Service Broker ahead of time, so I did kind of cheat. I'm writing in Python, not because it's the only perfect language, but- It's the only one you know. Because it's the only one I know well enough to live demo. And because there's a really handy library for writing Open Service Brokers in Python called the Open Broker API. So you can do an import of Open Broker API and then it does most of the work for you, which turns out isn't very much work, but it saves you at least 50 lines.  Here is a Service Broker, and all it does is return a catalog of some example services. Cool thing about the Service Broker pattern is that you can have multiple services in one broker. The Microsoft Meta Service Broker has, I think, eight or nine services in it now. And many of them have several different plans and lots of them take configuration parameters, so they really expose a good chunk of their entire Azure data services suite in a single Service Broker. So we're not talking about millions of lines of code here. So I have this. I'm not sure I'm getting enough out of the catalog though, right? I want to make my own choices. So if you're defining these plans within the catalog, aren't I restricted by what I can choose? Absolutely. Yeah. Freedom comes from constraints, right? And you can always write your own. Who's the famous guitar player who only ever had one guitar? I don't know if you have one guitar, but Tom Morello from... Tom Morello. He kept the stuff simple and, you know, but it's just pretty creative, I think. I mean, when was the last time we added an extra key to a piano keyboard? It's been what, hundreds of years? And yet we have new kinds of music produced every year. That's true. There's only 88 keys on that. There's only five API calls here. You could still play a lot of music with five API calls. Fair enough. All right, let's run this ridiculously short simple broker. And nope, it's running. It's on a local port. And I'm going to curl against it just so you can see that I get some JSON back. So I'm curling against the catalog API. And I got my services, service plans back with some descriptions and some IDs. And now I want to show you one of the cool things. There's no Cloud Foundry here at all. There's no containers. There's no Diego. There's no Bosch. There's no nothing. There's just one simple app. And I just curled against an endpoint. Now you can imagine, okay, well, I put that out in the real world. You got to use something other than curl, right? And that's true. And if you're using Cloud Foundry, then the Cloud Foundry Cloud Controller is going to go off and talk to those service brokers for you and pull the catalog and make it show up in the CF marketplace. And it's going to broker off those provisioning calls and that's awesome. And if you're using Kubernetes, the Kubernetes service catalog is going to do the same thing for you. But what if you're in a world that's not just Cloud Foundry and Kubernetes? It's actually a really cool command line client for the open service broker API written by the folks at Stark and Wayne called Eden. And you can just give it a service broker endpoint and some credentials and say, give me the catalog.  You can provision, you can bind, you can unbind, you can get service keys, you do whatever you would do with a service broker and you don't even need a platform. You can just deal with the services. It's pretty cool. Sometimes all you need is a MySQL database, which is an Oracle product run by Microsoft connected to a Google Cloud. All right, we're getting there. All right, now, here we go. So I sort of feel like you believe me. I'm not quite ready to bow down to Josh here, but we're getting there. I think we need to go somewhere with this. Service brokers are cool. Having other vendors make their services exposed in the open service broker API is cool. This is freedom, but most of our data is not in public clouds. Most of our data is in the world is in existing databases. It's in Oracle in the basement, it's in SQL Server, it's in Postgres instances, in some cases it's in DB2, it's in... Mainframes. Mainframes. I challenge you to write a service broker for every single thing in your data center. That's not an app. You can write a service broker for Active Directory, for the key card service. You could write a service broker that operated a internet-connected coffee pot. I actually think that provisioning is an important activity for a coffee pot. Could it write the tap to the wine cake? Sure. Service broker-connected wine cakes. Yes. So I think that, you know, I'm borrowing from other people here, I love quotes. The quote that comes to mind is Bill Gates. This is a service broker on every desktop and in every home, for every device. And I think I've showed you that you could write a service broker in 24 lines of Python. I mean, you probably need another six lines to implement provisioning. And you don't have to write these from scratch. Remember, there's a cute little library for writing them. In Python, pretty simplistic. There's also a very full-featured SDK. In fact, there's three very full-featured SDKs. There's the services SDK developed by Pivotal, that's open source. There's service fabric developed by SAP, open source. And there's Swisscom's confusingly named service broker project, which is for making service brokers. Also open source. Any one of those, and you pick your favorite language and your favorite set of partners, any one of those would be amazing for writing OSB APIs and connecting your coffee pots to the internet. Which, after all, is what the internet was built for. Thank you. That was good. There you go. I agree. We can take on that. All right, all right. We have some time for questions, a little bit of time for a few questions on the topic of freedom, or possibly on the topic of service brokers, or Julie Andrews. Or Julie Andrews. I will actually start singing karaoke if somebody doesn't ask a question, so. Yeah, go ahead. Yep. It's a great question. So here's the guiding principle. A service broker ignoring provisioning, right? You can actually make provisioning a NOOP, if you want. A service broker gives you an endpoint and a set of credentials. So the service that you're talking to can be SAS. It could be a multi-tenant cluster managed in your own data center. It could be a single-tenant instance that's created on demand. It could be any other crazy abstraction of those ideas. So as an example, there was a community member who used service brokers to create arbitrary network connections, overlay networks. So you'd create an overlay network and then you would bind it to apps and they would get a tunnel created from the app or they get reachability, essentially. ACL rules would be applied by service broker binding. You could, you know, you can make these various calls asynchronous. So you don't have to have it be something that returns in real time. So it could be remedy tickets. So you could say, here's my service broker and when I call provision, it creates a ticket to do a thing. Could be build a house. And then when it's done, it eventually updates the status and I say, oh, the house is provisioned. And then when I want keys for the house, I call bind, right? I think having a service broker that built houses would be super neat. You know, you can 3D print a house in like 24 hours. Why would you not have an OSP API connection to that? So it's robots all the way down. Robots, robots talking to robots, yeah. It's control plane. Yeah, so the service broker is only the control plane. It's never in the data path between the app and the service.  And it's a good question because there's a lot of places where the service broker API could evolve. And you might wanna say, hey, what I care about is a SQL backend. I don't even care which one. So here's a generic service broker for SQL and it's gonna take some parameters or introspect my environment or look up my credentials in some other data store and say, well, I've decided to give you my SQL because you seem like a nice guy. I don't know what logic could be applied, whatever you want, but it's out of the data path. So in a sense, and if you think, I mean, you go one step further and say, well, what about operating those services because the service broker might be talking to Bosch and standing up Bosch deployments. Come on, demand broker does that. They still then have to be managed, right? So there's an operator's interface to those services once they're running. Service broker doesn't care about that. Service broker, the separation of concerns is care and don't care buckets. So in the care bucket is, did I get credentials in an endpoint? In the don't care bucket is, how is the app actually connecting, talking what's the data path like? So is it SQL requests? Is it over port 3306? The service broker knows some of these things and it tells the app, but it's not in the way. I don't think we want it in the way. Now you can write, that's the other thing. The service can be as complicated as you want. So it might, and the broker actually could advertise its own address as the service. And so you could end up talking to the broker and it could proxy off to your data service if you wanted to. So you can have a service broker that also implements the service itself. That's not really OSB. I don't really know why you'd do that. I mean, I think it makes more sense to have a broker that builds houses. Yeah, it's a great question. So in the cloud foundry world, service brokers can be registered in two ways. One is globally by an administrator into the CF marketplace. The other is space scoped. So as a space developer, you can register a service broker into your space and it'll show up in the CF marketplace just for you. So if you write a custom service broker that exposes, I don't know, the post-grade, existing post-grad database you've got in your AWS account and you just wanna make sure your CF apps can go talk to that cleanly and have that separation of concerns around credentials, just register it into your space and then you can create and bind to those services and you're gonna have that service broker just for your dev team. Be aware, you can't space scoped, register the same service broker into multiple spaces. It doesn't work. Although you should file a bug on Alex, so it does eventually. It's not Alex's fault, I'm just picking on him. Yeah, it's a great question. They are, well, let's say they are. They're doing it frequently, right? There's, I think, despite having a lot of large organizations looking at using this open service broker API, it enables a lot of the smaller or newer ISV partners that are gaining ground from developers up in their organizations and these developers can easily pick and choose based on that common interface how they want to leverage those services. So the point is as our developer ecosystem grows, having this common way that they get used to interacting with the services through the broker will make it easier to then how do I figure out exactly which database or queue or house building service that I want for my application. So if I need Mongo today and I want to use Azure Cosmos DB tomorrow, by having that common broker, it's easier for the developer to reason about how they're going to do that or if it's not the developer, to Josh's point, the robot's all the way down. If it's a DevOps team or a configuration management team that's handling those through pipelines, it's easier for them to think about, okay, I just need to go make this simple call to do this. The developer, I assume his code is handling how it needs to from there because it knows it has the credentials and the connectivity to that service. I mean, yeah, to that point, you might not be switching from MongoDB to DocumentDB but you might be going from MongoDB Enterprise to an open source Mongo to Mongo run by Google or AWS or Postgres from the four or five different community vendors that sell managed Postgres tiles now. So the service broker, even if your app just wants Postgres, it doesn't have to know which Postgres. Is it clustered Postgres? Is it single instance Postgres? Is it Postgres as a SaaS service from someone else? Think of it another way. If you have a service broker that builds you a house and you just want to throw a huge rager, right? You're going to provision a new house.  You're going to deep provision the house because nobody wants that mess in their own house. Why would you clean that up? Sorry. It's not twisty. Yes, however, I'm hopeful that the OSB API team is going to collaborate now with APIs.json and a couple of these other service discovery JSON based API frameworks to say, hey, we don't need all of this to be new in OSB. We can use Swagger. We can use APIs.json. We can use other things that improve discoverability of an endpoint. So yeah, take your service catalog but make the metadata attached to it rich in this other format, right? The new DDI. Yeah, yeah. The number of catalogs and the number of marketplaces I think is going to proliferate because why wouldn't you? But there's no reason you can't have the description of the broker as separate from a running instance of the broker as separate from the services that can be exposed by that broker. And we'll see what happens. It's a bit like HTML, right? HTML is messy and horrible and decorated with JavaScript and CSS and arbitrary XML and blink tags. I use HTML as a reference because I've worked on several browsers. So just bear with me. It's horrible in all of these ways but it's good enough that it was like the glue in that separation of concerns between client and server that made client and server work for like a third of the time that we've had computers. So I'm pretty sure that this messy pattern of like, oh, it's BAPI and doesn't really have all the discovery we need and doesn't have enough metadata and what about billing and then who runs it and how many copies are there and that messiness is part of what tells me this is gonna work. Because it's not so strictly defined that it's been thought up ahead of time and it can't be adapted to different uses. Yeah. This talk was originally called when hell freezes over which I always ignore the title of my talk by the time I actually get there and give it.  So the payment for the ferryman is that billing and quotas and entitlement aren't in the spec yet. They're coming and Alex I'm sure will talk about where is the spec going so that this can be enabled but it's also that the SDKs, I know the any nine SDK does a really good job of dealing with billing and metering and entitlement. It's not something we've dealt with in the services SDK to my knowledge. We sort of, I mean, if you think of the SAS model you already have an account. You already have an account with Azure. Those account credentials are in back of your service broker call. And so again, it's kind of that separation of concerns. Control plane, data plane, money plane. You know, the money plane is TBD. Maybe I just made that up. I think we're good. Yeah. We're out of time and there's people coming in who want to hear other more interesting people. So thank you very much for your time. Thank you. That was fun.


Haha, dude, it's hard to see that stuff Especially when you have special eyes like I do special eyes. What do you look? I don't want to be up there. Anyway, I Just climbed over for the fun of the chat and we're waiting Binders ready chat Nurgh a gauntlet is a burner Lion I'm your gigante.  Hurry up. Hurry up. Give me. I see you. I didn't want to do that Me you called him a fiend. I'm gonna tell mama. We're ready. I will block all you stuff I got charges now. I had charges. We've driven up there. Good guy. Good. Good. Good. I want that piece Let me get it. Let me get it. Let me get it. Please. I want this Can I get it while climbing? Oh, it's lost forever Now we wait chat harvest some things let me pick this up. What's this? Yeah Yeah, let's see. Yeah Oh the fragments chat I don't even know what I need for the hammer. I might have enough I'm come I think I have one gym left I think So I remember getting two and I used one for the For the heavy bow gun Oh, really?  I've never done that I'm gonna fire your cannon. That's all right and yours How many I have in mind three four Okay, four Yeah, palakos can but we're in a group of we're in the full group. So no, no, there's no palakos here right now Why would you laugh at her? She was serious Sura macros isn't nearly as weak enough Are you fired a dragonator? It's over here Every time I've done this somebody else did the dragonator wasn't me your face isn't weak enough girlfriend Friend of the boat Oh, is this thing? Yay, I'm helping Wait, I thought I picked up I gotta turn this one around too Well, that's terrible. That's not a lot of money That's not a lot of money at all I feel shaman boozled That's the opposite of a lot of money Sell the carapace. Oh You can sell those They're not using anything What's that matter? What's up, piggy? How you been? It's been a bit. It's been a minute So you can sell the carapace. I did not know that But what aren't they way? Let me let me check. Let me add this hammer to my whistle.  Well, duh I knew that That seems like a normal mission though. I thought it was like bonus money Um Wrong place I really wish you could sort this alphabetical order. All right, so that's on page five I have 13 of these Uh, I don't see anything about I think the only thing I saw that could help you was uh, there was an armor gym one Uh, let's go to the upgrade So I need one gym Wait, did that not I didn't have to make that So dumb Uh, we will add this to wishlist Why?  No I forgot the name of the stupid uh charge blade It's not that that's a that's the lance The name of the charge blade chat It's a rarity six or seven. So I thought it was this That's kodachi. That is the um That is the the lance Am I going crazy? Yeah, I am See that's not it. That's the kodachi It's not the iron gun lance. It's not the mac does not the horn.  I don't see it Where's the charge blade chat? It's not white Does anybody see it? top left Sort by tree. Can you do that? Oh, thank you All right, so it'll be a it'll be a diablos weapon, right? I'm not trolling. I'm looking at the colored weapons. I see hammer I see horns I see a charge lance I see is this it? That's it. Okay I do I barely know what the icon looks like I barely know what the icon looks like Because I don't recognize these icons like you guys do I look at all these things And it just I know what the hammer looks like and I know what the bow looks like everything else is just like hieroglyphics to me Okay, so it just looks like a shield The um the icon Okay, um, can I meld a um, uh, zora magnarose jim somebody asked about that Probably it's gonna cost too much. I don't think you can 300 points it requires three golden gold waverian prints Should I do this we that we guys think how many of these do I have? It just needs one you have three. Okay Let's say 300 points 300 is the research points. So there's I've never melded before so there's there's research points and then there's melding points So when I first come in here We're gonna learn chat When I first come in here it says 300 points that is the green points And then there's melding points Okay I just do one of these and meld I love that's worth I mean how valuable are these gold waverian prints? You're only getting one gold a week. Oh those are gratis. I thought you had to farm those Oh Okay This is a song of learning I may have specialized but I can learn at the rate of an eighth grader chat an eighth grader who just had lunch Sino Giva shell Ooh fire cells I hate fire cells Oh my gosh, that looks amazing Yes I don't care about the stats at this point.   This thing is awesome 1144 attack. What is that compared to my? That's a little bit less than diablo negative 20 center for element is fire I'll think I'll think the oblo is Is the oblo is Was that impact or fire? I don't think impact is an element Maybe crazy I know there's a button to compare but I don't know how to do it here The element is blast. Okay Okay I guess I got to start thinking about this in non-war frame terms It's like cold and heat combined is blast But I heard this one does a cool little boom boom so All right, I'm going to be a charge blaze master chat I'm going to do it before we go to the moon First off, I'm going to recognize this icon Yes, I hit B on that didn't I All right. What time is it? It's time for another break My phone I have misplaced my phone 10 12 all right guys, we're going to take a break and then we're going to come back and farm something I don't know what maybe we'll do one of the event. Thanks um We'll farm something from the event. I'm going to learn this charge blade if it If it hurts me and it will chat it will I shall return let me go up to the to the uh To the gathering hub escalation point session id I mean, I'm sorry escalation point session and chat if you want to join our session on pc I'm going to sit here and toast while we take a break I'll be back guys. Thanks so much for watching three minutes. I'm going to get some more to drink Actually my water. I haven't touched my water and my coffee is hasn't moved. All right. I'll be back


Hello and welcome back to my YouTube channel, this is Daniel Rossell and this YouTube channel covers lots of things to do with living in Israel. These days, Israel's main international airport, Ben-Gurionier Tel Aviv, handles an excess of 20 million passengers per year. It's regularly rated to be one of the best airports in the Middle East and is renowned for its tight security. Traditionally, Israel has been a somewhat expensive destination in terms of flight connections. For a long time, Israel's flag carrier, El Al, enjoyed significant price protection on flights to and from destinations of significance to Jews and Israelis around the world. Think medium to long-haul destinations ex-Israel such as London, New York and the Far East, or in other words places where there were significant Jewish communities coming and going to Israel, or where Israeli business people might need to go on business. For leisure destinations, the market was dominated by El Al and legacy carriers, so airfares to and from Europe were relatively high. Things changed dramatically for the better in June 2020 when the European Parliament ratified the Open Skies Agreement with Israel, which Israel approved seven years earlier in 2013. The vote in the European Parliament was passed with a large majority, 437 in favour, 102 against and 147 abstentions. El Al lobbied against the move because the agreement directly threatened their financial interests. Some European political factions expressed concern about the agreement in light of their opposition to Israeli government policies and pushed for a delay. Fortunately, the Israeli government went with the greater public interest and Europe eventually also gave the agreement its green light. The Open Skies Agreement that Israel exceeded to enables all European airlines to operate direct flights to and from Israel to anywhere in the European Union. In reciprocation, Israeli carriers such as El Al and Arkea were allowed to operate flights anywhere to and from an EU destination. By guaranteeing this mutual right of access, Israel and the EU ensured that the bureaucracy required in establishing a new route would be far lessened. This made it much more attractive for ultra-low-cost carriers like Reiner, Wizzair and Easyjet to begin operating flights between Israel and Europe. As a result of the Open Skies Agreement, a vibrant market has opened up for flights between Israel and Europe. Given the high cost of accommodation and the high cost of living in Israel, it's actually becoming increasingly popular for Israelis to holiday in nearby European destinations rather than in Israel itself. Israel's location on the eastern shore of the Mediterranean also makes it a great place for exploring southern and eastern Europe. Cyprus is only a 30 minute flight away and Talia on the southern coast of Turkey can be reached in only 45 minutes flying time and Greece and Italy are only an hour or two further away. As a result, the list of airlines flying to and from Ben-Gurion is now a lot more diverse than it was only a few short years ago. Etihad and low-cost carrier Flight Dubai compete with one another on flights to and from Dubai, the product of the Abraham Agreement. Reiner operates flights between Ben-Gurion and destinations like Budapest, Brussels, Paphos, Cracow and Malta. Easyjet operates a frequent service to and from London's Luton Airport and Hungarian low-cost operator Wizz flies from Ben-Gurion to Warsaw, the Greek island of Santorini, Budapest and many other destinations. The Open Skies Agreement remains a powerful testament to the idea that foreign competition can provide a quality of life upgrade for many Israelis. It has also made Israel a much more affordable travel destination for plenty of curious Europeans. Thanks for watching, I hope this video has been informative, and please subscribe if you'd like to receive more.


Tako da. Zato je toga kompetija. Zato pravno tudi, da sem očeli, da sem ne ga poznala. Na tituljiéc smo razlučili 1. Všeč smo všli 2. Poživaj, da bomo, zelo, ovo.  Dončanje k dvej točnih vse je. Tudi pričaj sem se umožela, da sem posledal je tutorial na 4.30h, tudi katarina Kukić. Zelo je zelo vzvečenje v ovom delu, in vzvečenje vzvečenje na fri dnev, z Anani in Edwardem, je zelo vzvečenje v ovom delu. Tudi 3-5 svojov svojov, nekaj ne so vzvečenje v zelo, ali zelo se prišlično prišlično. Tudi je kubiko in električno funkcije. Zelo smo prišlično izgledati, da se vse, ki se vse izgleda, vse je, da se vse izgleda, kaj je zelo, da imamo dve vrste, kaj je vrste, kaj je vrste, kaj je vrste, kaj je vrste, kaj je vrste, In tudi je TORUS, da se tako pravimo, kaj smo obtežili, da je vse fundamentalne paralogrami za dv periodične funkcije, kaj je omega-1 in omega-2. To je bilo različ, da je omega-2, da je omega-1. Here we have two structures. Here on a cubic curve, this structure has been presented in the following way. So this is a circle b, and then this involution gives us a plus b. You remember that for a given a, there was also involution this tau, and then we map a to minus a in this structure. And here we have just a fundamental addition of vectors in a, if you want in a R2 or complex numbers in C, calculated modulo the lattice generated by omega 1 and omega 2. And then we constructed here very important elliptic function of order 2. So this is a Weistras p function, and we show that if you use the Weistras p function, satisfies a very important differential equation, which allow us to construct this map from this picture to that. Denoting this as a polynomial of degree 3 in p function, we have a map. Map is, so map phi from torus 2, let's say, gamma mapping z2 p of z p prime of z1. I mean, this is okay that this formula goes across this slide, because this is a map from this window to that one. And then the final statement was that this map is not only a bijection, it is an isomorphism of groups, isomorphism of groups. So in a sense, for example, if we have three points here in this picture, and we have three points, let's say, and w, such that u plus v plus w is zero, then we know that this map would map these three points here into, so you see, this is a plus b, that means that a circle b is tau of a plus b, that means this is minus of a plus b. So a b and minus a plus b are collinear, so three points here are collinear, if and only if, it's sum is zero, that means that three points here are mapped here in collinear points. And we are now going to use this very simple thing, but just before that let me make a few very simple remarks, but important. So, of course, here we are getting a polynomial of a specific form. Here we are getting polynomial of a specific form. It doesn't contain terms with quadratic terms, but of course with the appropriate change of coordinates we can always reduce general polynomial of degree three to that form. Maybe I can leave using this good example of Sergei, some exercises. So, talking about this, immediately we have an exercise to prove, so what was exercised last time was that we have these midpoints here, omega one, omega two, and omega one plus omega two, and here if you know these points e1, e2, e3, then if you remember exercise from the last time, we said that p prime is zero in these three points. So, from the last time we have that p prime of, for example, omega i is equal to p prime of omega one plus omega two is zero, so that means that at this point this is going to be zero, so that means that p of w1, p of w2, and p of w1 plus w2 are going to give us e1, e2, e3. So, e1 is, for example, p of w1, e2 is p of w2, and e3 is, for example, p of w1 plus w2. Exercise is to prove that these three points obtained in this way are distinct, so prove that e1 is not equal, e2 is not equal, e3 is not equal, e1. So, this is one exercise. Another remark, of course here there is some sort of choice, because we can denote p of omega one to be e2, p of omega two to be e3, and, for example, to use some permutation of this, and, of course, we cannot expect from an elliptic curve to know what was our choice. So, the question is when two elliptic curves are isomorphic, so when can we, in a sense, identify two elliptic curves, and the answer is given in terms, so we have this, you remember the picture of last time, so we have e1, e2, e3, and p infinity as four points where we made these cuts, where we glued these two planes or two Cp ones together, ramified over these four points. So, we have four significant points, which correspond to those points, which satisfy that two times this point is zero, and for four points, let's say e1, e2, e3, and p infinity in our last notation, what we can do as follows from the instructions obtained in the Sergei's lectures is that having four points we can calculate their cross ratio. So, we have a cross ratio of these four points, and as Sergei observed, cross ratio of four points is not uniquely determined, it depends on this choice of order, so we can get numbers like, and make mistake, but you will, lambda 1 minus lambda, what else, 1 over lambda, 1 over 1 minus lambda, and lambda over lambda minus 1, and lambda minus 1 over lambda. I think this 6 typically you can get, and then we can say that, I mean this is a theorem, two elliptic curves are isomorphic if this set of six values coincide for these two elliptic curves. Of course, there are some specific elliptic curves where you don't have six different values, but this is in a sense an additional story. So, what I want to say is that we should in a sense start accepting a principle that whenever you have something which is double cover over Cp1 with four branching points, it's elliptic curve. So, this is in a sense guiding informal principle. This principle, of course, can be formalized by Riemann-Hurwitz formula, which says if you have a holomorphic map between two compact Riemann surfaces, then the Euler characteristic of origin is equal degree of F times Euler characteristic of image minus something, which we call degree of ramification. So, this is a general story. So, what it is about, so we have something, let's say gamma somehow mapped to gamma prime. Here is gamma, here is gamma prime. A general fiber, if you take here some general point and consider inverse image, general this fiber has some fixed number of n points. This n is called degree of a map. This is a degree of a map. In our case, it's two. But it could happen that there are some points where you don't have so many points in preimage, like here, and you have some gluing, and these are ramification divisor, in a sense, counts. This number of points where you have this sort of effect and degree of this R is a total count of this sort of collapses of numbers of preimages above some points. So, in our case, since g prime is cp1 is of genus 0, this is elliptic curve of genus 1. This degree is 2 and this degree of R, since we have four points and we have this ramification, this is four. So, in our case, so we have 2 minus 2 times 1 equal 2 times 2 times 0 minus 4. So, as you can see, this identity is correct. So, this is a verification of this principle that whenever you have something, double ramified, some double cover over cp1, ramified at four points, this is an elliptic curve. Okay, so, now, let's, as a sort of example, use this in a sense, if you want to fully appreciate what we derived, we need somehow to apply this. So, one way how we can apply, we can use this fact that having three points in this picture with the sum 0 means that, in this image, they are going to collinear points. This information is enough to derive a very nice classical formula, addition formula for VHSP function. So, let me just, as a sort of example, this is addition formula for VHSP function. So, if I may allow a small remark, I see that all of us are a little bit today, in a sense, out of shape. I don't know if you had some interesting nights watching football or so. So, please try to follow, because I can easily make now mistake in the calculations on the board. I need your full attention in order to prevent some creation of new unknown formula. So, please, let's try to do it together. It's a simple exercise, but still it requires some sort of concentration. So, we have three points, u, v, we have three points, u, v, and w is nothing but minus u plus v. So, that means that take u and v and map here, you will get some, these two points that remind some line, and then that means that a point obtained as a map of minus u plus v belong to that line. So, suppose I will now, that the formula for that line is something like so, let me, something like this. So, p prime of z minus some a p of z minus b equal zero is satisfied for z equal u is satisfied for z equal for z equal u, v, and minus u plus v. So, when we plug these three concrete values then these three points belong to the same line here determined by coefficients one minus a minus b. On the other hand we know that these three points are also situated on the curve. They belong to this cubic curve so that means that let me just write this p prime of z square is equal this polynomial there.  And again if you plug these three values here we are getting this is also satisfied. So, so, this system is satisfied in these three points. So, from this relation I can exclude p prime and I can write here a p plus b square equal this four p cubed minus g2 p minus g3. So, now we need to calculate something so this is a so let's now try to rewrite everything in a just as a polynomial. So, u v and minus to b argument of p function we satisfy the following polynomial polynomial equation. So, the equation is four z cubed minus a square z square minus g2 plus 2ab I hope it's correct z minus g3 minus b square equal 0. So, this is a polynomial in z for which we know that it has three zeros we know three zeros. z1 equal u z2 equal p of v and z3 equal p of minus u minus v but we know that p is even function we can use it not use this in my mind. So, it's a p of u plus v. So, we know three zeros of this polynomial what would you suggest me to do? We have a polynomial and we know it's I mean we know we know these three zeros what we really want to do is we want to express this zero in terms of these two this is our ultimate goal. To get addition formula means to express p of u plus v in terms of values in u and v and this is our all the time the same dilemma so tell me what do you suggest me to do? If we have a polynomial and we know three zeros we can use what is the proper English term for this? Vieta formula Vieta this is in English Vieta in Italian I don't know why it should be in English Vieta formula saying that what? The sum of zeros is equal what? a squared over four sum of zeros so p of u plus p of v plus p of u plus v is equal a squared over four we are almost done we are almost done but not yet we need to calculate somehow a from here knowing that this is satisfied for u and v so we plug now here u and v and we are getting two equations from where we can calculate a in terms of p of u p of v p prime of v so what is a? A is how can we get? a is from here a is something like p prime of v minus p prime of u divided by p of v minus p of u this is a and finally we are getting the formula that we want to express to express p of u plus v minus p of u minus p of v plus two times p of v minus p of u p prime of v minus p prime of u square so please remember this formula if time permits in section 5 we will use it once so that's why we need it because we are going to if not use at least mention so this is additional formula for vajštas function we are getting it as a simple consequence of this of this situation okay and let me say just some remarks since we are here regarding situation of multi-dimensional billiards so for billiards and ponselet theorem we will see billiards in the within ellipse and for ponselet theorem in the plane we will see today or tomorrow that it is related to this addition structure on a cubic for higher dimensional situation for billiards within quadrics in dimension d it appears that relevant relevant objects are hyper elliptic curves so here in dimension d equal 2 so what we had here in dimension d equal 2 the relevant was genus g equal 1 this is elliptic curve in d equal d it appears that the relevant object is going to be hyper elliptic curve of genus equal d minus 1 so hyper elliptic curve in similar fashion as elliptic curve can be defined by its equation of the form y square equal p 2g plus 1 of x where p is a polynomial of degree 2g plus 1 or this is also true for for elliptic curves you can go to even dimensional and then here put p of 2g plus 2 we could have here also tell the story with the case of polynomials of degree 4 it just depends on this so here we have degree 3 because one of these 4 important points is chosen to be at infinity of course in a sense there is a subtle difference talking about elliptic curves of presentation in this way because in presentation in this way somehow this point at infinity is in a sense significant this significance is lost here because here in a sense all these 4 points are equally equally good but let me come back to this case of hyper elliptic curves of genus of any genus so the point that the genus is of course immediately bigger than one so in that case what I want to say is that still there are in a sense there exist there are these 2 screens one screen here is a curve itself of genus g so curve of genus g just schematically we can something like this so here is a genus 3 and here in this case object is here it was a torus and in the case of curves of high genus is torus of higher dimension so what is at the same time nice significant and in a sense transparent in the case of elliptic curves this is at the same time misleading in the case of elliptic curves so in the case of elliptic curves we have 2 pictures of objects which are isomorphic so we have here a curve here we have torus and double theorem says torus is isomorphic to the curve in higher general situation here we have curves which are not torus anymore not tori anymore but there exist some very important objects some high dimensional tori which play the role this torus here played here it's just a sort of a framework to understand that there is a sort of higher generalization which is going to be relevant for higher dimensional ok having said that let me also mention few few general important the theorems and I will formulate them for the case of elliptic curves and we are going to this way or another use so when we are talking about complex algebraic curves and elliptic and high elliptic curves are examples then there are two main classical questions so suppose you have some two collections of points let's say p1 pn and this is one and another qn these points belong to the curve you can also assume that some of these points have a multiplicity and that's why they use a sort of fancy name for this called divisor, just a formal sum where you can put some coefficients here if you want and here q1 but here there is no actual addition this is just a way of saying points with maybe some multiplicity and then the question could be given for example these two collections of points, is there a metamorphic function on gamma having these points as zeros and these points as poles this is one classical question and another classical question is so the existence of a function such that this is the notation divisor of function means zeros minus poles is p1 plus pn minus, this is just notation so the existence of such a function and the second question is, so this is the first and the second question is if you provide a sort of a set of points of some possible multiplications the question is what is the dimension of the space of functions having poles not worse than those prescribed here, so if I denote this by d, then we can say that l of d is a space of functions metamorphic functions such that these functions don't have poles worse than those described by d, this notation here is for the following so if we have some d1 which is some combination of n1p1 plus nkpk where pI are points on gamma and nI are integers, then we say that d1 is effective d1 is this is the notation and the name is effective if all nI are no negative so the answer to the first question is given by so called Abel's Theorem again Abel's Theorem and I will formulate this for elliptic functions so Abel's Theorem for elliptic curves so the answer is as simple as this so you have this n points one set of n points and another set of n points this plus here is just formal notation but the Abel's Theorem says since this function in the case of elliptic curve which is group, belong to a group then this is a real summation do summation in the terms of the group so calculate so having given p1pn and q1qn then do real summation in the group p1 plus let me do like this plus pn where here elliptic curve so gamma plus I want to indicate that here this is a real addition as described here and here so sum this n points as n elements in the group sum this n points as elements in the group and if this is equal in that group which means in this picture means equal as a sum of complex number up to the lattice and in this picture means here if and only if this is satisfied then there exists a function such that is this pp1 plus pn minus q1 plus qn so here this plus is just a formal notation saying we are taking these points and here the plus is real summation in this group structure so this is a Abel's theorem and this theorem measuring the dimension of this space is known as Riemann-Roch theorem it says that for the case of elliptic curve it says that dimension of this space is equal I will explain now degree of d so degree of degree of so I will explain here degree of d1 is sum of these numbers on I so sum of these coefficients this is the degree so this is for g equal 1 for g greater than 1 is little bit more delicate so for any g I can easily write the following the dimension of L of d is greater or equal degree of d plus 1 this I can easily say and I can say that for example equality is for sure so divisors d for which here is equality are called non-special and then for example if if degree of d is greater or equal d2g minus 1 then d is for sure non-special ok I am just mentioning this because it can have some relevance in higher dimension billiards but for us actually this is what is important so the dimension of this space dimension of this space in g equal 1 depends only on pure arithmetic how many how big this divisor is how many this is the dimension of the space it doesn't depend on geometry of these points can be can be scattered arbitrary there is no geometric component in this formula however in higher genera it can be important where what is the geometric position of these points because sometimes divisor of the same degree it can be special or non-special so here can be some some difference in this formulation that for arbitrary g dimension is greater or equal this expression here let me just mention this for reference this is a remaining equality but in a sense we are safe here because we know from here we know many gradients how to construct elliptic functions so maybe I can give again some exercise this is a suppose you have an exercise suppose you have you need to have at least devices of degree 2 so suppose you have 4 points on an elliptic curve and suppose that a1 plus a2 is equal I can maybe say b1 b2 b1 b2 is equal b1 plus b2 so construct a function such that divisor of f is a1 plus a2 minus b1 plus b2 so you need to play with vajstras function so vajstras function provides you LEGO cubes and you need somehow with these LEGO cubes to construct a function satisfying this ok now let us answer one more question and then we can go to direct and then we can go directly to Poncelli theorem so this is the question how to the so given cubic given elliptic curve this way or that way the goal is to describe points which satisfy the condition that so the points which satisfy the condition that now I need to think I don't want to erase now what I need later so let me think so definitely exercises I don't need so this no exercise no exercise no talk general anymore plan we don't need ok now we will see so the question now we are interested in is question so in that group structure we want to describe points such that nP zero so you sum several times point with itself and you end up at zero these points are some times called division points so can we here see these division points for example if n is equal to if n is equal to where are these division points for n equal to let me erase this so where are these points for n equal to do you see them the answer is they are there they are presented on the board this is omega one small omega one, small omega two and small omega one plus omega two they are already there so this is the solution of 2P equals zero 3P equals solutions of this we will call them flexes so the solutions are flex can we see these here those where 3P equals zero so for 2P equals zero we need to have here half half and then to sum for 3P what we need to do we need to take I think what one seventh or what one third two thirds one third, two thirds and then to produce these points this is how we can catch these points here but what we want to do is we want to somehow determine some analytic condition in terms of this equation here y squared equals polynomial of degree three in x so this is what we need to derive so suppose that nP equals zero what it means is that we can say nP is equal n zero n zero is in that presentation there is this point at infinity this is this point at infinity so we see that nP should be equal n times P at infinity and this equality means sum this in a group and this should be equal so if this is satisfied then by Abel's theorem that means there exist a metamorphic function having pole of order n at P infinity and having zero at P order n so by Abel's theorem that means there exist in infinity such that P is zero of order n this is translation of our problem using Abel's theorem so we see that this space is this one nP infinity so this is the space of functions having pole at zero of order not more than n this is space of functions this is the space of functions having so this is the space of functions metamorphic functions having pole only at zero of order not bigger than n what is the dimension of this space what is the dimension of this space so by Roch formula this is degree of this divisor so this is degree of nP infinity and this is n so take a basis take a basis f1 fn a basis in l infinity and then we can even derive a little bit more than what we need so suppose that instead of looking for one point P such that nP equal zero we can consider even more general to check if for given n different or some equal points is P1 plus Pn equal zero so take this points and so what we want to so if P1 plus plus Pn equal zero that means that there exist a function f in P infinity such that f of Pi is zero for i going from one to n but if f is in this space that means that there exist c1 cn constants such that f is equal c1f1 plus plus cnfn because if this is a element of the space it is expressed by the elements of the basis and we know so these are constants when we evaluate this at P1 to Pn we are getting zero so we are getting a system of equations we are getting a system of equations this is known basis and we evaluate in a known point so we are getting a system on unknown coefficient so what we are getting we are getting a system c1f1 of P1 plus plus cnfn of Pn of P1 is zero c1f1 of P2 plus cnfn of P2 is zero so on c1 of f1 of Pn cnfn of Pn is zero this is what we are getting so in order of such a system to have a non-trivial solution in c1 to cn we need to the condition is that this determinant calculated at P1 Pn of f1 P1 let me si f1 of P1 so c1 so what should go here what should I put here what what come on what is here what f2 of P1 very good let's see fn of P1 now is easy fn of P1 now it goes f1 is easy f1 of P2 f2 of P2 fn of P2 f1 Pn f2 Pn fn Pn should be equal to zero this is the condition so now these points P1 to Pn are going to coincide we are getting a limit case and for limit case we are getting Vronskian just to make it a little bit quicker is f1 f1 prime fn prime here is f1 n-1 fn n-1 calculated at this point P should be equal to zero so the condition that n times P is equal to zero P is division point if and only if this determinant is zero this is the condition now we can still use this picture we have had here because we can provide an explicit basis using this morphism we had here so how can we provide this basis so there are two now we need to we need to I'm just thinking if I can erase now this Abel's theorem suppose I can I need some space just for some small technical reasons we should now distinguish two cases when n is even and n is odd so suppose for first case first case if n is odd if n is odd I want to give you a specific basis or I want to ask you to give me a specific basis so what we know here is this map still here so you see p function goes to x so this is our x on the cubic side this is our y of a cubic side so what was the pole of p function 2 so x is of order 2 and y is of order 3 we don't have anything of order 1 we have x of order 2 and we have y of order 3 and we know that y square is equal polynomial of degree 3 x so if you know that x is of order 2 and y is of order 3 how can we provide the basis of this space which should be so we need to have 2m plus 1 function satisfying the condition that order at 0 is not bigger than this so first what any ideas x is of order 2 then we can go with x 2 this is order 2 so if we put here m we are still good because this is of order 2m and this is 2m plus 1 but we cannot go further with x now we can put y this is of order 3 we can put yx this is of order 5 and we can go on until here we need to be precise y times x of something what? this is where I need your help so what I can put here so y is 3 2m plus 1 minus 3 is 2m minus 2 so we can put here m minus 1 and now first of all we already have 2m plus 1 we already have 2m plus 1 we already have so we don't need to go further so this is the basis we could have for example go y square but y square is already a function of x so you cannot give us anything new ok so that means that we need to produce this sort of calculation we need to produce this sort of calculation for these particular functions and this is differentiation with respect to x so this is the differentiation with respect to x and the good thing with this differentiation with respect to x is that differentiating x on n with respect to x k times is easy so lemma d over dx j times of x of k and we are doing this at 0 even better is almost always 0 and it is we evaluated here now at 0 only at 0 it can be equal k so it is 0 if j is not equal k and if it is j equal k it is what? k factorial or j factorial if j equal k so it's good by the way and what is the now you can tell me for n equal 2m just for reference what is the basis for n equal 2m what is the basis for n equal 2m it's 1x x m y yx on what should be there m minus 1 is it m minus 2 m minus 2 ok so this is the basis in this case ok and now we need to calculate this there is no we need to do some calculation there is some tedious calculation thanks to this lemma thanks to this lemma I am now just referring to the first case this can be so here we have a here we have b 0 and here we have some c where a due to this lemma a is 1 1 2 factorial m factorial so this is m plus 1 times m plus 1 so it's a diagonal matrix it is a diagonal matrix which means that what is the consequence of the fact that this matrix is diagonal the consequence we are looking for the condition determinant equals 0 and this is a diagonal matrix with this positive coefficient so that means you see now we have to be very happy because that means we don't care about b we don't care about b we only care about c and of course c includes this now derivatives of y so if we suppose that so y square equal polynomial of degree 3 of x and now y is square root of polynomial of degree of 3 of x if we expand this in Taylor expansion plus and this Sergei say this is implicit exercise that c is of this form ok, there are some really coefficients which are not so important but let me write correct formula so and since we are talking about the condition so m plus 1 factorial is coefficient of the whole column m plus 2 factorial and so on so since we are looking for the condition when this determinant equals 0 we can forget about these coefficients and we can rewrite finally our answer in the form we can rewrite our answer in the form ok for n equal 2n plus 1 the answer is and so forget about these coefficients forget about these coefficients then we are getting we see that this formula this metric has a very specific form this is a very specific type if you just forget about these coefficients so this is so I'll write transpose of this matrix here so c prime so determinant is c2 hat c3 hat cn plus 1 hat then this one goes here c3 hat c4 hat then this one goes 1 higher m plus 2 until cm plus 1 hat then here one more cm plus 2 hat so the question is what is this one this is to check if you are still present what is this coefficient here so this is 2n the condition is this is equal to 0 so you see by the way this determinant has a very specific form what is this to be honest I am always confused because there are two types of matrices so one is Henkel matrix and another is teplet so one of these is Henkel teplet if you forget these coefficients then this one is one type and this one is another type this is Henkel that's correct so this one is Henkel and this one is teplet so how can we memorize this no way I think so this one is teplet I think that you should say the teplet is more important and then if you remember the teplet is more important then on the main diagonal is constant this is somehow I don't know other way to memorize this okay so what so this is so this one is teplet and this one is Henkel for this particular problem I think it's so this is in a sense a challenge is there a way somehow to explore in a more elaborate way why is this form appearing here and so on okay there was a question a few days ago about some polynomial entering into this game there was a question okay probably the answer was not satisfactory so personally so you see now we are looking for a specific c1 to cm that this function f what we get at some point has zero for the n so plug instead of f1 and fn the functions we have here these are some polynomials in x y times polynomial in x so when we do this here we will get polynomial in x plus y times polynomial in x so y is square root of this polynomial and what we want is in a sense we are looking for a solution which has a good approximation in certain point so this is the connection here is how these polynomials enter into a game so here we really get very specific polynomials okay now let me so this is for n equal to n plus 1 and now I can also just for the reference I should write this also for n equal to m so in particular we see that for n let me see so n equal example n equal 3 m is equal 1 what we are getting so m equal 1 we should put here m equal 1 we are getting c2 so all we have to get this is c2 equal 0 for n equal 4 c2 we should put here m equal 2 we get 3 so we are getting c3 equal 0 and for n equal 5 this is m equal 2 we are getting c2 c4 minus c3 square equal 0 so this is formulate an exercise is to you remember we had this curve yesterday and the point p 38 so can you somehow apply this machinery for this particular case so this is an exercise this is an exercise with this we somehow derived everything we needed in this section 2 and still we have some time to I think go to section 4 ok are there any questions so far ah this is no no this is going to be exercise next time so we still didn't come to connect this with we derived machinery but we still didn't so this is this is exercise yes not to derive fuchs but at least to compare ok so now we need to answering to Sergei's question now we need to see why do we need all this why did we use all this ok have you copied this formulas and everything can erase that is the point I plan to reach on the previous lesson so this is sort of this compatibility of our plans and our capability to realize the plans so now we are going to introduce one configuration this presentation now follows closely very nice paper of Griffiths and Harris of seven years ago or something I think it's 77 or 78 ah so so we are now looking this sort of configuration so given two conics B and C2 we are looking so let me at least say once again so we are thinking about now so we are thinking about two conics B and C2 and we are we want to code the situation where we have this tangent to C2 which intersect at B so tangents to C2 belong to dual of C2 so Griffiths and Harris suggest to consider this configuration configuration of lines and points such that line belongs to C2 star point belongs to B and there is one more condition that q belongs to L so this is a sort of configuration and as not exactly but similarly to what was mentioned today by Sergey here we have two natural here we have two natural involušen here in this set we have two natural involušen one involušen is so given L if this is L there is q1 and q2 there are two points and with one q1 there are two L L1 and L2 which pass to this q1 so there are two involušen associated with this set given line there are two points given point there are two tangents so the first involušen I denoted it is C2 star of L and q is L prime q meaning that L prime is another tangent C2 through and there is IB of L q this is L q prime meaning that q prime is another intersection and how can we look at this we can say this this configuration arises as alternative usage of these so what are we doing we are starting from a point and a tangent and switch the point so this is a new point with the first tangent then we keep the point but switch the tangent then we again keep the tangent and switch the point then we keep the point and switch the tangent and so on so this is how it goes so in this row of p or row of C2 star B is composition of these two involutions this is what it is now for example let me think so we can so what are the fixed points so let's think what are the fixed points of these involutions what are the fixed points of all of the first one when do we have a situation that in a given point we have only one tangent to that we have only one tangent to C2 so the fixed points of the first involution are those correspond to those q where we have only one tangent so where are those where we have only one tangent we don't see them in this picture this picture is misleading because it's too real too realistic we need to have something more complex actually it looks like this it always looks like this so now if this is B and this is C2 so we have this intersection points I just wanted to tell a joke but still we cannot ok so there are these four tangents and these four tangents are let's say L1 L1, L2, L3, L4 these are fixed points of the involution so actually we don't know what this P exactly is but we see that it can be seen as a double cover for example over B with four ramification points or it can be seen as a double cover over C2 with four ramification points depending which of these two involutions you choose on the other hand we know that conic so this is a Cp1 so we have double cover over Cp1 with four ramification points and using the principle I tried to convey to you at the beginning so this is like how it goes if it sounds like a duck if it moves like a duck and I don't know what else then it's a duck so since that is duck this is elliptic curve so using this principle we actually proved or scientifically speaking applying Riemann-Hurwitz formula we immediately see that Pc2 star B is an elliptic so let's look like this applying Riemann-Hurwitz formula we get that P of C 2 star B is elliptic curve and we know of course that elliptic curves have different appearances if they appear as cubics then involutions which have fixed points can be described by the following actually I mentioned this last time we get two points let's say T1 and T2 and we can define evolution from last time tau T1 as tau T1 of x is T1 circle x and we can define evolution similarly tau T2 of x is T2 x so this picture of having dynamics induced by two involutions in this screen can be seen in the following way we are applying two involutions all the time we are starting from some point x so tau T1 of x is this point here then we apply tau of T2 we are getting some point somewhere here applying again the first one we are going here then we from here applying this one and we are going somewhere come we are going somewhere here and so on so this is the way how we can see dynamics generated by two involutions so I will just finish with the following of course if we are looking at an elliptic curve as we did in this sense of a torus in this fundamental parallelogram that I will leave this as an exercise that so we are considering elliptic curve as a factor of a complex line by a lattice then there are two types of involution so if tau is an involution with a fixed point then tau of z is minus z plus b this is the form and if tau is an involution without fixed points so I will leave this as an exercise so if tau is an involution with fixed points then this is necessarily of the form in this presentation tau of z equals minus z plus b however there are involutions without fixed points and they have a little bit different presentation so here we have involutions here we have involutions with fixed points so in this presentation we can say that the first one Ib of z is minus z plus b and Ic2 star of z is minus z plus c2 then rho we are looking for of poncele which is composition of these two involutions is nothing but z plus c2 minus b if I'm not mistaken in the calculation so that means that these biller dynamics in this picture is going to be nothing but a shift for this element c2 minus b denoted as p and this shift doesn't depend on the choice of the point so that means if we want to have this poncele configuration to be periodic starting from some point close after n steps that means that that means that at that point we conclude that np is equal to 0 and then since np is equal to 0 if we apply this to any other point we conclude that rho n p is going to be identically equal not 0, it should be the identity identity so we are coming to this condition np equal to 0 on an elliptic curve and just 15 minutes ago we derived a machinery how to investigate if a given point is satisfying this condition or not to do that we need one more step this is to pass from given two matrices to a cubic curve to curve of the form y2 equal a polynomial of third degree in x and this curve is y2 equal polynomial t where this polynomial of third degree of t is determinant of tB plus c2 so this would close the loop once we show that this elliptic curve is isomorphic to that elliptic curve so in order to show that as I told you how we compare to elliptic curves we compare them as ramified coverings and we have these four points over which they are ramified if this cross ratio coincide then these two curves are isomorphic next time we will show even simpler thing we will show that actually we can realize this curve as branch over exactly the same B was