Welcome everyone, the students, the family, the faculty and the staff. My name is Richard Devlin and I'm the acting dean of the Schuler School of Law. It's been three years at the law school, two years in person, one year online. And particularly this last year has been a real challenge for all of you. But you have come through it.  And you will be the most emphasis classes, the class of 2021, the class of the pandemic. So I want to thank everyone for all your hard work. I want to congratulate you and encourage you to go forward. Decide the well-introduction of Unselfish Public Service and represent Dalhousie in the larger world, the larger community. Take care everyone, look after yourselves. Hello everyone, I am so excited to be finally standing here saying congratulations on finishing law school. I remember it just like it was yesterday, sitting in room 105 and having someone tell us to slow down and remember the next three years because they will fly by in an instant. And boy was that the truth. I am so excited to see what our class will accomplish in the future. Congratulations on behalf of the LSS. Class of 2021, you did it. And what amazing stories you'll have. On behalf of Student Services, congratulations on this incredible accomplishment. I wish you health and happiness on your path forward. Congratulations Class of 2021. Wherever life takes you in the next steps of your journey, please remember that you will always hold a special place in the hearts and minds of us at the law school as an inspiration for what is possible in the face of great challenge. Hey Class of 2021, congratulations you made it. It is a testament to your perseverance and strength to finish law school in a pandemic. Look if you can do this, you can pretty well do anything. You should be super proud of yourselves. I am really proud of you. I'm also really happy and excited for you. And I wish you nothing but the best in your coming career. Congratulations Class of 2021. On behalf of the CDO, we wish you all the best in your life and your career. Please come back and see us and share your stories with future students. All the best. I would first like to extend my congratulations to the Class of 2021 on all your success. It has been your perseverance and dedication that has brought you to this day.  You are the ones who will uphold justice in its many forms. You are really the change makers of the future. So please remember that there are many communities who have not seen this justice system serve them. So as you start your careers, do not forget your responsibility to make this justice system really this future more equitable for us all. Congratulations Class of 2021. Hey graduating class of 2021, you made it. It's been a particular challenge for you. And in fact, I think you guys have overcome challenges unlike almost any generation of Delaware grads. So very well done. Be proud of yourselves. We're certainly proud of you. Stay in touch with us. We're always interested to know what you're doing. Don't forget us when you're famous. Rely on one another. There's nothing like your classmates in the future to get you through tough times. And I'm sure you guys are one of the most resilient classes we've had. Congratulations and be excellent to one another. Hello everyone. As you know, I was Associate Dean for most of your years here in law school. So I'm familiar with most of you. I wish things were different and I could extend my congratulations to yours across your stage at Convocation. But I would like to extend my sincere congratulations to every one of you for a job really well done and please keep in touch. So here we are another Convocation and you are not here. We miss you so much, but we wish you all the best in the future and congratulations on surviving three years of law school. Congratulations class of 2021. You have thrived through law school during a pandemic. You have faced whatever life threw at you over the last few very tough years. And now it's time to actually take a moment and enjoy, celebrate and be proud of your accomplishments. It was such a privilege to be a witness to your journey inside and outside of classroom. Your journey of becoming these amazing, strong, compassionate and caring lawyers. The profession will be so much better with you in it. And of course, I feel compelled to conclude by sending a big shout out to all of you that I had the privilege and pleasure of teaching directly, of learning from directly, of working with in various capacities and especially to Section A. Forever in my heart. Thank you for putting up with all my techy murder jokes. Congratulations on your achieving this milestone in your careers. Even more so that you've done it in the face of this world pandemic crisis. I wish we had been together in the last year in the classroom. I wish I were with you today on your graduation day, but nonetheless, I join you in your celebration. And I wish you good luck and I sincerely hope we'll keep in touch. So have a great day. Enjoy this time with your families. Week one, questioning law school decision. That's life in law school. Footnote one, LRW has a period fixation. Note two, period. Got my two L course readings. Questioning. Supernote one. Pandemic in the second semester. IBID. Heard exams in my PJs. Note five. Year three in a pandemic. Note six, but also questioning. Supernote one. About to graduate. Note seven. Congrats from the library on three foot notable years. Hey class of 2021. I've taught at least two thirds of you. So I figured I'd pop on here and wish you a hearty congratulations. You've done it. Well done. It's been a joy to teach you and to know you. And I wish you all the best in your future endeavors. Congratulations. Hey guys, congratulations on finishing law school. I hope you guys have made a lot of great memories in your time with us. And just to my class, my ADR class, who are all graduating this year. It was really great to get to know you guys. And I'm looking forward to seeing how you guys change the world. On behalf of the Dalhousie Law Alumni Association and our 8000 plus law alumni from all around the world, I'd like to welcome you to our alumni family. Although we're not able to gather in person right now, we look forward to seeing you soon at one of our many events or celebrations either here in Halifax or a city near you. We deeply value our alumni family and will create many opportunities to keep you engaged with the law school after you leave. On behalf of all of us, we wish you every success and happiness. Congratulations. Hi everybody. It's Camille Cameron here. I really missed you this year. We've all missed you in various ways given the strange circumstances we've been through. But now you're here. You're at the end of your journey. Congratulations. You really deserve all that congratulations. Looking forward to seeing you as alumni. Can you believe it?  Boom, lawyer. Good afternoon fellow graduates, faculty and staff of the Schulich School of Law and family and friends. My name is Arthur Ferguson and I have the honor and privilege of delivering the valedictorian speech for the Class of 2021. I'd like to begin by acknowledging that we are in Magmaughey, the ancestral and unceded territory of the Magmaughey people. In making this acknowledgement, we must consider what it means in practice and in the context of today's events. This territory is under the Treaties of Peace and Friendship which was signed between the Magmaughey and Malassee people and the British Crown in 1725. These treaties did not deal with the surrender of lands and resources and in fact recognized Magmaughey and Malassee title and established a set of rules that was to govern the nation-to-nation relationship. As we all know too well, Canada did not uphold its end of the bargain. The result is a system that has and continues to be used as a tool of oppression and discrimination against Indigenous people. As future lawyers, it is imperative that we use our positions to actively seek to understand and ultimately dismantle the legacy of colonization that pervades Canadian law. Despite how far we have come, we have so much further to go. And with that, I would begin my remarks. To the faculty and staff of the Schulich School of Law, thank you for the opportunity to learn in a setting that was always open, welcome and honest. To acting Dean Devlin and Dean Cameron, thank you for your leadership over the past three years. To the family and friends of the graduating class, your encouragement and support has meant everything. Especially in a year where so many of us occupying childhood bedrooms, basements and kitchen tables. Most of you likely never expected to experience the final year of law school at such close proximity. I think I can speak for everyone when I say thank goodness it's over. And finally to my fellow graduates, congratulations. Today, for all of us, marks the culmination of three formative years together. It is the friendships that have been forged over this time that will sustain us in both our professional and personal lives for years to come. It is the people we accomplish this achievement with that we will remember long after today. In the time I have, I want to pose four questions to you that have helped me reflect on this moment. When former Senator Justice Murray Sinclair visited our class during the first week of law school, he explained that the purpose of education is to address four essential questions. Where do we come from? Why are we here? Where are we going? And who are we? I think the answer to those questions can help guide us as we leave the familiar routines of Shulik and confront the unknowns that our careers will inevitably offer. While truly answering these questions will likely take a lifetime, the past three years have certainly given me a greater understanding of what those answers could look like. The first question, where do we come from? When I walked through those doors three years ago, I was curious, curious about all of you, your backgrounds, interests, goals, and personalities. And I was curious about what law school really meant. My curiosity about all of you evolved into friendships. My curiosity about law school transformed into a passion. But it didn't happen overnight. When we began, I hardly knew what kind of person I wanted to be, let alone what kind of lawyer I wanted to be. I learned that while the career possibilities were myriad, the answers I was looking for were going to be found in the relationships I formed and the experiences I pursued. What I discovered was that you cannot effectively advocate for others if you do not understand where you come from and you don't have a desire to understand the experiences of others. Here I learned that the values you bring are as good as the skills you learn. The second question, why are we here? Over the course of this journey, I often found myself, I often found my curiosity leading me to uncomfortable places. But it was in these places where I learned the most. My first time in prison was one of these. I was there to give a presentation on the legal rights for prisoners, but I quickly realized I only understood how the system was supposed to work, not how it actually worked. I spent most of my time that afternoon listening to prisoners who understood the system in a way I never will. I have learned that simply because something is legal doesn't mean it is just. I have realized that part of what we have been taught is how much more must be done. Because for too many people, the law and the system it upholds has not protected them, simply because of the color of their skin and where they are from. Soon we will all be participating in this system. We have been given the skills required to challenge and change it. When you have the opportunity to make something legal a little more just, remember these skills. This leads me to my third question. Where are we going? Rosalie Abella, who will be retiring, Justice Rosalie Abella, who will be retiring from the Supreme Court of Canada this coming July once said, In a world that too often feels like it is spinning out of control, we need a legal profession that worries about what the world looks and feels like for those who are most vulnerable. I believe, and I hope all of you believe, that this profession is one of consequence and importance. It also comes with responsibility. We have a responsibility to use this education to improve the lives of those around us. Our purpose is to ensure that this system upholds and reflects our collective values. Our responsibility is to take action when it fails. Finally, we need to answer, who are we? Today, we are all members of the 2021 graduating class of the Schulich School of Law. This is not the ceremony any of us imagined, and it hasn't been the final year so many of us were looking forward to. It has been a year that has challenged all of us in ways big and small, easily described, and heard it put into words. But we are more than this year, and we have more than met these challenges. I hope today, every one of you is able to realize what you have accomplished and who you have become. I leave Schulich knowing the immense possibility that this class represents, and I am so proud to call all of you classmates, friends, and colleagues. Thank you for the opportunity to speak, and here's to good friends. I wish they were here.


Good afternoon. Welcome to Cisco's premiere sponsored session this afternoon session number four for us and this is day one of what has turned out to be a very exciting summit so far I want to introduce very quickly our speakers for our premiere session Lou Tucker our CTO and VP and VP He'll be joined with joined by Ken Owens also CTO and the clouds Cloud services group So with that I give you Lou and Ken and OpenStack changing the face of service delivery Great. Great. Thank you. Thank you very much I think what we want to do with this session today is really talk about what we're seeing in this kind of changing landscape and particularly how this affects The what we're calling the face of service delivery how services now are going to be delivered to users On cloud platforms, and I think this is a very exciting time I think you've seen this theme repeat itself in a lot of the other sessions that are going on here in the summit this week and in fact, I think it's I sort of start this this line of thinking by looking at the big disruptions that are taking place in our industry and what's making those disruptions possible right now, for example, if you look at Uber it's disrupting transportation Taxi systems Airbnb disrupting a hospitality and the hotel markets Netflix disrupting video delivery and Amazon disrupting traditional retail These are tech these are new disruptions that are happening because of the internet and because of cloud computing These are some of these very New companies got going very very quickly and are serving a worldwide market Because of the cloud and because of internet delivery that kind of disruption I think is important for us all to understand because in many ways the businesses that we may be a part of are going to be facing new competition from a lot of these disruptors and so we have to be able to disrupt ourselves and so we have to think about What is the purpose of cloud computing because originally from several years ago when Amazon really started getting going I actually was at a startup and using a lot of Amazon services And it was the greatest experience of my life in terms of being able to very quickly develop new applications and services And I could scale them up and down and all of a sudden we started talking about this dynamic API driven way to get resources And this was fundamentally changing the economics of computing because I didn't have to go out and rack and stack servers I could use somebody else's and through API's I could get access to them Run them for a couple hours and turn them off and therefore very much more accurately tracked sort of the demand curve and save an awful lot of money and so the first thing that people started thinking about when cloud computing was a fact that it was an Economic change that you could really be much more efficient But in fact, I think as time has progressed we've realized that becomes secondary becomes very secondary to the dynamic nature nature of cloud computing and that's when we've moved from this physical domain into the virtual domain So now we no longer think as much about how many servers we're running or how many racks and everything else Unless you have to be a service provider if you're an application or a service delivery Application you're thinking much more about virtual machines and about containers and this whole new world where these things become ephemeral You can spin them up. You can spin them down and very quickly deliver new services and Some people also talk about this and the age of kind of software-defined data centers Even traditional companies such as Bank of America is talking about embracing a software-defined data center What they mean by that is that they want to be able to look at their data center and use software to manage it instead of having System operators go out or network engineers and cable things and change everything and logging into machines and doing this They want to be able to do this through software that gives them faster time to market can dramatically lower their operating costs And they're looking then to have open systems behind this because they don't want to be locked into a single vendor for this and Services now they want to be able to span both Services they're running in their data centers and they're running perhaps somebody else's cloud and a kind of a service provider So in this model and wait one way to think about this is in fact Configuration which used to be out of a runbook you would you know have a system operator a login to some server and make a change Document the change go over to another system record that change So you could have compliance and everything else and now we've become to these kinds descriptive models expressed as code This means that you can repeat that operation again and again and again and know that hasn't changed and this is a much bet Much better way and this is the heart I think of all of that we're everything that we're seeing and changing service delivery because now we have a way to quantify Exactly what composes these services and how they're supposed to operate? So software-driven infrastructure now is really possible be based on policy So this is what IT organizations really want to do. They're trying to enforce a certain usage model a certain way of delivering a service Based upon policy rather than the configuration that they might have might have Might have applied so there's been a lot of talk about in particularly coming from Cisco where we're looking at things such as ACI Which is a way to express network policy Rather than what are what are how you want to be able to configure each one of these network systems so that you can talk About policies around this application. Is it internet facing or is it internally facing? Is it accessible from the internet should there be any traffic between this point this server and that server if there Should never be traffic between that that would be a violation of policy So you want to be able to express that and codify that as policy So there's a lot of work being done now an open source an open stack in other areas around this kind of group-based policy This affects very much the service delivery because now you can talk about very large-scale services and expressed as policies That you want to see happen The other thing I think that's happening is is quite interesting is that as a community We're getting together and deciding that it's better for us to work together in open source communities such as open Stack then competing for proprietary solutions our customers are getting too smart Our customers are demanding open systems our customers are demanding that we work together with each other And so in the past we've done that through consortia We've done that through you know getting together in standards bodies And now they're really asking for us to get together and do it through code So in something like open stack where Cisco has been involved since early 2011 We've gotten together and in fact the early neutron networking service that we got together We got together with 14 other companies many of which were our fierce competitors in the market And we decided we needed a network service in open stack to be alongside of the compute service and the storage service And so we work together and evolving neutron to what it is today to be able to be this kind of service that that is vendor neutral The customers can actually use get open stack from many different places And they have an API that they can count on and now the vendors have to compete on the implementation The vendors have to compete on the services they provide around that and the support they provide on that But the actual implement the actual API's we're not trapping our customers into that And so this kind of independence from vendors is something very important What that's also allowed is that now we have a large number of services that are targeting a much larger environment We know the the giants in cloud computing today via Google, Amazon, Microsoft have their own clouds We're looking for a counterbalance to that in this open community where there can be many many more there can be hundreds or Thousands of other cloud providers and and clouds that are being run inside of a company's data center. In fact, that was the original impetus for developing open stack was to develop the software to allow anybody to build their own cloud They can build it in their own data center They could build it and they could become offer their own cloud services So there's a lot of opportunities that being created because we've gotten together around this community-based software And in fact that we've seen and it's interesting if you go to the open stack org site we have a number of case studies and I really urge you to take a look at those because now there's a wide variety of industries and use cases that are being developed in OpenStack and you can find others that might be like your own business and why how they're Adapting open stack for their use so we're seeing things not just from sort of startups and Service providers, but also come big companies such as PayPal or Best Buy and even media companies such as Comcast or financial Services companies in Bloomberg. All of these are using OpenStack for different purposes So this has been actually very rewarding to see the number of different applications that are being developed on top of OpenStack Previously actually we just had a conversation about one of those new emerging verticals. I think which is around NFV Because we're seeing now service providers looking at a cloud platform Which has open API's and saying what we want to be able to do is start to virtualize our network services themselves And I think this is happening because there's another kind of perfect storm that that's developing here Several years ago. We started talking about SDN software-defined networking and and things such as open flow and now we're seeing that was happening at the same time as Cloud computing is happening and those things now are coming together and firming this kind of perfect storm where cloud computing and Software-defined networking network function virtualization is coming together at the same time and creating a whole swirl of activity Around this area as service providers are seeing here's a new way instead of providing You know different appliances that they would go to market with or buy from a vendor They can start defining these things again as software services get a speed to market and get much more Advantages out of the rest of their infrastructure So it was not surprising then in OpenStack itself when we defined a network service that we Defined it such that it would work with many many different pieces of hardware from different vendors And so that's why there's been a large number of plugins and drivers underneath neutron from most of the leading companies here Also outside of OpenStack. There have been other open source initiatives around open daylight This is where the community is getting together and again defining a software-based controller Whereby this can now be used with OpenStack because this is now a software way to define a lot of the network functionality that you want in the control plane directing directing the network to the future So this is this is a Picture that actually a number of us in the in on the foundation actually been working on there's a working group around Essentially OpenStack for the telco industry and this is where you can see a large number of different devices that have been developed Which have been hardened appliances to perform different network functions that are now being replaced by software And each one of these things are now becoming software that creates Requirements on the OpenStack platform for we want this to be carrier grade We want to be able to provide isolation of the resources. We want to be able to have non interference of those resources We need it to be much more real-time and all of these things now are creating Additional requirements on the OpenStack platform that again across the community. We're working together to bring those things about Interesting enough that we see the European standards bodies That's the then got into the picture here and started talking about publishing a spec and here they've defined the architecture For how they want to see NFV working in terms of that we have an orchestration manager We've got a virtual resource manager and other things in OpenStack plays a fundamental piece of this puzzle In fact another group has been started called OpenNFV And this is a group that is developing reference implementations based on OpenStack with open daylight of this kind of an architecture So in this architecture we can see we've totally replaced those hardware network functions with with virtualized services Those virtualized services now have to be carrier grade they have to be resilient They have to be able to scale and but this means that they can be much more rapidly deployed than having a truck roll bringing in new Equipment or trying to expand the number number of network services you can Previously have been done by hardware now you can rapidly bring those up as virtual machines And so this whole new model has coming about we're software as Mark Andreessen sort of said is eating hardware replacing it in a much more agile manner at the National Association of Broadcasters show In Las Vegas several months ago. We're beginning to see this happen now in another industry So it's not just in traditional web applications We saw in cloud computing on the kind of web services We had seen not just what we've seen then in the telco market with NFV now We're seeing it happen in the media creation and distribution And I think this is the one that's next in line and over the next couple of years We'll start to see a lot of activity in this space For example, if you look at the media and what's going on in this world today, there's a huge number of competitive pressures We're seeing over-the-top delivery of video services. We're seeing a lot of consolidation in the industry. We're seeing multi-screen Implementation taking place. We're getting our video and our and our things on many different devices So I want to be able to watching something on TV from a cable box I want to be able to get it on my phone and be able to get it wherever I am in the world All of these things are creating tremendous pressures and tremendous need for these industries to adapt and therefore They are turning to cloud computing for this purpose as well In fact traditionally what we've been seeing has been moving from video-based delivery services and these big switches With a serial digital interface Turning to now IP networks and so this is a big change because traditionally IP was not very good at this You needed to be able to have the delivery of video without dropping any frames You need to be able to have quality of service You need to be able to have dedicated bandwidth on all of these things in particularly around broadcast and ingest We're seeing this change happen now as we're seeing this move to an IP network and Moving it to the cloud So whereas previously we had these dedicated hardware switches now We're using general purpose switches and general purpose servers But we're now controlling them through software through controllers such as open daylight And this is I think again this kind of a big shift that's happening right now And yet we want to be able and this gives you simultaneously the same kind of a Guaranteed that you had once before but now you get a much more flexible environment So that you can now deliver it in many different ways according to what the consumer wants to achieve Classic example is home DVRs sort of like this because I think we can all relate to this We probably all have had DVRs in our home with a with a clock flashing on and off because we never could set it or whatever And we had to record our video and we had to hope that we received a disc fill up And this would seems fine because it replaced our VCRs Which is what we had in the past but now we're seeing these things moving to the cloud And there's a lot of very good reasons for why you want it up in the cloud one you're gonna have an infinitely large disc You don't have to worry about your disc filling up From the service provider point of view they don't have to keep all the separate videos for every single person there They can actually find ways to be much more economical and just mark the fact that you bought that and not even be able to Essentially store it for you individually Given certain digital rights Concerns around that and they also then can be able to deliver it to you whether you're in your home or not So this cloud-based delivery I think it's something that we're seeing come on very strong because it provides customers advantage and Advantages for the service providers as well. So this obviously now becomes a software model instead of the hardware model of the home DVR And this is the kind of thing that we're seeing so we want to be able to to have these kinds of new services being able to Delivered in this kind of a framework So when we do this we start to see again some of the traditional functions That we would have in a video kind of pipeline turning into software And here's where we're talking about it. It's kind of service containers that can be chained together I just press something There we go And what you have to do now so you're gonna start hearing a lot of people talking about orchestration Orchestration is how do you bring up these services? How do you scale individual services? What if you have all of a sudden the peak demand because of some event and you need to do one of these services expand that? To have a much more capacity For the increase in demand and you need to be able to have workflow managers that can that can move this through the system here so in Concert with that then we're seeing something called docker Kubernetes Mesa coro s this is the containerization of the world that we're seeing because developers now are finding Emulating a virtual computer by having a virtual machine is a very heavyweight thing to to carry around and it doesn't Provide much advantage to the individual developer It's nice because you can actually run a Windows machine on top of a Mac or anything else because we virtualize the computer But when we actually have a simpler environment We want to be able to talk about how do we package an application together and in the packaging of that application be able to run that as a Container anywhere you want to it's something that's tremendously advantage For the new application developer So I think we're seeing these new models now of developers of how we develop applications come about To enable these new kinds of services and increase again the speed of service delivery And for that reason I was to then turn it over at this time to Ken Owens to talk about what Cisco is doing also in terms of now containers There's many people and I think that Jonathan even in keynote Mentioned that some people think of cloud computing and containers as being a kind of war with each other or that you have to choose between virtual machines and containers My view is it's a lot more nuanced than that in fact that we're going to see all these things together What we're talking about particularly with open stack is is the orchestration in the management of virtualized resources and whether that be a container running on a virtual machine or a container Running on bare metal because an open stack we can also orchestrate bare metal It all becomes a single kind of plane that we can talk about as this new kind of cloud platform And so the age of like sort debating whether it's a container or virtual machine really becomes one of a developer choice Whatever they would like to be able to developing that's what we need to be able to support through that I'd like to turn it over to Ken. Thanks Lou So when you kind of think about what Lou was just describing why do you need a new model, right? I think it's not really so much a new model It's just kind of making open stack the platform that open stack was designed to be and so I kind of look at this from two Different areas right there's disruption happening in the enterprise where you're looking at how do you move from kind of agile? How do you move to agile development from your traditional waterfall type of development? How do you start developing these cloud components as services as Lou was referring to? It's kind of a natural evolution. I think from a developer standpoint to look at Here's how I like to develop my services How do I deploy these as services and so it's not much of a change from a developer standpoint from an infrastructure standpoint you want to offer services and so Open stack is a nice platform that allows you to create services and expose those services through API's and so it's a good fit between the two There's also this model of you know, like Lou was referring to how do you determine if you want virtual machines or bare metal or some other In between solution right and I think Most developers don't care right? They just want to get their application out as fast as they can they want to get it some feedback and they want To be able to modify and update their application based on the feedback they get from the market and so Kind of doing this hybrid dev ops where you can develop your application the way you develop it today Deploy it into an open-stat cloud see how the performance of that application is tweak the application performance to improve the performance Modify the underlying infrastructure if you need to to provide better performance It's kind of what I call this hybrid dev ops model the other thing that was seen is Digital disruption is definitely impacting a lot of enterprises today And so how do you kind of look at your your infrastructure as a platform instead of as a set of? Compute and network and storage and security constraints right? How do you kind of look at the commodization of infrastructure? Which is probably not the best thing for your business in the end if you're on low quality low-grade hardware How do you really differentiate your service in the marketplace if you you're building on top of a bunch of you know basic commodity hardware The other thing is around security and privacy so Containerization in and of itself is not secure But when you look at adding security and network capabilities to containers You can have a much more secure and a much lower fault domain when you deploy things as services So just you know a couple quick size into the what developers in the enterprise are telling us the sort of two camps You have your cloud native guys that are Either developing today for the cloud or for mobile or you or the traditional app developers moving to cloud native And you have your data scientists that are looking at how do they take the analytics that are going to help them make real-time business decisions Based on the application So the analytics guys are concerned more about structured data versus unstructured data and how you collect that Kind more like a Kubernetes model The cloud native guys are more interested in continuous delivery and mobile first development more the mazo scheduling model And then enterprise IT is sort of stuck in the middle of the two and and they don't really understand either one that well yet They will but right now. They're kind of like scheduling. What's that? You know, I don't need to schedule I'm more worried about how I provide security and governance and how do I connect you into my support systems? And there's a time figure out how to support DevOps or work in a DevOps model and so When you think about these three different, you know user groups I can't think of them in terms of personas, right? And so you have your developers at sort of the number one person I think about the most and they're really interested in how do I get this application from? Concept to production as quick as I can and not underline understanding under anything about the underlying infrastructure is their mindset You have the IT admins who want to kind of create environments that can be consumed as a service And they're trying to create this environment that the dev developers are going to leverage to take their application from concept to production And then there's a few other, you know personas you can think about like the CIO and the business owner But if you focus on the developer and the admin to use cases you hit most of the use cases that CIO and the business owner care about and then what do they want to do? They want to basically build their application deploy that application and then run it and so when you think about the build piece I think most of The philosophy in the in the developer environment today is don't make me or ask me to change how I like to develop So I have a framework I develop and I have platforms I like to use just let me continue to develop the way I like to develop From a deploy standpoint right now if you think about the biggest issue I think facing, you know open stack and other clouds today is that you have to write to those specific APIs and As Lou mentioned in his earlier conversation The latest version of code is the most stable and so if you are writing the open stack You have to write to the latest version of open stack every six months. You're changing your code or Potentially changing your code. No, I'd always change it But more than likely you're gonna have to change your code at least validate your code again And so when you think about how do you kind of develop your code once and Then just kind of deploy it in a CICD model afterwards. That's sort of the layer That we can write above open stack earlier Lou caught at the sandwich, right? So we're kind of looking at above the top part of the sandwich. How do we kind of create a platform? Interface that allows you to kind of write your code to interface with not just open stack but other cloud technologies And that's something we call mantle Mantel IO is the website and then from a run standpoint Most developers want to understand how their applications and those services they depend on are performing and behaving and so Creating a run environment that lets you from a single interface look at how your applications performing across multiple clouds across multiple services is a key part of the framework and then tying in some of the application intent policy stuff from APIC and from other Controllers in the environment and Tosca being able to give you a single interface that lets you look at your application From one console versus multiple consoles is a key part of the interface And that's what we call project shipped and so project shipped is a curated experience It's available at shipped IO Cisco shipped IO It's tied in with these components you see today. So we're trying to hit the Kind of the beginning part of the development cycle letting developers develop the way they like to develop with GitHub integration We have build packs for the most popular code In use today also Eclipse We're using vagrant to kind of run in a local IDE on your laptop, which is what most developers do anyway We're adding OpenShift and Cloud Foundry support in the next release of project shipped We're using drone for CICD today as well as Terraform to kind of give you an abstraction layer That lets you run across multiple clouds. And so today we support Amazon and Google Digital Ocean Cisco's cloud Cisco's MetaCloud and Along with Rackspace and VMware cloud so completely abstracting the underlying Infrastructure so you don't have to worry about writing to specific infrastructure APIs Although we don't completely hide them and so if you wanted to write, you know to an APIC controller We'd let you write directly to an APIC controller as well So we will kind of give you the a choice of taking the abstraction layer and not having to understand the underlying infrastructure Or if you want to really write to the underlying infrastructure We'll give you access to the underlying infrastructure APIs as well From the orchestration and scheduling standpoint. We're supporting both Mezos Marathon zookeeper and Kubernetes. So we're giving you a choice of all of those. We use console to give you name names namespace and tie it into DNS so each container gets an IP address You can route between containers in your services So it's a making network and security like a key aspect of every container And then from the service assurance side we're adding a lot of data analytics pieces that are being asked for by the data scientists community So things like elk the elk sack isn't is included log stash and zoom data for kind of Analyzing and understanding the different log inputs of your application and then like elastic so should give you You know this sociability of that of that data so We think about what mantle really is it's trying to give you that glue that every project has to kind of use today to Kind of build their application It's an end-to-end batteries included solution It's designed from from day one to kind of give you all the different components of the open source community in an integrated manner It's an extensive also as new components come out. We'll be able to add those components in very easily. It's an open source initiative within Cisco Well, it's on GitHub at Microservices infrastructure and Matto IO if you want to get to it directly from the website And what we're really doing is just basically taking what every Enterprise developers doing today on their own per team per project And so instead of having to kind of build a platform from scratch every time We're kind of giving you a basic infrastructure or microservices infrastructure as a service if you will That can run inside the data center and it can be deployed across any external cloud The architecture is pretty simple. It's Basically, we have control nodes and resource nodes the control nodes give you the basic, you know Scheduling services around How you set DNS up? How do you manage your resources? How do you manage your namespace the resource nodes that you do the on? the local destination scheduling We run it in a single data center environment and we can run across multiple data centers Which is sort of the the key thing that we focused on with Mantle and Today, I had to we support multiple clouds The the key thing I want to get to is the networking piece It was mentioned earlier and so to us networking is not just not just because was Cisco is important But networking is always important for any application, right? And so We're gonna make it a first-class citizen of containers today it runs each container has its own IP address Where we seamlessly integrate with different? networking protocols and to support BGP across the different container spaces We have basically created a unified platform for for networking across Kubernetes and workloads consuming meso services and from a policy standpoint we Releasing in the next few weeks with working with Congress. We're doing here in open stack We have a blueprint available to kind of let you do application policy intent, which? basically takes the whole definition of your application from a Set of like specific request to more of a I would like to have high CPU I need better networking or I need better such security Making it very simple to kind of define what you are looking for what you care about the most and then we can provide the best Solution underneath the covers to support that So actually I we wanted to leave some time for question So if you have any questions, please make your way up to the microphone But but actually can I had I had one question which is that if we look back at for example the shift? Diagram you've got very very large number of open source Systems that are there that must be a huge learning Challenge for many of the developers and they have you in in you're working with different developers developing these microservices How do they keep up with all of that? There's a lot of different things that they have to learn Yeah, they they they don't do a great job of keeping up with all the different open source projects And you think about open stack is a lot of open source projects within itself, right? So when you when you add the infrastructure pieces you try to figure out all the integrations to the back office and to the support systems that an IT enterprise has to support They they are really struggling to do this every single release And so trying to provide a single platform that is above the stack Right and integrating into different stacks that are below that API interface They they are very excited about being able to just leverage something that's open We're not locking them into any one technology We're not forcing them down a path first decision that they're making yeah They're gonna be working above open stack at the application layer Where they actually are creating the new microservices and everything right and therefore there's a whole lot of tooling that they have to learn There which looks very familiar to the tooling that open stack developers See it's all right hub and Garrett and everything else like that I think but it's done at that application layer and they don't have to worry about what what's actually running the platform itself Yeah, I think that it whether it's whether it's open stack on bare metal open stack on you know a KVM or it's You know AMIs it to develop but they really don't want to think about that They just want to write the code publish it Get this feedback from the community as fast as they can on how well their application meets the business needs Right and then make changes from that because again, they're they're in the business to make money for their company I'm in the business to make open stack better right right right right right in fact We should make just mention of two other open stack projects related to containers, which is Magnum and Cola And how does how does what we do with mantle work with with Magnum? For example, which is the the way that we are deploying containers So we've been working really closely with Magnum and we have two contributors on the Magnum project It's our work is tied in very closely, but we're again. What kind of? That's one implementation that we're trying to support but there's others that we also support okay It just just in terms of the audience. There's a Magnum is a project And a part of a big tent For open stack whereby we can you can bring up Kubernetes you can bring up me So and it's designed to be able to deliver it now in this latest release or whatever through even horizon You can start to bring up now this cluster that you need two of these containers And so immediately the the developer is only interested in working at the container level can do that And that can be run either on virtual machines or on bare metal nodes through ironic as well So we're seeing now again that usage of containers for microservices sitting on top of open stack Which can then support both virtual machines and containers because in a particularly I think then the the individual customer gets to make the choice there if they are actually Want isolation between all of the tenants? They perhaps need to run containers on top of virtual machines where we have the isolation the multi-tenancy provided by By the by the hypervisors Whereby if they are essentially running as a single tenant with lots of projects lots of applications lots of services They can start to think about using bare metal using ironic to run Magnum as well exactly nice Any other questions from the audience Gary? How are we doing on time? Okay, well, and I will be up here if you want to ask any further questions, but if not, thank you very much Thank you and can and We invite you to come down to the Cisco booth in the marketplace We're in space P3 come on down continue the conversation asked questions will be there As the comedians would say we'll be there all week. So come on down. We'll see you there. Thank you very much everybody. Thank you


I'm Scott McLeod, Director of Innovation for the ADA. Prairie Lakes serves about 45 school districts and about a dozen private schools in a geographic area about the size of New Jersey. We're a regional educational service agency that sits between the State Department of Education and our individual districts and schools. So today we're going to talk about what makes a quality STEM activity. We're going to highlight the importance of the design engineering cycle and go through a bunch of activities like robots and rockets and skimmers and whatever else we can come up with just to kind of show you the stuff in action and we'll sort of explain some things as we go along. Yep and we're going to put emphasis on the fact that this we're not the traveling STEM carnival right. We want to wrap this into learning, to deep understanding, to critical thinking and not just be a one-and-done experience for kids to get excited about but it has very little educational value to it. Right I think the challenge is that STEM stuff can be really cool, it can be fun, it's exciting, kids are engaged and enthusiastic to go home to other parents but like Mike said we don't want to be carnies right. We don't want to be like the carnival comes to town and it's fun and cool but then you go away and like okay back to regular business but STEM should really be a way of thinking, it should be a way of doing, a way of being, it should be connected to deeper learning material over time it should just be you know a one-time event. Right it's a great chance for kids to inquire about the world around them and explore it through hands-on activities. So thanks for joining us today or whenever you're watching it. Tonight, tomorrow, and we'll see you at the end. Thanks. Thanks. Here we'll see Colin engaging in the thrust wars activity and the object here is to get the balloon to cross the distance on the fishing line so students have to come up with the device that will hold the balloon on the line and possibly carry passengers. I don't do hope that the weight of this will balance it though and I think the most fun about making this we're not trying to get it to go but just making the initial design and all the useless features like the locking door. So by now Colin has realized that his device is probably too heavy to actually make it to the other end of the fishing line. It's it's got a lot of cardboard a lot of tape and you know but he's realized that as part of the design process even before he even tested and that sort of critical thinking about outcomes is what we want in kids. Okay Gabby so you have a new design here why don't you tell us a little bit about it and and what you changed? First of all I put things on to kind of push the air kind of through that. Second of all I put this in a different place before it was up here now I put it down here. Okay let's take a look. I didn't let it go at the right time other than it would have been. Wait let me try again I'll just pump it up. Lucas is working on the skimmer task and in this activity students are asked to create what looks like a boat to go from the start line to the finish line. It could be based on time it could also be based on how many passengers it can carry. In this case it's carrying paper clips. Okay Lucas you made some changes to your skimmer tell us about them. Yeah so I added just these because I thought they would help keep it from tipping over since it was falling towards the front and the sides so I put in like little rods to keep it stable there and then I added this to keep passengers from falling off. So Lucas's design has failed and he's now working on redesign like Colin had before so and that's what we want kids to do. We want kids testing out things collecting data getting feedback of some kind and then redesigning. If you really wanted to get fancy we could pull in an app like this one. So here's a skimmer from a different time some hit play here. This is a three sail skimmer and it's on pressure right now right I'll go back to the smoke. So you can see the three sails right there. There we go so you can see that first sail is the one that really harnesses most of the wind energy and the next two a little bit. If I go to pressure right you see there's that red that's piled up against that back one and then there's almost like a low pressure of vacuum here. Right. Afterwards. Right and okay got it and you can see there's a little push at the back and what looks also like a little bit at the front. Yep. For whatever that is. All right cool so again you can see the vectors of the flow and the energy push and so on. Very cool. Yep so a very cool app I think it's $2.99 sometimes it's free so. Thank you Mike. You're welcome. Here we have kids making water rockets out of soda bottles. The goal is to get these things to launch and go high. Lots of opportunities here that you'll see as they're designing and creating for developmental open-endedness depending on what their readiness is and what their interest is where they can test out different designs and make them look the way they want. So students are engaging in these activities it's important for them to record data so in the case of the water rockets we had altimeters that would give us velocity, apogee or the highest the rocket went and students could then take this data and make decisions using it as far as when they went into the redesign stage and then they would build and retest and take more data and just continue through this process. That data collection doesn't have to be fancy we just mocked up a quick data table with chalk on the sidewalk. Yep it doesn't have to be a google form on the iPad you can just write down a piece of paper the important thing is you're recording data so you can make decisions based on it. But if you want to pull in the technology they could obviously plot it and graph it using a spreadsheet program. Absolutely. Robots may be one of the first things you think of when you think of STEM and they are a valuable tool because it gives students the opportunity to not only engineer a solution but also have to program for a solution. Yeah so this time we're using the gyro sensor which is actually going to it measures the angle that the robot is turning. Okay got it. All right let's try it. Notice that one of the things that's important here is the ability for Mike and I to interact and try this as adults right as educators and so you know you can't just hand these robots to teachers and expect them to do amazing things with them they need the same kind of experiences and challenges that we've now set for ourselves we're saying all right let's make a track let's see if we can navigate the track let's understand how the different sensors work so if you're doing any kind of robotics work with elementary teachers they're going to need this hands-on play time right to figure out how this stuff works. Go ahead Mike can you set it going? Okay so now we're on round two where we're asking the kids to do something a little more complex so Mike what what what do we got going on now? Okay so now they have to program the robot to go the distance so there are no sensors involved. So they need to program the robot to go the distance and there's several ways to do that and we're using rotations so they had to figure out how far is the distance the robot's going to travel and then they also have to figure out how far around is it for one rotation of course that can be different depending on what wheels you're using okay so for these wheels they'd have to measure the circumference around okay and then divide and that's how many rotations they have to program into the robot. Okay so now would we have them just measure around the wheel or could we do something like calculated from the diameter or right so again depending on the age of the kids and what content that they're going to interact with in math you know you could have them use the formula you could have them just measure around. Robots are a ton of fun I think that's clear but we want to make sure we're pulling in meaningful learning too and so you know there's an opportunity here to pull in math in an authentic way and we need to make sure we're doing that otherwise this is sort of like a sideshow just as we talked about we don't want it to be carnival we want to speak a meaningful learning experience. The robots can take the role of rovers during the STEM camp students were asked to design a rover that could drive around the Mars surface and pick up dirt and take some measurements as they did so they were not allowed to see Mars they had to interact with it using a GoPro camera and a remote control for their rover. Notice how important collaboration is to all of this these students have to work together they have to generate ideas together they have to bounce ideas off each other there's no way that this is going to be successful if they're not working collaboratively to make this happen they're testing ideas they're brainstorming they're identifying potential solutions they're working together to make them happen there's no reason that students should ever have to work in isolation in a rich STEM activity we want them working together to solve problems just like scientists and engineers have to do in the real world right. Hey thanks for watching today we know you're busy and we appreciate you spent a little time with us we're just going to cover a couple things here at the end just sort of reinforce some things we talked about throughout the video. Yep and one thing we want to hit is that STEM isn't necessarily always going to be strong in all four areas science technology engineering mathematics but there are certainly a few things we want to look for in every STEM activity lesson unit whatever you may be doing. So for example it better be hands-on right so if you're running kids through textbook chapters or worksheets and you're trying to call it STEM you're sort of missing the point right so it should really be rich in terms of hand hands-on learning hopefully maybe it's got some inquiry in that you know that's what kids are looking for that's where the enthusiasm and engagement comes from right. We also want to make sure that we're going through the engineering cycle and design thinking with multiple iterations it's not enough just to go through once and have students create a skimmer or a rocket and then launch it and be done with it they need to design create and then test collect some data and be able to go back and decide how they would redesign and rebuild and then retest it again the more we can go through that cycle the more rich that learning activity is going to be right. Sort of drawing off the carnival point at the beginning of the video is that let's make sure that our activities you know they're fun they're cool but let's make sure they're connected to real learning hopefully some deep important concepts not just minutiae and that kids really get a chance to immerse themselves in math and science and engineering in ways that maybe mirror at least as much as possible what scientists and engineers and mathematicians do in the real world. Yep we also want students to be working in groups so they can collaborate and then we're thinking about how they can communicate out their work. We don't want this to be an individual activity because in the real world engineers work together to accomplish tasks and to build and to Yeah, we ran out of gas and add a lot of motivation to whatever we're doing if we make it a problem or challenge base. So if we throw out the students that they need to create something there's a problem how can we fix that problem how can we design something that will achieve a solution to a challenge that will push them to discover to critically think to collaborate with others in order to accomplish that task. So give them some kind of goal orientation don't let it just be something that you do in isolation science decontextualize give them something to strive for. And whenever possible we want that to reflect the real world what engineers are doing the real world too so keep that in mind too. Let's see what else is on our list. You can use tech appropriately and creatively throughout the process so you can use it for capturing and recording data plotting it out charting it out you can use it for sharing out to the public so it's a great way to capture process some pictures video electronic writing to you know accompany whatever you're doing their hands-on in the room as we capture their prototype in action so they can reflect back on it. Now sometimes when we have students you know send their skimmer down the aisle right or launch their rocket and then we go back and ask questions about they can't really remember everything that happened so if we can record that and then we can show them back sometimes some learning will come out of that and then when they go back to the redesign piece it's a lot more rich. Right photos are a great way to have them explain their designs or explain something that happened during the process where they can label it or voice narrated or something like that. Also we want to make sure that these are developmentally open-ended with lots of choice agency for students so for instance you probably notice with our skimmer activity we have lots of materials out we don't want to limit their choices as to what they can build their skimmer out of or build rocket out of or whatever they may be doing. We want to make sure they have plenty of choice plenty of options to try out. So Mike you want to talk let's end with process sounds good. So as they're creating things as they're designing things it's really not about the product they're making it's more about the process they're going through and we've talked sort of this whole session this whole video about the engineering process and going through those steps and that's where the richness of stem lies that's where the power of it lies is going through that collaborating that's where we're going to uncover those conceptual understandings that we want and so we really want to focus on the process and you know you saw the skimmers we made were not very good right but the learning that happened through creating those that's that's where the power is. So be sure you capture that process right okay thanks for joining us our contact info follows be sure to get in touch if you have any kind of questions or you want to share your own stories or you know anything else that you want we'll be glad to chat thanks see you. Dream big!


Psych2goers, welcome back to our channel. If you're new here, welcome!  Do you work very hard on how your skin looks or the type of haircut you get or how genuine and contagious your smile is? Do you tend to overlook personality improvements because of the non-tangible results that it might give? Today, we want to remind you that your behavioral improvements have as much or even more impact than physical improvements. To help you feel more confident, here are six signs that you have an attractive personality. 1. You have great conversational skills. Are you approachable, well-spoken, open-minded, authentic and honest? These are all signs that you are a great communicator and it's a good personality trait to have. Even if you're more of an introvert than an extrovert, having great conversational skills is still something that you are capable of having. As an introvert, you can still listen attentively and ask engaging questions and maybe act more so quietly approachable in your own way. Authenticity and honesty are also important in being a good conversationalist. This helps people to trust you and want to engage with you for how steadfast and true you are and how well you listen to them. 2. You have a positive attitude. Would you describe yourself as a positive person? If you are someone who has a generally positive attitude, then your personality is much more attractive than someone who would rather just complain and pout all the time. It's generally well known that optimistic people are more physically and mentally healthy and have more professional success than those with negative outlooks. People will generally feel more attracted to you if you show them more positive emotions such as love, kindness and happiness. 3. You're confident. Have you ever felt attracted to someone's sense of confidence and self-assuredness? Even more than their looks? Confidence is considered to be one of the most attractive personality traits. Your confidence can be used as an asset that shows how passionate and invested you are in the things that are important to you. Confidence is attractive because it shows that you believe in yourself and that you have a high level of self-esteem. 4.  When you have great listening skills, this will make your personality more appealing and attractive. When you're actively listening to what others have to say, this shows them that you care about them and what they say is important to you. For example, if someone tells you a really important story and you give them a general and non-committal response, then you're giving them the impression that you don't really want to listen to them and only want to talk about yourself. 5. You have a good sense of humor. Are you witty and humorous? Do you enjoy making others laugh? If you can make others laugh freely, take that as a compliment. Making people laugh can be hard because everyone has different senses of humor.  When you have a quick wit, you're using both your cognitive and emotional processes to come up with your jokes. So, you're also generally more intelligent. 6. You are open to trying new experiences. Are you willing to try out new experiences? Do you take more risks? According to Adrienne Furnham from Psychology Today, having an inclination to explore opportunities is one of the traits that people look for in a partner.  So, if you're willing to engage in new experiences, then you have two of the most attractive personality traits, flexibility and openness.  Did you relate to any of these points? Tell us in the comments below. Please like and share this video if it helped you and you think it could help someone else too. The studies and references used are listed in the description below. Don't forget to hit the subscribe button for more Psych2Go videos and thank you for watching. We'll see you next time.


This lecture is part of an online algebraic geometry course on schemes and we'll be about the relation between Cartier devisors and Vey devisors So you remember last lecture we showed there was a homomorphism from Cartier devisors to Vey devisors And we want to ask, you know, when is this an isomorphism or when is it injective or when is it surjective? Let's start with an example when this is not injective So if there are Cartier devisors can't always be distinguished by their zeros and for this We're going to take our ring to be the spectrum of k t squared t cubed So this ring is all power series of the form a naught plus a 2 t squared plus a 3 t cubed and so on Where you notice there's there's no term a 1 t to the 1 And This is a local ring and its spectrum has two points and it has one closed point Which which corresponds to the Ideal generated by t squared t cubed and it has an Open points that I'm going to draw as a sort of cusp because this is actually a Sort of germ of a cusp like if we put y equals t squared and Let's put y equals and we want y squared x cubed so y equals t cubed and x equals t squared you find that y squared is t cubed Which is the cusp here And we're sort of looking at this tiny piece of it by by working with formal power series Anyway, this is a local ring so the Cartier devisors are easy to work out so the Cartier devisors Will just be the units of the function field k which is the ring of Laurent power series Should be t to the minus one And we quotient out by the units of R Okay, so why aren't we taking an open cover of this? The reason is if we take any open cover of this one of the sets of the open cover must be the whole space So we may as well Just take the single open set consisting of the whole space and then Cartier devisors look like that Well the units of K Can be written as well that all none zero elements of k so you can write them in the form t to the n times Sorry, let's put a t to the n times 1 plus a 1 t plus a 2 t squared and so on units of R on the other hand Just look like things of the form a times 1 plus a 2 t squared and so on where here a is units of K So you see there's no a1 term there And So we see the quotient of the units of k by the units of R star is just isomorphic to Z times the field k where z comes from this n here and K comes from this coefficient there And on the other hand the very devisors are just isomorphic to Z because there's only one co-dimension one closed set so the map from the Cartier devisors to vey devisors has a kernel consisting of Corresponding to K and you can easily check that it's actually isomorphic to the additive group of K. I mean what else could it be? We notice that the element 1 plus a t has no zeros Or poles in some sense But is not a unit of R So this is different from the case we had when we The example we had last week when we were looking at the last lecture when we look at the affine line With its function field K X you notice that any localization of this If an element in the localization has no zeros or poles and it must actually be a unit and this this difference is why the Picard group to the vague group was injective in this case, but not in this case So let's have a look at a general condition when the map is Injective so suppose Let's take X to be notarian and integral Notarian is to avoid lots of tires and technical complications and integral is basically put in because I'm feeling kind of lazy So this means that X as a function field K which makes devices easy to think about then if X is normal then Cartier devisors are a subset of vey devisors. Well They're not a subset of a devisors what I mean by this is the natural map from Cartier devisors to vey devisors is Injective, but I'm going to be sloppy and say that means Cartier devisors are a subset of vey devisors So what does normal mean well normal means all local rings are Integrally closed in in in their field of quotients and Since we are assuming the scheme was integral all the local rings are domain So we don't have to have a headache one wondering about what this means Now what we do is We recall the basic theorem from commutative algebra Which says that if a ring R is normal then R is the intersection of all co-dimension Intersection over all co-dimension one primes of RP. So RP is the local ring at P And so what we do is we look at some any open affine of the form spectrum of R in X and Suppose a Cartier deviser is represented locally on X by G over H on on this open set X If its image is Nought in the vey devisors this means that G over H is contained in All local rings are over P and similarly H over G is also contained in all local rings are over P So using this theorem about normal rings. We see that G over H is a unit of R so is trivial as A Cartier deviser So what we're saying is that if a Cartier deviser has image zero in the vey devisors and The scheme is normal then this implies the Cartier deviser must be trivial everywhere. So The the kernel of the map of the map from Cartier devisors to vey devisors is Just zero And for example, let's look at the example we had we had this Ring k t squared t cubed and we noticed that this is not normal because t is in the in its integral closure T is obviously a root of X squared minus t squared equals naught which is an integral equation and Furthermore, we see that There's only one prime and RP is just k The ring of all formal power series in T. So R is not equal to the intersection over all P of RP so that shows where The example we had where where the map from Cartier to vey groups is not injective Fails Because it's not normal We can for this example The We can also ask about whether the Cartier deviser classes is contained in the group of vey deviser classes So in the previous example Although the Cartier devisors are not contained in the vey devisors every Cartier deviser is in fact At least the ones of zero degree are in fact principal So that doesn't quite give a counter example What we can do is we can modify the example slightly and find an example Where the Cartier deviser classes are not contained in the vey deviser classes and for this We just take the spectrum of k t squared t cubed This differs from the previous example because we're now just looking at polynomials not power series So this is just the coordinate ring of the cusp y squared equals x cubed with y Was t cubed x equals t squared Now the vey devisors are the same as for k t and you can easily check that the vey deviser classes are just zero More or less the same as for that there you notice there's a natural map from this Ring to this ring. So there's a natural map from the affine line to this ring with a cusp and Now that's sort of more or less identifies vey devisors of both But There's a Cartier deviser Um But sorry the following cart of Cartier deviser is not principal So I'm going to take the following Cartier deviser. I'm going to take the Cartier deviser to be given by the function 1 plus a t on the open set Where t is not equal to a to minus 1 here a is in k and I'm going to take it to be 1 on the open set Where t is non zero So here we've defined a Cartier deviser by covering the spectrum with two open sets and using different Meromorphic functions on these two open sets But it's not in the image of the function field k which is Rational functions of t Um, um, so This Cartier deviser is Not even principal so so in fact the group of Cartier devisors contains a Subgroup Isomorphic to the additive group of k in fact It is Isomorphic to the whole of k plus which I'll just leave as a fairly easy exercise Um, the next question we can ask is when is the map from the Cartier devisors to the very devisors and Isomorphism So we found conditions under which it need not be injective And we can also have an example when it's not surjective So here we're going to take Well, in fact, I'm going to give an example where the Cartier deviser classes to the very deviser classes is not on to and for this we're going to take the Um cone xy equals z squared so it's a sort of Draw it as a cone like this and We're going to take the deviser Look like this. So here's the deviser and this is the deviser y equals zero and This is a very deviser and we're going to show that it's not the image of a Cartier deviser Even near the point zero zero zero And so we just have to show I mean it's enough to show that it's not locally a Cartier deviser Near the point zero because and then we don't need to worry about what happens elsewhere And what we do is we look at the local ring of the variety x so x is big x is going to be the This double cone and we're going to look at the local ring at nought nought nought and it's got a maximal ideal Generated by x y and z rather obviously So let's call this m and we notice that m over m squared has Dimension three and this gen span by x y and z you notice this dimension is bigger than the dimension of the cone so it's a singular point And If we take the Sorry, so this isn't the idea why it was mortis. It's Generated by y equals Z equals zero So the ideal P Generated by y and z Has image y z In m over m squared which is Is two-dimensional two-dimensional vector space so it cannot be generated by one element So this ideal P is not principle Even in the local ring at zero. So so this So the very deviser Is not the image of Cartier deviser On the other hand we notice that if we take p squared, this is the ideal y squared z squared y z Which is equal y squared x y y z Which is just the ideal y so so P squared is a Cartier deviser Or at least it's the image of a Cartier deviser So geometrically what's going on is this here the Deviser y equals zero is A sort of double line Which is the Intersection of a tangent plane with the cone and if you sort of think about it You'll see the tangent plane meets the cone on a sort of double line And if we look at x equals zero It's again a sort of double line like this And if we look at the divisor of z equals zero It sort of looks like this and now you can see that the divisor of z squared is Equal to the divisor of x plus the divisor of y Which is not very surprising if you remember that z squared is equal to x y You can also Observe that it's sort of visibly not a unique factorization domain because this gives Is two factorizations of the element z squared you can either write the z squared or as x y and there's no way to Find a Write z as a product of Of other things so so so unique factorization doesn't hold in the local ring at this point Well, the reason I'm emphasizing unique factorization is we have the following rather simple theorem so suppose x is notarian and Again, I'm going to assume it integral just for safety then if all local rings of x are unique factorization domains then the map from Cartier divisors to V divisors is An isomorphism Well unique factorization domains imply that it's normal So we already know the map from Cartier devices to V devices is injective So we just need to show The map is onto and so We may as well pick some One dimension so Irreducible subset D of co-dimension one and It's enough to show that Cartier devices map onto this because the V divisors are generated by these things and what we do is we Pick any point X So here's our divisor D, which may or may not contain X And The local ring at X is a unique factorization domain and this implies that D is locally of the form FX near X In other words, it's it's at least locally the zeros of some function So we can find an open set U X containing X such that the V divisor is given by F of X And now what we do is we cover X by a finite number of Such open sets So we might have you know U X you why And you Z where is the divisor D? Here's X and here's Y here Z There's no reason why Z should actually lie on the divisor and on each of these open sets. We've got a function f X f Y and f C so that F X equals D on you X F Y Equals D on you Y and so on so the The the sets F F star you star define a Cartier divisor because The quotient of FX and FY on the intersection of you X and you Y must be a unit because And the the the only common zero is on DX and and the zeros cancel out and The All the local rings are normal because they're unique factorization domains So so if something has no zeros or poles, it must be a unit So we've constructed a Cartier divisor mapping on to this divisor D So to summarize the map from V divisors So the map from Cartier divisors to V divisors is Injective if X is normal and an isomorphism if X is also locally You unique factorization domain or its local rings a unique factorization here. We're assuming that X is notarian and integral So it's also worth remarking that if X is regular Meaning that all its local rings are regular local rings in other words X has no singular points then this implies X is locally Unique factorization domain So Cartier divisors at the same as V divisors So that's why all the examples we had have singular points if if a scheme doesn't have singular points Then Cartier divisors are essentially the same as V divisors Okay, next lecture. We're going to study the relation between Cartier divisors and the Pickard group of invertible sheaves