Before we start, looks like you guys had a good lunch. How was the lunch? All good? Also before we start, how many of us over here are Android app developers? Remaining are iOS? Can be Windows as well. There are a few phones there. So what we'll talk in today's talk is primarily for Android developers and also for any company CXOs here, CMOs here, marketing guys here, any marketing folks, very few. So the platform which I'm going to talk about takes care of two things. One, it gives you a beautiful dashboard or an insights and analytics for the CMO to understand how the app usage and the things are going inside in the product. And for the developers, it gives, again, a set of features like you can track issues, errors, crashes. And then you can also do custom logging based on certain use case scenarios of the user, how they use the app. So the whole idea of this is this is cross-pad form. So if your company tends to develop an app for iOS tomorrow or Windows Phone as well, they can absolutely use the same platform to increase the users. So what we're going to talk about here is this platform, the capabilities, and some use case scenarios. Why we would actually need this platform? So can we name one of the favorite apps that we use most often on our smartphone? Flipkart? Any other apps? Ola? Yep. Yep. Anything, no one uses Facebook, really? OK. So there are a lot of apps. And then apps, one most important thing, while you want to ensure that it's been used by users and you want to start monetizing it, is the UX of the app. That's, of course, one of the primary criterias. But there happens that in this smartphone world where you have tons of apps, 100, 200 apps installed on someone's one plus two, and then it becomes very difficult for you to keep track whether someone's using my service or he's going to pay me. For instance, I can give you a best example of wink music. How many of you use wink music? No one? Sawan? Sawan has a beautiful app as well. So all these guys let you stream music. The biggest problem these guys face is the users are not using this app as often as they should be using the app, which kind of hurts them in the subscriptions. So while you create an app, you would want to have a strategy where you can ensure that users always tend to use your app, either the reason could be that the user doesn't know all the features that your app has, or you are kind of throwing a lot of notifications there, because I have seen a few quotes on the stores where someone, I think this was with Snapdeal sometime back, where they started pushing 10, 15 notifications all within one hour, and people just want to delete this app, install this app, because they don't want such bad experience. So if you automate notifications, thinking it's going to be more engaging, it might not really help. So you would need some smarter solution over there. So we'll talk about that. A quick video. This has an audio. Let me quickly plug this. I get very annoying. And users might even delete your app. OK, let me just. You decide to get a little bit about the platform, how things work, use these devices. You've published an app. So now what? You have no clue if it's used often or simply gets forgotten. To keep your users interested, you start spamming them with push notifications. However, those could get very annoying, and users might even delete your app.  Well, now you have loads of data, but you still need to set up campaigns for every possible situation and send it to every platform separately. It just takes forever. Isn't there a better way to keep your users interested? Boom, introducing Azure Mobile Engagement. Azure Mobile Engagement is a platform that combines analytics with push notifications and in-app messaging campaigns. It monitors the activity in real time, like what pages are visited when and where. But it actually goes way beyond that. You can ask any question, like after how many visits a user is most likely to post to review or buy something. With this info, Azure Mobile Engagement helps you automate targeted campaigns that work on every platform and any device with an internet connection, even a smart TV. Finally, Azure Mobile Engagement also recognizes user segments for you. For example, the ones that are losing interest. You can directly send something really cool to win them over again. Awesome. Azure Mobile Engagement. Pilot your apps. Cool, so with this platform, we can actually keep a track of people losing interest, not logging in on our app, or maybe people using our app without logging in, without doing a social login, which means probably they're not sticky enough. For instance, I've seen a few apps which gives you more features, or it's more, you know, they only start pushing notifications when you sign in that app, and which the user might not even know about it that he needs to sign in, and he doesn't sign in, because, for instance, Flight Search's app, Clip Clear Trip, or Make My Trip. You can use these apps all the while without actually logging in the app.  I probably wanna target some actual users who use my app and give those deals to them, or maybe some users who haven't really logged in, I would want them to log in, and I'm gonna give some deal to them to make sure that they log in. Now, we will see all these data, and we'll also see how we integrate this. So, integration from a developer standpoint is pretty simple. We will see that a little later. The whole idea of this mobile engagement is actionable insights. Based on that, you take some action to get to ensure that your users are there on your platform. Dure them with some coupon codes or something, and they will be there in your app platform, and then a value-added push and communications platform. So, this takes care of push notifications as well, using all the platform notification services from either Apple or GCM or from WNS, that's Windows. So, it takes care of that, and it also takes care of some campaigns that your CMO would wanna work on. Like, you wanna do a quick survey with your users, right? Generally, how would you do that? Either you do it on a web, so direct him to the web, bad experience, or you can have a native interface within the app where you can get a quick survey. Now, with the mobile engagement, you don't need to build that piece. All you have to configure this is from the backend, what questions you wanna ask, and then push this campaign, and then you can select set of users you wanna push this to, like three days old users, or users who haven't yet signed in, or users with a particular tag on them, who have been looking for a particular product, for instance, Flipkart. If you're looking for a particular product on Flipkart, I can tag you with that product ID, or group product ID, and then based on, I can start pushing you offers on that particular product ID itself, and then I can run campaigns to ensure how successful was that. I wanna measure that campaign as well. It's not just about pushing a notification, then forgetting about it, but I would wanna track how many people actually viewed the notification, how many people viewed, and then took an action, went, you know, called an intent to actually create, or get into a particular product details page, and then went ahead to buy that. So I might wanna track all these things in a success scenario for my marketing folks. So you can absolutely do that with this. And then open APIs and easy integration is possible. So if you have a CRM system, or you know, already using that, and if you wanna integrate this with mobile engagement, so that when the sales actually happens in a B2B scenario, and you do an entry in the CRM system, that should actually trigger some notifications, or run a campaign in an integrated manner with this. It could do that. And then data protection privacy, that's something you would always see in a Microsoft slide. So we'll just skip that. Regarding the actionable insights. Now trigger actions, you can control this. We'll see this little, maybe in the demo itself, that will be more explanatory. And then push communication platform. So you can use this whole mobile engagement platform as a push communication platform as well. So even if you wanna push something like 6,000 pushes per second, the beauty here is, which we faced it with certain customers, is you can track those pushes as well in all three platforms, even on Android, iOS, and Windows, which means why you do a push campaign, and you wanna see how many users actually received it on a device, and how many actually clicked on it. All those details are captured in here. So if you wanna do this, where you wanna push a notification, doesn't go through, keep a check of it, repush, retry pushing the notifications. You can absolutely do that, and you don't have to create any logic to keep a track of that. So you can absolutely integrate with CRM, CMS systems. And data is well encrypted, so you don't need to really worry about the production of the data which you're sending on Azure, because this is something hosted, this is a service hosted on Azure. So your data goes to Azure, which is some cloud service, and then from there it gets pushed to the PNSS. So the data is kind of secure over there. All the communication happens in an encrypted channel. Okay, so the whole idea, use case scenarios, which I can think of for mobile engagement is maybe you can have a welcome strategy to start with. You can have something like when a user registers in your app, you can absolutely push, create a set of notification which will only go to the newly registered users. So you can set up four, five campaigns upfront of what you would want or measure as a success criteria of your app. For instance, someone who installs my app, he logs in my app using a social ID. That's one success criteria. If he doesn't do that, I will have another campaign which will get blasted on day three, which will ask them to sign in to use some new features which he's not able to use right now without signing in. So I will run that on day three for those guys who haven't yet signed in. For people who have signed in, probably I'll give them a week, and then I will push them a notification saying there's some cool products. Now, once you push this and they go inside the app, you can learn more about their user path inside the app, what kind of products are they using. And I'm gonna log that set of products and tag this user for that particular group product ID saying he's interested more in sports goods. So I'm gonna give more offers or run a campaign for only this set of users who are interested in the sports group. So you can absolutely do that without having to do even, once you've integrated the SDK with your app, you don't need to worry about the push campaigns. And one key thing which is not mentioned here is especially if you are working in R&D scenario, this is pretty simple. Wow, something just, okay, that's fine. So if you're working in R&D scenario and or you wanna use something like a trial of this whole service, this is available for free, up to 100 users, complete functionality.  If you're a startup, you can absolutely test this out. And the pricing goes something like till up to 10, 10 lakh or yeah, something like one lakh users and above. It starts per active monthly user. It charges you around 60, that's the charge. But that only happens once you exceed 100 users up to one lakh. And if you go beyond that to 100 lakh, it becomes 40% and then becomes 30% and then it becomes even lesser. So it goes really less, the more users you have. But for trial purpose, 100 users, it's absolutely free. Okay, so you can absolutely manage the whole marketing portfolio with this. So generally we have seen some issues where you have a set of developers and then probably there's a bunch of marketeers and marketeers want something, developers don't know what they want and the whole integration kind of takes much longer time to build. So using this tool, you have a beautiful UI for the marketeers. All the developers have to help them to integrate this SDK into their app and then marketeers can completely take care of the engagement strategies and so on. Okay, and quickly, personalize customer experiences. So the whole idea of using mobile engagement is you can constantly keep a track of the user experiences as well in terms of user using which feature in your app and so on. So it completely gives you an idea of what features your users are using more or which ones you need to really work on and where are the more errors coming from? What are those device types? Which platform of Android is it running on? And so on. So you will get all those details and you can absolutely work on top of that to improve and of course also personalize customer experiences. Okay, so now quickly go and let's check out the dashboard that we have. So it also helps in boosting your user retention. So there's a beautiful dashboard which you'll see here which shows you how many users have installed the app and use this app, total number of monthly active users and users who joined for the first time versus users who were there the last month and so on. So it gives you that. And based on this, you can quickly track the evolution of the different controls in your application based on this and of course you can increase the ROI. Okay, so here are some example scenarios. As we said, so on the first contact, then anchor to a win-win scenario, demonstrate value through conversation, create a tree-like campaign. Now the whole idea of this is you try and connect with someone and once you connect with that person, it's easier that you will get him engaged in using your app. So that's all what we want to do here. So demonstrate the value through conversation, maybe you push notification, here's once back to notifications, you get to know him more and then you push him another notification and then finally he buys something from your service. And then you can have a create a tree-like campaign, as I said, or else kind of statements where if the user did this or no, if yes, then go get him to this campaign, else get him to this campaign and you can absolutely build a tree kind of campaign here. So here's one scenario, as I was talking about. On day one, we connect to Facebook, remember your last video, this is something like an experience which all of us give while we let you sign in an app, while you sign in the app, we'll kind of keep all the cues of videos that you're watching so that if you start watching the same video on a different device, you can still start, have a common list of videos that you have liked and watched and so on across those devices. So you tell about this feature to this guy, he still doesn't connect to your app or probably he's already connected and we'll take care of another campaign with that, but we can talk about another feature which is one click sharing your favorites with your friends. Now if he still doesn't connect, we go back on day 10 and say, discover your friends and favorites. That should be the most important feature your app offers. Without signing in, he will not be able to do that and you can track whether each of these level one or level two or level three campaign worked for him or not and if it didn't work for him, then probably yes, he really doesn't want to use our app. So otherwise we would have tried thrice to get him on a platform and if he is there, ask him, try and upsell the product like or not to help suggest great content and then also this helps boost your ratings. What happens is people who use your app more often, best use case example I can think of is OnePlus. The OnePlus fans who buy their phones, someone who's waiting for that phone and he really wants to buy that phone, probably would rate that phone well. I'm not saying the phone is bad, I also have one. The phone is pretty good, but you would see the reviews and ratings to be like extraordinary which is because they have a good fan base and they kind of involve or engage with their fans so the whole social media is flooded with OnePlus fans. So in that scenario, if now OnePlus asks for a rating or a review, they certainly are bound to get better reviews on Amazon than Nexus 6P, right? And they do have a better review than Nexus 6P. So that's probably because they are loyal users of that platform or a service and similarly if you know our users constantly using my service or my app, probably he's the one who's gonna give me five stars else he would not be using my app, right?  So you can absolutely run a campaign for people who have logged in and used your app for the past 10 days or bought some stuff, services from your app. Those are success criteria for your app. If they have surfaced that, then probably you wanna run a campaign for them which is rate my app on the store, right? So you can run those things to boost your activities. That certainly will give you better store ratings which is very important as well. Boost the hidden gems. So there are a lot of great features maybe overlooked by users. This happens with all of us because we are kind of one track. So when I constantly use one app like Flipkart, I kind of think Flipkart is better than Amazon. And this happens with maybe all of us. And but from the Amazon side of things, they might have a cool feature or maybe they had a feature called QR scan before Flipkart had that. So if they would want to run a campaign saying, let's push this new feature because our competitors don't have it and let's kind of lure some people to actually try our service so that they can really appreciate our services. So in those scenarios, you can actually expose your services which are actually overlooked by the users so people don't know about it. I have seen a lot of complex apps where I didn't realize that these are features supported especially from a tagging and notification purpose. You generally don't realize it till you actually start using the app or you start engaging with the app. So something like apps sending you in alerts and stuff generally is overlooked. So if there's a news app which kind of personalizes your news and then pushes those news to you on your personalized timings, probably I would love to have that app but probably I didn't know that this app has that feature. So you can absolutely expose those stuff here by running such campaigns. And this also promotes its value to active users and this is absolutely very important stuff. All the apps which I've written absolutely have crashed in most of those platforms. So I would need something like an app inside or Google Analytics to keep track of that. Now this guy does that for me as well. And this will tell you all the details that you would require, what kind of issue that was. And you can absolutely use this and you can even create a campaign for people who have had reported a crash. And then sending them a notification saying, sorry for the bad experience, we are working on it, we are looking into the crash reports and we'll get back to you shortly. Now if you give this kind of an experience in an app, this is pretty awesome. And you can absolutely build this with mobile engagement here. Okay, so this is how mobile engagement works. Basically your app will have RSDK which will push all the data, usage data from the client onto the Azure mobile engagement platform. And post that the CMOs, the marketers, the developers will have all the access of APIs to get the data and then analyze the data, get the insights into dashboards form and so on. And then you can also use this as a push platform using push communications or doing an in-app messaging. Now the cool feature of this is you really don't have to build something like an in-app messaging system for the app. The SDK will do that for you. So you just use the SDK there, add some permissions and then you would be done with that. Okay, now these are some concepts which, or maybe some APIs which we have exposed from the device and users side which captures data of the device versus the users. A user can have to device and devices are unique. And then there's something called sessions and activities. Sessions is when you open an app, till the moment you close that app or go out of that app or the session is on. So session is that period till what time you use the app for. But if you wanna track certain stuff within that, at that page level, you can do that with activities. So you can start an activity on start of a page like when user starts to buy a product. You might want to select this product, look at a details view, then click on add to cart, then go to the cart and buy it. Now you can do two things here. You can start one activity and then you can also have a job which you will see later on. Now jobs are very helpful in tracking. Now these are time-based, just job. So while you start a job, you'll have to end the job and it's ideal in scenarios where you're doing a payment. And users or developers would want to track because this has happened with me actually where we tested it on 3G. 3G internet is still pretty good. But when this app went into remote locations where even the low internet bandwidth of 2G was available, things were not working well. The JSON packets which were coming in were a little big for that internet and then there was some time-outs which were happening. APIs were failing because there were like five, six different calls happening, different services which was printed on each other. So now since that happened, for such scenarios you would want to keep track saying we have tested or benchmarked our APIs for this much of delays and stuff. Beyond this, we don't know because it's a third-party service that we are using so we don't know whether we can actually expect that kind of time from that particular API. So if you want to keep track of that, you can set a job, start time, stop time and then report it back onto your service saying this took so much of time. And this also helps you understand your user experience of your users because if he's taking a time of five minutes in buying a product, probably he's not going to buy it from you the next time. So you've got to keep track of those things by starting and ending a job and then you have things like events over there where you can pass on certain contextual data at that point of time from that particular page to the backend service. For instance, someone started to purchase something. So before you start a job, you can send an event saying purchase event or add to cart event. So that at least you can, in the BI, you can see how many people actually went through this event of adding a product to the cart in a day. So it's one million, 10 million and so on. So you can get those stuff here by logged in as an event. An error is basically for tracking errors. Like you started a job, the job was running, it went after 10 minutes, it returned some error from the service and then you might want to log that error for you to actually research on that. So you can put that as an error, saying API error, put the details in there and send it, call this particular API for error and then it sends it onto the server. Similarly, crash, you don't really have to do anything. Crash will take care of, you know, on itself. So even if you do some things like divide by zero or anything which makes the app crash, it still keeps a track of that, of those details. Now application information, platform installed on, all those data are right there. So now quickly, in just 10 minutes, let's have a quick demo. So I'm gonna share my screen and here we go. This is the portal on Azure and this is what you see here is what I have created a new mobile engagement project here. So all you have to do is click on create, give application name, platform, application resource name, location where it's gonna be hosted on and the collection name, which is mostly like a company name that you have. Once you have created it, you would get the details, the connection info. You would get something like the SDK keys which you would need. You have the connection string which you would have to pass to the SDK so that it identifies which cloud repository it needs to send the app data on. So this is gonna be a little important. While we have this data, we can actually quickly get started with the app. So here's a very basic app. Now this is just a two-activity app and in that we are just passing one message from one to the other one using an intent and then we have used some libraries in here which we would show. All you can see here is right at the beginning in the activity we have extended engagement activity right here on the top and then we have added like three lines for instantiating the mobile engagement of APIs here. So engagement configuration API and then engagement configuration, we have to set the connection string. As I said, we have to copy that connection string from the cloud and then just to authenticate whether which backend are we sending the data to so this gets right from there and then instantiate it. That's all we do. Now once you do this, we should be good to go in terms of instantiating the mobile engagement in this app. Now beyond this, if you wanna put some stuff like of course you would have to add things in the manifest you will have to add few permissions because mobile engagement requires them. So you would see a few permissions added over here which is not specific to mobile engagement but mobile engagement is gonna use those so you're gonna have those permissions in here because as I said, as you saw, it does a lot of things, it does notify you, it does notifications in background and stuff like that. Now once it is there, what I'm gonna do this, do here is I have an Android emulator. This is a Visual Studio emulator by the way and it's free of cost and works on without Visual Studio as well. So now I'm gonna push this to that device. Yes, emulator, okay. Let it push. In the meantime, we can actually look at the backend. Now this is where the whole thing is. This is how the dashboard looks like and what you can see here is you have analytics. So now while it's deployed, it's just a basic message. I can, it's just to invoke my SDK which is done now. So now I can actually go ahead and create a campaign and stuff. So let it load my portal. In the meantime, while it does so, what I can do is I have app here as well. So here's the project. My app mobile engagement which you created on the Azure portal appears here in this mobile engagement portal. Now you can go inside this. Right now you can see total users is one because it's only I who have been using this for now. Now you can see your users' details, your active users, retained users, who were users last month, even now, and tracking per source, retention. It gives you a retention matrix which a marketer would love to see. Then number of live sessions you had in your app, like number of people who opened up your app, you know apps like Uber and Ola use this a lot because they got analyzed on the search, the search rates based on this. How many users, which areas are actually using my app? So they actually keep a track of the session. I'm not saying they're using this product, but they got to use a track of sessions so that they can know about how many users are actually trying to book my cab from this region so that they can actually go ahead and do a search. So this guy kind of tells you live session counts. You can see those counts. Then you can see the activity details, those activities, which it's actually not the same as Android activity, but yeah, each of those pages is kind of an activity here. So you can kind of compare this to the Android activity, but it's the same name on Windows and iOS as well, where there's no activity there. So this is the user path analysis, and this is very important because this will tell you the different areas of your application, different pages of your application. For instance, I have done a poll. I have done something like an announcement, and it shows you all these things. Right now it doesn't have data because it's only single user, but the moment you have more data here, it'll start pulling up. So basically this tab called analytics will pull up data after 12 hours or so. You'll see, start seeing more data after 12 hours. It's not live monitoring here. Live tab is different, which I'll show you, but this will show you connected lines in terms of showing you the user path, how he uses the app, and then you can run different campaigns for different users with different paths. So you can find some users who actually open your app by notifications, and some users who actually start your app from your main activity. So then you have events and so on. This is the cool part, which is monitor. So while we have deployed the app, let me just check that out. So I can quickly go ahead and start that one. I think this was the app. So while we have started this app, it should ideally show one active sessions, and I would do that on my phone as well. So this shows you number of live active sessions, and so on. So let me start this up. Now this is highly dependent on the internet connection. So only if my data is going there, so it shows up one here now. It should show up two now, because I have started my phone app as well. Now it will show you live sessions, total number of live sessions here. Similarly, you can do a lot of live monitoring by activity, which activity people are using most often, or with jobs people are using more often by different names. I haven't set up a job name inside my app, so it doesn't show up anything here. Events that you have errors if you have, and similarly crashes that you have. So you can pass all these details, and one cool stuff is the alerts. So I have done is, I have another interface for this, where in this project, I have a UI. So let me share that UI with you. How do I get that here? Okay, so you can see this. It's my one plus device, and here's an app which is running on it. Now you can see I can start a particular activity. I can end that activity. I can actually crash this activity as well. This activity can be, I am on a particular page while doing that configuration stuff. It crashed, so I will get that crash report. My app didn't crash. I can crash my app as well. So I click on this, crashes my app. Now I can actually, in real time, go back and check the crashes, and you would see the crashes. Let's check them out. Details. Okay, so while it loads up, it would bring up that crash, which I did just right now, in here. So I can do a real-time monitoring, and what I can do is I can configure it with an alert. So I can set up an alert here, saying whenever I can set up an alert condition, I have set one already, which sends me a mail whenever there are more than five errors. So when I get greater than five errors, I get a notification via email saying, you know, there are more than five errors. So I can come back and check out what those errors are, and then update my app. You can set up a new alert on different criteria. So it gives you a lot of criteria. Either it can be number of active sessions. So once your active sessions goes beyond a big number, you can create an alert for that, active jobs, because you know that I can't do maybe more than 1,000 concurrent processing. So I would probably keep a track of whenever my total number of active users go beyond a point, I might want to look at my back-end scalability and so on, so you can set up a job here. Similarly, errors and crashes are there as well. So you can set up a crash. You can set up saying threshold for detection is gonna be greater than 10. So I'm okay till I have 10 crashes per hour. You know, I can set up minimum detection rates, saying keep looking for it every 30 seconds or minutes or hours, whichever is, you know, better for me. And then drop a mail to my email ID. So I can just put my email ID here and it would configure whenever there's 10 crashes happening in a window, not in a window of 30 seconds, but yeah, I think overall it would let me know about that. So it would at least have 30 seconds of gap in letting me know. So kind of by the time you get an email, it's like one minute. So that is what it would tell you by setting up an alert here. Now other cool stuff here is the campaigns that you can do. So you can very easily go ahead and create probably a reach campaign. Now in this reach campaign, what happens is you can push notifications based on either you have an announcement, I create the welcome kind of thing which probably would have come up here. First time user. So let me push one here. So I can go ahead, create a quick poll, create a new poll, and you would see number of people who actioned on it and stuff like that. Test poll. You can mention a category, sales or something and then go set up. Well, I would wanna send a system notification. If it's not important, I can live without that in app notification as well. I would do this anytime, title, test. Just quickly let me fill this. You can upload an icon as well. So you don't hard code here in the device level. Yes, no. Right, so this should create a question for me. Are you using Android? So similarly, whatever makes more sense for the business, you can just quickly go ahead and create some quick questions here, quick polls here, and then expect the users to go ahead and you can write some expressions as well to filter out your target audiences based on tags. Now I'm gonna push this to everyone. Who's using my app? Go ahead, create it. It creates this campaign or your campaign is done. Test poll, one, two. While it does, you can activate this guy. This takes a few seconds and then activates it and then it sends a pushes notification. So one sec. While this pushes, I would, I have already got this notification here. Let me show you that. So you see this test here which I just created so I can tap on it now. While I tap on it, you see that poll came up and actually it went because this app triggered something. So that's how I can actually go ahead and create a quick poll without having to code that in my app because my app was a basic simple activity but there was something called as a poll page. So if you look at the SDK, SDK drags and creates a lot of, you know, XAMLs, AXAML over there. We are sort of out of time. Okay, so I'm done with this now. If you guys have any questions while the next speaker comes up. Yeah, I don't think we have any time for questions either. Okay, I'll be outside anyway. So if you guys have questions, you can come back. You're the Microsoft Booth, I'm assuming. Yeah, okay. Okay, thank you guys.


Yeah, I don't go to you first all day today. Um, well listen, I think they're both really good. Uh, I I'm not good at pigs. I'm not mean either gonna give my picks out like it's tough Like I think it really depends on what you need like Palo plays Really on the box a lot. You see as a lettuces them right now Jalen Green obviously more of a wing perimeter player So honestly, whatever my need is, I mean, they're both really good But I mean, I don't think it's a stress to say that Palo Bancaro is much more further along Right now than Jalen Green was a year ago and that probably speaks to Palo Bancaro came in the league You just see his physique his body type like he's ready to go. He's a man already Um, and he can play against really any player in the league. There's obviously some stuff He needs to work on defensively But he's he's there from a from a skill standpoint a physique standpoint. Um, like he's got all the tools


you want to start? I don't know, maybe if Zach wants to introduce this thing or I can do it, I mean, I'm more than... Oh, you can do it the way. Yeah, absolutely. You can do it. You can introduce our joint venture. This is a very nice thing and I want to start by saying that for me it was a very nice surprise when Emmanuel contacted me about doing this thing, this thing together and I find this initiative of a Met Associates seminar from ICP, very, very interesting, bringing all this community together regularly. I think it will be very, very nice. So thank you for thinking of us and this occasion. I'm very, very glad. And what makes me even happier is to have today, besides the catchy title, Certainty vs. Uncertainty, my friend Emmanuel Carnero and a friend of the Brazilian Met community who even a few hours flight away continues to impact what happens here. So I'm very, very happy and it's a joy for me to introduce Emmanuel Carnero from ICP who's talking about Certainty vs. Uncertainty. Please. Thank you, Edgar. Can you all hear me fine and see the slides? Yes. Okay, I have some people in my screen here. I hope you don't all don't close the video because it's nice to see some faces here. It's a bit solitaire on this side of the screen. Feel free to interrupt me at any time and have a conversation. The point here is just to have a little bit of fun this afternoon. Thank you, Edgar, for the kind words. It's very nice to be in touch with all the Brazilian community that is joining us today. Thank you very much for your initiative to have this amazing webinar and analysis and PDE in Brazil that started in the very beginning of this crisis.  So we're kind of launching, as of a few weeks ago, our math associates seminar which is supposed to happen also once a week to gather our community of a CTP of associates of visitors of friends around the world. So I welcome everyone this afternoon here and some, most of you I know, I have met and the ones that I haven't met yet, I'm sure you have the opportunity to be here at ICTP at some point in the future as an associate as a visitor or participating in a conference. It's a pleasure to have you all here. And it's good to be in touch with everyone. This is the first time I come back to ICTP after three months. So I feel very happy today here in the office. The topic today for the talk. It's a bit, it's about Fourier uncertainty. So the type is certainty versus uncertainty. And I'm going to talk a little bit about this new joint work that I've been doing with the Oscar Chessada Herrera, which is from IMPA, is from Costa Rica. So I guess all the Latin American countries today will be very well represented here. Okay, are we ready to start? Good. I will divide, let's see if I can divide my presentation in three parts. Okay, I'll do a brief introduction on the Fourier uncertainty and recall some of the main facts. Then I'll move to a little bit of this particular instance that I want to discuss and the previous works. And then the third part of the trilogy, I want to talk a little bit of our point of view on the subject. By the way, again, feel free to jump in the conversation at any time. So my first slide is a little photo of the guy that's going to be the main actor in this afternoon, Joseph Fourier. And this is a memory that I have from my first trip to Paris. So the names on the Eiffel Tower, it's a very nice way that the French have to pay a tribute to their scientists. So there are 72 names in the Eiffel Tower and Fourier is one of these names on the west face of the Eiffel Tower. So let's talk a little bit about just to introduce the topic and to show you a bit of the relevance. A very basic example, I guess it starts with Fourier series. When you have a function, let's say, can you see the little hand moving here? And I do this. Yes, we can. Very good. So if you take a function f periodic function, say the in the torus minus half half and put in here the interval, you can do what we call write it as a Fourier series. You can take any integer coefficient k and compute what I call f hat of k, which is just the integral of the torus of f against the exponential into the minus 2 pi i kx. My exponentials here will have this normalization. And then it's known fact that under some conditions you can write your function f as the sum of these sines and cosines with this Fourier coefficients. And this is, I mean, not of people, not everyone remembers, but this is present in our everyday life much more than we think about. Every time that we talk on the cell phone, we're sending a signal, somebody's just cutting this Fourier expansion of my function into some number of frequencies. I give you maybe 30, maybe 50, the first 50 signals. And then you get on the other side and you have to reconstruct the message. And the way the reason we do it is that our ear, I mean, it doesn't make a difference if we send just 30 or if we send the next thousand. It doesn't make a difference to our ear. So it's much more cost effective to just send a little bit and reconstruct the message on the other side. Same thing happens with the TVs. A few years ago, I remember when I was a grad student, there was this boom of LCD TVs and then the definition of the TVs went from 720 to 1080 and then full HD, ultra HD. And then after a few years that stopped. That stopped essentially because the definition achieved the threshold of what our eyes can see. So it doesn't make sense to subdivide the screen into more pixels. So every day we're kind of playing this game of giving partial information on the Fourier transform side, receiving partial information on my function side. And with this partial information on both sides we try to optimize whatever our problem requires. We try to do the best that we can with the partial information on both sides. And this is essentially what the Fourier uncertainty phenomenon tells us. So this is my proper definition. So let's take an integrable function f and I'm going to define for the rest of this talk the Fourier transform f hat at the point c. C is in my rd. rd is my Euclidean space. It's just the integral of f against the exponential e to the minus 2 pi i x c. This is the inner product. You know, you start with an integrable function. This is a well-defined and continuous object f hat. Plancher-Elst theorem says that this is an isometry in L2. So you can extend this to an operator in L2. And these words that you may have heard in literature so many times, Fourier uncertainty, it appears in many different occasions. And to me, it essentially says that one cannot have an unrestricted control of a function and it's Fourier transform simultaneously. Okay. Now, this may take a lot of shapes and you may find this phenomenon in very different makeups. But essentially, this is what it's what it means. So let me give you a very first and basic example. It's kind of trivial if I give you the whole function say suppose we're in dimension one and I say that f is a Gaussian into the minus pi x and I ask you, can you have on the other side f hat of zero being the number c of your choice? And then you may say, well, no, not unless you're very lucky because I gave you the function completely. So if I gave you f completely, I'm already giving you f hat. So unless you choose the c here to be one out of luck, you would be screwed. So there's no, not any room to work on the other side. On the other hand, if I give you the function f and say that it's a Gaussian, but just outside of the unit interval and I ask you, can I have them f hat of zero at the other side being the number c of your choice? And then you have room to make that work. The answer would be yes. Okay, so if I give you less information on one side, you can ask for more information on the other side. So this is basically what it means. Of course, the topic is very, very broad and studied in mathematics and physics. It has their own interpretations in physics since beginning of 1900s. A quick check in Amazon, you find books about it, you know, and books that are just entitled the uncertainty principle in harmonic analysis. This first book has 600 pages, for example, there are people whose area of research is just uncertainty principles in free analysis and so on and so forth. Okay, so this is a big topic where we're presented now with daily activities. Let me give you, say, a few of my favorite examples of Fourier uncertainty principles. Okay, so this is perhaps the first one by Heisenberg around 1927, where he says that the L2 norm of a function on the left hand side is bounded by a constant that depends on the dimension times these momentum here.  So if you look at the right hand side, you will see that the mass of f and f hat cannot be too concentrated near the origin, right? Otherwise, these integrals on the right hand side would be small, they cannot beat this one. So this is the first manifestation that you cannot concentrate too much, the integrals, the mass of f and f hat around a point. Let's see. So Hardy around 1933 had this other version of the uncertainty principle.  Okay, so f and f hat cannot both have a Gaussian decay too much. In particular, you cannot, from this result, you cannot, you see already that f and f hat cannot have both compact support. Okay, so you already know, at least to go out from this talk knowing that f and f hat cannot both have compact support. Let's see. Another one that comes up in our study here this afternoon is this one by Ameren and Berthier from 1977. If you have two sets in RD that are of finite measure, then the L2 mass of f is controlled by the mass of f in the complement of e and the mass of f hat in the complement of f times a constant that depends just on the sets and the dimension. In particular, this other result here also implies that f and f hat cannot both be, for example, compactly supported. Let's go to more recent examples. Well, this is actually an old one. There's actually more than an uncertainty principle. This is what's called an interpolation formula. Okay, so there are these formulas that allow you to somehow reconstruct your function given some particular set of data. So here if a function is in L2 such that the Fourier transform is compactly supported in the interval minus half half, then you can recover your function f by just the values of f at the integers. So f of x is equal to f of n times this sink functions here. So if I give you the values of f at the integers, you recover completely your function f. In particular, if the values at the integers are zero, then your function has to be zero. There's a recent one by Danilo Radchenko, he was a postdoc here at ICTP, Marina Vyazovska, that they can recover a Schwartz function. So suppose you have a Schwartz function that it's even, then you can recover f from its samples at square roots of n and f hat at square roots of n, where n is a natural number. And this a n of x are interesting Bayes functions. There is a recent one, which is also, I like it very much, this is one by Mateus Souza and João Pedro Ramos from last year that says that if you have an alpha number between zero and this one minus square root of two or two, and you give me a Schwartz function that vanishes f and f hat at the plus or minus n to the alpha, where n is a natural number, then the function has to be zero. So you see, the idea here is that if I ask you too much about f and f hat, then you probably end up with just the zero function. Okay, so this is good for my introduction. Any questions so far? Are we good? Let me move into this part called Sign Fourier Uncertainty. So in 2010, there was a nice paper by Burgan, Kahane, and Klosell, where they put together a certain phenomenon for the Fourier transform and give applications to a problem in algebraic number theory. So they were interested in this particular paper and giving bounds for discriminance of number fields. And they somehow connected this to a problem in Fourier analysis, which is essentially they want now, the original uncertainty principle of Heisenberg tells you that you cannot concentrate the mass of a function and its Fourier transform around the point. Here, Burgan, Kahane, and Klosell, they want to investigate how much can you concentrate the negative mass of a function. So suppose now your function has a positive part and a negative part, they want to understand how much the negative part can be concentrated. And the setup is the following. So let's take a continuous function f in rd from rd to r, and I'm going to say that this function is eventually non-negative, well, if f of x is bigger than or equal to zero for sufficiently large x. And I'm going to define this r of f, or you can read radius of f, as the infimum of the r's such that f is non-negative from that radius on. Okay, so it's kind of the last sine change of my function f. For example, there is an example here, I just took a polynomial 10 degree 10 polynomial times a Gaussian. So this point here in black, so this is an even function, this point here in black about two points something is the last sine change of my function. So the r of this function would be this point. So they investigated the following problem. Now, consider the following family of functions. So you are in dimension d, and I take a function which is l1 of d, not zero, a function that is continuous, even real value such that the Fourier transform is also integrable. So you have function and its Fourier transform, both integrable. So they're both continuous, they're even, and they're real value. Now, I'm going to assume on the third line here that f and f hat will be both eventually non-negative. So at infinity, they will become non-negative. And the competing conditions is that at the origin, they are non-positive. So f of zero, which is just the integral of f hat is going to be non-positive and f hat of zero is going to be the integral of f, it's just going to be non-positive. So you see the tension going on here. f is eventually non-negative, but its integral over the whole space is non-positive. f hat is eventually non-negative, but its integral is non-positive. So there's a compensation here, there's a competing thing here. And they define the following quantity. So I'm going to use the calligraph a to represent the class of functions and I'm going to use this math bb a here to represent the constant that I want to investigate.  So the product of the last sign changes of the two functions. Why do I take the product? Well, it turns out that this is the right quantity to be investigated because this is invariant under dilations. If you have, if you want to take your function f and rescale it in order to put the last sign change very big, then you will do the reverse operation with the Fourier transform and this product remains invariant. So dilation here will not help you. So this is actually the natural quantity to be investigated with the product of the radius when these two functions, when each of these functions becomes non-negative. And what they do is that actually this quantity here, this infimum, is bigger or equal than a certain constant, a certain positive constant. So you should see this as an uncertainty principle. This infimum cannot be zero. You cannot make the negative mass be very, very concentrated in the origin in this sense. They actually give these nice bounds that this quantity here, depending on the dimension d, is actually bounded by above and below by, it's of order square root of d. And then the constant here is one over two pi e and the constant here is essentially one over two pi. But the message is that it's something away from zero. Okay, here are some examples. So I just take a function f of x sum of two Gaussians minus another Gaussian. You can see here when this function becomes non-negative and this other function becomes non-negative. And what I want is just the product of these two radii. And you can see that the competing conditions are verified. Both the integrals of both of these functions are non-positive. So, okay, so let's now think a little bit about this problem. So the first idea that Bruget and Kahane close our head is that you can actually reduce the problem to just look at eigenfunctions. You don't have to look at the whole set of f and f hat. You can do some interesting symmetrizations and just talk about eigenfunctions. So recall that we started with this family. You have f and f hat integrable and continuous, non-zero, non-zero functions, such that they are both eventually non-negative and their integrals are both non-positive. So what you do is you can play this rescaling game, because if you take f delta of x to be f of delta x, then the Fourier transform has an x over delta. So when you play this rescaling game, you can align both of the radii of f and the radius of f hat. So you can just assume by rescaling that the radius of f is the same radius of f hat. Then you can just sum the two functions. So when you consider this function h, which is just the sum, well, let's just take a look at why this function w has a radius which is less than the original f. And the reason is that, well, if you are past this radius r, this function is positive, this function is positive. So obviously you will sum and you get something positive, non-negative. So the last change, last sign change of w is certainly less than or equal than f. One interesting and crucial observation is that this function w is not zero. This function is not zero. And the reason is if w were zero, you would have f equals to minus f hat. So you would have f being essentially eventually non-negative and eventually non-positive at the same time. So f would be eventually zero. The same for f hat. But when I say eventually zero, I say that it has compact support. So f would have compact support and f hat would have compact support. And then your original function would have to be zero, which is a contradiction. You started with a non-zero function. This function w that you construct like this is also non-zero. So it's a function in the class and it does a better job. So if you start with any function in the class, you can always reduce it to an eigenfunction that does a better job. So that's the moral. So you can, well, there's even another symmetrization that if you want, you can actually average over the group of rotations of the Euclidean space to make your function radio. We are not necessarily going to use this right now, but it's right here. There's this additional one if you want to take a look. So sometimes I will consider them this, this eigenfunction problem. So this eigenfunction problem, I'm going to label the same thing, calligraph a plus 1d with a star. And this is just the set of functions f, continuous and integrable, real value. Now such that f hat is equal to f, so it's an eigenfunction with eigenvalue one. Now the two conditions are become the same. So now the condition is just that f is eventually non-negative and the integral is non-positive. And the thing that you want to minimize is just the infimum of the radius of such a function. So here's an example of some of two Gaussians minus another Gaussian. This is an eigenfunction with eigenvalue one, with a certain radius here being something between 0.6 and 0.8. Now let's take a look at the proof of this result of Bourgan, Kahane and Closel. The proof is actually quite simple. If you take now a function that is an eigenvalue, an eigenfunction, and you set r to be the last sign change, the radius of f. Now I'm going to denote x plus as the positive part of a function and x minus as the negative part of a function. So the integral of f is just the positive part minus the negative part. And you are assuming that this is bigger than zero. So this negative part somehow wins. It's bigger. So therefore this is the string of computation that needs to be done. The first inequality is the Hausdorff Young inequality. The l infinity norm of the Fourier transform, which is the same as f in this case, loses to the integral of f. The integral of f, the l1 norm of f, is just the sum of the positive part and the negative part. But this loses to two times the negative part because of the competing condition. And the negative part, as we are setting the problem, lives in the ball of radius r. And here you just use a basic holder inequality. So it's two times the l infinity norm of f times the volume of br. And you end up with the conclusion that the volume of the ball, br, has to be bigger than 1 half after you divide by f infinity. So this is how you get the lower bound for r by simply using Sterling's approximation. It cannot be very small. Let's see. There was further work on this in 2017 by Philippe Gonçalves-Gioggo, Leveri Silva, and Stefan Steinerberger, who produced refined estimates in all dimensions for this and also studied and proved the existence of extremizers. So these principles, they proved that in fact, in each dimension, there is one function that is an extremizer that does the best job possible. Okay, let me move now to a related or a dual silent certainty principle that was introduced by Henry Kohn and Philippe Gonçalves in 2019. And here the idea is to allow another eigenvalue, the eigenvalue minus one. So everything that we did was for this s equals to plus one. Now I am going to allow s to be minus one. And I will start with the following family. Again, now I'm going to baptize a s of d, s could be plus one or minus one, your favorite sign. These are the functions which are integrable and continuous, even revalued such that f hat is also integrable. Now the last condition is that f and s f hat are eventually non-negative. Okay, so when s is one, it's the original problem. When s is minus one, you want f to be eventually non-negative and f hat to be eventually non-positive, if you want to think like this. And these are the competing conditions. So if a function is eventually non-negative, its integral will be less than or equal to zero. And this guy is eventually non-negative, its integral will be less than or equal to zero. So you have perfect competition here. And you define the same way, this constant a s of d as the infimum of the square root of the product of the radii of the last sign change. You can play the same trick that we did before and reduce this problem to eigenfunctions. You may assume that f hat is equal to s f above. So in the case of minus one, you can talk, you can think about minus one again functions. And gonsalves and con, they actually prove the same sort of bounds for the plus one that Bourguin, Kahane, and Cosell had obtained. They prove for the minus one. It's bounded above and below by square root of d and times universal constants. So here's an example when f hat is equal to minus f. So you have really an eigenfunction here that does the job. You see that the last sign change here is actually a little bit bigger than one. My next example has two functions. This is not even taking an eigenfunction. You have f as this sine square pi x over x square minus one and f hat as the Fourier transform. So it's a band limited function. The Fourier transform is compactly supported. And you see that the last sign change is one. And here it's one, two. So the product is one. This is actually an extremal example. In dimension one, you cannot do better than the answer being one. We will see in a minute. So let me talk a little bit about sharp constants. You see, I showed you the, I showed you a problem. We defined a class and I showed you the proof. The proof is not particularly complicated. But what is the most beautiful part in these problems is that one can find sharp constants and extremizers in very, very few cases. You can do numerical computations to try to estimate what your constant is. But in very few cases, people have actually found the exact values of the sharp constants. For those of you who are new to this, so this, these problems about finding the sharp form of a functional in a quality analysis, they are very, very fundamental problems, beautiful and highly non-trivial problems, historically speaking. To me, they are almost like the, among the royalty of problems in analysis that one can think about. In this particular case, these constants were found just in four occasions. So this paper of Conan Gonzales is a paper in Inventionals in 2019. It's a very nice paper. They actually found the constant for the original problem of Burgan-Kohane-Kozel in dimension 12. It proved that a plus one in dimension 12 is square root of two. And they also recognized that the new dual problem that they proposed with the eigenvalue minus one was connected to the study done on the sphere packing problem. And as a corollary of the things that were done for the sphere packing problem, we'll see in a bit that there is a free analysis problem there too. So this, I pose as a corollary of the paper by Conan Gonzales in 2003, the paper by Vyazovska for dimension 8 in 2017, and the paper by Kon Komar Miler, Rychenko and Vyazovska in dimension 24 for the sphere packing. And you get the sharp constant in dimension 1, 8 and 24 for the minus one eigenvalue problem. And so these are four examples of some of the best papers in analysis and number theory over the last decade. These are three papers in the annals. This is a paper in Inventionals, for example. It is very hard to find. It is conjecture that in dimension two, the value of this constant in the eigenvalue case, this is not minus two. I'm sorry, this is minus one. It should be this number, four-thirds to the one-fourth. And the actual original problem of Burgain, Kahane and Coloseu in the real line turns out to be very difficult. And by now we have a conjecture that the value of the sharp constant should be this number. This is a very nice conjecture in this recent paper by Felipe Diogo and John Pedro Ramos in 2020, where they do this and they do this extensions of this plus and minus one sign uncertainty principles to a general framework, operator framework, very interesting applications to spherical harmonics, handcuff transforms and others. Let me give you just an idea, or I'm just going to browse for those of you who like a little bit of number theory and for you to appreciate the difficulty that it's behind finding these sharp constants. So I'm going to take this example from the paper of Henry Cohn and Felipe Goncalves showing that this constant for the eigenvalue plus one in dimension 12 is square root of two. So when you want to show that this constant is square root of two, you need a tool to prove a lower bound, you need a tool to prove the extremality, and then you need a tool to construct a function. So the extremality in this case comes from a very nice Poisson summation formula. You start with the z in the upper half plane, you have this Eisenstein series that we call E6, it's just this series here. The nice thing about this series is that it has a Fourier expansion, which is one minus sum, no negative coefficients Ck, which turned out to be a multiple of the sigma five, sigma five is the sum of the fifth powers of the device, okay? This turns out to be a nice modular form, and this is the functional equation that it satisfies. And once you have a functional equation like this, you have a Poisson summation, you can apply what this means for a Gaussian, and then you can extend for the whole class of radio shorts functions. So this leads to a very funny Poisson summation formula for radio shorts functions in R12. So you see, for every radio shorts function in R12, this formula in blue here holds, you know, it's a bit strange and different from other Poisson summation formulas that you might know in the sense that you have negative coefficients here. But this is exactly what we need, because if f is equal to f hat, you move this guy to the left, you move this guy to the right, and then you have f and f hat at zero on the left, and this quantity is supposed to be non-positive, and you have the sum of all the rest of f's on the right at the square roots of 2k. So if you are past this square root of two, which is the number, everybody will be non-negative. And here's the competition. You have a non-positive number equals to a non-negative number. Essentially, both sides must be zero. And this is, there is a little bit of technical work here, but this is essentially how you prove that this number has to be bigger equal to the square root of two. So you found the right lower bound, and you actually, from this Poisson summation formula, you actually found a hint on how to construct this function. This function should be zero at this value and at these values too. So there's a lot of zeros at this point square root of 2k. And this whole machinery developed by Viazowski for the sphere packing problem and her collaborators tells us how to construct certain functions with prescribed data of f and f hat at nodes of interpolation, like square root of n and square root of 2n. And just to give you an idea, this is, I'm going to tell you what's the extremal function in this case. We take z again in the upper half plane. You have the three classical teta series here, teta zero zero is just this one, teta zero one is this one with alternating factor and teta one zero is this one. So the classical teta series, and you have this 24th power of the Dedekin eta function, which we call the discriminant function, delta of z. So consider these four functions. And then you let a function psi to be this guy, teta zero zero to the four, teta one zero to the four, multiply by teta zero one to the 12 divided by delta, and put your f to be this function. Magically, this function will be your desired function. It has the zeros where it has to have, there is no negative where it has to be, and there is no positive where it has to be. It looks magical, it looks like witchcraft, it is a little bit, but there is a reasoning behind this thing. So once you work with modular forms in this case, once you put the eigenvalue conditions into play, the eigenvalue conditions that need to be satisfied, what you have is essentially the certain spaces of modular forms of a certain degree is a finite, finitely generated space, it has certain generated in this case by five elements. And when you put the conditions of the eigenfunction that you need, and some decay conditions at the cusps, you essentially solve a system and find that this would be the only function possible in this five-dimensional vector space that could do the job. Then you plug in and prove that it actually does the job. Okay? So when you see it for the first time, it's a bit magical when you start to read and understand, there is a lot of insight and what goes here. But bottom line, these are, have been very hard problems. The sphere packing problems are even harder ones. So this is a little bit of a story on the sphere packings. It has been solved in dimension one. It's obvious in dimension. So this is a problem of packing spheres in the space to cover the highest possible density. In the plane is when the circles align their centers as an hexagon.  This was, the score was about 90% of the space and the score was about 75%, 74% of the space. This is the density I'm calling delta three. And the connection to these problems on sign and certainty comes from this, this problem proposed by Conan Elkis and independently by Dimitri Gorbachev in the early 2000s. Let's call it the linear programming extremal problem for sphere packing because they found this way to give bounds for the sphere packing via a problem in Fourier analysis. So their problem is this, this problem here. Consider a continuous function which is not identically zero and such that let's say G and G hat are even real value and integrable functions. G of zero is equal to G hat of zero. This is positive. Now it's a bit different from our original problem. So G hat here is positive everywhere. So it's a positive definite function. And G is just negative outside of a radius R. So whenever you can construct a function G with these properties for some radius R, your density of sphere packings in dimension D is bounded by this number. So the idea of course to get the better and better upper bound is to find this R as minimum as possible. And this is exactly what they did. So they realized that they could construct functions like this by taking, you know, you can take a base, Hermit bases or Laguerre bases of polynomials times Gaussians. You can put the first 100 coefficients and try to optimize. And they were getting functions in this paper by Conan Elkis that we're doing a very, very good upper bound, very close to the density of the best known packings in dimensions eight and 24. In dimension eight, you have this E8 root lattice. In dimension 24, you have this lattice called leach lattice. So they were getting very close. And they conjectured that in dimensions eight and 24, one could find a function G exactly with this solution. And this is what the Vyazovsky did in dimension eight and Konkou-Marmille-Rachenko-Vyazovsky did in dimension 12. This problem is even a bit more complicated than the sinus certain in the sense that whenever you solve this one, you note that I put here a function G because if you define F to be G hat minus G, then this becomes a function in our class A minus one of D. It's a function that it's a minus one eigenvalue and it's F, it's eventually non-negative. Okay. So the solutions come exactly from these two papers. Okay, I hope I am not overly technical so far. So the last 10, 15 minutes here, I want to talk a little bit about the different point of view that we are looking into this that I'm calling generalized sine free uncertainty principles. Okay. So we have discussed the problems related to eigenvalues one and minus one. So when we started to think about this, Oscar and I, I remember we call you that this is all that I'm going to talk about in the last minutes is this based on this joint work with Oscar Quezada Herrera. We discussed, we started discussing this last year, maybe in August or September, and things got a little bit slow because now there are kids in the picture. But our original idea was how can we, you know, find suitable problems with the other eigenvalues? There should be somehow sign uncertainty for the other eigenvalues. It should somehow be related to odd functions instead of being related to even functions. And then our goal moved or evolved to a situation that we wanted now to investigate the situation where now the signs of f and f hat resonate with the given genetic function p at infinity. So I'm going to give you a given genetic function p and I want the signs of f and f hat at infinity to be the sign of that function p. And I'm going to put a competing weighted integral condition for you to have a nice problem. So everything that happened before will just now be the case p equals to one. And I am going to make an adjustment on the definition of actually eventually no negative functions that I'm going to allow now for functions to be actually measurable. You're going to take a measurable function g. I am not going to identify functions that are equal almost everywhere anymore. And I'm going to say that the measurable function is eventually no negative. If it's no negative for sufficiently large x, but for all the points and not almost all the points. Okay. And I'm going to define the radius of g as we defined before. The infimum of the radii are such that g is bigger equal than zero from that moment on for all the points, not almost all. Okay. So now let me tell you a little bit about this function p that we are allowed to put the very minimum that we can ask for this function p. So p is going to be a measurable function such that it is locally integrable and it's going to be even or odd. I want to take even functions to talk about some eigenvalues and odd functions to talk about the other eigenvalues. So this little funky tau here will be zero or one according if the function is even or if the function is odd. So these conditions p1 and p2 are the minimum ones and I will assume this throughout the rest of the slides. Let me assume two other conditions for the moment, right? So I'm going to assume this condition p3 that p is annihilating in the following sense. If you have a function integrable, which is a continuous eigenfunction of the Fourier transform, suppose that p times f is eventually zero, then the function is zero. Okay. This is a bit, you have to digest a bit this definition, but this happens for example, if the set where p is different from zero is dense, okay. If the set where p is different from zero is dense, then you essentially get that f will be eventually zero. So we'll have compact support. You cannot have an eigenfunction with compact support. And the condition p4 is that p is homogeneous. So suppose p is homogeneous, there is a degree of homogeneity that I'm calling gamma and gamma has to be bigger than minus d because the function has to be locally integrable, particularly integrable on the origin. Okay. So let's start with these four conditions on p. With these four conditions, I can pose the problem. So the problem is this. Consider the following family of functions. Now there is the dependence on this sine function p. p will be at the same time the weight and the sine. So I'm talking about functions which are l1, continuous, real value, and they are even or odd according to p being even or odd. Now I'm going to ask that f hat pf and pf hat are all integrable. I'm going to ask these integrability conditions. And I want pf to be eventually non-negative and this pf hat times s minus i to the power to be eventually non-negative. So these functions are eventually non-negative and the competing condition is of course that their integrals are non-positive. And you can play the same game, define this minimal constant aspd to be the infimum over this class of the radius of this guy times the radius of this guy. So it's a little bit to digest. It's in principle it's not clear that this class is non-empty. It's already a nontrivial problem to verify that the class is non-empty. But let's try to move on. Let's forget that for a minute and let's try to move on. So the first things that I want to do is to play the same trick that we did before to reduce to an eigenvalue problem, to an eigenfunction problem. So you can do scalings. So when you do scalings, since your function is homogeneous, the sign of the function doesn't change with scaling and then the integrability doesn't change. So this uses the homogeneity condition before you can align the radii. You can assume without loss of the generality that the radius of pf and the radius of this other guy are the same. And then you sum the functions as you did before. You consider w to be the sum and this will be non-zero. This will be non-zero because of that annihilating condition p3. And therefore you will be producing a function that does a better job. The radius of p times this function is less than or equal than the radius of these two guys. So when you use these things, you reduce your problem to an eigenfunction problem. So I'm going to put the star here to denote the eigenfunction problem. And this is now. You have a sine s, s is one or minus one. You have a function p, which is both the weight and sine. And you have dimension d. I want functions which are integrable continuous, real values such that f hat is given by this. So you see here all the four possible eigenvalues appearing. So f hat could be s i to the tau f. I want pf to be integrable as well. And I want pf to be eventually non-negative, but it's integral to be non-positive. And the question is find the minimum radius such that that phenomenon can happen. The previous problem was just the case when p was identically equal to one. So when you arrive at the eigenfunction problem, this eigenfunction problem makes sense even if you don't consider those conditions p3 and p4. So you may forget those conditions and just write with the full generality that their function p is just locally integrable and that it's even or odd. And let me make an observation here that if you take two functions that are different, that are equal almost everywhere, take for example a function p1 which is one everywhere and take a function p2 which is one everywhere for all x, let's say we are in dimension one, except on a sequence of points. And in this sequence of point you change the value to be minus one. And this sequence of points is a sequence of points going to infinity. Now any function f that belongs to this class for this function p2 will have to have product of ef times p being non-negative. So then at these points it will have to be actually zero. So this function will have to have zeros at these points an that you change the sign. So even if two functions are equal almost everywhere, they do not generate the same problem. Even if your function p is zero almost everywhere, you might be talking about a non-trivial problem here. So this is vastly unchartered territory. So this is the problem. Let me give you some examples of what we did. So let me recall the problem once more. You have an eigenvalue problem. You have a function p of your choice, just locally integrable and even or odd, a dimension d and a sine f. You want a function which is integral continuous and which is an eigenfunction in the appropriate sense. You want pf to be integrable, pf eventually non-negative and whose integral is non-positive. And you define, you want to find the minimum radius. For example you can take this function in d variables p of x1, x2, xd to be just x1. You can just integrate it against the first coordinate x1. That's an allowed function. And we will show for instance that the solution of this problem with the plus one sine in dimension 22 is 2. Here's another example. Take a function p of four variables given by this crazy thing here. It's a polynomial of degree 4 in its variables. This is also a function that you can plug in here. And we will show for instance that the solution of this problem with the sigma plus one in dimension 4 is square root of 2. Okay. I am a little bit past time but I think I got your attention a little bit with this. So I'm going to allow myself if Edgar allows me to go five more minutes to show a bit of what we did here. Can I, Edgar? Are you still there? Okay. I'm assuming you are. Well, I will say randomly. Yes, you can. Okay. Good. So this is a bit of our paper on this topic has two parts. In the first part we follow the classical path to obtain the sine uncertainty principle. The classical path meaning that we try to put the original and the methods of Bourgan, Cajun and Closel into a very minimal and genetic framework to make things work. Okay. So this includes two sorts of results. First, there are results that guarantee that the class of function that I'm talking about is non-empty. Okay. There is another type of result that guarantees that my function p is admissible. Admissible in the sense that I have some sort of inequality that replaces the Hausdorff-Young in that sense. I have an equality to leverage this norm of, this L1 norm of pf. These two ingredients when combined they will yield the sine uncertainty principle. So this is an example of a non-empty class. So assume that your function p, remember everything that I talk about from now on or all the time. I'm always assuming that p is locally integrable and that it's even or odd. These are the conditions p1 and p2 before. Now assume that your function p is, you can integrate against any Gaussian. When you multiply by a Gaussian, this is integral. Now assume that p is a harmonic polynomial, harmonic homogeneous polynomial of degree L times a function q, which is eventually non-negative. So you can take any homogeneous harmonic polynomial of degree L and you can take any function of your choice q, which is just eventually non-negative. Then the class is non-empty. So this is a lot of examples where the class is non-empty. In particular the examples that I gave you. Now, admissibility is what comes in here as a replacement to the Hausdorff-Young that one had before. And it essentially is a condition like this. Maybe I don't know if I can highlight here. I want that my function, I will say that my function p is admissible if for any eigenfunction of the Fourier transform I have that pf in the L1 norm controls some Lq norm of f. So there are some results like this in the literature where you have with the two norm controlling the two norm here where p is a quadratic polynomial. But these are kind of rare. This is not very tricky. This is a bit tricky to find. But I'm going to call that this is the admissibility condition. And another one of our results is a sufficient condition for admissibility.  So suppose there is a lambda such that the set of points such that p of x is less than or equal to lambda has finite lab bag measure. Then this function p is admissible in this sense. But this is a sufficient condition for admissibility but not necessary. There are functions which don't have sub-level sets of finite measure which still have an inequality like this. And one example is that function p of x to be equals to x1. Just the first variable. So when you put here p of x to be x1 there is this inequality here when q is equal to 2. So morally speaking the first theorem is just if you have a class that is non-empty with a function p that's admissible then there is the uncertainty principle holds. There is a constant c star such that this is bigger than c star. C star depends just on the function p on the dimension d and the exponent of admissibility q. And this is as I told you pushing the techniques the previous techniques to a limit or a near limit. But even with this you can generate a lot of examples you know. There are a lot of examples where the class is non-empty. Anything that has bounded sub-level sets or sub-level sets of finite measure is also admissible so you get in a good shape. The final result and this is probably where I'm going to stop. It's actually different so this is the second part of our paper here. It's a different mechanism to obtain the sign uncertainty. It's a different method than the one conceived by Bougain, Cajun and Closet in the beginning. And we call it dimension shifts. So there is a way that you can relate two different dimensions to relate the uncertainty principle in two different dimensions with two different functions. And this is very powerful in some cases. I'm essentially going to stop my talk here reading with you this this statement. This statement is the following. Take L an integer bigger equal than 0 and R of L or tau of L will be 0, 1 is just the congruence class of L mod 2. So start with the function P in RD plus 2L that is radial. Write P to be P0 of the radius. Now consider a function P tilde in RD which is of this form. P tilde is the radial version of P times a harmonic polynomial H of degree L and a function Q which is even non-negative and homogeneous of degree 0. So the first part of the theorem reads the following. If the original class for P is non-empty in dimension D plus 2L then a certain class with another signal here for P tilde in dimension D is also non-empty and you have this inequality here. The thing that you started is bigger or equal than this thing. In particular if this thing here is bigger than 0 if you can prove this then you're in good shape for this one. So this is particularly useful when you can use this construction to fall in the previous situation. So this is what's most surprising, this last two lines. If in some cases you can come back, you can do reverse engineering in this formula. If P has a bounded sub-level set, this is bounded meaning that it's containing a bounded ball. It's not just that it has finite measure. If P has a bounded sub-level set, if Q is equal to 1 and if H is this polynomial x1, x2, xl, it's in the orbit of this polynomial with the action of the orthogonal group then you can the equality here above holds. You can come back. So there are two quarter-layers of this result that I want to mention and to finish my talk. First, suppose that you start with a function P in that dimension. Let's say dimension 100 going dimension 100 and you have the function P which has a singularity. So this is useful when P has a singularity. So let's take P to be x to the minus 20. So if you have a function with a singularity x to the minus 20, it's very hard and it's not even possible to find those admissibility inequalities when you have a decreasing weight with a negative power x to the minus 20 dimension 100. So what you do here is you take this construction, this, you take this H to be just a harmonic polynomial with two variables of degree l and you take this Q to be this function here. Modules of x to the l the same degree and then you take sine H over H. When you multiply your H will cancel, you'll just have x to the l to add to the singularity at P. So if you start with the gamma singularity, you have gamma plus l, you lower the dimension, that's the price you pay, you lower the dimension, but you get a better singularity.  You are in position to use the theorem before, but now in dimension 60. So you're good. So you find an uncertainty. So this is how you shift the dimensions to get the uncertainty. And of course, this equality case when the polynomial is harmonic in the orbit of this guy is a very surprising result because then you can connect those four cases where we had the solutions, the solutions of this coming from the sphere packing and this new solution coming from the paper of Conan Gosawis and those four solutions, they generate other 14 sharp constants. And this is the new sharp constants in this, that all the sharp constants that we know about this problem. So you take just r is just a rotation is a guy in the orthonom or orthonomal group, you can just apply it in the in this polynomial x1, xl and you can take dimension 8 minus 2l. So a with this sign, with this polynomial in this dimension, it's going to be square root of 2. This comes from one of the results before. So you have here three cases plus five plus nine, you have 17 cases, but three of them are the dimensions eight and 24 of sphere packing in the dimension 12 of Conan Gosawis. And then you kind of lower the dimensions and you get some polynomials here. So the crazy example that I showed you a few slides before of a big polynomial in dimension four was just this case here, where you take l equals four. So if you have l equals four, you are in dimension 12 minus eight, four, and you have a polynomial which is x1, x2, x3, x4, and I just apply the random rotation on that guy to make into that very big polynomial that I showed before in the answer is square root of 2. Moreover, we have a mechanism to transfer the extremizing functions that we had before into new extremizing functions in these cases. There's a there's a way to to write explicitly one from the other. Okay, I guess this is time for me to stop here. I hope this was not overly technical, but I know it was a little bit, but it's a bit of an answer. My talk is a bit of an uncertainty principle itself. I have too many things to tell you and too many details to tell you, and I can only do one or one or the other nicely. Thank you very much for your attention, everybody. Okay, so thanks a lot, Manuel. I think it was a great talk, really, and so if anyone has any questions, I invite you all to please raise your hand in or write in the chat if you have a question, and I will call up on you to lend you to your mics in order. So, let's see. If anyone has a question, just write please. I may just just say one thing and not really a question, but Manuel, any time you want to ask me for four more minutes or 40 more minutes for you to talk about mathematics, don't even ask. Thank you. Thank you very much for this very next talk. Thank you.


Rotterdam. I'll talk about my work through education institutions while they are one inside the app that I work for in Rotterdam and where we've been in the efforts to work with Floss and hybrid and experimental publishing so I hope one of those places is the publication station It started initially years ago inside of the microphone. So, in English it started something with the antenna. So, the publication station was essentially a print workshop for many years. An all-of-print individual decoding academy in Rotterdam where there was and still is letterpress workshops, silkscreening workshops, lithography, resort print, etc. And in the last two years there was an intention to move more towards electronic digital publishing. This is an image of our wiki. It's a little conversation address but we can quickly go there. But we have an idea, so publicationstation.philipial.htr.anel slash wiki. And try to type that in this amount of time and get the price. And essentially we've been building or like we have an analog print workshop, also building the electronic publishing resources, so wiki, our own server. And recently there's a device lab which is a place with many, many e-readers and art forms where students and teachers can test out publications and forms and formats, etc. So, one of the goals that we set on from the beginning last year and a bit quite influenced by the research that was done by the digital publishing toolkit project that was called Friedler and he mentioned. So we see the similarities of the icons. Oh, this is a bit, yeah, it's not expecting that. It is smaller. One of the idea, well, the principles that we try to do with our efforts in hybrid publishing is ask ourselves and really try hard to see if we can, with our own needs, publish across different formats. And it should be done quite cheaply and without much fuss and without hiring external companies and outsourcing the work. And essentially the idea that was mentioned by them before that there's this multiple formats that write from one single process and one single source. One of the interesting things about that is the connection between source and output that should be in both directions, it should circulate in both directions. So what happens is that ideally this should foster a dialogue between content creation process and design processes so that one and the other can talk to each other and then graphic designers wait for the content to be finished so that they can start producing what will become the book. And another interesting thing, I'm really sorry for the resolution but maybe I'll try to give you the slides in some other format. But another effort was also to try to find out other formats. And this is what you're seeing, it's a book trailer. This is something that Michael Hortons of the Russell prototype and it was the idea of a script that opened EPUBs and looked out the metadata and all the images and let them out as an animated gif so we can get a quick overview of the book. So this is also a little bit, I think, an answer to the presentation that we had on the first day on Friday. The students here that were talking about their use and difficulties with using free software or quicker work and why they use it and what's a little about it. Why do we use force in these processes is mainly because the tools that we're using are really doing stuff that I don't see proprietary software being able to do or even interested. So, for example, Bandog, the Markup Conversion Wheezy print that is quite an interesting layout tool, I think by the library, that works from HTML to PDF using CSS and reading comments as well with the CSS specification, make files as kind of recipe books, open source fonts, et cetera. And of course, another interesting aspect is how to integrate in the same process of various contributors and collaborators so it becomes quite important there and this is something that is still not completely able to communicate with bachelor students and students that are collaborating it as a project but we see it as incredibly important. And this idea of small pipeline kind of workflows where programs that do one thing like Bandog that converts and then Markups and Wheezy print that produces PDF in the end that just do one thing and do one thing well and can be recombined with some simple recipes and expanded and changed, et cetera. So, some of the projects that, no, it's going to be tweaked open text. So, one of the projects that we work on is still working on. It's an online magazine about social design and we decided to work with a Wiki as a... Oh, thank you. So, one of our ideas was to have the contents of that magazine on the Wiki so it could be edited by anyone, any collaborators from any part of the world. But Wiki's favorite rightability instead of readability is not the best format for reading on the screen. So, I decided given that there's a web API for the Wiki why don't we simply build a frontend based on HTML templates and then using Bandog you use those templates and you check the HTML that's converted and something that looks like this on Wiki then very easily can look very different like this form. And then, yeah, we continue using this formula to run other projects. It's a fairly simple formula but another one was the Media Design Graduation Show catalog where you have Wiki pages again with the student words and the info boxes and images. So, the same kind of Wiki page when it was placed on a frontend then became something, this is starting to break the website, something like this. So, this doesn't have a lot of content and maybe it hasn't been updated because the ad place goes through the script. And there's also the refugee phrasebook which was an interesting project that dealt with, yeah, essential phrases for refugees and it was translated from many, many languages and see how we could do this on a print publication and work with students in doing so. Which was quite interesting challenge especially working with fonts and other alphabets and non-Latin alphabets, the device lab, et cetera. So, yeah, thank you. So, that's how we try to get these projects going and try to stimulate hybrid publishing, experimental publishing, thinking about what publishing is in a broader sense. Well, we try to use the Wiki that I showed you previously as a site of documentation where every single process, every single experiment that we do is documented and hopefully used by others. I think that the same process that we're using putting Wiki content onto another frontend was used by hackers and designers of the year and they used the same ideas and a bit of the same code and did this similar thing for themselves but of course they changed it and really made quite an improvement out of it. Then another important aspect that I think is incredibly relevant is this collaboration between tutors and students and external guests that work together on projects and as peers and not as different levels of knowledge. Yes, and the integration of flaws in the curriculum. So, for example, this year I studied teaching first year graphic design students, web design, so it's at HTML, CSS, and JavaScript. And the reception for the majority was quite good. But of course this all comes up from a lot of blood and tears and nothing is really easy. But I think we, well as educators or as developers or as artists and designers that use free software and alternative tools for doing our work and building those tools, I think we should challenge the assumption that artists and designers should be kept away or are afraid of text-based interfaces of code programming and maybe try to find ways on how these two universes could work together and find points of contact and interest. Because my reaction in teaching 18-20 year old students web design, they are very open. They don't take it as, I don't do this, I'm a graphic designer, I work between design and rules training. No, they go through it. You open a text editor and they start writing HTML and then you write in CSS and you have to do it with an interaction in JavaScript and from there on I think we can also pay for those for more interesting work. I haven't said that. I'll just move slightly. So this is in the same city that and the Pittsburgh Institute which is the post-grad institute inside the field of academic is next year in 2007-2016 starting a new program called Experimental Publishing and I invite you all to visit our website that we are building the curriculum and developing what will be the program next year but of course this program doesn't come out of nothing in having so many people here that either were part of the what was once the network media and I'll go back to the chronology and at some point became a media design and experimental publishing. I think it's quite interesting to refer this but just to give you a brief overview of the core of this program that moved I think it started in 2002 correct by Femke, as network media masters. I think Michael also became a part of the program into teaching quite early on. At some point Florian Kramas continued the program and in 2012 it merged with another program and became media design and more recently now we're having experimental publishing and what will be experimental publishing. It will be a master's program where we want to take the notion of publishing as a medium not stick perspective and really look into it at these different dimensions technical, social, political what are the implications of an act of making something public whatever it means if it's via whistle-blowing publishing fancy culture what are the reasons for these forms of publishing to exist today and what are their social sphere and what are their relevance and importance and challenge actually the notions that are taken for granted nowadays about publishing about new digital and what's going to change and really try to disrupt and play with these existing notions and try to not stick it for granted but actually experiment with it and then of course building a lot on the tradition from media design and network media where flaws, tools are very important and code are an essential part of the work process and I think it will be quite an interesting and challenging program so the applications are open I think until one of May I'm not wrong for European so if you know anyone that's interested in sharing the self promotion we'll be also very welcome to answer questions later now about the program front to approach the Michael Montau there's also some current students so yeah, I'll wrap it up here thank you