Ant, are you going to get a handshake? Great. I don't know if. Thank you so much. I woke up this morning and I saw Remkis' check-in, and it said something like, welcome today to a word camp. Or as I like to call it, how am I still standing? So that's pretty appropriate for the day. So yeah, let's get started. Last month, a few colleagues and I, we visited a potential client here in Europe. And it's a brand name you all know. It's not uncommon for our business to meet with these kind of larger clients because we deal in enterprise work place. That's our space, right? And the purpose of this visit was a two-day workshop to refine the requirements that this business had and for us to be able to better explain how we would approach the execution of this project. Just click right here. And at the end of the second day, the project lead, who's also the product manager, asked me if, in my opinion, WordPress was the best solution for them. That's a bit of an awkward question because I'm obviously here to sell WordPress. But, you know, I was a little taken aback. But however, a question like that that demands trust has to be answered with honesty. I'll come back to that question. We all have our own personal WordPress origin story and it's great because of how wildly different they are. Not only have we all come here from very disconnected or unconnected sort of backgrounds, but even the work we end up doing in WordPress itself is very wide-ranging. My little story goes back almost 10 years when I was looking for a CMS through which I could template the designs I had created. Even though I had created my first website in 1995, it took a while before I turned my hobby into a full-time job. And once I did, I started creating small business websites. I continued building my theme shop. And, you know, I was just trying to figure out how to bring value to businesses.  And I have a lot to thank WordPress in that regard. It took me a long time to realize back then that WordPress itself is not actually a product. Sure, it's the actual piece of software, but if you want to attribute growth and success, then the real product was and is WordPress.org. A place where developers and site builders meet to exchange plugins, themes, reviews, ratings, feedback, discussions. And this WordPress ecosystem, that constellation gave me, you know, personally the feeling that with all of that, you could sort of achieve anything, right? You could build almost any use case without ever touching code. And in a weird kind of way, it felt like this unfair advantage, you know, compared to other businesses and agencies in the space who are selling other CMSs. And I think many other people felt that power, right? That with WordPress, you can do anything, right? And the realization I'm making now, however, quite a couple of years later, is that the work I was doing then isn't necessarily competitive or isn't necessarily considered competitive or affordable anymore by today's terms. The environment has radically changed, right? And continues to do so. But the key point here is that, at least for me, is that WordPress, as a consumer-facing brand, has become less relevant. It's transitioning from being this all-in-one solution to being this tool in larger technology stacks. It feels like it's out of the spotlight and, you know, given this reduction in exposure, I'd even go as far as to argue that we've reached peak WordPress and that that moment has come and gone. But then, and I can see it in your faces, you say, but no, WordPress is at 31% market share. It's the highest it's ever been. How can that be? I'm not disagreeing with the fact that WordPress is installed on 31% of domains. But to say that WordPress powers 31% of the web is stretching the truth. The CMS isn't necessarily the epicenter of the web anymore. You don't really have to have all your functionality inside of that. You know, you may use it for a bit of content marketing or a part of your website. You may use it just for a couple of landing pages. The use of WordPress today is very diverse. We need to think differently of how a product as mature as WordPress will continue to grow alongside this rapidly evolving web. To me, there are three general trends, I'd say, that stand out and are changing how WordPress is perceived and being used. As I mentioned before, I think the plug-in ecosystem is where a lot of the growth occurred. But now the web is kind of catching up. There's a lot of other solutions out there which are contributing to the kind of solutions and websites or web experiences or digital experiences we're building. Let's take this oversimplified visualization of requirements, right? On the left, we have simple websites. You know, that could be a couple of pages, contact forms, 14 or social media links, just a regular small business website. And on the right, we have more complex applications, e-commerce and what have you. And if we assume that WordPress can do anything, which it can, then the color blue represents WordPress. It fills the entire spectrum. Let's go back eight years or so. As we were, you know, as this plug-in ecosystem was doing very well and we were starting to, at least I was at the time starting to build a lot of WordPress websites. White represents the sort of solutions which are likely to be maybe more cost effective than WordPress, but still bring the same kind of business value, right? So on the very left, we might have the YOLO websites, right? So that's when you redirect your domain to your Facebook page and you kind of call it a day. And then you might have one or two website builders and that was kind of it. There wasn't that much action back then. In the middle, you have this sort of, yeah, you have some maybe some more specialized kind of website builders, maybe for like real estate agents or other kind of industries. And you also have this evolving market of software as a service, right? Specialized tools. And at the very right, you have enterprise, right? So like Magento or whatever, just being there, as usual. So in this scenario, we have WordPress, which is still very dominant, right? You see lots of blue and it fills lots of spaces and it's still a very useful and very consumer facing tool. If we fast forward to today, we see a lot more players enter the market, right? We have website builders that have TV ads. Some of them even have Super Bowl ads. We have plenty of website solutions for specific industries. We have lots of specialized software as a service, essentially providing more depth and more UI than a lot of WordPress plugins can. And we still have the enterprise, right? And WordPress can still do everything, but is it the most competitive solution then? That's the question. If we wanna take a slightly more scientific approach to this question, we can look at consumer trends over time. Here we're comparing WordPress, again represented in blue, against the mixed bag, Shopify, Wix, MailChimp, HubSpot, just various online solutions which are targeted to the business. We see here that WordPress in 2011 experienced somewhat of a peak for interest and then started to become less relevant to consumers directly, whilst other solutions increasingly grew in visibility. If we add up those other four companies in 2011, it's no match for WordPress, but in 2018, just those four companies alone already exceed the interest of WordPress as a topic. What's changed? For one, we're all using a lot more specialized software, right? We might have managed emails in WordPress someday in the past and then further down the road, we might have said, okay, we're gonna start using MailChimp because it has a lot of features. And now we're so far along the road that even MailChimp isn't enough. We might have one tool for newsletters. We might have one for transactional emails. We might have one for e-commerce and another for abandoned carts. And the list goes on, right? Specialization is increasing. Second, SaaS solutions are able to expose more complex business logic and solutions to non-technical users. So someone in a marketing department doesn't have to call a developer, you, me, or someone else to make a change, right? They can make that change directly in a tool which is completely closed off of code or anything like that and is geared towards business users. And to be honest, that makes sense, right? I mean, I look at a couple of use cases when let's say a friend comes to me and says, hey, I need a small business website or whatever, you know, what should I do? I mean, you'd think I'd say, well, you know, go on WordPress but I don't wanna deal with their problems, right? I don't wanna do tech support for them.  So they're even defended against that, which is good because then there's also full documentation, full support and all these kind of things. So this layer of consumer facing solutions have really come a long way. If we were previously, you know, people who assembled websites for businesses and the technology was WordPress, then we kind of maybe had this hybrid product management role, right? Because we were managing email or marketing or landing pages or conversion rates or analytics inside the WordPress site and now we might have a more specialized role that's geared more to the core of WordPress. So I think that this generalist role is fading, right? We're now either WordPress specialists or marketing consultants or product managers and a few people are all around us. But that even doesn't scale to very complex projects where there's an increasing amount of specialization. And this is nothing new to the web, right? 20 years ago, we were all web designers. You know, this half of the room used Dreamweaver and this other half said, yeah, I know we write, we know we're cool and we write perfect, semantic, clean, HTML4 inside notepad, right? And that was it. And now it's such a diverse room. There is no such thing as a static role, even in WordPress, right? So we need to evaluate our roles and how we interact with WordPress and what is the value we bring. Another big topic or trend that we've seen in recent years is when products and services mature, platforms are starting to be built around them, right? Modern platforms are the applications that manage to connect two or more players and facilitate the exchange of value between them. So Airbnb, for example, might be the largest facilitator of apartment rentals, but doesn't actually own any apartments, right? It connects the apartment owners, right, the producers, with the guests, the consumers. And that is essentially what we've seen so much of when we hear the word disruption today. It's often about new platforms shaking up old industries. And this is also a large part of WordPress. And platforms in almost every sense always have two things in common. They own the transaction and they own the customer. In many ways, WordPress.org was ahead of its time in that regard, right? I mean, when I joined WordPress.org, there's already tens of thousands of plugins. The amount of interactions or transactions that occur on a daily basis are insane. Examples of transactions can be downloading a plugin, leaving a review, rating a plugin, rating a theme, providing feedback, or whatever. These are all valuable things that happen on a singular platform. WooCommerce is another great example, right? But this is with financial incentives. It's a marketplace where developers can earn revenue from connecting and providing solutions for shop owners. And we don't only see platforms evolving around plugins and themes, we also see them evolving around development work, right? This is Codable, which many of you know. Having grown their company alongside WooThemes, WooCommerce back in a day, and now being the sort of job work marketplace for the broader WordPress ecosystem. And this creates more confidence for the buyer, right? Let's take these two examples. Both these people on one of these job platforms has five stars. One has, I think, five reviews, four reviews, four finished projects, sorry, and the other one has 653. The same applies to plugins and themes, right? You decide in a moment which plugin you'll go with or theme or even developer based on very simple things. The platforms around hiring WordPress developers does lead into another segue, which is this topic about the commoditization of development work, right? In the future, it's picking the developer, or a developer, sorry, the same as picking your Airbnb host or picking which restaurant you'll visit in Belgrade this week. I mean, I know myself, I'll open my phone and I'll see 10 restaurants that are close by and I'll take the one with the highest rating. And that's, I mean, it's extremely lazy, but that's what I do. And a lot of people buy like that. So platforms like these are natural evolutions of our industry. It's normal that development work will face increased price competition as the market matures and as the product becomes more stable. And when, so we've talked about plugins, we've talked about themes, we've talked about the sort of job marketplace, so development work. Hosting is another big topic inside WordPress. I mean, just looking outside in terms of who the sponsors are, we can see that that is where a lot of the revenue is. And part of the reason I think that WordPress has lost some of this consumer facing appeal is because we don't have enough consumer first or business first WordPress platforms in the ecosystem. Many of these existing WordPress platforms are just hosting companies, right? Strong onboarding, strong UX, an extremely well curated marketplace of third party plugins for core businesses and marketing needs. These are the opportunities. If you want to reach customers directly, one click install and access to a C panel dashboard, it's not good enough anymore, right? You have to provide a fully integrated solution. And most of us aren't building platforms, but are affected by them in some way, even in WordPress, be it plugin theme developers, agencies, freelancers, or any of us others. The theoretical answer to safeguarding ourselves against these layers that own a lot of the business is to own the customer relationship, but that's not necessarily the easiest thing to do anymore, especially as WordPress is also getting pushed deeper in the stack. And the conversations that happen at the very beginning happen somewhere else. So in practice, it's again a different story. We have to settle for the next best option, which is diversification. Not attaching your inbound business or sales or however you generate revenue to a single platform, be it a theme marketplace, plugin marketplace, WordPress job marketplace, or any other kind of platform. We have to diversify and find different ways to earn revenue. The opportunity here for us is to learn how to market, sell, and talk about WordPress in a manner which speaks to the business value. The third and final point I want to talk about are digital experience platforms, or DXP. If you haven't seen it yet, it's being pitched as this sort of next step or evolution of CMS, mostly a marketing activity so far. The theory is that by using one of these platforms, you can reach all your customers across different mediums, devices, whilst also doing so in a highly personalized manner. In other words, content is tailored to the user's profile, habits, behavior. One of the key marketing points of DXPs is also that it's not web-centric, right? So in most marketing material, that's visualized with smartwatches, voice hardware like Alexa, digital signage, and more. The CMS, however, remains a big part of DXP, but it isn't necessarily the driver, right? It depends on the vendor, but the sort of wording that is taking center stage here is AI-powered, intelligent, multi-channel, personalized experiences at scale. I can't even make these things up. In the above slide, the company is Acquia, and many of you know this company as the enterprise vendor of Drupal, and they've been very successful. For a long time, it's just been a CMS, and that's been the marketing, but now it's sort of shifting to this digital experience and trying to own the entire digital strategy of various companies. This is Cycor, another prominent enterprise CMS, but you also see this digital experience language coming in. I'm personally still trying to figure out if DXP is a marketing play to try and displace CMSs. I'm seeing a lot of big talk, but the case studies and detailed implementations are somewhat lacking. I can see how large retailers have built their own custom solutions, let's say like Walmart or Target, which are able to provide very high degrees of personalization, but I don't see how a DXP will plug and play into your business if you haven't been doing these kind of activities before. It requires a lot of prior business intelligence. However, I do believe that DXP will succeed in its marketing message. It is being marketed very well after all, and that enterprise will start spending some money on it. And I like the above landing page because it's telling us what DXP is today without kind of realizing it. You'll see in the bottom left this says content management combined with deep customer insight. To me, that means CMS plus AB testing at the very lowest, and at the highest, it's something that's very abstract, which might just all be marketing. Regardless, it affects us. DXP is a concept that's specific to enterprise today, which not everyone in this room works with, but the underlying motivation here is for solutions to become smart and intelligent. Instead of using if-and-else statements, we're going to start using large sets of data to predict what the user wants or more directly how to increase engagement, conversion, and overall business value. If we tied us back to my first point about how plugins are losing value, in many cases, this is because these SaaS solutions are also becoming more intelligent. They're also utilizing large pools of data to make smarter decisions in an automated manner. I see this as an opportunity within WordPress and one that I hope we will see a lot more innovation in 2019. The idea of creating microservices or tools embedded inside of WordPress to help create a more intelligent CMS. You see decoupled and head-to-less WordPress projects are quite popular these days and we do a fair few of them at Human Made. But in terms of vision for WordPress, the software itself, I don't see the visual or presentation layer as a big priority. That feels like it's someone else's specialization or responsibility. Instead, I think we have to focus our attention on automation, machine learning, and more predictive workflows for the end user, especially in WP Admin. If there's a specialist tool for every business need inside a tech stack, then WordPress specialization is content. Ingesting, creating, editing, analyzing, manipulating, enhancing, distributing content. And that's where I feel our attention should be in helping creating a smarter WordPress. Yesterday, when Matt was asked about what is his favorite feature of Gutenberg, he talked about the copy-paste feature. I think that's a smart solution because another way to have done that would be to say, okay, there's an import button, you press on the import button, you paste your text inside, and then you say, okay, what format am I importing from? So maybe you select that from a dropdown, and then you click import and it pulls it in. Instead, the system automatically tries to detect which format you're using and then adapt it and import into Gutenberg by itself. That's a smart solution, it saves people time, it creates value. At Human Made, we've also been playing around quite a lot with images and media. This is an example of us using the recognition API from AWS to automatically tag images inside of WordPress. So if there's a large amount of images inside of WordPress, this is quite useful, all the way from editorial to SEO use cases. And the third example I have is from Amado Vice, who does do quite a lot of work in this. I'm not sure what's more impressive is if it's the actual smart feature he's created in terms of spelling correction or his actual facial expressions in the bottom left once they start kicking off, what you will see in a moment. So now he's typing some text, he's very surprised. He's thinking about it. And then maybe he'll accept that and he changes that. This is quite smart. And that's a small thing. There's a lot of ways that can grow. So if you're a plug-in developer that's feeling the pressure of SaaS solutions competing with you, then this is where you need to go. If you're a hosting company who's offering a WordPress-first solution in terms of marketing and you want to improve your platform, this is where you need to go. And if you're already working on these kind of things, then talk to me because I'd love to hear about it. We already have a lot of initiatives that you've been made and I'd love to see what you're working on. Coming back to that initial question last month when that company asked me if WordPress is the best solution, let me tell you what I told them and what you should be telling your clients. WordPress is the best solution. Everything I talked about today are opportunities. They're good problems to have and we're in that situation because we've built great solutions and we have an even greater community. Measuring WordPress growth through market share is a distraction. The real growth opportunity is in being able to drive WordPress up the value chain. It's about creating a smarter WordPress. It's about interoperability and having WordPress play nice with the ecosystem at large. And it's also about value perception and how you market, sell, and talk about solutions using WordPress. How WordPress will grow in the next year will be nothing like the previous years. We need to adapt and take risks. We have to innovate and then we have to do it all over again. That is the future of WordPress. Thank you. Thank you very much, Noel. We do, in fact, have time for Q&A. So let's bring them in. The way that we've got them arrayed here today is we do have a couple of mic runners around. So if you really need a mic runner, do take a wave. There's a few mics at the back and we've got the stands here, just while people are sort of cogitating and thinking those through at the moment. You pointed to Ahmed's work and things. Are there other people, other conversations? Because you're not the only person thinking about this by a long way. One of the best, definitely. But are there other places where this conversation happens? Where you, like Ahmad and other people who you've noticed lately have really got good visions for different directions. Perhaps we've got our point of view, because we're in human-made and other people are in different segments as well. It's not about, it's less about point of view and I think it's more about perception in that people may feel limited to trying and solving things through WordPress or through the old WordPress ways, through PHP and things like that.  You have clients that may benefit from needs like this. There's someone like Ahmad sees that in a new project and says, I can do this. So it's more about your own personal vision and sort of breaking through barriers to be able to see what kind of innovation we can really drive beyond a more linear CMS. Yeah, okay, I appreciate that. Do we have a question there from, oh yes, yes sir. I know, this is Bo. Thank you so much, great presentation. I have a question about the digital experience stuff which is something I've seen quite a bit, like especially in the enterprise. I was just wondering, do you think the potential clients or customers who are below enterprise, you know, like spending maybe, if you talk about hosting or product like 50 bucks a month or 100. Are they there yet, like to get that message? Do you think they're susceptible for that specific marketing spiel or is it just, you know? I think so, no, I don't get that question. I think so because to me DXP is a marketing exercise. Like I said, I haven't seen mind blowing implementations. There's a lot of A, B testing and there's a lot of, if customer bought product A then showed them product A again. There's, you know, maybe payment providers like Clarna or others, you know, large platform players have a deeper vision of these things. But like I said, it's not a plug and play thing. I think there's an opportunity for companies that are completely integrated like Shopify to be able to do things like that and to be able to just automatically do those as part of their USP and as part of their sort of value proposition towards prospective consumers. But I do feel it's gonna be a good chunk harder for people who wanna do custom implementations. But for right now, I feel like it's mostly a marketing exercise until I see otherwise. Thank you. Pleasure. Thank you very much. Another question from Odessa? Yes, please. Hi, thanks for the presentation. It was great. So going back to your graph before where simple versus complex sites, where do you think the biggest area of growth is for WordPress going forward? I'm biased, you know, so I work in the enterprise. So I think medium-sized businesses to enterprise is where it's at. I think there's a lot of consumer-facing solutions at the bottom end, which are, it's not even so much about there's a website builder for businesses at large. There's a website builder for restaurants. There's one for doctors. There's one for lawyers. There's massive saturation at that point. And I guess also from a price perspective, there will be increased pressure. So if you wanna be able to continue to drive customized solutions, I guess you have to try and market yourself towards that midfield or that upper end, because. And how do you do that? Do you market your tool towards the developers that are building those sites, or do you market it towards the enterprises themselves? Is this in regards to your plugin specifically? No, just yours. Well, I think for an agency, a lot of that is just whatever has worked for you before, but scaled up in a much more, I'd say, aggressive manner, a much more outbound manner where you're trying and you're experimenting a lot more. Different markets react to different approaches. But if you try small things, lots of different small things to see what works and what sticks, I mean, that's the lean startup, right? It's going down that road. But otherwise, I'm sure there's channels, right? Like platform channels and all that. So there's G2 crowd, there's clutch, there's various review platforms for agencies. But again, you have a tie-in to platform sense. That sort of diversity you'll have to seek out for yourself. And if you're in any way possible, or you have the possibility to find your own independent channels, go for it. Thank you. Pleasure. I think we have another question there. Yes, please. Yeah, hi, thank you. Just wanted to tell us about something about the security in the future of WordPress, so... Oh, I'm the wrong guy. You're the wrong guy. I'm the very wrong guy, yeah. Okay. I'm on the marketing side. I make everybody feel great about stuff. I don't want to worry people about security. But there's definitely, I don't know, it seems like there's a lot of blog posts around it. There's a lot of documentation and a lot of activity, I'd say. But it seems like everything's nice and secure, seeing as WordPress powers 31% of the web. Okay, okay, thank you. No worries. There is a part of the marketing message, isn't it, when we go and talk to clients. And it has been, I think at the moment... Nope. We try not to. Okay. No, no, no, I'll say one comment to that and it's that I try not to use it because I feel like it's an easy way out in meetings. So if you're not able to talk about what you bring as a solution and the value you bring to the solution and you actually have to fall back to talking about a tool's statistics, then you just have to re-evaluate your business. Kasper? I think that's the gentleman was first. Hi, thanks. Sorry. I just wanted to ask an advanced customer question. Like, you know, in all small, medium businesses, the solutions are vast and you can try for your own and look what WordPress fits you. But when you go up to the enterprise businesses, they're all comparing not to WordPress. They just look if the solution is right for them. It doesn't, it doesn't matter at the cost. It just look concentrate on the solution. So when you're coming to a customer that big, you must have a vast value, something to offer more than it's WordPress and it's good that it will fit you. So what are your ways to market? That's a great question. I think part of the challenge with DXP, and obviously DXP is great from a marketing perspective because it speaks to that, right? It's like, we are your digital strategy. Take all our products and implement it, right? We also were the best. Yeah, yeah, exactly. But the thing is there's a lot of best in class solutions out there. I think WordPress as a CMS is best in class when it comes to managing content. And there's a lot of other tools that are out there that work equally well. And from a DXP perspective, sure, you can combine different tools. You can combine WordPress. You can combine A.B. testing analytics, various marketing tools and everything together to create a best in class, I guess, bundle because that's what the DXP does. The challenge is in marketing that and being able to go to the customer with a concise offering in that regard. Not to show them fragments, but to show them one. And that's our challenge, I think. That's something I'm learning too right now. For my market is mostly concentrating on the open source, open code benefits. Like it's adjustable, you can move all the other benefits, but... And Acquia is like that. When you're going to a real value, I can't shine in a way. Like I know what I can do. I know what I can offer. But in a WordPress equivalent platform, it's not that simple to explain. No. For enterprises. And that's something we have to learn. How do you bring that constellation of different tools under a single banner and a single offering to C-level executives who are trying to understand your value proposition in two lines? So I'm learning it too. Thanks. Pleasure. Yes, of course. Okay, yeah. Thanks, Noel, for the presentation. I still marvel at the timing that we heard the M stock before and how to move all this forward. And then you come and say we hit peak WordPress. I thought that was amazing. I'm sorry. No, that's good. So you said you're coming from the enterprise side of obviously I've been in touch with the consumer base a lot through customer support. So I'm wondering, do you have any message for the micro agency, the freelancer who caters to the do-it-yourself side owner? Basically, it's not enterprise level, but it's let's say smaller level in terms of revenue. And I can imagine that person would be very worried to hear like we hit peak WordPress. Do you have any message for them? Yeah, no, of course. So there's two things. One, I'd like to preface that with a story, a very short one when I was talking to Boe yesterday. And he's building a platform, the CF platform that he's been building. And he's getting a lot of feedback that it's just not as responsive and doesn't have all the notifications and all these things that modern web apps do. So there's a lot of pressure to deliver even the smallest of micro interactions to try and match what the largest platforms are doing. So there's a lot of baseline works that needs to be done even on the front end to even provide something that feels and looks like it has the same sort of authority and command as other solutions out there. So there's a continual catch-up. On the second side, to answer your question, I think it's the role specialization.  Are you actually just a really good developer and you're not supposed to be in this whole like marketing business side of things? Are you a product manager? Are you able to pull together all these different solutions and put a strategy together to measure KPIs and conversion and engagement? You need to decide kind of what you want to become. For us at HumanMade, we're always trying to push forward. Sure, we're an agency, but we do so much more because we're curious and we have to be to be able to keep up with the web. That's just, it's almost a given. It's the sort of, and maybe it's like this sort of feeling of entitlement, I guess, to stay in this sort of static space. Like, oh, this is how I build websites. I'm gonna do this for the next 10 years. No, like it's just never been like that. We've had a very nice, you know, we've enjoyed a lot like I said before, it felt like a superpower having WordPress and all these plugins like seven, eight years ago, but times are changing. We have to adapt. Thank you. I'm gonna try and squeeze in just one more. Boss? Well, we have to have boss. Okay. Another short question about world domination. A lot of the future of WordPress seems to be branded in the Gutenberg project. My question for you as a marketing guy, what do you think about the Gutenberg brand? And second question in 2025, will we be at a Gutenberg camp or a WordPress camp? Yeah, you're killing me. 2025. Shit. I think Gutenberg is exactly what we need right now.  So even like that smarter WordPress stuff will hook in really, really nicely into that. I don't know if we'll be at a Gutenberg camp in 2025. My hope is obviously that for WordPress we keep focusing on content and are able to do an amazing job at that. Maybe, I mean, maybe we'll be at an AI camp or a machine learning camp or something else at that point. Maybe, you know, the code base or the amount of code that creates value inside of WordPress or rather business value exceeds the actual foundation. Maybe there's a significantly larger ecosystem around that then, I don't know. But either way, whatever it is, it's gonna have to match the business value that other tools are putting out today. Thanks. Pleasure. Okay. Well, I think you've finished it off the morning beautifully. Thank you very much, not far. Thank you. Thank you very much, Milky Way.


What's up guys we're Rewire and Varsky and we are here on the wonderful Plymouth Ho for Ocean City Sounds 2018 People down here are up for a pie. That's what I do. I live for the pie. So I think we're gonna get on The crowd is probably the best crowd I've had throughout this summer run I've seen lots of sweat and yeah, I just want to get people to sing along and have some fun Hello, we're Bastille. We are so excited to be on Plymouth Ho finally Woody's been willing to play here since we started as a band He actually brought us to come see it to be like one day children. This will all be a gig or something I think there's a sort of in a good way a kind of wildness and a madness to people down here Anytime you reach the kind of far reaches of the country North Southeast or West people are pleased to see it. Everyone's super happy generally speaking loud enough for jumping around Hi, I'm Pete Tong and this is Ocean City Sounds. This is Ocean City Sounds This is Ocean City Sounds


was turning around as I went to get her ring. But what is going on with Scott? Welcome back to our channel.  So I know as you can see by the title, I'm going to be throwing Mariah's ring in our pool, but there's going to be a huge twist that you're not going to want to miss out and you're going to want to know what's going on. So this right here is Mariah's real ring. As you can see, made to perfection, this is the ring that I propose to Mariah with. And this ring is a fake identical replica ring to Mariah's engagement ring that I bought for her. Marvel Squad, when I tell you identical replica, it is literally down to the band, down to the diamond, the exact same look, size, and feel as her actual engagement ring. Comment down below which one you think is the real one and which one you think is the fake ring. I'm not going to tell you right now, but I know. There's a small detail that changes both of them. So what's going to be going down is now that I have an identical replica ring to her engagement ring, I'm going to be finding a time to go out near the pool and getting kind of annoyed at Mariah. Somehow getting her ring, maybe I'll just like, you know, ask for it and then I'll get annoyed and then throwing in the pool. But what she's not going to know is I'm going to have this fake ring, the one going in the pool. But that's not all. Before you skip this video, I have one more twist. When I show Mariah that it is not her ring that I threw in the pool, that it was a fake ring, I'm going to pull out this ring.  I must have thrown you real. And that's where Mariah's second reaction comes in. So she's first going to be pissed that I threw her ring in the pool, then she's going to think it's a prank, which it was.   To be clear, one more time, I have two fake rings and one real ring. They all look alike. Mariah is going to think that I threw her real ring in the pool. And you already know, this ring costs a lot, these rings cost like five bucks. She's going to be pissed. This is the ring that I propose to her in Hawaii with. Oh my God, what am I doing? I should just buy a casket while I'm at it. Guys, this girl is going to kill me when she finds out that this whole thing is just set up. But I just ordered some Starbucks so I'm going to go wake Mariah up. We're going to have maybe a nice breakfast or something and then we're going to get this prank on their way. Guys, if you love our videos and you love our channel, make sure to hit that thumbs up button. It helps us out so much. It lets us know that you guys are loving the videos. And most of all, if you are new to our channel, make sure to hit that subscribe button because we have crazy prank videos like this all the time. Vlog videos, we have so much stuff coming up. This is just the start. But let's get going. I'm going to put this ring back in her room and we're going to get Mariah outside by the pool. This might be one of the craziest videos I've ever done. Finally, I thought I heard someone up there. I literally woke this girl 10 minutes ago, guys, and she was in the bathroom that whole time. What are you doing? Back, girl. On your phone? No. Probably. No, just not waking up and taking all my eye boogers out. Snappy, what are you doing on the counter?  What? Get off, bro. No kiddies on the counter. I love you so much, okay? Clean day today. Cleaning the house. Is that what we're doing today? Yes. I want to go swimming today. Swimming. Yeah, I want to go in the pool. We have not been in our pool in like how long? I've been in our pool. Exactly. Since your sister was here. That was months ago. Tell me if you guys had this problem.  You swim in it for the first couple months and then like you kind of like use it every once in a while in here and there, but it's not to the extent that you thought you would use it. Nobody wants me to have it though. Yeah, yeah, I love having a pool. I think it's more of like an amenity. So when we have people over, we can all have fun in the pool. And when I'm tanning. Oh yeah, and when you're tanning. You can go in there now because the cage is half broken still from the hurricane, so. I almost don't even want to put the cage back on, but then all the bugs that come in. You look cute today. Thanks. Yeah? Wanna go sit out by the pool or something? Yeah. We should eat. Can I eat first though? No, we should eat by the pool. When have we done that? Let's just dip our legs in. I bet you the water's like 85 degrees. I'm not gonna try the pool. I'll go after though. When have we done that though? We've never done that. No, we don't have ability to do that, but. So, first time for everything? Yeah, I'm gonna do that today. I should be eating. I said eat by the pool. You said we can do that today but dad, she's eating. I don't want to eat by the pool because of buggies, you know? No, there's no bug channel. It's morning. You're being very difficult right now. Get out of the pool. I just woke up. I know. And by the way, girl, let me see your finger. Where's your ring finger at? Oh, why? I just woke up. Where's your ring? Can you please go put it on? You do this every time. Can you please go put your ring on? You'll wear it today? I love when you wear it. Even on weekends. I wear it every day. I just woke up. I have to shower myself still. I'm gonna go get it for you. Okay. Damn, even with the bagel, that looks so much better. Oh, baby, am I happy? I am. Ever since I proposed to you, I've been the happiest man in the world.  Even when you scream at me. I even want to look the ugliest ever in the morning. We both look ratchet today, but you look good. You're killing me. Oh, so I do look ratchet. No, you're killing the bad head look. I like it. Are you gonna put words in my mouth? I didn't say that. I didn't mean it like that. I didn't mean it like that, though. I meant like, you know, look it. I got Mariah to eat her breakfast out here. We've never done that, so why not start today? And do I know what another sad part is? Since our cage has been kind of destroyed after the hurricane, we can't let our cats out anymore. Otherwise, they're gonna be roaming our yard and we don't want them to like run under the fence or something. Not there's holes under the fence, but like, I'm sure they can find one. Cats can pretty much get into anywhere, right? Are you good? Are you miserable right now? No, I'm just a dryer. Oh, yeah. We had a long night. Look how good the pool looks. Mm. Do you have something? Crystal clear. I do want to go swimming. Do you want to go swimming? Not really. No? Mm-hmm. I want to swim. If I have eyeballs in my eyes, I'm so sorry. And dye your need, would you ever go and swim? Like if you had to swim, would you? Yeah. Yeah. Our pool's hot. The heat is on, right?  Yeah, it's like probably 90. Which is perfect, because that night gets cold here. But here, you can see the spots, like the screen there, the screen there, and that's about it, but the screen is gone. So our cats would just be able to like, you know, run straight out. Oh, he's trying so hard to get out. The door gap is like, just barely open enough where he can get his paw through. He wants to come up, this is why it feels so bad.  Snap. I know, but I don't want to take the chance of him like running under the fence or just like getting into like, bugs and stuff, you know? I need to clean this outside area. Cause they loved it out here. We definitely do need to set something up where they'll be able to like, run around, you know? You need just like a cleaning crew and come here and clean this whole thing up. Yeah, or we could just get a pressure washer and it's done.  I know, we need to clean that. Speaking of cleaning, does your ring need to be cleaned up? What was it? It looks nice to me. I know, we don't know like, when are you supposed to, are you supposed to clean your ring like every six months or so? Cause like, you don't even notice it's getting dirty until it gets dirty. I don't, but this doesn't even look like it's really getting dirty. I take really good care of it. I take it off whenever I shower, I sleep, I put it right back in its case. I actually want to get a really cute jewelry box. It's kind of dirty. I think it's kind of dirty. I mean, it's not that dirty. It's actually like rinse it off. It's not dirty at all. Did you take chlorine? Will it clean it? Why would you put it in the chlorine? I'm gonna stick it in the pool and see if it cleans the ring. No, you're not. No, you're not. No, you're not. No, you're not. Why would you be stupid? Don't, baby, please don't do it. Babe. No, don't, don't, don't, don't. Why would you rinse it out? I just want to try it. No, I don't want to put it in the water though. Right, it's not a big deal. No, please don't do that. Why would you do that? Please. No. Let me just try it. I don't think chlorine's gonna ruin anything, right? Why would you put a nice ring by a chlorine water? I'm sure, yeah, you could probably swim, I'm sure you could swim up there, but why would you, I don't want you to. Okay. Why don't you just drop it and move it? All right, well, I'm just gonna try, I'm just gonna try cleaning it because it needs to be. It's not gonna get hurt. It's not gonna get hurt. It's not gonna get hurt. No, it's not gonna get hurt. Right, no, no, no, no. It's fine, I will get it. No, don't take the scimmer, get in there now. Why?  It's fine. I'm serious, please, please, please, please. Let me film it, it's funny. It's funny.  Why would you actually do that? Babe, I'm kidding. Why would you? I had a fake ring made for like, exactly like your real ring. Do you see? Oh. It's not my ring. Give us not my ring. Wait, that's not the real one.  Babe, why would you do that? Babe, I got you in here, should I now? Did you actually make me do that?  Babe, it's yours, babe, go get it! I hope you just had a scratch on it. Wait. That's the fake one. No, it's not the fake one. Wow, it's in the deep end. Go get it. It's actually my ring. Yes. Did you put a fake ring on me earlier or something? I just dropped the one on like, brown. I hope that's not, that's not your real one. Babe, that is my real one, go get it. Switch the bottle. Go get it. Wait, let me see this one.  Babe, go get it. Go get it, you didn't get it though. Oh my God. Go get it. Just go and get it. It'll be funny. Babe, please stop.  Please stop. It's not a big deal. Come on, let's go check my ring box. Babe, that is your ring. Mariah, that is your ring. She's checking her ring box. Oh my God! Go in the pool. She's checking her ring box to see if that was her real ring.  But it's kind of funny. Why are you not taking a serious ring? That's our ring. That's a... Go get it. I messed up. Are you up right there? Yeah, that's your real one. That's not funny. Wait. Is that a fake one? No, this is your real one. You can tell. Okay, give me the fake one. Why would you do that, actually? Can you tell now? Yes, why would you do that? It's a funny bill. Where'd you get those rings at? Amazon. It's literally the exact same ring as hers. I genuinely thought... A little different. But it's a little thicker. Even though I knew you were filming, I genuinely thought that was my ring. I know. That's not funny. Oh my God. She's never gonna let me have the ring out of her sight. This is a reminder to never leave your ring out of sight. Baby, you just gave me an actual freaking heart attack. Why would you do that? That's like funny. I thought, wait, buying one engagement ring would be funny, but buying two fake ones? You're not gonna get it because that's how I can use that if I'm a bro. I know, that's another bonus you can use these when you're out. Look at how dope that is. Compared to your real one. Yeah, but it's small. It's a little bit smaller, but you can't tell. Why'd you get a two? Did you get a two-carat on there? I typed in three-carat. Yeah, that's like... That's a 3.2 though. Oh. So maybe... See how close it is? Look at the band. Oh my God, baby. Right, I love you so much. Why do you think I wanted to eat breakfast out of you?  So when you put the ring on me, that wasn't my actual ring? No, it was. Oh, we switched it out. I switched it after you gave it to me. Okay. I do have like little two. It's kind of... Don't scare me like that.  The first one you threw in, did you think it was your eye-catching? Yes, I did. I did. Because I showed you it first time and I threw it. Like, I swear at everything. I literally thought you did that and I was being pissed off. I was actually getting so mad at you.  And then I showed you the other ring and you're like, wait, this is not it. Yeah. This one's a high profile. Yeah, yeah. See how it sits up? But this one says, I like that it sits down. I like this one too. And the diamond's better. Mom's gone, I hope you enjoyed this video. Make sure to give it a huge thumbs up. We got this girl.  Oh, that's funny. Good morning, babe. I at least got you Starbucks. Does that count for anything? No. Accounts for nothing. I hope you guys enjoyed this video. I literally been pointing this for weeks. I've had these fake rings literally just brewing in my office, in a box Mariah would have never even known. And I've been waiting for a type of video to film this with. Oh, this was the perfect one. And the fact that I got two of them, I knew I was gonna have to get two of them. I knew I was gonna use them both somehow. Whatever.  You got me really freaking good. I can't, not again, my heart was, babe, my, I'm actually having a panic attack. I don't think you realize, I was actually having a panic attack. I'm surprised you weren't yelling, bro. Cause she's so nice.  Mom's gone, I hope you enjoyed this video. Make sure to give it a huge thumbs up. Make sure to subscribe if you are new. And turn on those push notifications so you never miss out on dope videos like these. And we will see you in the next video cause I'm about to go beat it bad. But until then, peace. Love you guys.


Yeah, so thanks a lot and thanks for the invitation and give me the occasion to talk in this seminar So as the title indicates This is going to be about an I think number theory and a distribution and with a lot of harmonic analysis and this is About joint work Currently still ongoing with Arthur Fourier and Raviya Frazane and we hope to have a manuscript. That's more or less Complete and readable in the in the next few weeks about what I'm going to talk about So let me begin with something that I think everybody loves In mathematics not only number theory, which is the Euclidean Fourier transform and then we'll discuss about geometric versions of that and then generalizations to different situations so I I start with the simplest case so the free transform on the Real numbers so formally speaking we have a function from the real numbers to a complex numbers and then we define another function by integrating f of x with The kernel exponential minus 2i pi xy With respect to x of course there needs to be conditions to make this work But in principle, this is the classical Fourier transform and this is one of the most important objects In mathematics not only pure but also applied mathematics Let me summarize a few of its formal properties and then a few of its more refined and attic properties So just of course free inversion which is that you can recover the function by More or less doing the same thing up to a sign in the kernel Of course again, this may or may not make sense point-wise, but in principle, this is only something Which we feel is true Then very important second property is the planche-relle formula which tells us that in the L2 space The Fourier transform is an isometry And sometimes depending on the normalization it might be isometric up to a scalar but with this normalization that I've written down here It is exactly an isometry Another very important Property which looks a bit more complicated at first, but it's really an essential one is the compatibility with convolution Which says that you can identify The product of two Fourier transform as a free transform namely of the convolution product So the convolution products and x again formally because it might not exist To the product to the integral of the product of f of y g of x minus y with respect to y So these are more or less formal Almost algebraic properties and it has lots and lots of other More analytic properties. So for instance Things like stationary phase which I'm not going to write down in Detail, but many of you know what stationary phase does it allows you to give asymptotic formulas Functions defined as Fourier transform or for other oscillatory integrals really Let me mention another one Which is Absolutely essential in many many analytic applications The uncertainty principle which again takes many many forms and I'm not going to write any in a specific way But roughly speaking. It's a more analytic way of saying that the Fourier transform of a delta function Is the constant function one? Okay, except that this does not always make sense point wise It might make sense distribution wise and in a smoothened form this becomes This can be transformed into various forms of uncertainty principles Many many other things which I don't write down and it has applications, of course This is definitely not an object of pure beauty that people look at from far away and don't try to do anything with it and Here I'm not going to even try To write the applications You can fill in the blanks with whatever is your favorite application So there's a famous saying of Martin Eichler that the fifth operation is modular forms I think the more proper way would be to say that the fifth operation of mathematics is really a Fourier transform In one form or another Okay, so this is the classical Euclidean Fourier transform and It has lots of generalizations So we're going to talk about finite fields soon, but the first thing I want to mention is the really general form of Euclidean Ability and commutative free analysis is Pontiwag in duality So you start with a locally compact group G Then maybe a function on this locally compact group Then you have the group of characters so continue some morphisms from G to the unit circle and You define the Fourier transform Which is now a function on this group, which a priori is different from G by doing something similar But integrating with k of x instead of exponential 2i pi x1 Okay, and these are lots of similar properties and For instance, this is what gives rise to the million transform so maybe the case apart from the Real line that's most commonly used although the Positive real numbers is isomorphic to the real line the million transform is a version of that okay so that's the classical Fourier analysis and Now what I want to discuss the really the proper beginning of the talk is Starts with a letter of the lean to cash done in 1976 Which you can find online if you Google for letter of the lean to cash down you will find a link to copy of it It's quite an interesting reason you observe that there is a So let me write it as a slogan first. There is a geometric incarnation of the Fourier transform Over finite fields and I should say over additive groups of finite fields, but let me say over finite fields And I will discuss a bit later. What was the lean's motivation? It's for magnetic number theory and it goes at the more precise form of this Slogan is that so we are given finite field K and then I will always write K bar for an algebraic closure and KN for the intermediate extensions of degree n inside K bar That's just notation. We're going to use we fix an additive character of K and It's going to be useful to think that we've also fixed an isomorphism of C with some QL bar where L is does not divide the characteristic of the field but you can just think of the complex numbers for To understand a bit what's going on at least Then we need an extra object, which is I'm going to call a coefficient object and It can be different things. So there are more or less equivalent ways of viewing it. So you can think of it as a Galois representation of the field of rational functions In one variable of a K you can think of it as a an elliptic shift or something even more complicated but concretely If you Want to think really in terms of concrete and attic number theory properties this as a trace function and This is really what is of interest to an attic number theory at least and in fact it has trace functions for every extension field KN So complex valued or QL bar valued functions Defined on KN and here and we may be given example So for every polynomial F in K of X There is such an M with TM X Again, yes, I'm going to lose Hopefully have enough space equals Psi of the trace From KN to K of f of X Okay, so these are somehow some of the simplest Trace function that correspond to some of the simplest of these coefficient objects Okay, this is all data which Is a priori given and what they'll improve or observed So I say observe because really given the formalism that E and got and the the algebraic geometry school and Developed in the previous few years. This is really just an observation that things fit well together So there exists another coefficient object. So this is the geometric Fourier transform Okay, so it's again of the same kind You can think of it as a Galois representation essentially or as an analytic sheaf or something like that and the point is that its trace functions Are the discrete Fourier transforms of The trace functions of M so with some normalization which I'm not going to put in right now But we sometimes is convenient. So it's the classical discrete Fourier transform with Y Psi of the trace of XY So let's look again at an example to see immediately what this tells us So if we know that we can construct the objects that I described in the previous example So very simple coefficients where the trace function are exponentials of polynomials Then this gives you new objects new coefficient objects color representations or something with the property that the Corresponding traces. Let me just do it over the base field and let me even take K equals FP and Psi of X is the standard additive character, which I write as E of X over P then this Discrete Fourier transform becomes Over FP just a classical additive exponential sum. So E of f of X plus X times Y over P so the kinds of sums which come up all the time in an antique number theorem and I Think it should be said at least for me that the fact that such a thing exists. So as I said Given the formalism of algebraic geometry that existed at the time. This is really not so hard It's just putting the formalism correctly together But the fact that there is such a geometric for a transform is something quite extraordinary if you think in terms of modular forms because the These coefficients you could also think that their trace functions are eigenvalues of a cooperators on On on the on the GL2 or on some GLR of the field of fractions of the finite field K Then you're constructing a new automorphic form with completely different Behavior where the eigenvalues are the Fourier transform of the given eigenvalues that that's quite a remarkable Remarkable thing Okay, now what the linear already observed in his very first letter is that this is indeed an analog of the Fourier transform, so we show the version of inversion. He showed the versions of the Planche-Rell formula Which takes the form that? Essentially takes the form that if M is irreducible Then the same for the Fourier transform and Conversely if you think of Galois representations for instance There's something very important is the convolution also works and this is also mentioned in the first letter of the link So I'm not going to write it here because I'm going to discuss it later and now you might ask about the further properties that I mentioned quickly above and about applications, so it's Something quite important in application. There's a version of stationary phase because it takes a very different form than what you might be What one is used to in the niting number theory, but no more showed that properly interpreted there is a version of stationary phase for this geometric Fourier transform On the other hand as far as I know, I mean, I don't know of any good analog of The uncertainty principle, I know some statements which look a little bit like uncertainty principles But I would guess that I don't There I would guess there must be some because this is such an important fact of the classical Fourier transform But I don't really know one for this geometry Fourier transform and for applications Well, I would say that There's many of them In a niting number theory So the lean mentions in his letter so apparently it was an answer to a previous letter from cash done but he had been seeing about it because of applications to estimates for some trigonometric sums and you can imagine that indeed as we've already seen with this geometric Fourier transform you get an algebraic formulation algebraic understanding of Classical families of exponential sums For which you may be able to use some tools from a algebraic geometry to say a lot of things and indeed This is what happened so you have Estimates by combining with the Riemann hypothesis Over finite fields You have a quick distribution statements which again I will discuss later in more generality, so I won't state And so on I won't state the result right now But it's not that so there's applications So Lomo also use that to Give new proofs of parts of the veil conjectures and a product formula For the signs of function equation for epsilon factors, okay, and something which For me and Fouvri and Philly Michel has been very important so our all series of works That we did in the last six or seven years Depends essentially on this geometric Fourier transform Okay, and there's an IT application so there's the This will function in arithmetic progression There's some other Level of distribution for Primes in arithmetic progression and so on and so on all kinds of analytic facts now depend essentially on the on the Fourier transform and and of course there are others and again I'm not trying to be exorbitant They are certainly less at the moment and the classical Fourier transform, but it's only is a very important tool In number theory Okay now What Really is the starting point of the work that Arthur and Ravier and I did is The question about what about other groups? So like there's a contraging duality for groups other than the real numbers Try to think about and we can see there are other groups with their own characters that we might want To to consider and for which are exponential sums which arise parameterized by such characters Which we might also want to study so Here the oops in question would be Technically commutative algebraic groups of a finite field, but let's take the simplest example would be we look at The invertible elements of care of one of its extensions Then we have multiplicative characters And if we have functions defined on the invertible elements of k or of its extensions we can construct so for coefficient m Not only can we construct the discrete Fourier transform? additively, but we and look at multiplicative character sums So some of chi of X Then the trace function of X Okay, now X is in K and star and so this come up in application Now we view these as functions of chi so This is really what should be called a discrete mailing transform and these come up naturally also in applications At least in I think applications now for an individual individual chi We still have the Riemann hypothesis of the linear which can give in principle Can give good estimates sometimes it's hard to prove but In principle it gives you an understanding that's often sufficient for applications But now what happens when chi is varying if you need to understand What happens on average over over chi like we know that We have a good distribution properties from delins Fourier transform. Do we have something like this for the discrete mailing transform and And examples of questions like this arose in particular So Evans has this question to Katz concerning sums like this so over over Fp so where X bar is the inverse of X modulo P and Various sums of that type arose in questions about quantum chaos by Rosensweig and Rudnick I won't write the sums that arose but they are somewhat similar maybe to the Evans sums and In a series of work of Keating Rudnick and a number of other You also have sums of that kind that arise naturally for occasions and also Sometimes with more than one character. So you have okay d times where these fixed so in that case the group the underlying group is not the Multiplicative group of a finite field. It's a product of actually many of them and of course you could consider also the product of additive group times a multiplicative group and in some cases you could also consider an elliptic curve the K point of an elliptic curve of an ability of variety and then the characters of that and and write at least write down some exponential sums And ask if you can say something about their properties Okay, and so the first thing I want to to discuss although it's not literally necessary for for the later discussion, but it's something which maybe is we're seeing is that Let's say for for this multiplicative character case There is no analog of the lens Results, so you might think to be more precise. So you might think because we know for the Duality three four five a billion groups that the group of characters of k and star is isomorphic to KN So functions defined or characters of KN are star are just the same Via such an isomorphism as functions defined on can star. So you could think maybe so given M Maybe there's an Maybe I should write M twiddle now. It's not an M ad like the lens one. There's another M twiddle again or On the multiplicative group. So the beyond k bar star with trace function After some isomorphisms Equal to the discrete man in transform. This would be very optimistic But that's a way that you would be able to generalize the lens work very easily Okay, and the fact is this is not true except in trivial cases Let me explain one reason why it's not true because it actually is quite nice And it fits with some of the things we we did with Arthur and and Ravier in more Complicated and fancy circumstances. So I don't really know who first made this observation I mean, I would guess that Deline must have known that there's definitely a version of it in a paper of boy Ashanko and Greenfield, but this something that was known much earlier than that I think the argument we we used to show that it doesn't exist is maybe new So the thing is that if if there was one so if m tilde existed It would have an L function like Galois representations or sheaves have L functions so this would be one way of defining it would be as a formal power series So exponential of the sum of t to the n over n and then you would have the sum over the characters And this would be the points of the algebraic variety on which this m tilde is supposed to exist and then You would have to have the sum of the corresponding traces which would be these these k of x times the trace function of of m and this would be a Rational function because that's true for any reasonable coefficient Galois representation sheaf whatever and This would have some kind of function equation now you can compute this just using Standard discrete for analysis over k and star. I mean you can see here Exchanging the sum you're going to get the size of k and star times the value at one of the trace function At the origin of trace function using this you will easily see that it is indeed a rational function But this is not it does not have a function equation except in digital cases okay, so To summarize the situation at that point so we have naturally arising at least families of exponential sum Let's say over multiplicative characters, but also over other Which are not additive groups We suspect or we might hope that we can understand them statistically We can do experiments. I mean at least let's say the epsons are very easy to compute and events did that and checked that it seemed to be distributed to some measure and Now the question becomes Where does it come from? So this counter example shows it cannot come from Simply applying the Linsik distribution serum somewhere. So it has to be something different that's happening so What can we say? So what can be done? I should say so as I mentioned these questions were They are natural and so both Evans and Zef Rudnick asked Nick Katz about their respective examples if algebraic geometry could say something about it and Katz was able to to find a solution For so for K and star in about I guess close to 10 years ago now and What we have done So with Arthur and and Ravier is so we generalize the approach of Katz Which is based on on convolution plus the ten I can formalism to any Commutative so here I'm going to give the technical connected Connected commutative. I should have said algebraic group For any finite fields, okay So in particular You have the multiplicative group which corresponds to this Multiplicate invertible elements in K bar or K n bar you have You could have the additive group to some power times a multiplicative group to Another power this would corresponds to having Some sphermatized by D1 multiplicative characters and D2 Elements of the field defining additive characters But you can also have elliptic curves or you could have a billion varieties more generally all products of these so the Kissing is we have something that's that's completely general So I won't try to explain how the the theory really works. So I mentioned the key properties the convolution So the point is if you look at the definition of convolution on an arbitrary finite group There is no character that shows up in the definition of the convolution of two functions Which means that the formalism of algebraic geometry will be able given two objects two coefficient objects to construct essentially a third one whose trace function is the convolution of the trace functions and Once you do that, I mean that that's something that again is that At the level of what the linear observed in 1976, but what cuts realize is that combined with Terrican formalism, which I won't say anything what it is Is a way to then associate a group To any coefficient on the multiplicative group any coefficient object on the multiplicative group And then he was able to exploit that and again the Riemann hypothesis to prove an equidistribution theorem for some over over the For some sparmatized by multiplicative character And what we've done is generalized this to in such extent that we can now do this for any of these Commutative connected algebraic groups and not just the multiplicative group or the additive groups So we are the precise statement Of the general equidistribution theorem that results Okay, so it's really the Very general equidistribution theorem for any kind of Expansion sums that you can write if they are parametrized by the characters of such a group. So Let's do this So we start with the data. So k k bar k n As above so finite field algebraic closure L is different from the characteristic of k is a prime number M is a coefficient object Which concretely you can think of as given by its trace function, which have to be some nice functions And again an example would be for any if you have a function On the sorry, I should define the group first. Otherwise does not really make sense So G is connected commutative algebraic group over over k Example to keep in mind for the case of multiplicative character is the multiplicative group Then you have a coefficient object on M and again A simple example that you can always think of is If you are if you have a function on the group With values in the defined line. So just a polynomial function of the coordinates of your group essentially And then you can find an M with trace function given by psi of f of x So psi n of f of x we are saying is composition with the trace Okay, again psi is a fixed additive character of k But this this side doesn't mean that we're just working with With the additive group. It's just a way of constructing coefficient objects Then what we show so there exists the following So there exists an integer r non-negative So maybe I should say to be precise for Not to claim that I understand what I'm writing here But so the the correct technical words For these coefficients objects for which the theory really works Of which these are special cases You have to have a perverse leaf And the way I'm writing it it should be of weight zero And uh semi simple Yeah, semi simple But that's just so that I write the words once Okay, then and so that the theorem is correct It's precise and correct. So there exists a non-negative integer There exists a group Which I'm going to call h sub m Side glr And you can think of it as glr of c So it's some group reductive group With the following property so Let first k m Be a maximal complex subgroup Of this h m So for instance h m could be the old glr of c Then k m would be The unitary group of size r over c Then let mu m be the probability r measure So the invariant measure on this compact group k m With total mass one Then the discrete mailing transform or Fourier mailing transform associated to this coefficient object on this group uh as sums As complex numbers become equally distributed Uh essentially like the trace of a random element corresponding to mu m Of k m so Write it first in words and then the precise statement. So the discrete Fourier transform of the trace function of m on g of k n Are distributed When n gets large Like the trace of g Where g belongs to k m is distributed Like mu m Okay, and the precise statement is the following So for every function f On the complex plane Continuous and bounded If you look at the following average So it turns out if you just look at a single finite field uh You might not get convergence. That's something that's well understood. That's already arose in the theory of the lean And we go around that by doing an extra Cisaro average That avoids uh having to make unnecessary assumptions so Okay, so we average First over the given finite field Uh the values of the function f add these discrete mailing transforms. So these are just the sum a chi of x t m x k n So just the discrete Fourier transform that you might Expect for the given group g Okay, so and here is a chi Then we do these extra Cisaro average to avoid complications And these converges to the integral of f of the trace of g with respect to this measure mu Over k m So this is the statement. So again Summarize a little bit So we have a finite field. We have these extensions We have some group on which the characters define interesting exponential sums Like the multiplicative group Some kind of coefficient object whatever that is but for our instance exponentials of polynomials And then these become always equidistributed at least in this Average sense with respect to a very special kind of measure Because it's a measure on the compact the trace uh on on the compact group Okay, and so Should there be a normalization here That uh t the Fourier transform of t. Are you not should you divide by the square root of the number of terms? Right. That's actually that's hidden in this Uh asking that the coefficient is a perverse shift of weight zero It turns out that this automatically gives you the right normalization It divides by the square root of the dimension of the support of the trace function That's actually quite quite efficient and quite nice Because it could be that the trace function of m is supported on a smaller dimensional sub variety Like for instance the diagonal in in a product And then you don't want to divine by the square root of the size of the group But by the square root of the size of the diagonal and the Assumption that you have a perverse shift of weight zero whatever that means automatically incorporates this the right normalization Even even in the sum of a chi Yeah, it's exactly so there's a normalization here in the average of a chi i divide by the number of characters So that's that's you have to do But in the sum of a chi to to define the discrete Fourier transform And you have this m which has this property of being a perverse shift of weight zero You don't need to do the normalization by dividing explicitly by the square root of the number of terms Okay, so as I said, so delin proved this in the case of the additive group to any power really or even any unipotent group commutative unipotent group Which occur in some analytic applications through vid vectors and things like that uh, and cats Did that for gm? and Plus some abelian varieties and some and some coefficients With some extra conditions So here we have a really general statement Which is really quite nice. I think So they're still work to be done to when you want to apply it because you need to compute Uh, especially this group hm and this And this integer are but at least in principle This is saying that in the realm of this kind of standard type of Expansion of some parameterized by characters of algebraic groups Uh, there is always some a priori equal distribution One might need to make it explicit, but there is always an a priori one And that's that's a I think it's a first step towards kind of Uh, a more general form of delin's geometric Geometric free transform We cannot quite have exactly the same thing, but we're having some of its aspects That that still makes sense now In that is work you don't need to average over the The degrees which you have in this chaser, right? Yeah, but in cat's book, he always assumed that the geometric and the arithmetic monodromic groups are the same And if we had this assumption, we could also remove the cesaro average But we don't the point is that it's quite hard to compute these groups and Uh, if you want to make this extra assumption, that's something that's could be very difficult to check So here we have the advantage that we have a statement with no assumptions at all concerning the monodromic groups, whether the arithmetic or the geometric ones are the same or not But if we knew this a priori, so sometimes if we compute the group We can see that they are the same and then we can also remove the cesaro average Okay, so I don't want to to go too much over time. So now maybe I should say a few words about applications and then I want to discuss questions so applications So There are some to well Equal distribution, of course, if you can compute the group you get equal distribution And and we have various examples where we we can do that. That's Okay, so I should say maybe explicit equal distribution Uh, so we can improve quite significantly some of the results of Of the the Keating-Rudnik type. So maybe in particular results of all Keating and and Rodity Gershon So the point being that their results is about a question where the natural parameter space is multiple multiplicative characters And they were able to deal with it by using Katz's theory By specializing all but one of the multiplicative characters and letting just one vary But if you do that then you get worse results than just looking at the very the family of all the parameter parameterizing characters and applying our results And uh Again, we hope to have more and maybe even better example or better applications Later, but at least the the series there now and the applications can follow Now, uh, as I said, I want to discuss a bit about some questions Um, so maybe the most important one is okay. So what about a more geometric? Maybe I'll just discuss that one to avoiding too much time more geometric version in the general case Okay, so we saw that literally speaking if you want the uh, the discrete main transform for instance would be corresponding to some Gallery representation. It doesn't work because there's this L function that does not have the right properties um But still we have we have an L function and this L function as Is a rational function. So some properties are there. We have a distribution. We have convolution. So It's still tempting to think that maybe there is more. Maybe it's just optimistic But maybe there's still something on the other side And here there's a little hint that I find very very interesting. It's raviere who pointed out to us Uh, if you look at the delin theory, so let's say Viewed as being exponential sums Over ga so Fourier transform of the additive group And here we would have the the ones over Over gm now cats And others had noticed kind of results Which is really not any formal As far as I know any really formal statement, but there's an analogy with Uh differential equations demodules so linear polynomial linear differential equations the the kinds of things um Which Such that basal functions for instance are solutions of now I'll do it this way. So if you have Now exponential sums over gm They are still an analogy with linear differential equations over c star and and this as I said has at least been used By cats, uh many times now So what we want to do is we want to take, uh, the mailing transform now For differential equation, there's a very very well defined being transformed And what's the image of the main transform for differential equations? You get different equations difference Okay, it's very easy to give a concrete example. You take exponential minus x that solution of linear differential equation Then here you get the gamma function And it has satisfies gamma s plus one equals s gamma s Okay, now we are wanting to do the discrete mailing transform of exponential sums over gm or things like that and what we're asking here is Is there something here and therefore we should also ask is there an analog Can we go from difference equation? What's the kind of finite field analog of difference equations? It's clearly not an obvious thing because the the link between linear differential equations and exponential sums is of course not Obvious either But it's very tantalizing to think that something is going on here I mean at the very least we can ask the following question now Suppose we start with an object over gm where we see clearly what is the corresponding od Then we're going to have a different difference equation from mailing transform on the top It will have some kind of tanic and there is a group associated to that Is it going to be the same as the group that cuts associates to objects over gm? And then we can try to do the same with other Other groups instead of gm and again This is something which is very tantalizing and for the moment. We don't really know what what's going on And I think I will stop here, but I might be to take questions, of course


Coming to you from our GIS studios, I am Anicia Antoine and this is your Midday News Brief. St Lucia is looking to strategically position itself and highlight its many offerings as tourism industry representatives met in the much anticipated Florida Caribbean Cruise Association FCCA conference in Puerto Rico. The conference allowed industry personnel to network share ideas, examine best practices and establish business initiatives with top-level cruise executives. The St Lucia delegation headed by Prime Minister Hon. Alan Chastney and Tourism Minister Hon. Dominic Fede seized the opportunity to update industry partners on plans to commemorate the Caribbean Equine Cultural Festival happening on December 12th and 13th in St Lucia. The two-day luxury lifestyle and racing entertainment event will be held at the newly commissioned Royal St Lucia Tourist Club in the southern town of Viewfort and will feature the inaugural Peter Cup that boasts one of the largest horse racing prizes purchased to date in the Caribbean. St Lucia is being promoted as a must-see for investment and leisure. The St Lucia Sports Academy has become a home away from home for 37 students who have enrolled there. The School of Excellence for Sports is the first and only boarding school on the island where students will concentrate on sporting disciplines including cricket, football and track and field. Academics will also feature prominently within the curriculum. For the school's principal, Delia Charles, the initiative by the Government of St Lucia and the Ministry of Education is a welcomed one, one that she believes addresses the thinking that one size does not fit all when it comes to education. From dawn to dusk, the students are engaged both physically and mentally in activities that will prepare them to gain sports scholarships abroad. The St Lucia Sports Academy opened on September 11th, 2019. With a history of rampant flooding in times of inclement weather, 12 communities in Castree Southeast are now set to benefit from significant investment to help reduce flooding and its debilitating impact. On Monday, October 21st, 2019, area residents gathered for a second time to receive official updates from the experts responsible for the execution of the DVRP-funded EC6 million dollar slope stabilization and drainage projects. They include reinforced concrete retaining walls, basket retaining walls, masonry retaining walls and rip wraps. Glory Cedar trees will be planted along the river banks to help with slope stabilization. Communities in Castree Southeast are marked to benefit from the flood mitigation works at Country Village, Otso, Badu Shosei, Bel-e, Bekso, Wavin-Pwaso, Mac, Crownlands, Labe'i, Sarat Gap and Air Bar. The projects are scheduled to be completed by air end and present a collaborative effort between the Department of Infrastructure and the Department of Economic Planning through the Project Coordination Unit, PCU. This has been your Midday News Brief. Thank you for watching.


When his dad almost fainted at the sight of the ultrasound, this nurse was shook, saying it's one in 15 million. When Bethany met Tim Webb, she knew things were different. The way he smiled, his passion for life, and lofty dreams were all something that drew Bethany to him. Thankfully, Tim felt the same way about her, and after two years of dating, he popped the question. They had no idea that their lives would be turned upside down just a few months after their wedding. Once the joyous fog of their nuptials lifted, Tim and Bethany started to talk about what the future looked like for them as a couple. Would they buy a house right away? What about pets? Would kids be a part of their plan? Bethany told CBC News that she was willing to start trying for a family right away, but Tim wanted to enjoy their time as newlyweds for a bit instead. My husband definitely wanted to wait a year. I was fine with whenever it ended up happening, but then we found out we were very excited, only thinking we were having one. Once Bethany realized that she was pregnant, they raced to the doctor and ordered an ultrasound. What happened in that doctor's office would be something that they would both remember for the rest of their lives. After positioning the ultrasound wand over her stomach at just the right angle, the tech looked at both of them and announced that they were expecting not one child, but four identical quadruplets. I could not believe it. I was in such shock. Honestly, I thought the ultrasound tech was joking. I couldn't believe it. It was a good thing that my husband was sitting down because he almost fainted. Neither Bethany nor Tim had a history of multiples in their families, and since they conceived naturally, it was even rarer. After doing a bit of research, the couple discovered that having four identical girls occurs in one and thirteen to seventy million pregnancies. Thankfully, the rest of Bethany's pregnancy went smoothly and they welcomed four identical twin girls named Abigail, Michaela, Grace, and Emily into the world. With four newborns to care for at home, Bethany and Tim were lucky that the community rallied around them. People from their hometown even started a fundraiser and made sure that they didn't need anything. It's really amazing because there would be no way we could do all this by ourselves. Their neighbors, family members, friends, and complete strangers did their best to make sure this overwhelmed couple was taken care of completely. Without their help, Tim and Bethany knew that things would be so much more difficult. While the thought of having four children is something that Bethany and Tim never contemplated before their daughter's birth, it's something that they're both coming to terms with now. She told CBC News, I think my husband has gotten used to the idea that there are four babies, but I don't think he's gotten used to the idea that there are four identical girls. There will be plenty of arguments, bickering, broken hearts, and blunders in the years to come, but they both know it's worth it to raise four precious girls. The web home may be chaotic for the foreseeable future, but they wouldn't have it any other way. Fortunately, the girls are growing each day and are as happy as can be.


My name is Chris and I'm a recovering drug addict and alcoholic and I'm here in this video to tell all of you to quit enabling Bobby Burns What is up everybody? This is Chris from the rewired soul where we talk about the problem But focus on the solution and my channel is all about trying to help you to improve your mental health And what I like to do is pull different topics from the YouTube community or pop culture in general and try to teach you How to improve your mental and emotional well-being So if you're into that kind of stuff make sure you subscribe and ring that notification bell So this is kind of funny Those of you who know me you know my beautiful girlfriend Tristan and we've been keeping up with Bobby we've been keeping up with Bobby ever since the Shane Dawson series and For a while I've seen this kind of spiral and I'm like telling Tristan I'm like do not let me make this Bobby Burns video until this weekend I got other stuff to do just don't let me make this Bobby Burns video until this weekend because we saw What was going on Instagram, which I will share with you in a second and just now I was just finishing up a different project for some side work and While this video was rendering I'm just scrolling through the comments on Bobby's new Instagram post and I'm just like oh, no Oh, no people don't understand what enabling is So let's talk about this and again who knows if Bobby Burns will ever see this video Who knows if a bunch of Bobby Burns fans will ever see this video But if you have somebody in your life whether they are a drug addict or alcoholic or not This is a good thing to kind of learn about when it comes to enabling because I don't think a lot of people Understand what enabling is so before I get started Here is a disclaimer that a lot of people are going to disregard and leave angry comments anyways But figured I do it anyways. So first thing is I'm glad that Bobby is safe I'm happy for him. I almost died multiple times and yeah, I'm glad that he's safe Second thing is I am not saying that Bobby is a drug addict or alcoholic I'm going to talk a little bit about self-medicating though. All right, so anyways for those of you who don't know Bobby Burns You know, he got a lot of YouTube stardom from you know Shane his collaboration with Shane Dawson Bobby Burns made a hate video Shane Dawson made a series called confronting my hater They came together came became good friends Bobby's channel blew up and he's been kind of on a downward spiral ever since I've done some videos in the past about His mental health and if that so make sure you check out my channel if you want to check out those so anyways most recently He posted a picture of him like kind of bruised up and he said that he got into a car accident now One of the things that happened was Shane moved Bobby out to Los Angeles and Shane ended up buying Bobby A jeep okay, so Bobby got into a wreck in this Jeep and he was all bruised up Well, apparently there was a bunch of comments saying like oh my god You ruined Shane's Jeep like Bobby's already been getting a lot of flak for kind of like ruining the opportunity that Shane Dawson gave him I don't know if I can necessarily disagree with that, but Like I mentioned this in other videos like I'm not a fan of Bobby's music But I'm glad that he's finding his own Creativity but like from an outside perspective and I might do a whole another video on this Like his mental health is deteriorating like I'm a week or two weeks ago He did a video and it was sponsored by better help and he says that he uses the service and Here's my thoughts on it. Okay with all the better help hoopla going on. Here's my thoughts on it either a He's not really using better help and working with a therapist or B He is but he's just not listening to his therapist and that that's like just purely my opinion I might be totally wrong Healing your mental health takes a while, but following his journey. It doesn't seem like it. Well, anyways After Bobby post that he got a lot of hate and he just made a new video, which I'll show you right now Yo, so I just wanted to hop on here and say shout out to all the people who are so angry at me for breaking Shane's Jeep I got a car wreck.  This is Like seriously I Got a car wreck It's the first car I've ever been in my entire life I was completely sober go look at the comments on my last picture if you guys don't understand what I'm saying Most of the comments are like five hundred six hundred likes.  I was close to dying today and all anyone gives it about is the fact that a Car that another youtuber bought me was damaged That's That's pretty I'm okay I'm okay, but that's like that's So, yeah, is it messed up that people are saying that yes, okay? Like is Bobby playing a victim here? Yes, okay, and I'll talk about both sides of this. Okay, so I'll say this Working in addiction recovery and I've been clean for six years my substance of choice was opiates If you haven't gotten the memo, there's a massive opiate epidemic in the United States I think the last report was over seventy two thousand people in the United States alone died of overdose, right? Like I know plenty of people who have overdosed and survived I am somebody who almost died from my addiction, right? But here's how you talk to somebody who is screwing up like this, right and says hey And this is what I would say. Hey, I'm glad you're alive But you need to straighten up. All right, like it's both of those things like I work I work in a field where life and death is real, right? Like I have had over 70 people die in the last three years alone. I don't play around with this, okay? So like when somebody overdoses, I'm not just like well glad you're alive. Let's just forget about the behaviors So one of the things about we mentioned is in his video was that he was sober and I call bull on that I call so much bull like are you serious and the fact that people are going to Pretend like he was okay So in case you don't follow Bobby Burns on Instagram, the only thing he's been posting about for a while now is getting high That's all he's posted about and let's not forget his Instagram story Showing us what happened before his accident Yo, so I just wanted to hop on here and say shout out to all the people who are so angry at me for breaking Shane's Jeep I got a car wreck.  This is like Seriously, I was on the opposite side of this I would say okay well those Instagram stories are pretty far apart. He might have sobered up But this is somebody who is perpetually getting messed up and under the influence and we're supposed to believe That when he got in a car accident This was the one time the one time he was sober like that is Being willfully ignorant. Okay. This is how you might be enabling somebody in your life. All right Oh, well, they were sober when they did that like come on you guys. You're better than that You're better than that. There are there are so many people who die from not just addiction Accidents while being under the influence that a lot of it was because people lived in in a state of denial The people in their lives lived in a state of denial and I don't know I don't know where everybody's been But look Mac Miller just passed away. Okay, we had people people like Little peep passed away. We have people like Michael Jackson Prince We had celebrities commit suicide, you know, like just so many things so many things and One of the reasons for this and one of the reasons I have a mental health channel and try to increase awareness and decrease the stigma It's like everybody just normalizes it Like when I'm looking at those comments of people defending Bobby, all I see is positive reinforcement Why is Bobby going to slow down or change his behavior when people are co-signing his BS? You know, like something that breaks my heart something that breaks my heart is when I see People pass away and everybody is surprised except for people like me Who see this stuff coming from a mile away, right?  All right So it seems to me from an outside perspective that Bobby is self-medicated with weed, okay He is somebody who wasn't messing around with this stuff He might I think he mentioned like he used to smoke a little bit and I wonder by the way This could be a whole nother video I wonder if Jordan feels any guilt about that because it seemed like from what I saw it seemed like Jordan smoked She got him into it and Bobby just kind of ran with it, right? But if you go through his Instagram you go through his videos all the more about rolling bloods and that like people have been messing With Bobby like saying like all right, dude. We get it. You smoke weed like that's all he talks about That's all he talks about but like on a more like on a serious note like again Like I'm extremely empathetic like I've been there. I've been there I feel like he is self-medicating with the marijuana. All right, and like that. That's not good That's not good. I've done other videos about is marijuana addictive and all of that So I know some people might think I'm making like this false equivalent talking about People have passed away like listen. I live in Las Vegas. I voted to legalize weed.  She got into a near-death car accident Maybe about a year or two years before she passed away All right, and she was driving drunk and then she died at 24 years from alcoholism. All right So like you guys like we can't just keep minimizing this stuff like I'm legitimately making this video Because I care. Yes, I give tough love, but all of you out there need to quit enabling this behavior Sorry, like that's just the way it is Do not be surprised if he continues to spiral downhill because people are defending his behaviors All right, but anyways, that's all I got for you with this video if you like it give it a thumbs up Let me know down in the comments below.  I don't care Thanks for watching and thank you so much to everybody on patreon who is helping support the channel And if you're new make sure you subscribe and ring that notification bell because I make a ton a ton of these tough Love videos. All right, and if you would like to help support the channel and support what I'm doing here Make sure you click the top on that icon right there. All right. Thanks so much for watching. I'll see you next time


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