The emergence of all these kind of thinkings about system thinking or regenerative thinking, living systems thinking is really something that emerges from us looking for solutions for what is not working. And because I think really think that we are more mature as a societies, we are like at the moment of taking responsibilities about what we are building.  Lucia is an expert on platform-based business models. She has been around the collaborative economy for more than a decade. Later platform economies helping organizations of all kinds in the adoption of these new models, bringing the platform mentally to the company and building a relationship of real guidance and support. She's been an advisor for DeCathalon. She is also an expert evaluator of the European Commission under the H2020 program. Professor at business schools such as ESADE and or ESADE, ESADE and an international lecturer and speaker. She has participated in the design of public policies with the European Commission, the IDB, CIPPEC and some local governments such as the General Latate de Catalunya. And she knows the ecosystem very well and what is happening internationally. Has a wide network of trusted international collaborators and she is the president of We Share an international think and do tank that explores how the platform economy is impacting different sectors, markets, organizations, citizens and society in general. As an independent consultant, she helps organizations in designing platform ecosystem strategies with boundary less the company behind platform design toolkit. A methodology to help public and private organizations to design their business models from a platform ecosystem point of view. She was always exploring the design of regenerative ecosystems and network strategies to impact a scale being part of an international forms of communities, connecting people, projects and ideas. Luthier, welcome to Inside Ideas. Thank you, thank you for having me. It's so great that you can make it and I just wanna give our listeners a little heads up. Luthier and I know each other because she's contributing a wonderful portion to a collaborative book, Menu B that I've been working on for years now on global food systems reformation. And she writes about regenerative platform ecosystem models and how they affect our lives and how they're better models. And so that really leads me to my first question for you and for you is how did you get here? How did you get to this point, Luthier? So, well, as you said, I started with the collaborative economy 10 years ago. We saw this movement as a way of really putting the center at the citizen in the production of value and self-organizing in communities under some kind of interest or values. And I was always interested in understanding how these platform business models could generate a positive impact in society and create a more fair, yeah, societies in general for everyone. So, because of that, it was for me, it was like a natural evolution, finding or I was starting to listen this work about regeneration that we were talking that is obviously not new. But for me, it was. But in any case, it's supernatural for me. It resonates a lot with me and it doesn't seem like something strange for me in any case. So it was because of that that I started to research about this term, about this movement, about this kind of designing is a mindset as well as the platforms. So I was trying to understand the intersections between platform design and regenerative design in order to merge both and trying to really develop or design platform strategies for regeneration. That is the thing, the focus that I have at that moment. That's beautiful.  And then maybe some aha moments to what you've done with platform models in the past or says, wow, I feel like I'm in the right place in the right direction, so to say. So I wanna know how have you been and what is your learning or what are the new things that have come to the surface with all the world crisis and things going on that are learning lessons for you? So the sharing economy started really like with a strongly need of organizing ourself, of really change the way of how we produce it and how we relate each other. So for me, it was very interesting understanding that at the end it was like the immunological system reacting to something that it was happening. So clearly institutions that started at war with a specific goal that it was to maybe to facilitate us or to generate trust in order to relate with these institutions and mutualizing for sure some kind of services for us, for citizens. But at one point, of course, during the COVID lockdowns, we realized that we can't for sure to, well, it was like the realization that the systems or the system, how is it at that moment, is really not working. And the most impacted ones, according to me, of course, it was education and health, that the base of our societies. And it was like, well, we need to do something with that. So if we have the tools and we have the knowledge and the experience of how we can really self-organize and coordinate value creation through this platform, business model that is not a technology or just a technology because it's also a tool, but it's about all a mindset, how we can really create these communities of people under some kind of interest and values and coordinate among us in order to obtain what we need from each other. So the reality is that platforms started also to develop the same capitalistic and extractive business models. So each time that started like a more communitarian, I would say, or more fair business model, it was another trend inside the platform economy that was coming back to this idea of more extractivist and commoditizing, not commoditizing products and humans, commoditizing one side of the marketplace. So at the beginning, it was the sharing economy. And when we realized that everything, it was putting in the same bag because of, because it wasn't like a very clear definition about what was the sharing economy. For me, it is still a kind of the powerful of this movement. But the fact is that started with the sharing economy, a lot of the companies started to use it as a distribution channel and trying to get a piece of the pie. And then appeared some platform cooperatives that it was like the reaction of the system in order to fight against more capitalistic platforms. But then started with the gig economy that it was worse because at the end it was the commoditation of the producers. And because of that, the reaction or the pattern that we are seeing is that it's emerging this creator economy that is coming also from the passion economy. That is basically if you have a passion, if you have a knowledge or experience or something that you can use or you can deliver in a platform strategy or in a platform environment, you can generate incomes from what you know or you have the experience. This is super relevant because there is like emerging a lot of applications and platforms in order to help you to run your own life, to run your own business. And this is super relevant because with the We3 and with the DAOS, we are seeing like how we are like aggregating or yeah, I would say aggregating again, people, these creators, economy producers. At the end, all of us are creators. It's not that it's just one part of the citizens. We are all creators. We all have something that we can bring to the market. So and now they are like joining communities built on blockchain in these DAOS that are community ownership with shared governance and is distributed as well or decentralized but has other kind of characteristics that are making easier and easier to communicate and to coordinate between us in order to achieve some common goals. And it's very interesting because at the end it's agent-focused strategies. So the most important is to create the space that is the definition of platforms, create the space for the producers and consumers to interact and to transact between them but about all to create value. I absolutely love that. And I have to, you threw out several different economic models or types that are out there. So we wanna get into the depth and substance and we wanna go deeper but I wanna kind of unpack a little bit more of the terminology to get people up to speed where we're going with some of this terminology so that they can be on the same wavelength. Really we saw the emerging of this, you know, the worldwide web was supposed to be decentralized and distributed for everyone kind of a thing. And then we really got into after the internet was established, you know, a lot of file sharing peer-to-peer networks, Napster and many other ones that were out there and the big push of peer-to-peer really was born and again, another crisis. So around 2008, there was a global crisis and a lot of people reacted, you know, with this creator, with this gig economy and started creating and self-organizing and generating value and what they could program, code, create to do. And a lot of this kind of emerged but in that process that you just described, you mentioned numerous types of models, numerous types of economies, you know, shared value, gig economy and so on. I want you to help us to understand and where they're all going. What are all those one model? Are they all separate models? Is it one that we should say, hey, I'm living and breathing the gig economy, that's the economic model I'm working in or can you be working on the gig economy on a regenerative economy or circular economy at the same time or are they all different? How do you understand that? And what do you see in this development that kind of defines some of these terms that you threw out there to us so that we can understand them a little bit better and how that process is now evolving? Yeah, so the macro term that we are using is platform economy. So all these businesses, communities, whatever that are using platform mindset and technology in order to coordinate value creation. This is the common thing between all of them. The only, the difference is that how is concentrated, I would say the benefit captured by the platform. How is the 10% or the 10% that is captured by the platform itself or how is distributed among the participants in the platform? So platform cooperativism is that. So it's community-owned and they share the governance, the decision-making. In the case of the gig economy is the platform that is capturing most of the value and the producer is a kind of commodity inside. So it's the same that you are using or you are taking an Uber car or another. So the driver is very little difference. So they are not putting the focus on the producer. They are putting the focus on commoditizing this kind of service. The other way is the creative economy. The creative economy is focusing completely on the producer. So, and this is something that platforms do really well because they introduce this idea of having multiple producers serving personalized items, services to these consumers that are looking for something special, something that needs to fit with what they are looking for in price, time, for whatever. So the focus is on the producer. If the focus is on the producer, they are trying to empower them. But at the beginning of the creative economy that it was more about focusing on content or influencer content generation economy or this kind of environment or context, it was like repeating the same pattern of the most capitalistic platform. So the YouTube, the Facebook, Twitter and so on trying to capture most of the value. So because of that, they started to use their own applications, platforms, tools in general in order to run their own businesses on their own rules. So, and because we have all these applications of platforms and tools, they can do it. This is the thing for me. They can do it because the availability that we have of these tools. So it's also a pattern of a bundle, a bundle, bundle to a bundle to bundle. It's aggregating, aggregating, aggregating, aggregating. And this is a very interesting pattern because we are at that moment in a moment of this aggregation with a lot of applications or platforms and tools in order to help this booming a creative economy in helping them to generate incomes. And it's super interesting because at the end, I am sure that it's also a big opportunity on putting or setting the base, the rules of what we want to build. So if we need to create a new platforms for all these creators or all these Web3 users, so we can use it, the lens of regeneration in order to create a really, really a change on how we coordinate value creation and how it's impacting in our environment, in our societies in general. So I think that this is something that system thinking and for sure living systems is bringing to this idea. It was the theme that I was most interested in in understanding how really nature works or living system work in order to apply it as a lens for design regenerative platforms or platforms strategies for regeneration. This is, I think that the most interesting thing. I really love that and that's why we initially connected. So because I like your way of thinking and systems and how you use the model of regeneration as the way that the world has really always worked and that it's one that can go on pretty much indefinitely. One thing that we've discussed before is that platforms are systemic dynamic models. They're the systems you approach to solving some of business and organization and life problems and issues and kind of a model for the world that have worked pretty well. And as you've mentioned and it's really in the heavy in the gig economy and hardware and software and developers use platforms a lot. And as we know, and you can tell us a little bit more about this as well, platforms are being used in the millions around the world. People are running or in the platform model system. Boundaryless.io is basically already got 70,000 people and organizations who are using that model of platform model as well. It's something that really works well and we can see it when we get into these, the worldwide web and technology where things are systems of systems, different plugins and apps and this way of looking at that. But can you tell us a little bit more that we might not know what are the real numbers in the world of kind of people using this platform model? How many are using it effectively? How many are using it not effectively? Kind of as a commodity and kind of where we sit today in that aspect, can you have some of those numbers? Yeah, so well, platforms are very different between them. So we can compare Amazon with Apple or with Facebook or with AB and B. So they are very different in the way that they operate. In any case, the common thing is that they are using the platform like the tool and also the mindset creating this space for the producers and consumers to interact and designing for relationships, putting the elements in order to generate trust in order for the interactions to happen because if there is no trust in the ecosystem, there are no interactions. And so I would say that there are some numbers out there talking about the seven of 10 enterprises or companies are platforms, already platforms. I think that so I don't know if it was Robin Chase from Fibcar that told something like if everything that could be a platform will be a platform because it's a more efficient way of organizing value creation is more transparent because of the ratings that people, bidirectional ratings the users and consumer giving them in the platform and they are more economic because once you have the structure, the infrastructure so the cost of aggregating new producers or new assets whatever in the platform is almost zero and also because they are using network effects. So more people using the platform is more valuable for everybody in the platform. So because they are leveraging on network effects is more scalable. So yeah, I would say that I don't know if everything will be a platform but most of the business models out there will be a platform. So. They run on that, yeah. In Europe, I've heard numbers of 28 million people in Europe are working in the platform economy and that the seven of the top 10 most valuable companies in the world are running a platform economy. I actually think globally it's a lot more and people don't even know that's really kind of what they're running and what they're on because it's just a model that works. It's one that's almost autonomous and if it's set up right, if it's applied correctly it's one that addresses all the facets of a complex organization. And the reason I kind of ask about that is really is the systems view of life approach is the systems dynamic modeling the way the limits to growth wrote about it in 1972 and MIT came up with the systems dynamic modeling in the world model three at MIT, Don Alamedo's, Dennis Meadows, your grander Steve Barron Jr. wrote the limits to growth and they were all creators of this world model three that use systems dynamic modeling to kind of model our world. And so it's really platforms are a systems dynamic model that's really been there for a while. We're calling it platforms and it is very successful. It's a direction in the way that the world works in this complexity science and systems science. But now we're throwing something much more powerful into the mix. And that is we've discussed regenerative ecosystem platform models. And so not only does it have this systemic and the system science in it inherently now it's adding in the ecological phenomenon of symbiosis or regeneration. And so I maybe you could say, you can address or I can, how long this has been around? Because the last five years I've been asked to speak on regeneration probably 20 or 30 times. And everybody seems to think because I have a lot to do with agriculture and food as you know that I'm gonna talk about regenerative agriculture but I always talk about regenerative economies and regenerative models that have all the things that we're discussing here today and how is that evolving? How did you come up to it? And can you explain it a little bit to us? Yeah, for sure. Yeah, so I think that we are more mature as a society because we can't or we think that we can trust on institutions we need to really to organize ourselves because we need to because we want a new model or a regenerative model. So I think that the emergence of all this kind of thinking about system thinking or a regenerative thinking, living systems thinking is really something that emerges from us looking for solutions for what is not working. And because I think really think that we are more mature as a societies. We are like at the moment of taking responsibilities about what we are building. So for sure that more people have more awareness about what we really or how we can include nature and natural environments in the equation of when we are building businesses. So and all these people now is inside companies. So most of the people that connect with me are people that are at the same level of thinking, looking for that kind of solutions. Maybe they can't put it on words or they can put labels on that that I prefer not to put labels because when you put labels you are like a constraint in the thing. But in any case, I think that this is because we are more mature, we want to be adults in this adult we need to take responsibility of our ads, of our business. And then we are really learning and how to do it. So yeah, I feel that it's a very, very organic and natural evolution. I've heard you say in the past that you call it almost the second renaissance. And that's interesting because really one of the first scientists, one of the first artists, one of the first people to really write and talk about regeneration was Leonardo da Vinci. And what I've seen, and I mean, I have a big stack of books here, I mention books all the time on the podcast that talk about Fritz Hof Capra or talk about that are written by Fritz Hof Capra or talk about different Italian authors, different Spanish authors or that there's these cultures or communities that very much have this almost a Latin renaissance or a renaissance in general of where these things have been around for a long time. This is not new, it seems new. It seems like it's something that's trending and it's going beyond sustainability, it's going beyond ecosystem services, it's going beyond platform. But it's really not, especially since you say it's the second renaissance, right? So it kind of explained that to us to let us know or help us understand how this is not just a new thing, the new trend that we're jumping on. Yeah, well, the renaissance was characterized by that. It was like a moment of really of intellectual, cultural creativity, new ideas putting in question how the things work at that moment. So we are like living at this second renaissance that some creator or economy, as friends are talking about and I am really, really agree with that because it's like we are living in this moment of creativity, of being aware of that we are creators in this world and that we have the tools in order to do it. So I think that it's a moment, a very interesting moment in the story. We are in the, as Lisa Genski, one of my friends and wealth leader, she said something like we are in the moment of the not yet, we're in the moment of the no more and in the moment of the no yet. So we are in this middle moment, but I think that we are more close to the not yet. And yeah, so we are creating new things. We have the tools in order to have this autonomy to create. We can coordinate because we don't want to do it alone. And we are joining communities with the same interests in order to do new things or to create new things that it was one of the reasons because of the collaborative economy or sharing economy in the United States was born because people wanted to really to stay in community again, because it was really, no, it was so isolated. I am a freelance and it's like I don't want to be alone. I don't want to create alone. I need my tribe and I have my tribe in Gwisher. That is like, yeah, the community that I joined two years ago and I'm still there. Like now I'm the president of Gwisher Spain because we are an organization that is distributed. We are an independent organization and nonprofit at a thin tank and two-tank and we are a distributed community. We learned a lot about how to manage a distributed community. That it was no easy at the beginning, how we can manage the sessions, how we can create the space for everyone to show the potential, how we took decisions, how we organized everything. And it was super interesting. We have a lot, yeah, a lot of learnings about that. So I think that this is an interesting moment of the story. Well, I think it's really interesting because you educate organizations, corporations, and especially at the EU level on the age of 2020, you're educating them on some new models, but there's two aspects in there that I'd like you to tell us about, which I really love because it gives me hope in this world of broken systems and kind of a dis-ease around uneasiness with some of the systems in the world. And the biggest one is really in government and bureaucracy and organizations that are bureaucratic or in politics where it's very bureaucratic or even in governance where the models, this regenerative platform model is one that really gets you in into those barriers of entry that disaggregates the market. And obviously the biggest one that's very complex is food systems. And so I would love to kind of hear about how this model really helps in those situations, especially foundations, NGOs, those who are really trying to offer planetary services or make the world better in some respects for people and planet. And then also if you touched earlier on DAOs, which are decentralized autonomous organizations, and that's also very similar to what you're just talking about, which is another form of model. Could you tell us a little bit about those positive experiences or how you talk to people about that and what's emerging, what's coming out of that? Yeah, so the first thing is that at the beginning it was more like traditional companies trying to understand how to really move their industrial or more linear pipeline business models into a more ecosystemic or platform business models. It was at the beginning. Because now it's a mainstream platforms, there are more governments or public institutions that are very interested in understanding how they can use it in order to really generate a major impact and also to understand how they can keep being relevant in the market or in the society. Because if they can help us evolve, if they can't really take care of us, that is not a question of taking care because I don't like paternalisms in institutions, but in any case, I don't know what is going to happen with them. So it's interesting because they are started to talk about platforms and about regeneration and about that and so it's easy to understand. So the thing is platforms really make sense in the more bureaucratic environments and also in the more fragmented ones. So if you have an ecosystem, people interacting in some way that they are already doing something, you can think about whatever. So what you want is really to facilitate them to connect them, to create the space for producers and consumers to interact and creating some kind of mutualized services in order to help them to evolve. The platforms are really hotbeds of learning. So if you can focus on learning, so and you need to ask yourself how the system is learning. So if you can really go deeper on that and you can really design a space for learning, you have a very powerful space for the communities, for the producers and consumers to really to evolve and to using their capabilities for sure to find the place that they need in order to show their maximum potential and their uniqueness. So yeah, public institutions have a real, they have a public serve, so they need to really to serve to the citizens. So I would say that, yeah, it makes a lot of sense to develop platform strategies and more and more that they have to be really regenerative for territories and because of that, we talk about this idea of, well, the pandemic, that we realized that we couldn't really to trust all international value chains. It was the past week, Larry Finn that is one of the CEOs of BlackRock in his letter to the shareholders talking about the war of Russia against Ukraine is really, it's the end of globalization. I think that, well, I think that they sell very well this idea of globalization, but it's not working at all. And I think that we need to come back to this idea that of course ideas or like things could be globalized, but all production needs to be really done in the place. So with this idea of how we can connect with each other in order to evolve our thinking and to learn from others and how we can really develop the strategies in the place in order to be more resilient and to fragile and so on and really regenerate the social issue and the community issues and this, you know, good direction is also creating this cat. Yeah, it's not just the Ukraine war, it's also happened in Brexit during COVID. So huge problems, supply chain, but everywhere's had supply chain and global trade issues during this time where people aren't working the jobs, they're not there to harvest the products, they're not there to run the ships, they're not there to do the things that are necessary.  And then the second main reason which you also brought up is we for too long have commoditized products, whether it's farm food, food, agriculture or whether it's cars or computers or whatever it is, we're turning products and things into commodities and trading them on the stock market. And it's basically a way to cheapen products, cheapen and if you cheapen anything, whether it's food or a product, you're actually cheapening life because someone's being paid less, farmers making less money, the producers are making less money. That means people get lower wages or they go without insurance or they're working under slave labor conditions and things. When we turn things into commodities, it really hurts society as a whole. And one of the biggest examples is in agriculture is the Brexit, the United Kingdom produces five times the size of the United Kingdoms, it's food elsewhere overseas. So five times the size of the United Kingdom is the amount of land that they use elsewhere to produce commodities that then get shipped to the United Kingdom for them to thrive and survive. And that's a model in times of pandemic economic downturns that has a limit to growth and has an end to crash. And we've seen the ripple effects of that there and all over the world. And that's just the two examples that Ukraine and the Brexit. And so I love that you address that. Just before we go further, I want you to maybe if you don't mind touch on the Dow, the decentralized autonomous organization, kind of what's emerging, what your thoughts or feelings are there on because that's also can be a very regenerative model as well, it's just still emerging and being kind of how do we make that work because a lot of people are like, they don't really know how that works or what your thoughts or feelings are there. But then I want to put you on the spot after that with really getting into some core principles, if you don't mind departing some wisdom to us on what are some core principles that businesses and organizations can use around regenerative platforms? So there's some certain steps or things that they should be looking at, thinking about to move forward with this concept and you help them to do that. So I don't want you to give away all your wisdom but if you can kind of give us some guidelines as well after you talk about that, I'd appreciate it. Okay, so yeah, well, thinking about value change, you know, very well, that is very linear, right? So we need to really to start to thinking about value networks instead of value chains and becoming or moving from this idea of linearity to an idea of more circularity. So those represent that. So there is for sure who is or a group of people or one individual that is creating the space for others to create value but the difference is that it's not more like this owner of the strategy or whatever, the person who developed the space, creating content or whatever inside the community is everyone creating value. So, and I think that this is something I'm doing like right now on a course on regenerative economics and for sure the most interesting thing is to connect with the other people in the ecosystem and trying to learn from each other and to develop new things. So the DAOS are doing that. So are creating the space in a more circular way for everyone create value in the ecosystem. So this is super relevant. Yeah, and the other thing is that what I'm seeing is that for sure is a super good tool. You know that there is also a very interesting to think about how they are generating the trust in the ecosystem because well, this is an idea of from Sangeet Chaudhary that is one of the relevant thinkers about platform systems and that. So in the industrial era, you the trust was generated by the, or because of the contracts between the two parts, the two sides of the enterprise and the consumer. In the case of the platform economy, it was more about the terms of services that the platform needs to put in order to really to manage the interactions between the entities and also to solve conflicts or whatever. But this I think that this is the more, very interesting thing in order to understand how we can apply the regenerative lens. This needs to be included in the mission of the company. This needs to be included in these terms of service, how you are gonna develop your products and services. And it was also referred to reputation. And in the case of the DAOS, it's more about these protocols, the software itself, but also the consensus model, how you are gonna create new blocks in the ecosystem and how it's gonna be that. And I think that there is the interesting thing, no? How you can add the lens of regeneration in order to really create something that is regenerative. The other thing is that, and it's because I love this idea of managing two worlds that it's about that the platform strategies are more about looking at the outside, looking at the ecosystem, what they are doing and how you can facilitate it. Creating some kind of total asset services in order to help them. In the case of the DAOS or in the case of the regenerative economy, it's more about, it's a movement or it's something that it needs to come or to emerge from the inside. So you can really create regenerative products or services if you are not regenerative inside your company. If you have or you are running a traditional company with the same silos, thinking about value chains and thinking about, you know, or some kind of toxic environments and because of that, we have the big resignation. If you are not really thinking about how you can create inside your company really the conditions for everyone or everything to drive and to flourish, you are not gonna be capable of creating products and services that are really regenerative. So it's a bigger circle of inner and outer.  And because of that, I would say that the most important principle in order to apply a platform strategy for regeneration, it is olism. It's thinking about the one system that everything that you do on one part of the system is gonna impact the other part of the system in a positive way or in a negative way. And this is the most dangerous one because there are a lot of diversity in how can they evolve. And then this is that, I would say, the main principle to think about as a one system, as a whole and trying to design for the benefit of all the entities inside this ecosystem, I would say human and non-human stakeholders represented. So how you can include the voice of nature, the voice of heritage or the voice of whatever, you need to take into account that is something that is contributing in your ecosystem. The other is about potential that this is something that platforms do really well, that is really thinking about how you can empower each of the entities in the ecosystem to show their uniqueness because of that platforms like Etsy are very relevant because you have a lot of creators, producers that are doing very interesting, beautiful things with their hands and all them are personalized. I mean, there is no copy on that. I mean, there is no two things that are the same. And so platforms are doing this very well to leverage on the potential of the entities in the ecosystem in order to generate a really interesting market for everyone. The other thing is that you need to intentionally design for development. So this is not something that it happens without not putting an effort on that. You need to really design for development because platforms are these hotbeds of learning. You can really create the spaces for them to connect. It could be through offline events, it could be through a chat, it could be through whatever you need to do but the platforms that will be are and will be more successful will be that that are really creating communities around them that they can learn from each other in a peer-to-peer relationship. And yeah, I mean. I love it, I absolutely love it. That is so beautiful. And you also gave a wonderful TED talk very much on platforms and very similar. It was in Spanish though. So not all of us speak fluent Spanish and understood it. Can you, do you mind please summing that up a little bit for us and what some other lessons are that you really wanted to convey in that TED talk and bring across to us all to kind of to know about platforms and regeneration. Yeah, so it was a form, I have done a lot of conference on lectures and so on but the TED talk, it was something really different because I was not talking that I am used to talk to CEOs or whatever, people that are starting businesses and so on. It was more for the citizens. And this was difficult for me to understand how to really try to resonate with them. And what I did, it was like, for me it was like a very personal journey developing the content for the content for the tech because it was like, yeah, I wanted not to, because you know what, if you had started to talk about system thinking, living system theories or whatever, I mean, you need to use words that people can understand and can resonate with. So I wanted to go to my personal experience when I was a premium from my first baby. At what moment of this period, I needed, it was like something very strong need of taking care of everything, of creating the space of painting the room, whatever I needed to do in order to really to create the space for the baby to evolve and to be well. So it was like my idea of how to really connect with this idea of return national, that is taking care. And it's something that I think that resonates a lot with us. It's taking care of ourselves, it's taking care of the people who is around, of our families, of our friends, are taking care of the environment. So if we really connect with who we are, this is something intrinsic in us. So the thing is that we just need to remember, we have passed a long time in this linearity, in this isolated or silos, whatever, and with no content with nature. So we need to understand how to do it or we need to learn how, yeah, what is our role? As humans in this, or for many a scenario, even it's not me, you know? So yeah. Even about reconnecting, so how do we reconnect? We've been disconnected from nature and our infrastructures or societies are kind of distanced from our planet and our environment in many respects.  And so you wanted to make sure that you created that. And I think that's something that we need to put into to our life as a model as well. And so that really is great because it brings me to the hardest question that I have for you today. I ask all my guests the same question. And it really, I kind of ask it in two ways.  And it's not the swear word, it's what's the future but there's a more interesting way to say it which we've all been experiencing lately and that is what does a world that works for everyone look like for you? Yeah, well, I would say that it's very similar to some ideas of SolarPunk that is ecology mixed with technology, serving humans in order to achieve what we need to do, to really to create a more collaborative society, one that is fair, where everyone could be or could have health in any, I mean, at any level. And the one that we can treat with respect life because we are one more in this web of life. So we really need to design as everything matters. I absolutely love that, that is so beautiful. And it's so true, I'm glad that you shared that with us because if we don't have an understanding or a feeling what a world that works for everyone looks like, then we don't have a path, we don't have a model to work towards, we don't, and we're kind of going to be taken by some of the models that we've had and the past or currently are operating on that are making us uneasy at the direction that our world is going with wars and other issues and problems. And so I really love that. And at the core, it's everything that we've talked about today and what you talk about is really, it's a new model. It's a regenerative ecosystem platform model. It's a better model for life, one that continues to work for everybody because everything you mentioned is included in that model. It's the basic needs of humanity that we come back and we tell and we agree with each other that we're never going to let humanity get below a certain level ever again that we're all part of this bigger family. And so I really like that. And I love that you're empowering people and organizations around the world with your teachings and your advocacy and your consulting that you do to help them to make the shift. And I know it's not a process that will probably ever end or you've reached your goal. It's one that's a process that keeps going over time. That's a great thing about platforms and regenerative models. They're always evolving and growing and learning and the web gets stronger and in places where it might not be weak then it's built up in other areas that are more important. And so I really like that open-mindedness and the tips and tricks that you've shared with us here today. I have three more questions for you before I'm done and that is really where do you see your evolution going in the evolution along with these regenerative platforms? What do you see before the end of this year and what do you see before December 2030 when we're supposed to reach the Paris Agreement and the sustainable development goals? What's your hope? What's your vision? What will you be working on? And what can we expect to see on the horizon? Yeah, what I'm working hard at that moment is to really to create some kind of framework in order to help whoever is interested. About all companies or enterprises that have a platform wants to really to include some little ends of regeneration and the ones that have regenerative projects that wants to become or understand how to really to create a platform strategy. Because I think that there are like these two worlds sometimes when you are in a more social innovation environment I don't know, it's not a general theme but I would say that they are not very used to technology. So what I want to bring is that some kind of new ideas or ideas or this mindset in order to help them to really to leverage on platform strategies to scale their impact. So in a community ownership way of thinking. So my idea is to work on this framework. I am working hard on that. And yeah, and I would say that I am a learner and it's my position in front of that. I would say that I am trying also to connect with my with whatever is my ancient wisdom in order to really to generate something that is honest. That is not something that is going to be or what people could use it as they want but my mission is really to to build the capabilities inside these organizations in order for them to think by themselves and to change the way they see or they look at the world. Great. What have you experienced or learned in your professional journey so far that you would have loved to know from the start? Wow. I learn every day. So it's very difficult for me to say if I were, you know, I don't know I am a very extreme curiosity person. So I'm always trying to learn new things. So for me it's like the infinite. It's because of that regeneration also is something that resonates a lot with me because it's not like the goal. It's more the journey that you are starting and step by step you are understanding more and more how you can be regenerative or you can embrace regeneration but because of that what I'm doing is trying to really to enjoy the journey not to look at the final whatever goal. Thank you so much for letting us inside of your ideas. It's been a sheer pleasure and unless you have some questions for me or there's anything we forgot to discuss I really appreciate your time and I look forward to working with you more in the future and the release of menu B because your contribution is fabulous and I think a lot of people will really like it and be able to see how it can apply into their business models whether they're a farmer a food producer or whether there used to be a commodity trader that they can see a model that works better for the world and for them. Thank you very much for inviting me and for creating the space for to share and to contribute to this new change of paradigm that we need to. You're most welcome. Thank you so much. Have a wonderful day. Thank you very much. Bye bye.


And today, I will only speak about 2D. And tomorrow I'll speak about 3D. So yesterday, I just did a small thing here. So let me recall very rapidly just the 2D part of it. So if you remember, I said here, Gevray, here I said Sobolev. And they put gamma was 0. And here I said gamma less or equal than 1 half. And so here there are two papers. So there is a paper by Bedrosian and myself. And that corresponds actually to mu equals 0. And then the other paper was also with the v call. And that's mu larger than 0. And the Sobolev case was done by Bedrosian v call. And I think it was dark at Princeton or Maryland. I don't know, Wong. Anyway, so I'm going to talk today. I'll explain a little bit these two results. So that's the one you have, the mu equals 0. So OK, so maybe I can recall very rapidly, since maybe some are new faces. But what we started last time. So I'm writing the equation. I mean, we start with the Navier-Stokes equation with the viscosity, because mu equals 0 will correspond to the Euler case. And we are looking at perturbation of the so-called Quet flow. So this is like a stationary solution. So today I am working in the case where I am in the torus times r. So x is in the torus, y is in r. So then if I write down my solution as the Quet flow plus the perturbation, then I can write an equation for u. u will be solving the following equation, y dx u plus u grad u. So that's the equation satisfied by u. So think of u as a small perturbation. So I'm trying to understand this with a small initial data. And I'm trying to understand the long-time behavior of this. OK, so yesterday I ended by telling you something about a change of variable. And OK, so I mean, if you look here at this equation, like the most problematic term is this term, because this is a linear term. And what I explained last time is that there is a change of variable that allows us to get rid of this. OK? Did you consider an additional term, u2 minus u2z? Yeah, OK. In two dimensions, that term is not, sorry. Yeah, no, it is. Yeah, you are right. No, no, here I should be writing it since I'm writing the equation on u, actually. But then it disappears when I take the curve. Yeah, yeah, you are right, so it's. Sorry? The two you put up are correct. That's correct, also. If I'm consistent with my notations, yes. Yeah, actually, it's true that in the 2D case, we never write the equation for this. We immediately write the equation on omega. So yeah, so I said when we are in two dimensions, actually, we don't work with the velocity, but we work with the vorticity. Like tomorrow, I will be more working with the velocity, actually. So immediately, the interesting quantity is the vorticity, so it's dx u2 minus dy u1. And then we can write the following equation for the vorticity dx omega plus u squared omega. So then we have this equation. And it is actually a closed equation on omega because you can recover u by the following non-local cooperation. Now, what I would like to do is, and I think we started doing that yesterday, is to tell you about to split things into linear effects and nonlinear effects. So the first three things I'm going to talk about are three linear effects. So the first one that I described last time was a change of variable that allow us to get rid of this term. I'm going to write it on the nonlinear equation. I mean, I did it last time so that we still keep this guy. And that was the following change of variable that we went through last time. So that's the natural change of variable to do. And if I write down capital omega to be my small omega, then the following equation becomes dT capital omega. Then the good thing is that this term disappears. And I get grad per phi omega, laplacian l of omega. So that's the equation we get. And laplacian l phi equal omega. And here, the important fact is this grad l, which is the following gradient. So I think I explained last time that the gradient in x, y of small omega get transformed in this gradient l of capital omega. Capital x is no longer correct. No, it is. Or y and phi messages. It's OK.  You just go around. And actually, I think that's, I mean, since you asked this question, it is really behind the terminology mixing that we use. Basically, if you just think about the transport equation. So if you look at this transport equation, and let's think that you start with a small perturbation. So it's periodic in x. Let's say you start with a small perturbation like that. This is your perturbation at time 0, let's say. So then let's say this is like y equals 0. So what's here will not really be moving. But then if you look at later times, this will be tilted, then more tilted. And then if you look, then it becomes, start becoming something like that. Because of, exactly because of the periodicity. And OK, I mean, it's not difficult to see. You take a weak limit. The weak limit will be the average in x, actually.  If you take the following equation, let's forget about the vorticity. I mean, you can solve it. So it's just the initial data, the y stays, x minus ty. So that's the exact solution. And this converges weakly when t goes to infinity to the average in x. So we said x is periodic, so it's just the average. OK, maybe I can write it down like this. The average is 1, let's say, so it's x. So you go to a function of y. Particular, that is even consistent with the fact that you go back to a shear flow. I mean, at least in the inviscid case, you go to a shear flow, not necessarily the quet, but the perturbation of quet. But you go to something that is still a function of y. So in particular, I mean, you can see it at the linear level. Let's say at the linear level. I forget about this term and forget about the viscosity. So if you forget about the nonlinear term and the viscosity, then you have dt of capital omega, which is 0, which means that nothing is happening, because all shear flows are stationary solutions. OK, so actually, another interesting fact about this that I want to maybe mention is that's one of the reasons actually why, at least as of now, our result, we only know how to do it if we perturb quet. I mean, we are trying to understand how to generalize it to other perturbation, not of quet, but perturbation of other shear flows. One interesting fact is the fact that if you take Fourier transform of omega, let's say small omega, you still get a transport equation on the Fourier side. And that's, I mean, there is a Fourier side explanation of this picture that one can understand, which is the fact that you are sending information to high frequencies in eta, actually. So if I take omega hat, so remember, here I'm taking x, y. It becomes k eta. I mean, it may be confusing, because afterwards I will be using k eta for the Fourier in capital X and y. So the eta I'm writing here will be different from the eta I'll be writing afterwards, but it doesn't matter. So y dx becomes k d eta. I mean, there may be minus, I forgot whether it's a minus or not.  So what's happening here is that for k different from 0, so for k equals 0, nothing is happening. So you are keeping the mode k equals 0, nothing happening to it. But for k different from 0, what's going on is that the information is being sent to high frequencies. And OK, since you are taking your omega initial hat in some space, let's say some sobolev or whatever space, so then you start seeing the decay. You start seeing some decay when t goes to infinity, but for k different from 0. So this is what is called mixing, which in a sense, one can try to understand it as, the way I understand it is more or less like dispersion, but you are replacing the Fourier side and the physical side, basically. OK, so this is the first linear effect I wanted to tell you about, which is the change of variable. So this is the first, I mean, it's a simple thing, but still you start seeing some interesting features from it. So now this change of variable has two, I mean, you can see has effects here. Instead of Laplacian, now you have Laplacian L. And here, instead of Laplacian, you have Laplacian L. And each one of these two Laplacian L will give us some linear effect that will be important. So the next linear effect I want to tell you about is so-called enhanced dissipation. So this is an effect when, for viscosity, it is strictly bigger than 0. So it is a linear effect. So let me just write down this linear equation on omega. So we just have dt omega equal nu Laplacian L omega. So now you can see what is your Laplacian L. Your Laplacian L has a t in it. So if you write this on the Fourier side, you will have, OK, so that's, don't get confused. The eta there and the eta here are not the same. There is a relation. I mean, it's a small exercise one can do. But here I'm taking the Fourier in capital X, capital Y. So then this gets minus nu k square plus eta minus tk square omega. Make sense for everyone? So then, I mean, you can solve this. You can solve this.  Basically, you can write omega tk eta equal exponential minus. So we can integrate between 0 and t. We should keep the nu here k square plus eta minus ks square ds times your initial data. Of course, the initial data, the initial capital omega is the same as the initial small omega, because at time 0, x and small x are the same. So this is the formula you get. So then, I mean, what is the interesting fact here is the fact that we have a t here, t square, and you integrate it, it becomes t cubed. So then you are, I mean, there is a small calculation one can do to prove that uniformly in k and eta, at least when k is different from 0. So I mean, there is an important effect when k is different from 0 is the fact that this term will give you more viscous effect. So basically, I mean, it's not a difficult calculation to do is that your omega hat of tk eta for k different from 0, one can prove that uniformly in k eta. Of course, you can already see that here there are times where you may have small problems when t, when this starts becoming 0, you can lose some uniformity. You have to be careful about those times. But what one can prove is something of this type, something that depends on the initial data. So somehow you get a decay of this nature with some constant c. So that's why the name enhanced dissipation, because now you see dissipation over time scales of order nu to the minus 1 third. So here we have dissipation over times like nu minus 1 third.  Normally, the heat equation gives you decay over times like nu to the minus 1. And so this is a very important fact. I mean, here we are more thinking about very small viscosity, so if I'm thinking that my viscosity is like 10 to the minus 30, this means it gives me a decay over times like 10 to the 10 rather than 10 to the 30. So that's how you should understand this. So it gives much stronger dissipation for non-zero frequencies. And by the way, this is really the type of, I mean, it is enhanced dissipation by mixing. And this is really what you have when you put sugar in your coffee and you stir it. This is more or less the effect, like the reason you try to rotate your fluid is that you want to enhance the effect of the dissipation. OK. OK, so what this happens for k different from 0, OK? Of course, like 0 modes, for k equals 0, 0 modes will just experience the regular heat equation dissipation, right? There's no enhanced dissipation for the 0 mode. OK. So again, this effect came from this Laplacian L, like the Laplacian that became Laplacian L in the new coordinate system, OK? So now there is another effect that comes from this one, which is actually an inviscid effect, not the viscous effect. So the third linear effect is so-called inviscid dumping. And the reason this is, I don't know when the terminology came in, but I think it's around the 70s or the 80s. I mean, the terminology came related to the Landau dumping in plasma physics, even though historically, like Landau dumping is, I think, around the 40s. But the first instances of inviscid dumping go back to a paper 1907 by Orr. I think it's somehow a paper that was forgotten, but I think now it's came back as a very important paper. And OK, so what's the inviscid dumping? I mean, the whole thing is really in this formula. I mean, just write down the formula and you can see what's going on. So if I go to the Fourier side, so I look at my phi hat. So I think I have to put a minus, OK? So if I am, OK, let me write it down like that. If I ignore the viscous effect and I ignore the nonlinear term, so if I ignore both nonlinear term viscous effect, then omega, then this will be just the initial date. I can't forget about the t, OK? So now what's happening? So if you look here at this and you look at it a little bit, you can observe two facts. You can observe two facts. Like if you look at t large, when t goes to infinity, your phi will decay like 1 over t squared, at least for k different from 0. So if k different from 0, phi hat will decay like 1 over t squared, OK? So that's the first thing you can observe. The other thing you can observe is that if k and eta have the same sign, before you get this decay, you can have some growth. So there is a transient growth before the decay, right? So after possible transient growth, OK? So all this is just in this. All this is just in the denominator. Just look at the denominator. Think of this as being constant. I mean, just you are drawing the function 1 over 1 plus x squared. If you look at the function 1, it's just like it goes up and down, right? So I mean, again, here all what's going on is this is like the picture. If I draw the picture of this, let's say at least thinking of this is time, this is how it goes up. I'm going to draw it in the case k and eta have the same sign. If I take the time here eta over k, then I can start here. This is where I reach my maximum. So then my function does this and then decays. So here it decays like 1 over 1 over t squared, OK? So this is what is called, the reason this is called inviscid dumping, so it means that the stream function will decay like 1 over t squared. But also the velocity will decay. Now the velocity, remember, I think when I made the change of variable, I also said that I'm going to use the following capital X, capital Y to be the velocity in the new coordinate, and that this will be the grad per l of phi. I think I explained a little bit that. And this means this is dy minus t dx is a minus here dx of phi. So then on the Fourier side, this means that your u hat of t k eta, OK, modulo of sines, OK, let me not think about sines, so then I can just put absolute values, times phi hat. But then phi hat, we wrote it there, divided by k squared plus eta minus t k squared, OK? So what does this imply? It implies that u1, so for k different from 0, u hat 1 will decay like 1 over t because of this.  And u2 hat will decay like 1 over t squared, OK? So here I'm not being very precise. In this sign, I mean, it's not uniform in k eta. All those things are not completely uniform in k eta. You have to worry a little bit about it.  Maybe I'll erase this because I'm not going to use some. So the exercise says the following. Decay costs regularity. This is really the decay costs regularity. And this is an important time. Yeah, decay, what I mean by decay is decay in time. So decay costs regularity. Basically, what you can prove is that, so here the sign different means k different from 0. Different means the Fourier. It's just you take u1 and you remove the average. So if you look at this, so here I'm not putting a hat. So I'm taking this in a physical space. So I'm looking at it in HS. You can prove that for the linear problem, so I'm just taking the linear problem without viscosity. So this term is not there. This term is not there. So it's just this formula with a fixed omega hat, which is just omega hat initial. So this exercise is related to this formula, just this formula. Take this formula with a fixed omega hat initial, fixed initial data. What you can prove is the following thing. 1 over t, so I'm using the Japanese bracket, omega initial in HS plus 1. Same thing if you look at u2 HS plus 2, or HS. This you can get 1 over t squared omega initial in HS plus 2. So if you think a little bit about it, you are losing three derivatives in this. I mean, if you want to get the 1 over t squared decay, you lost three derivatives. Because normally u is 1 derivative better than the vorticity. But to extract the decay, you are losing here three derivatives. So here there are three derivatives lost to extract the decay. Three derivative loss to get decay. OK, so for those familiar with Landau-Dumping, for the Vlasov case, in the Vlasov case actually the decay here, the amount of decay depends also on the regularity, meaning that if you put more regularity, if you put Gevry, if you put analytic data, you can get exponential decay. If you put higher Sobolev, you can get here and s, depending on the regularity you have. So that's why the Landau-Dumping from this point of view is easier in terms of decay. So this is a completely linear effect, again. And the interesting thing about this effect that it is actually independent of the viscosity. So I wrote you here the formula with putting here the initial data. OK, but I mean, even if you put this, this will be a term that's just helping you, actually. So we will still get this, even with the viscosity, but the point is that all this is uniform in the viscosity. So you can see here, at least, I'm trying to do both with viscous term, with the viscosity and without the viscosity, but you can see at least there are three different timescales in this problem. So there is a timescale on which the inviscid dumping happens, which is independent of the viscosity. So that's one thing. So then you have another timescale, which is the following timescale, which gives you this sort of enhanced dissipation on this timescale for non-zero modes. And then there is also the zero mode itself. The zero mode itself will decays at the regular timescale of the heat equation, like nu minus 1. Any questions, maybe? Before I move to the nonlinear effects? Yes? What? Not equal, this one. What? Human. It's not equal. Oh, so not equal means the non-zero modes. So maybe that's like, there's a notation. OK, so maybe that's like, there's a notation. Maybe I can, and actually that's the reason why the two is up, the one and the two are up, is exactly because we use, in the paper actually, we write it always like that, to talk about the Fourier. When we do the Fourier transform, we put the k here, eta here. And then this guy, we distinguish the zero mode. This is just the average in x. OK, let me write it without the Fourier. So this, what I mean by this, is just the average in x. And this is nu minus the average, OK? I mean here, I wrote things in the physical sides, because then I wanted to use the Sobolev. But like most of the calculation are really done on the Fourier side. OK, now let's go to the nonlinear effects. And then I hope I can state a theorem, or maybe next, I don't know if I'll have time. So nonlinear effects. Yeah, yeah, still have time. So nonlinear effects. OK, so what do I mean, what are the nonlinear effects? So if you look at it, nonlinear effect means this term.  It is a transport term. And the first thing you can think about is that, it's not that bad, because I just told you about the fact that this guy is actually the guy I wrote here. So I have decay. I have decay of it. I mean, there is decay, but modulo the zero mode. The zero mode is not decaying, right? The zero mode is not decaying, so I need to do something with the zero mode, right? So that's the first thing you can think about. And it immediately requires actually to make a nonlinear change of variable, right? So basically the first thing we can see from this term is that we need a nonlinear change of variable, OK? It turns out, I mean, this was one of the parts where we spent a lot of time to think about what was the right way of making the change of variable. I mean, there are many possibilities one can think about.  So that was the first thing you can think about is to do, you can say, OK, let's do a Lagrangian change of variable. It turns out that's not the right, I mean, that's not the best thing to do. And what we ended up doing is actually just get rid of the zero mode, right? So basically what we do is that we take x to be x minus ty minus the integral between 0 and t of, OK, so this guy, u0, 1. So u0, 1 is just a function of y. Doesn't depend on x. That's one of the advantages that I'm doing the change of variable here with something. In x, it is just a translation, syds, which is actually the piece that doesn't decay, OK? Of course, there is normally, I mean, you should be worried about this guy. I mean, this guy decays like 1 over t. So the decay of this guy, it turns out, I mean, is critical. But actually, in the first version of the paper, we had a logarithmic correction because of this guy. I mean, we thought that we need a logarithmic correction, the statement of the theorem. But then there is a very subtle cancellation that actually told us that we don't need that logarithmic correction. There is some sort of cancellation. But I mean, it was not important in the proof anyway. But somehow, the advantage of just doing this rather than doing the Lagrangian change of variable is that this is just a translation in x. And it has advantages whenever you look at this picture, which I'm going to talk about again. OK, so that's the change of variable. And it turns out that also took us a while to think about that we also replace y. So instead of using y, we make the following change of variable. We replace y by this quantity, by the quantity that we put here. So this is the change of variable we make. So here, I'm making the change of variable. I'm trying to explain things when u equals 0. For a reason, so that at least you understand the nu equals 0. Tomorrow, I'll be more talking about nu larger than 0. And actually, there is some subtlety in how we change this, in the case nu not equal to 0. We have to take into account a little bit some dissipative effect. So our change of variable will not be completely just transport type. OK, so this is the change of variable. The hidden reason for this is exactly to keep these operators. I mean, what you see here, I want to keep it in a nice way.  I want to have something that looks like this when I invert my Laplacian. I'll explain it. I'm going now to explain this. So when you make this change of variable, what happens in your equations? So what happens in your equations? Now, my Fourier is xv gives k eta, OK? So now, the Fourier is really in this xv, in this nonlinear xv, OK? So if we do that, actually, the new equation you get is the following. Let me write it in some, OK, let me use the following notation, f of txv is my omega in txy. And it corresponds to what I was calling capital omega. But now, since I'm using a different change of variable, maybe I'll use a different notation for it. So this is the equation now we get. And I'm going to only insist on u equals 0, OK? We get the following equation, dtf plus u, I mean, the u here has nothing to do with the u I was using there. And it's a slightly long calculation with change of variables and so on. The whole thing is really change of variables. And the u looks like this, OK? It has 0 dtv. I'll explain a little bit some of the notations. That's v prime, grad perp v, projection different from 0 of phi, OK? So this is what we get. I need to explain to you a few things. So v is a function of t and y. I mean, just based on the change of variable. I'm just going to tell you about it right now a few things, but then we'll forget about it. So v is a function of ty. Again, one of the advantages is that I'm not mixing y and x here. I mean, it's very important in the proof. It's very important in the proof. So what I call dtv is just dtv. What I call here dtv is this notation is you make the change of variable, you write it in the v coordinate. So it's really dt of v of ty. So you see that there is a change of variable that takes y to v. So you take the time derivative in y. So this is a time derivative in y, but then you express it in v. I mean, that's a notation. That's a term that comes. It's not difficult to see. I mean, whenever you compute the dt, the dt in y and the dt in v are not the same. And there is a change of variable that comes. And it is this term. And what I call here v prime is more or less the same thing, but with y derivatives. So the v prime is like a dy of v, but then expressed in tv. Again, OK? Just, I mean, OK, this is at the technical level, but it's really a nonlinear effect. I mean, the reason I'm trying to insist upon this is that there's a nonlinear effect coming from the change of variable and the choice of the change of variable, right? So I think I should have insisted on this. Actually, I said one thing, which was maybe this u1, it's true that the u1 decays like 1 over t. But I explained to you last time that in 2d, this happens in 2d in 3d, it doesn't happen completely. In 2d, what you see here is not the u1, but it's the grad perp of phi. Actually, the grad perp of phi decays like 1 over t square, because I'm not seeing this term there.  So basically, you come back here. You come back here. We got rid of the zero mode. So now this guy actually decays like 1 over t square. The derivative doesn't lose me more because there's no t in it. So this actually will decay like 1 over t square. There are 10 pages of calculation to actually also prove that this guy decays 1 over t square minus epsilon. So at the end of the day, when I wrote the equation in this way, I managed to rewrite it with a velocity that is integrable, like better than 1 over t decays, actually 1 over t square minus epsilon. So that's the effect of two things, of the change of variable, removing this non-decaying piece of this interesting sort of, we call it like some sort of null condition, like some nice cancellation between the terms that have growth.  So this is a good form to work with. Any questions about the stage? If you put exactly capital U, the whole thing, yes. You don't put this. And then you put an x dependence here. So then it mixes all the issues that it mixes x and y. And I think it completely destroys this. So actually, I didn't say one thing here, which I think is good, because in the linear case, I had Laplacian l phi equal omega. In the nonlinear case, that will not be the case, right? Because my change of variable is nonlinear. So normally, you should be asking me, what is this phi? How you relate phi to f? So the way you relate phi to f, so let me erase this. I'm just going to forget about these guys from now on, because I explained more or less how that comes. And you can ignore that. So the phi will satisfy some Laplacian, what we call Laplacian t phi equal f. And the reason we make this and this is that I'm trying to make this Laplacian t look very similar to the Laplacian l. So there is a whole technical part, what we call elliptic estimates, that try to say that Laplacian t behaves like Laplacian l. If I make different change of variables here, this will not be the case. OK. Maybe I can just mention one instance where, yeah, let me write down the formula for it, because then at least you can see why we make the, I mean, I can write down the formula for it, but then we'll forget about it. So Laplacian t phi is dxx phi. So that doesn't stay the same, because you are just translating in x. You are not doing anything. But OK, let me not go write you the whole thing. But the whole thing here, I mean, you have some coefficients. Let me not write them down. So basically, by making the change here normally, there is a term which is like v prime square. I mean, there is another term, a mixed term. But I like this expression to stay the same. dv minus t dx. If I don't make the change of variable, if I keep y here and not v, there will be some v prime somewhere. And I will not see this piece exactly as it is there. OK, sorry, I think it's 1 plus v prime. No, no, no, it's v prime, because v prime is like 1. So think of this as being small. I mean, this is small, so v prime is like 1. So v prime is like 1. So this is a perturbation. It's a small perturbation of Laplacian l. OK, so this is the first nonlinear effect, which is the change of coordinate. So now the next very important fact is the so-called nonlinear resonances. In the plasma case, it is called echoes. I mean, we can call them also echoes. And let me try to motivate this. And I think the motivation is, I mean, I insist more on the motivation, because I think it explains why there is Geoffrey, actually. OK, so we have this equation, dtf plus u grad f equals 0. I'm just taking the inviscid case. And let's say I'm trying to prove some energy estimates. So the problem is about global estimates. And I know that my u, which you see there, will decay like 1 over t squared, right? So forget about this one. I mean, it's more like this one. Let's say you try to prove Sobolev bounds. So if you try to do Sobolev estimates, so then you apply s derivative to your f, OK? So we are all used to apply Leibniz's rule. And you just worry about two extreme cases, either all derivative heat here or all derivative heat here. So then you get a term like this.  And you have a term like that. Now, if you try to do Sobolev bounds, you multiply by dSf, multiply by dSf. And then you get dT of dSf squared over 2 integrate. What do you get? Plus dSu dSf squared f. So you have a term like that. Let's say that's equal. And you have a term like this. Oh, sorry. OK. Let's put absolute values everywhere. So the other term you can integrate. It turns out like the u you have here actually is not divergence free. But it's not a problem. I mean, that's not an issue. Plus something like gradu, let's say dSf after you make the integration by part and so on. OK, let's put these two guys. So the second term is very good. Actually, this term is perfect because the derivatives are hitting f. So these are the high frequencies. And I said gradient u, I said that u decays. But I have to lose, I have to spend some derivatives. But it's just a gradient of u. So if I take s large, I will still see that this decays like 1 over t squared. Actually, because of the dt v minus, OK. But this term is fine. It is integrable.  Now, if you look at this guy, if you look at this guy, actually, f doesn't decay. So this guy doesn't decay. And this guy will decay. But to decay, you lose derivatives. So if I say that dSu is, let's say, if you put this, I mean, you can put this in L infinity, let's say. And you put this in L2. And you put this in L2. So if I want to say that dSu in L2 decays like 1 over t squared, at the estimate I explained there, then I have to put f. f is like the omega in Hs plus 2, OK. That's how I can, if I want to get the decay, right? If I want to get the decay, I have to lose derivatives. And then it doesn't close.  And so I mean, from this sort of heuristic, it seems that the problem is completely lost. I mean, because you are losing two derivatives. I mean, usually in these problems, if you lose one derivative, you hope that you can close it in analytic. If you are losing two, nothing you can do. OK, so that's how starts like you have to be more precise in the analysis and then stand where the loss happens and so on, right? So that's now what I'm going to talk about. Next thing is, yeah, let me take like seven minutes or maybe more, 10 minutes, 12. Oh, great. So then I can tell you the whole non-resonance thing. OK, so actually the whole picture, I already draw it. The whole thing comes from this picture. The whole thing comes from here, right? The whole thing is really hidden here. Because like the u, the u is what? The u you see here, that's like the grad perp of phi.  OK, so this term can be problematic at these times, OK? Can be problematic at these times. So that's, OK, now you let me go fast. I mean, I'm going to skip one thing, which is I just say that we are going to use para-products. So we are using the bony para-product. There is a small heuristic that I can explain why we do it. But it's more or less in the formula you see there. Like whenever you take derivatives, I mean the guy you put on it, derivatives, is the high frequency. The other one is low frequency. So basically, I'm trying now to draw a toy model that will mimic the maximum possible growth coming from this term, right? So all this, all what I'm doing now is more like heuristic. But somehow it will give me what is like the maximum loss of regularity I can get from this. So notice I'm saying loss of regularity instead of growth. Because as you saw there, I can play between loss of regularity or decay. So this term, I mean for those familiar with, I can write it down as, I can split it in two pieces, u high, f low. So I mean for those who are familiar with para-products, that's really like a para-product. For those who are not familiar, think about a product as a convolution on the Fourier side, right? And then when you write down a convolution on the Fourier side, let's say if I say that I can decide which one of them has higher frequency. And if this is low frequency, if this is the low frequency, that this is more or less the term that you'll be seeing more like here. So that this term will be OK. The problematic term is when the high frequency is on u, because that's where you start losing, right? So basically, this is the important term that can give me growth or loss of regularity. So then what we do, OK, so then I look at it and the u high is actually coming from the f. Because the u high, remember that the u high was this grad perp of phi. So a good candidate for the right heuristic is that if I write down, so this is like a toy model. I mean, we write it as a toy model in the paper. So here I'm writing the product as a convolution on the Fourier side, OK? And I'm only taking into account this term. So this is the low frequency. There is a t k minus l eta minus c dc. Actually, here precisely, actually, I wrote only one piece of this scalar product. I mean, the most important term in it. OK, so this is the low frequency. So this is the term I said here doesn't decay. That corresponds to this guy. The guy you see here corresponds to this part, more or less. And part of the loss of the derivative, for instance, is here. You see a c here. In the denominator, you have a c square. But when t is exactly equal to c over l, this term is not there. So you are not gaining those two derivatives. And you are losing the derivative, right? If you are. But then if you want to get decay, then you have to lose even more derivatives. So basically, this is where the loss comes from. It comes exactly at the time when this lc talks to this k eta. And you are at times where t is like c over l, OK? I mean, this is like low frequencies. You can think of this as fixed equal 1, let's say. So now, how do we draw our toy model for this? So the toy model, I'm going to jump a little bit. I'm going slightly fast for one. So it turns out in all this, I can think of c as being equal to eta. There's no big, nothing very dramatic happens between c and eta. So let me ignore the integral in c. I mean, it has to be proved. But think of c as being equal to eta. That's the first simplification, c equal eta. So then, and let me ignore this term. Just think of it as a fixed constant. I mean, it becomes like some sort of kappa. So again, I mean, here eta now is fixed. So you can think of this as a system of ODE's. It is a system of ODE's that couples all the modes l, all the modes l. Only modes l different from 0 appear here. And what we are trying to do is what to find is what is the maximum growth of this system of ODE. So at the end of the day, there are absolute values everywhere, because I'm trying to find the maximum growth. Another reduction that one can do is that I just need to look at l such that l is less than square root of eta. It's not difficult to see, because if l gets large, this term starts helping you and start really killing some of the loss that you are getting. So really, the difficulty is when l is less than square root of eta. Now, one can observe this system of ODE and try to find what is the largest possible growth. How the largest possible growth can happen. And the largest possible growth can happen in the following scenario. So eta is fixed. And I'm taking k to go from square root of eta down to k equal 1. I'm starting from some k0 square root of eta down to k equal 1. And then I'm looking at times eta over k0 eta over k0 minus 1 or the weight of time eta. So remember that if I look at the mode k0, which is this one, it will grow at this time, eta over k0. So that's where that mode, the mode k0, the effect of it becomes stronger, because that's where this guy disappears.  Then at later time, k0 minus 1 will talk to k0 minus 2 and so on. So then we write down the solution of this ODE. And we see what is the worst possible growth. And then you find out that the worst possible growth is like exponential c square root of eta. And that's how you see that this is the Geuveret type of things you have to take. So basically, since this scheme gives you this growth, you will take your initial data such that your initial data is somehow eta to some power s, where s is larger than 1 half. So then even though you go through this growth, it's still fine. Maybe I should stop here. I didn't give you a statement, but I think at least I try to explain what was the next time. Yeah, right. Yeah, OK. Any questions? Questions. No, maybe I have to just so that I can clarify how I have mine. So we are really interested in the detail because of the anyone who don't feel as long as people think. I know just a bound in a two set. We really want the detail because when you integrate, you have to go to deal. So basically, you want that f converges. Right, so what you want, you want that your f of t converges to some f infinity when t goes to infinity.  Like you want time derivative to be integrable, but to get time derivative to be integrable, you don't want to lose derivatives there. Because saying that f converges, saying that f converges, it means that then you are looking like, you start looking like linear, right? Good. Other questions? So, can we show, yes, bigger than one half is a yes. So we are working on that now. I mean like I'm with a postdoc, you dank. We are trying actually like, so you see in this toy model, I mean I got rid of many things. I put absolute values. I said, okay, let me just look at this. But does this really happen? I mean the question you can ask, because this at the end will give me this growth. But then the question is, does this really happen? Are there cancellations that can make this?  Yeah, whenever now I start looking at this sort of growth, the way the growth here is computed, is that you put absolute values and then you have some large ODE system, but then you say, okay, I mean I can simplify it. But then the question is, okay, does this really happen? So then you go back and you try to put energy in this mode and you try to follow it, how it goes through this mode, and you try to make sure that the growth really happens. To justify that the jewelry is really necessary in the proof. Or it can be a very different exponent. No, no, yeah, but here I think it is the right one. The jewelry we have is the right one. Okay, thank you.