scared of in the next 30 years? Yeah, we people are very lucky because we are in the area that the world is in a very big transformation because of the technology. I think we read a lot on the books that 200 years ago when the first industry had a lot of great people. And then 100 years ago, electricity, revolution came, a lot of success people. So now the technology comes. This new technology will also create a lot of success people, interesting careers. But honestly, every technology craft revolution come is going to create a lot of social problems. And my view is that first technology revolution caused the first world war. Second technology revolution caused the second world war. Now we are in the third. So what is going to happen? People worry about artificial intelligence. People worry about robots. People worry about a computer and data and privacy and securities. People start to worry. But whether you worry, it will come. If you don't worry, it comes. So the thing is how you change yourself. So I think next 30 years, the world is going to be a huge change. If there is a third world war, I think the world war should be a war against disease, a war against environment pollution, a war against poverty together. We should not a war against ourselves. But if we do not see clearly that using the new technology to solve this problem, if we do not unite, align together, human being going to fight each other. Because every technology revolution will make the world unbalanced. When something unbalanced is coming, somebody used to be number one, suddenly become number two, and somebody was number three, and then become number one, it's all about comfortable. So we people, if you are 30 years old, my view is that when you are 20 to 30 years old, you should follow a good boss, join a good company to learn how to do things properly. When you're 30 to 40 years old, if you want to do something yourself, just do it. You still can't afford to lose, to fail. But when you're 40 to 50 years old, my suggestion is that you should do things that you are good at. And not do things that, oh, that's very interesting. I want to try something new. It's a bit dangerous more. When you're 50 to 60 years old, spending time training and developing young people the next generation. When you're over 60 years old, better stay with your grandchildren. But this is normally general like that. Not everybody would like it, but most of the people would like that. So 30 years old, we people next to 30 years are the luckiest period. We are a lot of challenges, a lot of things are happening, and with our knowledge, with our physical power, it's time you need to change. You have to think we all, if you want to be successful tomorrow, it's impossible. If you want to be successful a year later, it's impossible. But if you want to win 10 years later, you have chance. That is why I think. Wonderful, and moving on to a topic which is kind of impossible to ignore after 2017. So a third of all the people in leadership positions in Alibaba are women. But the picture elsewhere is a bit more bleak and a friend of mine recently said something that has really stayed with me. Women are tired of empowerment programs, and we want power. In 2016, Venture Capitalists invested $58.2 billion in male founded companies, while women received just $1.46 billion. And as 2017 showed, it was a year of sexual harassment scandals, the gender pay gap still exists, and unfortunately, the list does go on. So how do you think we can move beyond tokenistic gestures towards genuine gender equality? Yeah, that is very interesting. A few years, or like four years ago, there is a journalist came to our company, and when he leaves, he asked me, question, why there are so many women in your company? And I never realized we have so many women in our company. I said, anything wrong? Today, we have a 49% close, 49% of the employees of our company are women. That's a lot for a high-tech company. It's not on purpose, we're high women. We think they are great in helping us to grow. Alibaba is an e-commerce company. E-commerce is a service industry. To serve people better, you should have a service heart. We find that women doing much better jobs than men. And the second is the last century, people compete because of muscle. This century is not muscle, it's the wisdom. And next is, I believe, if a person want to be successful, you should have a high EQ. But if you don't want to lose quickly, you should have a high IQ. But if you want to be respected, you should have high LQ, the Q of love. So those three Qs put together, a lot of men, they have a high IQ, but low EQ, a very tiny LQ. Women balance-wise, they're the best. If you want your company to be successful, if you want your company to operate with wisdom, with care, women is the best. In my company, we notice that women, they care others much more than men. Even our site, today on shopping site, e-commerce, we find women buy a lot of things, but they buy for their husband, their parents, their kids. Men only buy for themselves mostly. Because you care, that make you the difference. So, we have 49% of the colleagues in the company are women, and close to 37% of the senior management are women. We never feel anything wrong, and women, they sacrifice more when they love it, they believe it, they continue to do it. So, I think one of the secrets, the source for Alibaba to be successful was successful in the past 18 years, because we have so many women colleagues.  So, if you try to say, now we got too many men, so we have to put some women inside, that's a disaster for both sides. So, what would your advice be to companies that operate a little bit more on that quota-based system, but seem to be not making progress or not having such kind of representation? When you have a quota-based, you must have a lot of women, you will never reach that. I think if a company full of men, trust me, this company cannot be sustainable, cannot keep on growing, because just like a steel, if you don't put what inside, the steel is too tough, it's easy to break. So, when you have a woman and a man in the same company, when something happened, the women are the, for example, one of the value for our company is that the number one value is custom number one. Inpray number two, sharehold number three. This is everywhere I go. Wall Street did not like my point, but this is what I said, you know? If you like us, investing us, if you don't like us, please sell our stocks, don't buy us. We believe custom number one, inpray number two, sharehold number three. But when men start, when we start to compete, men suddenly get very angry, and you know this stuff, and women always come up and say, la, la, la, la, la, la. No competition, let's think about custom number one. Most of the women, they believe in the first day dream or first day love. Most of the men, easy to forget. I'm sorry, I say bad about men. This is statistic, this is statistic. This is why our company worked the balance. And it's naturally happening. It's not like we, so a lot of people have a tendency, men is better in doing things. But we think that even if men is better, we still have to hide the women inside. And then the women have enough women, women will hide people more. So this is how we did it. So I don't know how many high tech companies in the world like us, 49% colleagues of women, we feel proud of that. As you should be, and I think it's a very strong message to send out to, like I say, other companies that have a range of reasons for why that's not the case, because clearly you've been able to build the most successful one with that being so. So you just came off stage after giving a talk about digital e-commerce and what needs to happen for small businesses and entrepreneurs to benefit from globalization. At a time when globalization once again is being questioned, is it really delivering for the most vulnerable, is it delivering for people in the global south, et cetera. So what is your message to people in this room, kind of about globalization really? I think globalization cannot be stopped. Nobody can stop globalization. Nobody can stop trade. And I believe if trade stops, war starts. Trade is the way to solve the, to dissolve the war, not to cause the wars. I think globalization did a fantastic jobs in the past 30 years, enriched a lot of countries. But of course, cost a lot of problems, right? Young people did not have opportunities. Small business have not opportunities. Developing countries were neglected. But it's only 30 years. It's a baby. It's a growth, you have to improve it. If you do not improve it, then you kill it, it's easy. Most of the time, to kill something is much easy. So my beliefs, it's our generation's job. Today, we have a better technology. We will have a better knowledge about globalization. And it's our generation's responsibility or opportunity to improve it. Tell me, how can you stop globalization? Now artificial intelligence come, robots come. My grandfather worked 16 hours a day. And he said he was very busy. We work eight hours a day. We said, we are very busy. Our children may only work three hours or four hours a day, for three days a week. I bet they would say, oh, we're very busy. Then when you work only three hours a day and four days a week, what are you gonna do? You're gonna travel around, right? You cannot stay in the home for a week.  Because of mobile, how can you stop it? It's impossible. So thinking about that, the only thing is to improve it, to make it simple. Global trade should be simple, should be modernized. Should have given inclusive. Everybody had the opportunity. So my strong message is that, well, I feel sorry for WTO. They did a great job, but in the past 20 years, when you put the 200 government ministers in a one room, how can they agree on something? Right? Because I don't like your country. No matter how good suggestions you have, I say no. But business people, even if we put 200 business people in a one room, we will surely agree on something. And this is what I believe. The next generation of globalization should be inclusive. Should create opportunities for young people to get involved. The first globalization in human history was controlled by few kings and emperors. Last 30 years, globalization was controlled by 60,000 big companies. If you're not among the big companies, if you're not in the big, powerful countries, you don't have a chance. The next 30 years, I bet we will have 6 million or 16 million or 16 million companies get involved in globalization. And I'm sure we will make it happen together with a lot of people, maybe with you. With the mobile phone, if people still use the mobile phone 30 years later. With the mobile phone, you can global buy. You wanna buy something from Kenya? You just click. You wanna buy something from Norway? You click. Global buy, global sell.  Today you can sell across the world. And global pay, global delivery, and global travel, the only thing you have to bring is a mobile phone. No, not even a passport. This is gonna happen in 10 years. And this is the message. Let's catch this opportunity. If you complain, the other people catch it. If you catch the opportunity, embrace it now, you will be the next Alibaba. People like me, I was born in a very poor family. I never got a great education. I failed all the examinations for what reason, I don't know. But later I realized, I don't have money, I don't have technology, I don't have a lot of good backgrounds where we have a rich uncle or something, no. The only thing I competed with my people, the young people, is let's compete for 10 years later. This is what I believe 10 years later will be happening. So everything I do for that goal, I know 10 years later this thing is gonna happen. So prepare for that. Because I know if I compete with him for next month, no chance. So this is how my message is, it's a challenge, but it's an opportunity. And it's the opportunity for people like us. This world, the most difficult thing is to convincing a successful people. Ah, when you tell him, this is a great opportunity, he's, nah, nah, nah, nah, nah, forget it. I've been doing this for 30 years. But for people like us, we're looking for opportunities in order to survive. So we will do anything to be creative. So this is the message. 30 years, it's an opportunity for us. It's a challenge for those people who are 60 years old. If they're 60 years old, not discrimination, but it is tough for them. And maybe says something about how we need to look at government as well in terms of more young people entering and doing policymaking. So if there are these six or 16 or 60 million businesses and we're all kind of hyper-connected, on a more fundamental level, do you think consumption is making us happy? Consumption? What do you mean consumption by making us happy? Oh. Through consuming, are we becoming happier through being more connected? Is that genuinely driving- Yeah, of course, consumption will make, without consumption, definitely we're not happy, right? But if there are consumption, may not be, the thing is that for me, people like me, people say, Jack, you are very rich, but I don't have time to spend money. And I know this money does not belong to me. When I have a one million, two million, it's my money.  When you're over one billion daughters, that is the responsibility, because the society trusts you, Kim, manage the money better. If you think that is your money, you're getting to trouble. This is what I always believe. That's the, how can a person have a, you know, 20 or two billion dollars? No matter how hard your work, you should not get to two billion or 20 billion. The money people put on you is the trust and credit you have. So you should spend money better than government. You should spend money better than the other people. Spend money on the right places. So, consumptions, yes. And I can service industry. Let me tell you one thing. The AI robots is gonna do a lot of jobs because in the future, machine, but there will be, we have to create modern service industry. Service industry is gonna help. Service has got a lot about consumptions. So, doing the consumptions, service industry has huge hope, but you have to do it uniquely. I think from that slightly philosophical point, we can turn to the floor for questions. So, if you could please introduce yourself, say where you're from, please keep your questions succinct so we can take as many as possible and relevant for the whole group. Great. So, would you like to go first? Hi, my name is Parv. I'm a global shaper from Chicago and you just talked about philanthropy and I'm curious if you could talk about how you see your role as changing in the giving space, particularly given the fractured world we live in. Yeah, I think philanthropy is not about helping others. It's about helping yourself.  There's a big earthquake in China and there was a big earthquake in Japan. So, we donated, I forgot the exact like $20,000 to Japan, 10, $15 to China and people say, why we donated that little? Because we have a lot of money. And my point is, no matter how much money you donate, 20 million, that money compared to that is tiny. You cannot change a lot of things there. The thing you donate because you change yourself, when you change, the world is changed. You have to change yourself, right? And the next thing is that philanthropy is about action. It's about involvement. It's not about giving money. It's about wake up people's consciousness. So, I think, to me, I grow up, my globalization is growing up in this place. Year 2001, when I was nominated as a young global leader, I had no idea what is Davos, where is it, and what is about a young global leader. But I came here, I learned, I listened about these corporate, global citizenship and globalization and all things. I grow up. I understand, this is, to me, it's like a philanthropy. And this has changed to me as I talk throughout, that every time I come to Davos, I wanna spend time with young people. Because I benefit from young global leader, early days. When I listen to Bill Gates, Bill Clinton and George Soros, these guys, at that time we don't have that many people. Very poor facilities. Well, very cold, I work like a, 13 minutes driving in a bus coming here and watching. But, I think it's good. But this is good, this is the tuition we pay for a grown up. And now, to me, I think spending time with you benefit me, and also I wanted to tell you my true experience, it's a philanthropy. It's not necessary to use money. So, by charity, you cannot change the world. But if you have a philanthropy heart, you can do a lot of things. So, I find this, well, unfortunately, a lot of people have a business heart here, making money, but do things like a philanthropy. What we should do is should be a philanthropy heart, but have a business skill. I hate a lot of people, do the opposite. They wanna make money, but they talk like a philanthropy. And they do like a philanthropy, waste a lot of money. So, this is what I mean.  Care others, the first day they join. I don't like a lot of companies, make a lot of money, making pollutions and counterfeit products by the end of the year, don't need some money. This is no good. Business, the philanthropy idea should be buried into the business model. That's, yeah. Philanthropy of heart, I think, a message that young people can take forward, please. Hi, my name is Iman. I'm a global shaper from the Jeddahub in Saudi Arabia. My question is, as a young boy, did you envision that your life would be like this? Did you have intuition about this? And if so, do you think the belief helped in achieving such great success? As a young boy, I never thought, as a young boy, even today, I never thought I would be here. Honestly, I never thought. But as a young boy, when I look back, every problems I met in my, when I was a kid, benefited me. I've failed so many times. People probably know that I'm applying jobs for so many jobs, over 30 jobs, all rejected, not even got a chance. 24 of us interviewed for a KFC job. 23 accepted, I was the only guy rejected. Six people went for, looking for a police job. Five accepted, I was the only one. My cousin and I applied job as a server in a four-star hotel in my city. We wait a long queue for two hours. He was accepted, I was rejected. So my mother looked at me, oh. You know, because, but I know this is a training course for me. Before my 30 years old, I'm a failure. People lose, but I never give up. I think there is an opportunity there. And then I was a teacher later. I graduated from university, I was a teacher in the university for six years. Number five, because number five year, I was elected the best teacher of my university, elected by students. So I said everything I told my students are the things I learned from books. What if I go out and spend 10 years, experience all the things, whether fail or succeed, I go back to teach again. That was my original thing in doing Alibaba. So I never thought I could be rich. I never thought I would be successful. I never thought we could survive for 18 years. And then we made it. The only thing that today we should do is share the experience and know how, especially share the mistakes, suffers with others. My thinking is that, you guys remember, if you wanna be successful, learn from the other people's mistakes. Don't learn from the successful stories. Successful stories they make, don't listen to that. There are a lot of reasons behind it. Just like, I remember the first time Harvard Business School came to us, say, Jack, we wanna write a case study for you. Yes, years 2001, or 2000. They came, they spent one week and they write a report and I read, no, I said, this is not me. And they say, this is you. I said, no, this is not us. He said, this is you. So they make up his sign. And the case study go up and they start to teach in a lot of universities. The next five years, they invite me to go to the case study and they always find a competitor of my company. And after every case study, Alibaba will die, that company will succeed. All the students agree. And actually every five years, all the competitors die, we still survive. So how can you study this kind of a successful story? Learn from the mistakes, the other people. No matter how smart you are, you will incant these mistakes. You learn from mistakes, not because you will be able to avoid mistakes. You were able to, when these mistakes come, these suffer comes, you know how to deal with it, how to face it. The book I want to write, if I want, if I can, is Alibaba 1001 Mistakes. This is the most treasurable things that in my life. In my life, it's not how much we achieved, it's how much we go through the tough days and mistakes. And this is what, if you step to think now, it'll be good. So if you're asking me when I was a boy, I never felt about that. Never. I was lucky. That's going to the back of the room, yeah. Fan Lin, YGL from Shanghai. I know that you recently spent 100 billion RMB in deep tech research. I just wonder how you think of the future of deep tech and how that's going to influence humanity and creativity. Yeah. Thank you. I think technology, as a company, we invest 100 billion RMB, which is like 150, right, of 15, no, yes, 15 billion US dollars on research or research for the high tech. As a company, I hope in the future we gain our profit because of our technology, not because of our size. So we have to invest in technology, but I don't like this kind of, there's a debate. Artificial intelligence, big data, is going to threat the human beings. I think artificial intelligence should support human beings. The technology should always doing something to enable people, instead of disabling people. When steam machine comes, when cars come, people hate it. Today we think cars are good. When the automobile, when first it was designed, people hate, especially this horseman, right?  So I would say people like us, when we have the money, we have the resources, we have the young, all the excellent young people, we should spend money on the technology that enable people, empower people, make life better. So this is what we want to invest. And I think artificial intelligence, no matter how artificial intelligence is good, human being in the future competed with the machine on knowledge, you don't have a chance. Computer is always going to be smarter than you are. When there's a car, forget about a whole round faster.  They never forget, they remember everything, they never get angry, they calculate it faster. But computer can never be as wise as a man. What's the difference between smart and wisdom? My view is a smart people see something the other people don't see. A wise people see something he pretend he did not see it. Our smart people know what he want. Our wise people know what he doesn't want. A wisdom, the person with wisdom knows, I don't, only you know I don't want this. Then you know what do you want. So I think this is the money we invest to support human beings, not supporting Alibaba. This lab is open to the whole world. It's not supporting Alibaba at all. Wonderful. Hi, I'm Saad, global shipper from the Islamabad Hub. I really want to know what you think about leadership. Is it something that comes from the heart when it comes to decision making as a leader? Or is it something that you have to think through your mind as well? Or is it a bit of both, like as a leader when you make decisions? Would you go with something that you really feel about from your heart? Or would you try to have a mix of your mind and your heart when you make decisions? Yeah, I think you are, it's a good question I've been thinking about that. First, it's your nature instinct. There must be, have something that you're different. Second is you need to train. You should be trained through all the tough experience, but still positive. I found out some great leaders in the world they are always positive. They never complain others and never complain. And they look at the things in a different view, like normal people. So I think people in my company, they, at the beginning they don't like me because I'll always think about 10 years, five years. And then after we're working together for three or five years, they find that, hmm, you're right. Then we got the credit rating. And as a CEO, one of the jobs where everybody's happy, you have to see the unhappy things. When everybody's unhappy, you have to see the happy things. So leadership is nature, but you have to have a train and learn. And I got my leadership sub-upgraded in doubles. I see so many, you know, Ali Enter financing, how many people here know about Ali Pay? Thank you. Ali Pay's decision was made here. I was thinking about Ali Pay, you know, but I was not there to launch Ali Pay because in China, if you do financing without license, you will be in jail at that time. So I said, I went to the banks, can you help us do the e-commerce on transaction? No, no, no, no, no, no. No banks would accept it. So if there would be no Ali Pay, no financing, the e-commerce would go nowhere. So I was, year 2004, I was here. Listen to a speech by two state leaders about leadership. And this guy said, leadership is about responsibility. You believe it, but the people don't believe it. But if you think it's so critical, you should pay any price to do it. So that day changed my mind and I called back to my team, say, let's launch it within one month. If somebody has to go to the jail, I go to the jail. Who would be the second one following me?  That was called the leadership determination. And year 2004, I made the decision here, I called back. And now today, the Ali Pay launched. And it's so big, over 800 million people today using Ali Pay globally. This is called. Wonderful. Shall we take one from YGL to be fair? Any questions? Yeah, please. Hello, my name is Pablo from Costa Rica, YGL. And given your perspective of caring and being a trustee of all this, what do you do for self-care and also maybe within your company? Because early on, often you can cut through sleep or other things in a silly way and realize that if you plan for the long term, one also has to take personal care, be there mindfulness or prioritizing better. And how are you doing this and how do you promote that in your company? Well, I'm lucky that I have 18 co-founders. And most of them are my students. When in my apartment, I told them my dream, my mission, and the video tape, funny thing is that when Ali Baba launched for the first day, I said, that's a video tape, everything. Every important meeting that we join together. The purpose is that someday, if we fail, we give the videos to the people to research why we fail. If we succeed, give the videos to the other people. So we have almost a whole house of video tapes. So the first video tape, I speak for two hours and those guys look at me. Okay, so that's what do you want to do? I'll say, yeah, this is what I want to do. We, I was the first, I know nothing about technology. I know nothing about management. I know nothing about, but the only thing is that you don't have to know a lot of things. You have to find the people who are smarter than you are. My first way is always find people who knowledge on computer, smarter than I am. Accounting, smarter than. For so many years, I always try to find the people who are smarter than I am. And when you find so many smart people, my job is to making sure the smart people can work it together. And then if smart people can work it together, it's easy, the vision they will believe.   The people who believe the vision, but they don't work together. So very important is finding smart people. And my job is two things. Making sure they work together. So the culture is important. Long term, you should be ideal with great strong vision leader, but you should also have to survive today. If you don't survive today, you're gone. So the best way to promote your company is not to you. It's not me as the CEO. The best way is your product, your services, your employees. The best product of your company is your employee. So I spend a lot of time, early days, for the first 2000 employees of Alibaba, I speak everybody for one or two hours when they're joining. I said, I will never promise you will be rich. I'll never promise you will be promoted, but I promise you'll suffer. I promise you will have a terrible life, right? And all the terrible things I promise when joining this company, right? But good things. If you promise good things, you are misleading them. Because I cannot promise myself. We know this is the future, but we are not sure that we will be there. But if we don't work together, we all will lose our job. So that's the thing. And they do always say, and I think I've experienced this myself, if you knew how hard it would be, would you do it? And it seems for you, a key part of leadership is leaving the ego at the door and being willing to hire people better than you who you can then kind of empower to carry out your vision. Wonderful, yep. Hi, I'm Iman from Global Shapers at Kata Hub. How does your experience of being a teacher influence how you run the business today? And do you have any message for those who are working in education space? Thank you. How does your experience of being a teacher in early days influence how you run the business today? Yeah, let me tell you one thing. I never thought I could be a CEO. I never thought later I'd become a good CEO. One of the things I learned is from teachers. As a teacher, very important, the character of a teacher is the teacher always expects his students better.  This is all teachers want. If you don't have this kind of thinking, it's a terrible teacher. A good teacher always wants the other people better. You don't want this teacher in jail, that teacher is a bankrupt, that teacher is, you know. So as a CEO, I trained myself is I always want those people join the company do better than they thought. And everything the company should do is making sure the good environment to train him to be more positive. And everything will do. This is a teacher. And teacher does not mean I know better than you are. Everything I know better than you are because I learn from others. So a teacher should learn all the time. A teacher should share all the time. A teacher should always expect the other people better than you are. And by the way, education, it's a big challenge now. If we do not change the way we teach, 30 years later we'll be in trouble because the way we teach, the thing we teach our kids are the things past 200 years is knowledge-based. And we cannot teach our kids to compete with machine who is smarter. We have to teach something unique that is machine can never catch up with us. In this way, 30 years later our kids have the chance. I hope I answer your question. It's a very difficult one to answer, but what are those skills that you think we need to teach? If it's we're moving away from knowledge, what are the key things? Value, believing, independent thinking, teamwork, careful others, these are the soft part. The knowledge may not teach you that. That's why I think we should teach our kids on sports, the music, the painting, art, to making sure humans should be different from, everything we teach should be different from machine. If the machine can do better, you have to think about it. Hi, this is Hashem Al-Mousawi from Manamahab Bahrain. Talking about education and the fourth industrial revolution, we have been talking about people losing their jobs or they are not qualified enough with the skills needed. What are you doing for the people within your company to get up to date, to get continuous learning, to know what's the technologies needed for the future? Thank you. Okay, luckily I don't have to do a lot of these kind of things in my company because we are fighting every day for the future. We know if we do not work hard, Facebook, Google, and Amazon, all these are gonna do better, right? So this is why we are globalized, it's a competition globalized. But we know that it's not about technology competition. All the people today, not like 18 founders, we only have two and a half people engineers. Today we have more than 25,000 engineers. They are all smart people. They do a much better job than 18 years ago. But the only thing I wanna make sure, these people have a front of their heart. Because the data we have, the technology we have, if they do not have a great heart inside, we are gonna make disaster for the world. Google, Facebook, Amazon, Alibaba, these companies, we are the luckiest company of this century. And if we're lucky, we have the responsibility that we should have a good heart to do something good. But on the other thing I wanna say is about that making sure that everything you do is for the future. For example, you know about the single stay? Single stay shopping day, November 11th is the day we created. Last year that day we sold 25 billion US dollars for one day. And for the first minute, over 70 to 80 million people rush inside to shop on the mobile phone. So you have to making sure the system is good enough. We have to finish 270,000 transactions per second. Can you imagine that? 270,000 transactions per second. If we do not have that kind of computing capability, the whole system will crash because we have several hundred minute people waiting to pay the money. So this technology, why we do this for crazy for one day because we want to test the technology. This technology should be inclusive because we know today in China we create probably like a hundred minute package delivery. We believe a day. Five years later, the whole world would deliver one bit of packages per day. Think about what's the technology? Are we ready for that? So the single stay is to test this technology and to be inclusive. So making sure the market drives them, not the boss drives them. Thank you, everybody. Thank you, Jack. Jesus Peda from Mexico, from the Monterrey Hub, Global Chaper. We have been talking a lot about our generation and I really liked the conversation about three years before, three years after. Actually, I have been hearing these days that we haven't had this snow in 30 years here. So the question is going to a quote of Professor Schwab. He said a couple of months ago to the Global Chapers that we're the first generation that is going to suffer about climate change and we are the last one that can do something about it. So with your teaching passion, what are the three things that you are saying to educate the 30 million employees or the 100 million employees or everyone who's looking at this video to develop an environmental mindset, a spiritual mindset, a non-fragmented world mindset? Okay. First let me say this. Every disasters, the environment disasters as a refract of human heart today because when we need it too much, we destroy a lot of things. I think the human being today, the past 200 years, because of knowledge-based technology or knowledge-based training system, all we want to do is get more. We want more things. We go to the moon, we go to the Mars, we go this, we're always outside looking. We human being never inside looking. If you don't have the inside looking, you will never be wise. If you're not wise, you never know what you don't want. Today, human being get much more than they expect they should have.  So this is why the big data, the data technology is trying to understand the inside of human. What is data? It's the human behavior. A lot of people, trust me, the machine will know you better than you know yourself. Have you seen some movies? Some gamblers? You know, they're great. They look at the eyes. If you got a good card, this eyeball will shake. But normal people don't see it. The machine see all the details. Machine know you better. So what I would say is that in a data period, human being next to 100 years, people, human will know, try to understand. Inward looking. We will see a lot of ugly things of our self. When knowledge based art were looking, you see the ugly things of the others and good things when looking inside working. This is Chinese philosophy. Chinese people, our believing is always inside working. We're supposed to have inside checking every three times a day. But not really. If you can really do that, you can do that. Then you'll be good. So for environment, you're right. If we do not make action, take action, all the money you make will spend in the hospital. Medicine, our kids will be in big trouble. So let's fix it. And fix our heart, fix our know-how, and fix our belief. When you have a belief, when you have a religion, when you have a valley, it's better. So I think this is what I concerned, I mean, this is what I think the young people should always have, the valley, the mission. If you have something you believe in your heart, that would be different. This is what next to 30 years education you focus on. We have time for one last brief question, Karuna. Thank you. This is Karuna from the Portless Hub, Mauritius. And my question is, you talked about your workplace culture. I'd like to have one or two examples if possible of how do you motivate your employees? And I'd like to throw in the last question is how much do you sleep? Because this is such a controversial topic and I keep hearing different versions. Thank you. What am I keeping the different versions? Sorry, I've heard CEO sleep on the three to four hours and whereas I've heard you need at least eight hours. So yeah, I'm a bit divided and I'd love to know your secret. Thank you. How do I motivate my incentive or my colleagues? You have to find the people that can mobilize themselves. It's impossible to encourage a negative person. So the people you hire, the people work together. They have to be. Or we have 65,000 employees now. You cannot hire all of them that positive. But at least the people work with me. My management, leadership, several barriers, they have to be positive. They have to know the incentive of the others because I cannot incentive everybody. I'm making sure the culture. So I think it would be very painful for me to talk to my vice president and he need to be incentivized and mobilized every time. This won't work. If he does not know how to incentivize or mobilize his people, it's better. I don't think he will be the vice president. He should be a good engineer, good designer, but not a good leader. A good leader should know how, but not by only money. Most of people incentivized or mobilized, not because you give them a lot of money. You give the respect, trust, appreciation. And correct, hard to fold advice. When you come to my meetings, internal meetings, you will be shocked because we are not like a lot of the other Chinese company. The boss say everybody listen, take notes and go down. We are like a war room. We make decisions based on whose voice is louder. Early days, you know, we fight it, we talk and then making sure everybody speak up. So this is also a way of mobilizing an incentive. There are a lot of ways and different people have a different way to send incentive. So this, how much time I sleep? I have a problem. I mean, I don't, I sleep not very short, but not good. And I don't eat a lot. This is something that I was curious. I don't eat a lot. I'm a very pickle on the food. And I don't know, sometimes I would love to see what the things that make me work like that. But for sure, that if I suddenly give me one month's vocation, I'll collapse. It's, I think there'll be something in my heart, maybe the mission. At the beginning it's a mission of 18 people. Today we have 65,000 people. They all believe the mission. So if I give up, I can give up my position, but I should not give up the mission. If I give up, everybody say, hey, no. So, and then especially to the Alibaba at this size, people continue to say, for this size, the growth rate will be tiny or down. So what's the 10, 20 years Alibaba will look like?  How to do it? So I spend a lot of time thinking. A lot of time thinking. And I love to think. Good thing is I have a good team now. I think I do the talk, they do the work. Yes. On that note, I think, I'm sorry we didn't have.


Good morning everyone Let me make an announcement first that the talk of Alexis Vassar on Friday Which was announced at two o'clock will be at 1 30. So it's going to be a talk between 1 30 and 2 30 So you have more time to get to Venice or something, right? So I'm We have now the next talk today the first talk today That's again Laurent de Villette and he will continue with his third lecture now on collisions in plasmas delanda equation, please Okay, so thanks a lot Let me recall a few things that we saw yesterday first if you remember we made a sort of general discussion of the contributions of various Colleagues to the Entropy method and then we Turned to the definition of the Boltzmann equation and After a certain number of computations and after having used a large number of assumptions We ended up with the abstract form of the Boltzmann equation which is written here on the blackboard so before Writing a parametrization of the of the operator and It's something that anyway we saw with Daniela yesterday. I wanted to Discuss a little the remark which is that there is an obvious Class of steady states for this operator which are the so-called Maxwellians So what I will call the Maxwellian systematically In this series of talks is a function which is the exponential of Second-order polynomial in the variable v but as v is a vector This is not just any kind of second polynomials which would give you a general Gaussian It is a second-order polynomial in which the matrix acting on the second degree Of the polynomial is a matrix which is the Constant times the identity so in that case we really speak of Maxwellians and This class of functions can be seen as a subclass of the Gaussian functions It's exactly the subclass of the of the Gaussian function which have a covariant matrix which is a Constant times the identity so it's a way you should you should see this so let me explain why this specific functions Play and a very important role in the theory The point is now suppose that you have this m of v defined like this and let's try to see What is the result of?   So I can write this Also like this So this is true of course only If you look at this integral and this is exactly m of v prime times m of v prime star And as you can see You get that this quantity here is equal to zero when f is n, okay so you get this class of Steady states for the equation df over dt is equal to is equal to this quantity here Okay, what's a very important remark for the for the rest of the of the lectures Let me now Present the Two typical parameterization For the Boltzmann equations. So this is another remark if you wish it is possible Recast the formula which is here which is quite abstract because of the direct masses inside The integrals let's see the term Which I called q both Be so the be here is referring to the Cross-section be here. Okay, so I parameterize the operator by This parameter here, which is something which is sort of a free parameter in the in the operator Okay, sorry. Anyway, I can recast this a term would be equal to the integral so now I Have an integral over three n In dimension three n and I remove a set which is of dimension n plus one So I will have an integral over a set of dimension To n minus one The one will be over our n and the other one will be over the sphere s and minus one So this gives you something which is of size to n minus one you still keep the same notations for the The part which represents the gain and the loss term and You remove the direct masses and you just write Be here and now be as we will see so let's call it be one if you wish It will be a function still of the modulus of the relative velocity But the second term will now the second variable will in fact be the Absolute value of the scalar product between the Direction of the relative velocity and this parameter omega which now lives in this sphere as n minus one and if you Remember the formulas presented by Daniela yesterday. This is exactly identical once I Write the definition of v prime and v prime star in terms of omega and this can be done in this way So this is usually the formula which is written down in textbooks on the Boltzmann equation and one can check that These formulas are really exactly equivalent to saying that V plus V star is equal to V prime plus V prime star and the same for the Kinetic energy So more precisely there is equivalence between both two things and the fact there exists a parameter omega on the sphere such that this It's not very difficult to see first. You write things in the reference frame of the center of math and then you say that two vectors have the same modulus if there is a symmetry Which goes from one to the other and this symmetry is exactly the symmetry related to the to the vector omega But actually I will never use this formula. So this is just to make the connection With with the talk of Daniela because I will really now look at the specific case of the dimension 2 Which is not the physical case but which is a case in which it's a little easier to do the computations So I would like to use a parameterization which holds only in dimension 2 for the same for the same kernel So when n equal to another Parameterization, which is actually much better behaved from the point of view of math and it's the following So it's a still the same formula But now since the dimension is 2 the integral is taken over R2 for the variable V star and the second variable will be chosen in S1 so in the in the circle and one good way to parameterize the circle is just to write an angle between Minus pi and pi so let's call it theta then the rest is identical and here Usually we take a parameterization of the cross section Which is such that it is the Absolute value of theta which appears in the second variable of of b which I call here b2 And the parameterization is given by the formula which I think It may be easier to understand if you are familiar with the with using The center of mass reference frame so the center of mass reference frame Means that you look at the average of the velocities before collisions. So it's v plus v star over 2 So once you are in this frame The two velocities are v minus v star over 2 and v star minus v over 2 So if you look in this frame the collision Happens with just two particles arriving like this And then they will get out like this And you will use the angle between those two lines to parameterize the collision So if you do that it just amounts to say that you use a rotation of angle theta Taken on v minus v star over 2 and actually this parameterization is Is probably the best unfortunately it works only in dimension 2 because it's not so easy to parameterize The sphere in dimension 3 With something which has the properties of a group When I say it's not so easy it's a euphemism So the the point here is that you have all the good properties with this parameterization because It's something which is linear in terms of dv star gives v prime v prime star And it is something in which the parameter appears in a very simple way Whereas here as you can see the omega Appears in a quadratic way okay, so this is Something which makes the computation usually quite easier And it's a reason why what I will present to you now Is presented only in dimension 2 because it's a little easier to do the computations But I promise to you that All what I will now say is still A true in dimension bigger than 2 provided that you accept to do the heavy computations Okay So let me do now By the way or those who wish to see another slide This is already written here. So let me now give a definition The definition of the so-called grazing collisions So if we consider that the Boltzmann equation in itself is a definition is actually the number 5 We say that grazing collisions are collisions such that the Velocity after the collision is close to the velocity before the collision which Also is equivalent to the fact that The same holds for the second partner So grazing collision is a collision in which the velocities are Changed by a very small amount And as you can see looking at the formulas that you have here This is equivalent to saying that v minus v star is almost orthogonal To omega so all of this is equivalent to V minus v star Is almost orthogonal to omega if you use the omega representation written here Or if you prefer it's also It also corresponds to theta which are close to zero But if you take theta is equal to zero You will just get v plus v star over 2 plus v minus v star over 2 which is equal to v Okay, so Another way of looking at grazing collisions consists In writing that theta is close to zero That's okay The remark Remark which is actually due to I think both Lando and Chapmanic and Colling So this is this is something which was understood between the 30s and the 50s is that For plasmas All collisions are in fact grazing collisions So in plasmas the main difference that you have Between for the particles with with respect to a neutral gas Is that the particles are charged so you have like electrons and ions And when they collide They collide because of the Coulomb potential And actually the Coulomb potential has a very long tail That is you can feel another particle from very far away and because of this You can You can guess that The in a in a typical collision with the Coulomb potential the Amount of velocity which is exchanged during the collision is very small and so This remark which was done let's say at the at the physical level Has deep consequences At the level of the operators so for plasmas all collisions When I say all Well, it's not really a mathematical statement. Let's say All collisions are grazing So this means that when you're looking at a plasma Basically, you should take The Boltzmann operator. Let's say this one in dimension two And you should perform a limit when theta goes to zero Inside the operator V prime and V prime star depends on theta through those formulas And you try to let theta go to zero So let's try to do this that is Theta go to zero so let's do the computation when N is equal to two so let's forget Forever the general case and let's concentrate on the case of dimension two If you let theta go to zero You see that you can write V prime as V plus V star over two plus V minus V star over two cos theta plus V minus V star over two orthogonal sin theta Where I use a sine orthogonal for The rotation of angle pi over two Of a vector, okay So by definition This is just this is just this and now if you expand The cosine and the sine Thanks to let's say Taylor expansion Um You see that you get V plus V star over two plus V minus V star over two Minus V minus V star over four Feta square Plus V minus V star over two orthogonal theta plus Something which is of order theta to the free When theta go to zero goes to zero, okay, and you can simplify The first Some here and get exactly V which is not surprising since we saw that This parameterization is chosen in such a way that when theta is equal to zero Then the collision Corresponds to no collision at all if you wish Okay, if we do the same for V prime star The result is given by the following formula Which I will not fail but it's Of course, very easy to check With a formula which is very close And now let's compute at the formal level F of V prime F of V prime star Sorry Using those formulas so this amounts to have F of V plus V minus V star over two orthogonal theta minus V minus V prime star theta square plus This times the quantity which is here and when you perform now they When you expand F in Taylor series around V here and V star here You can believe me That What you get is the following So I will try make no mistakes here So you have terms in theta You have terms In theta square Let me write it rather here It's F of V plus V minus V star gradient F of V theta minus V minus V star over four gradient F of V theta square Plus This is a sum over components of V minus V star orthogonal A and J this is divided by two And here I have The second derivative of F with respect to variables I and J theta square over two and you have Let's say A formula which looks like the same and which starts with F of V star But the rest has more or less the same shape So let's have a look to to what happens In the in the kernel We just computed this times this when theta is close to zero The first term is F of V times F of V star which will cancel With the F of V times F of V star which is here The second term Will be made of things which are proportional to theta But the point is that here the cross section depends upon theta in an even way and because of this The term with theta will give a contribution which will be zero in the integral Okay because of a parity argument And so what happens is that only the second order terms are not zero and You can check that Here you will have Things which look like the quantities that you see here so As you can see Inside you will have things Which look like this one or the same With gradient F of V star so you can expect to to have inside the result Terms in which you have first order derivatives of F And they will be multiplied typically by V minus V star This is cannot product And you will also have terms which look like this, okay now If you take The tensor product Of the orthogonal of two vector is exactly as taking identity Minus the tensor product of the corresponding vectors So you can also expect to have terms Which would look like this identity times V minus V star to the square minus The tensor product of V minus V star With itself We do the contracted product with the gradient of the gradient of F Let me say that this is just a shorthand notation for the matrix Which have the component V minus V star square delta ij minus V minus V star i V minus V star j And the contracted product consists in taking the sum over all possible components Of this quantity so if you don't like Let's say the abstract algebraic notation You can use the components and you will find this Okay, so I will not perform this computation because it would need like Alphanar more actually But I will write for you the result on the slide So maybe let's look at this slide When you do this Asymptotics, which is sometimes called the grazing collision asymptotics in the Boltzmann kernel you end up with the formula Which is written here and which is called the Lando operator for collisions in plasmath So the how does this relate to the computation that we have done here? Just have a look to this a here This a is made of One part Which is a number and one matrix And this matrix is exactly the matrix which is written here Up to the multiplication by z square Okay And this is applied to gradient f of v And then you have a divergence here. So this is a second derivative which Corresponds to the second derivative that you have here as for the first order terms You can See that they arise from the part of the operator which is here in which you have just one derivative Because this one can be absorbed by an integration by parts So you can believe me that when you do seriously this computation Yeah, I will It's just in the not in the right order But I will it's promised So what you get at the end is really this and what I will do now is just write it on the blackboard because we will Need this definition In the sequel so this is definition six Wish so The Lando operator defined by q Lando and of a parameter psi, which is a given function It is quadratic. So we typically denote the operator with The notation q of f f This is the divergence v minus W usually we do not use v star for the Lando operator. It's rather the common notation is rather w We have this pi of v minus w which is a matrix And this matrix is applied to the vector Which is obtained by Taking this kind of symmetric form in which both f and the gradient of f Are appearing so here pi of v minus w This is The projector on to The hyperplane which is orthogonal to v minus w And its components are given By The identity so delta ij minus V minus w i v minus w j divided by The modulus of v minus w to the square We can you can just Take it as a definition and in fact it's the Way It's really as a definition that it was proposed by Lando in 36 in his first paper on the Lando equation That is Lando made no link Between the operator which is defined here and the Boltzmann operator It just wrote it directly like that And the link which I sort of Let's say discussed a little Is something that appeared Basically in the book of chapan and calling but actually not the first edition is a very old book And it's in in an edition from the 50s that for the first time this link is established Unfortunately not in the vocabulary of math so you really have to To to read it carefully in order to understand what they really do but basically it's a computation that I have Presented to you so one way to make this computation precise is To actually Define A sort of rescaled version of the cross-section in the Boltzmann equation. So you define So you take the Boltzmann equation in dimension two and you define a new cross-section by concentrating the The collisions on the collisions which have very small Angled theta that are the grazing collisions. So you do it in this way. So here you divide theta by epsilon you extend This quantity by zero for theta different from Which is not between minus pi and pi And you put here one over epsilon to the free and the free here has nothing to do with dimension It is obtained by just putting one over epsilon in order to have something which is Of integral one let's say or a fixed integral and then the one over epsilon square which remains Correspond to the fact that the two first terms are disappearing In the computation. So if you since you go to order two You have to divide by one over epsilon to the square in order to get something which is finite at the end So this is the right way to do let's say the mathematically the computation which is done here and Let's say the proof That you have this property here that you can go from the Boltzmann equation Operator to the Boltzmann to the Landau operator for a given f It's something which was written down. I think first at the mathematical level in the early 90s There is a paper by Brigitte Lucca and Pierre de Gaulle and also a paper by myself on this topic But the real hard proof is actually Uh Can be found in in a series of paper by alexander and villainy So I think there are actually two papers which were written around 2000 in which They do not only prove that for a given f this is converging To this but that for the solution of the equations You have a convergence of let's say f epsilon towards f And they do it in the setting of renormalized solutions of Lyons and diperna So I will not say much more about this, but let's say the the the hard mathematics is done in this in this paper here Okay, so this is more to convince you that you can get Lando out of Boltzmann My my lectures are really not on this topic this time But at least looking at this at this Uh Link between the two equations you can guess something It is that since the maxwellians were steady solutions of the Boltzmann equation They are still steady solutions Of the Lando equation because the Lando equation is obtained as a limit of the Boltzmann equation of the Boltzmann operator. Yeah so it's uh I don't remember what is the exact hypothesis in in the results, but it seems to be Yeah, but uh in some sense when you do this you sort of forget About the fact that they are charged particles and you can look for any kind of b and any kind of c And at the mathematical level, of course, you can decide what are your hypotheses? So the important point is that at the end it should contain The Coulomb case which corresponds to c equal to 1 over z And if I remember it is a case in at least the second paper Of alexandre and villani so it contains the interesting case But of course for the mathematical point of view you can try for Any kind of cross sections well, so uh, it's not it's not a surprise that the maxwellian are still steady solution of the Lando equation, but I think it's important to Check this directly On the Lando kernel, so I will do that immediately Because this is really the first The first step to the entropy structure of the equation. So in order to do this, let's Let's look at the quantity gradient m over m when m is a maxwellian This is just the logarithmic Gradient of m and since this is an exponential in the logarithmic Derivative, of course, it disappears and then you just have to take The derivative of this or the gradient or the let's say the derivatives Of this and as you can see this is a second order polynomial So the derivative will be actually an affine function Okay, so gradient m over m It is something Which can be written like if I'm not mistaken b minus 2c Uh scarer product with v This is affine.  Okay, and so this is m of v m of w times This quantity at point v minus this quantity at point w What I get is just minus 2c v minus w here If f is a maxwellian This is just Uh something which is a multiple of v minus w But the operator pi Here is exactly the projector the projector onto The orthogonal of v minus w. So when you apply it on this vector here, you get zero Okay, so this is a way of seeing that maxwellians are steady solutions of the lando equation without knowing anything On the Boltzmann equation. Okay Let me at this level say one word about the This trend operator here So, uh, let me first say that this is not only, uh An object, uh with which mathematicians are playing It's really what is used in numerical codes when you want to Understand the effect of collisions in the plasma So it's something Which is really, uh of Interest for the for the physicists And, uh, let me add that it has a structure which is, uh, quite fascinating Because, uh, if you are familiar with parabolic equations or let's say with elliptic equations Then you can see that you have a structure Which is quite, uh, uh common that is, uh You have the divergence Of something times the gradient Plus something times the function.  So it looks like The usual, uh, think you can find But they, uh, uh, the point is that Those coefficients are obtained through an integral of f in the variable w So they are themselves depending on f So it looks more like let's say a nonlinear diffusion But usually in nonlinear diffusion you look rather to things which are let's say local like laplacian of f square And here it's not laplacian of f square. It's like laplacian of f times f convoluted with something And so One, uh, way of looking at this is just to forget everything about kinetic theory And to try to use parabolic theory to to get, uh, some insight And there are a lot of works which are been done in this direction And then there is a second, uh, way of looking at things Which is to, uh, keep the symmetric form that we have here As you can see this is extremely symmetric And, uh, this means that somehow you forget about the parabolic nature, uh, of the problem And you really base your analysis on on on kinetic theory So there is some in some kind of, uh, two ways of looking at the same objects Which are completely completely different Uh, and let me say that what I will present now which is the Entropy structure of the equation is rather based on the kinetic way of looking at things And, uh, it's not so obvious to use the entropy structure if you look at it from the parabolic point of view Uh, anyway the main difficulty as we will see, uh, when we are looking Precisely at the most Important case from the point of view of physics That is when the the function psi here is z to the minus one and this corresponds To the case of plasmaz Is that somehow those two terms, uh, are, uh I mean the the drift term is not really dominated by the Uh, by the diffusion term So the symmetry when you take this Potential here the symmetry Uh, plays a very important role in the equation And trying to use a diffusion without looking at the drift for example leads nowhere Okay, so this is just a general A general remark, uh You can see this in some sense as some kind of interpolation between the Boltzmann equation And the Fokker-Planck equation that I presented during the first lecture From the Boltzmann equation it retains a quadratic nature It retains a non-locality And from the Fokker-Planck equation it retains of course the parabolicity Nature of the first term and the drift that you have in the in the second one Uh, and from both it retains the fact that you have a set of steady state which has to do with Gaussian functions Okay Well, all of this is a little, uh fuzzy, let's see So let's come back to, uh more practical, uh practical things Um, the, uh Next step consists in, uh looking at a weak form of the operator So here I will not write things down. I will just use slides The as usual in PDEs let's say or in the theory of integral equation actually the same It's quite important to understand what happens when you multiply the operator by a test function and you integrate So this is just, uh The definition of formulation if you wish So here if you take the Boltzmann kernel and you multiply by a test function phi Typically what you obtain is a formula in which you have the term which is appearing in the Boltzmann equation This f of v prime f of v prime star minus f of v times f of v star and this is multiplied by phi of v and integrated Okay, then what you do first is to make a change of variables which transform v in v star and v star in v When you do that the phi of v becomes the phi of v star Okay, and if you take the first formulation And the second one and you divide by two you will get just one half of phi of v plus phi of v star at this level Okay, once you have done that You use another symmetry of the equation which consists in changing v and v star in v prime and v prime star So this is a change of variables Which is quite easy to perform For example, when you use a parametrization, uh that was Uh presented by Daniela or the parametrization with theta in dimension two But when you do that, it's easy to say that you have a Jacobian which is one So it does not appear in the in the integrals And the phi of v plus phi of v star becomes phi of v prime plus phi of v prime star And of course here things are reversed That is v v star becomes v prime v prime star v prime time v prime star becomes v and v star and so the minus here Makes it that you have to change the signs at this level Okay, so by doing two changes of variables in the In the in the big integral here you end up with this Uh Formula here by the way, sorry about the this this is uh, this is not true. It's a different It's a different integral here. It's integrated over over the totality of the variables. It was lost in the So let's let's look at a direct consequence of of this formulation of the of this weak formulation Let's now take phi equal to one v i or v square over two So this corresponds to looking at mass momentum and energy On the world if you wish that is instead of looking at it for just one particle You just integrate over all the particles entering inside the kernel So when you do that if you take phi equal to one It's clear that you get one plus one minus minus minus one and you get zero If you take any component of v You will get here, uh, let's say v i plus v star i minus v prime i minus v star prime i And this will be equal to zero since you integrate On a set in which you have this direct mass here And of course the same will hold For the v square over two here because of the direct mass that you have here, okay So the first consequence of the weak formulation is that you have This global conservation of mass momentum and energy Which is no surprise because this is something that you imposed at the microscopic level at the level of each collision In the kernel. So you will recover it when you integrate over all possible velocities And now let's try to do the same for the Lando operator. So the Lando operator Is written here Okay Now you multiply by a test function phi of v and you integrate So as you can see here The first thing you have to do is an integration by part Which transforms the divergence here in a gradient on the test function So it's exactly what will happen at this level You will have this gradient phi of v which will appear And then you do As in the Boltzmann equation you do a change of variables which transform v in w and w in v So when you do that You you get gradient phi of w And if you look at this formula here, you see that this is anti-symmetric with respect to v and w But if you if you change v and w you get exactly the same Except it's a different sign.   The first one is this one And the second one is this one Okay Now let's try to see what happens if I take again phi equal to one vi or v square over two If you take phi equal to one, it's clear that you get zero because gradient phi is zero if you take Let's say any of the components of the momentum v Then the gradient will just give you Quantities which will be constants. So you will get a constant minus itself.  So this gives you also zero Actually, you could get this formula directly from the Limit that you have From this link that you have between the Boltzmann equation and the Landau equation. Okay But it's better to to get it directly from the formula of the Landau equation And now it I still have five minutes to present to you The entropy structure itself That will be the main object for the two last lectures so This is the so-called h theorem of Boltzmann. So in the case of the Boltzmann equation It was proven by Boltzmann himself around 1860 and it consists of noticing that If the in the weak formulation of the Boltzmann equation There is something which happens if you take phi of v equal to log f of v So takes just take phi of v equal to log f of v So you replace here phi by log f What happens with the log is that you can transform log f of v plus log f of v star in log of f of v times f of v star So it's exactly what you will find in this slide Here I used log f of v as The test function and here you will get log f log of f of v prime f of v prime star minus log of f of v f of v star And you see that using the log is the only way to get here something which depends on these products Any other function would not give you that it will still give you Something which you cannot relate to the quantities which are here. Okay And now the the the observation of Boltzmann is that It's as if you had something like x minus y times log x minus log y And since the logarithm is an increasing function.  This is a conservation of mass And you get exactly the quantity which is here and so you see that this is Negative okay, what you have obtained is the entropy structure for the Boltzmann equation that is uh This entropy This is an entropy So the proposition is that H is an entropy For the Boltzmann equation The entropy dissipation Which I will call d as I did in the first Lecture is given by this formula here, okay So this is the entropy structure for the Boltzmann equation I don't know if I still have one minute maybe At least to announce what I will do Tomorrow so the point is that Since the Lando equation the Lando kernel is a limit of the Boltzmann kernel You can only expect that by doing the same transformation that is you multiply by log f you integrate You will get something positive, okay This is a direct consequence of the link between the two but actually It's better to see directly on the on the kernel. So if you remember the weak formulation of the Lando equation Which is here What happens if now you put log f instead of phi? As you can see you will get gradient of log f of v which is gradient f over f And so you end up here with Gradient f over f of v minus gradient f over f of w And you have here a symmetric matrix Which is taken on The same vector as a quadratic form And so in order to see that This is positive It is enough to know that this matrix is definite positive or at least semi-definite positive And since it is a projection It is in fact semi-definite positive Okay, so you get that this is also bigger than zero and The proposition which is here is also true as a consequence for the Lando equation Finished by saying that Since now we have the entropy structure for both the Boltzmann and the Lando equation The point is to check that this entropy structure is strict And then to try to produce an entropy-entropy dissipation estimate Which if It exists Will lead to Convergence theorems for the Boltzmann or Lando equations when the time t goes to infinity And this will be The point of the two last lectures That I will present tomorrow


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