 And the idea of the panel was more or less to discuss about your thought about new mathematical theories within your fields that you think we should invest on from an industry perspective if you had. So before going that, what I did rapidly for the audience, I went, there are two things I did, is look at what we call basically the important mathematical problems that were posed in the last century by Hilbert. As you know, there was the 23rd problems of Hilbert in the last century. And then I looked at what were and which are by the way shaped all the mathematical research around the 20th century for people who are interested in that. As you know, many people try to at least solve the growl during the 20th century. It turns out that in 1998, so a couple of years ago, there was also the will to define what or the 20th Hilbert problems of the 21st century. So I think it was the mathematical union of mathematicians who asked Smales to put into place what he thought were basically the most important problems that we should solve within the century. So I was quite surprised to see that many of the problems, I mean some at least of the problems that are there are in fact problems which are quite useful for our ICT business basically. So these are the problems and then I went into looking exactly at the PDF he wrote. Just to take a look. And it turns out that if you can see, and this is also maybe I don't know if it's a change in the mathematical I would say research realm of the things they're doing. So the Riemann hypothesis came as before. The Poincaré conjecture is still there, it has not changed. And there's problems of complexity coming in, which I think are very important for us as you know algorithmic people where we're trying to find some solutions with low complexity. So does P equal NP is there at least? There's also here integrals zeros of a polynomial of one variable, which from my point of view, we haven't still found any urgent need but maybe there we were in. Height bounds for geofing curves, I encourage you to look at them. So another thing which is quite surprising is that these problems are also quite understandable. I mean you can take a bit of time for research engineers, which is not the case of the things I found in the Hilbert problems. So for some reason I don't know if the level of mathematics in industry has increased or there's a change but in any case it's something that is readable at least in terms of understanding what they're talking about. The solution is another question but at least we understand what the guys are aiming for. The infiniteness of number of relative equilibria and solicit mechanics, which also was not of big interest for us. Distribution of points on a two sphere. There are some applications I can spend on time but not here where we can find some application of this conjecture. Introduction of dynamics in economic theory, which is also something very important. It turns out that this is a practical problem for people looking at equilibria stuff and things. As you can see here, extend the mathematical model of general equilibrium theory to include price adjustments. It's the first time I can see that I would say strong conjectures or strong problems are put into place like that where there's practical meanings, where price adjustments are there. Linear programming problem, big issue for us as you all know, still for industry in trying to solve some optimization. And here you can see is there a polynomial time algorithm over the real numbers which decides the feasibility of the linear system of inequality AX higher than B. So I'm quite surprised that at least then the closing lemma, I'll finish all this. One dimension dynamics generally hyperbolic. Centralizers of the of the formalism, the morphism. Hilbert's 16 problems still there. Loren's attractor. Navitz-Stokes equations still there. Right, not still there, but at least a big issue. Solving polynomial equations, so wait a minute, the Jacobian conjecture. Solving polynomial equation, as you can see here. Can you solve a zero of n complex polynomial equations in unknowns be found approximately or on the average in polynomial time with a uniform algorithm? You have to know that these recent years, there's been a lot of work on this. And they found some solution on average, I think. There's a couple of solutions now, but not on a deterministic point of view, but on a probabilistic point of view for solving these things. There's been a couple of work these last year. And one which is also very interesting is the fact that also artificial intelligence kicks in. And at least basically in what they call the most important problems of the century with respect to what is the limits of intelligence. So I don't know how this is going to be solved. But it's interesting to see that there is a will to look at the limits of all the intelligence we're putting also in devices in terms of these, okay? So the aim of at least this first presentation here was just to show you that the fact that I would say the community of mathematician is working on these problems is assigned that there's also some economic flavor on how we can exploit these things in the next year. I'm quite surprised that these things are now at the level of where we also want them then they can be used at least or they can solve some problems that we have actually now. So this goes back to my questions that I'm going to ask the panelists because I'm not here to the presentation. In your fields basically, the three of you, what do you think are the main mathematical problems or directions we should invest on? Or if you have some ideas. So I can start with you. I know you're going in two minutes. Well, let's start with you if you have some ideas on. Well, I mean, I can make two comments on this list. Yes. Which we just saw. So I mean, something which I mean, from all the problems which are up there, one which is closest to my expertise is certainly the one on the Navier-Stokes equation because that's on a partial differential equation. I do think that this is a very good problem. But I'm not sure whether that's kind of the most relevant problem from the point of view of fluid dynamics. I think in fluid dynamics, one of the issues is better understanding of turbulence. And for turbulence, for instance, it might be actually more important to understand the route towards singularity formation in Euler's equation rather than ruling out or constructing a solution in Navier-Stokes equation. So sometimes I think that kind of focusing a field on one particular problem is not always the right thing to do. The second remark was this problem on economics, which you highlighted. I think by now nobody in economics would put this up. I mean, this general equilibrium theory of Aero and Dubreux is, I think, I mean, I had some contacts with people in theoretical economics in Bonn is, I think, completely out of fashion. And nobody would think now of extending that to a time-dependent case. People rather think of mechanism design, game theory, kind of these subjects. So I think, partially, some of these subjects very quickly date, right? I mean, some of the subjects which Smale put up there, I think, probably already completely dated. So one should be also careful there. So do we have some new problems coming in by a new Helbert? Were there any 20 problems posed? There were seven problems by the Claymat Institute. Yes. And there were seven problems. So one million dollar problems given by the Claymat Institute. And so I think you had the Riemann hypothesis. You have a Poincaré conjecture. You had Navier-Stokes, P versus NP. And then you had the Hodge conjecture. It's more on algebraic geometry. And the Bertens-Wünnitzen-Dyer conjecture, more on number theory. Maybe I'm missing some of them, but I think maybe it's... Young-Mills theory. Young-Mills theory. Okay. So in fact, those problems that were put at that time in 1998, there are a couple of ones which were out now. I mean, I wouldn't claim any authority, but it's my gut feeling that that might be the case. Okay. Okay, in your case, Jean-Claude, do you think in your field basically, what are basically the new needs in terms of research in mathematics to solve the issues where you are? I mean, in the list that you showed, maybe two items, but not exactly what I saw very quickly in the paper. There is maybe the one on the points on the two-sphere, which has to be generalized, obviously, for us, for the n-sphere and even Grassman manifold and even Flag manifold and complex one. This is one problem that could be very useful to tackle non-current communication in many settings. And then maybe the one related to the Diffant-Tin approximation. I mean, but not exactly this problem because, in fact, this tool of Diffant-Tin approximation appears in our area when we wanted to address the problem of, let's say, multi-user coding, especially, for example, one thing that has become almost classical now in our area is the computer forward, which, in fact, is very... I mean, its performance is very related to Diffant-Tin approximation, a multi-dimensional Diffant-Tin approximation. So there is also the problem of interference alignment, but not the one that has been popularized by... By, sorry... Jafar. Jafar, yeah. But the other one on the real line and all the variations on it. And so that's... I mean, among all these problems, this is what I see. Then there are some other tools, I mean, some other mathematical problems that have to be solved for some other applications in our area. There's the one that we discussed with Emmanuel this morning. For example, this problem of solutions for nonlinear integral equations. Not only solutions, but especially some superposition principle, and, okay, so... Yeah, so the idea of here what I wanted to do is basically to come up, but I mean, we're not gonna come up from the panel, is the same thing as Hilbert did in the 20th century to come up with a couple of applied mathematical problems for engineering, that we could come up with a list. I think this could be good stuff done by the engineering community where they would highlight the 20 problems of mathematical engineering that should be solved to help engineers because the ones which were, as you said, were pure maths, what would be the applied problems or at least the applied mathematical tools which should solve for conjectures to move forward in our disciplines, yeah. Well, to the list, I would add, I mean, all the open problems in multi-user information theory. It's amazing that the interference channel or even the relay channel is still open questions. So, I mean, information theory is the basis of the industry, a big share of the industry revolution from wireless to devices, to memories, to everything. And the fact that some of these problems... So, I mean, information theory is not very developed in maths departments generally speaking. And so, I think it deserves... I mean, the multi-user version of it and a lot of problems which we'll discuss a bit later would deserve to be as important if not more. I mean, physics is changing, the world is changing. So, I mean, we have a new set of questions to look at, and so I think we should include these questions as being the very essential problems. And so, say, multi-user information theory would be a good example. The second one is high-dimensional geometry. So, I think it starts showing up as being a very important aspect of the quantum channel series, essentially high-dimensional geometry, but cognitive sensing is as well. And so, I think we lack understanding of very basic questions in high-dimensional geometry. So, these would be questions that really come from a very practical motivation of modern economy, right? And physics, I would say physics, because physics are things, physics of data, physics of communication. So, physics goes beyond what we have learned about fluids and things like that. There are still, I mean, all these problems, but it seems this list is apart from the P versus the NP story. It makes as if the world had not changed at all in the last 20 years, also. But, I mean, so, no, isn't it? Isn't it? Yeah, yeah, yeah. One comment following what you said about multi-user information theory. There is this, also, another problem coming from group theory, in fact. For example, you know, these non-shannon inequalities from information theory that has, I mean, an equivalent encoding theory. And, you know, for example, the aim is to go beyond the, let's say, England-like bounds. And for this, we need to consider some groups, some very specific groups. We don't know which ones could be... I mean, we have some example, but we don't know which properties they have to verify to be some good candidates in order to build, let's say, good network codes. Yeah, so, I didn't want to quote it too early, but, I mean, large network and network mathematics, I don't know how to phrase it in terms of, I mean, either graphs or whatever stochastic network of whatever kind. I mean, everybody has this definition. But this is the essence of reality nowadays, of social networks, of communication networks, wireless network, YLAN, Internet, all social interactions in terms of these subjects. I mean, how is it that we didn't even see the world graph in the... But understanding the mathematics of graphs in anything that goes beyond what has been done, and there is some structure in there, as it starts showing up in the work of Aldous. So, I mean, it seems to be absolutely fundamental. And so, I would find this list very interesting, but, I mean, sort of oblivious of the fundamental evolution of our society. I mean, the world has been changing in a very dramatic way. And if mathematicians don't want to address this reality, I mean, I think everybody loses mathematicians on one side, but also the community. Because it means that because these things are sort of man-made, you can't do math on them. But there are structures which are fundamentally present there, right? But could you formulate a question in this area? Because it's more or less evident that it will be a topic for the future, and a lot of people will work. But it's maybe more difficult to isolate a single question, which will be the center of activity. So if you have an idea, I would be very interested. Yeah, in the first set of questions, I mean, we quoted the Relate Channel and the Interference Channel. Okay, and we can quote a few others, which have been open for a while and for which there are some results. Yeah, because the solution is mainly in the question. So if we can formulate in a nice manner the question, then we did half of the work. And I think this is the main problem when we try to list mathematical problems that can solve our engineering is to find what is the mathematical problem in the right manner. But I agree with you that the basic of the networks would be the Relate and the Interference, then you can add it up. Well, when Maxwell wrote his equations, would anybody have said, please think of how to formulate Maxwell's equation? I think things don't work that way. So I just, I mean, would say things. That would be my answer. Yeah, okay. I think also that these nonchalant type inequalities, these nonchalant type inequalities for more than three variables is very, I mean, something very important, because it's not easy to understand where they are coming from. And when the number of variables increases, so when your network increases, there are many more and so. Okay. So another point, the question was another question of the panel was about are we asking too much to mathematics? Well, I put in this question, I didn't bring the document, but I went through the document of age 2020, which is basically the vision of the EU for the 2020. And you just run the one term which is progress and the other term which is innovation. And if you look at the progress of the society, you look at the number of the word progress which is put in that document is very few. If you look at the number of times innovation is cited in it, it's incredible. Which means that we're pushing much more and more people to have results in a very quick manner in the sense of advancing the impact on the economy. And I know there's been some studies, I think. I don't know if it was done by Cédric or who about the impact of mathematics on the GDP. It turns out that, so I don't know if it's true. You can confirm afterwards. You have to know that 15%, so mathematics have an impact on the French economy of the order of the GDP growth of the order of 15%, which is quite huge from what I saw. I mean 50% of one discipline having such an impact is quite huge. The question and if you see also in this H2020 we're looking not more into doing a progress of the society but having really innovation behind. So do you think and I see also by the pressure that we're having to get more and more results rapidly from mathematician that can be translated into something. Do you think that we are overdoing it towards what we think of what mathematician can bring us? Or do you feel comfortable with that? As an academic, do you feel comfortable with that? Did you see a change in the fact that you are more and more connected to industry now than before in your work or not? Although you are always in Inria. He's not maybe the best guy. I should ask a guy from US to answer this. For my own work I'm not able to answer because I'm really not more and more involved with industry. But you get some requests now that mathematics should have more and more impact not on doing the progress of science but having an impact on innovation. Because from the documents I see this is exactly what we're requesting more and more from mathematics. I mean it's an issue for finding the funding of an institute like IHES but I don't, I mean it's from my point of view I have to define the idea that it's crucial to maintain a very high level of theory to be able to develop a very high level of application. And so, and it's not something completely admitted easy but it's a kind of thing I have to advocate. Okay. Would have the same view and I would spell it a bit differently. I think so especially young people should be encouraged to take risks and to do things which will be used for 15, 20 years after they do it and if you put the bar in terms of popularity with whatever metric and if you insist that people collaborate with industry through external matters you just kill the system. Okay. Maybe I have just one small story. It's not exactly about industry but the relation between let's say high theoretical mathematics and applications and some years ago maybe six, seven years ago I don't remember very well. I was still in telecom at that time and I was contacted by Leila Schnupps who wanted to to renew I don't remember exactly what it was. It was a European project on you know Galois theory, La Grotandeek so that's something very, very theoretical and they asked her I mean the probably the one who were responsible of let's say financing this project. They asked her in fact to find some applications and so she went to me to ask me if I wanted to be part of this but finally I said yes but of course it was something that I didn't understand very well and so I didn't do anything but at that time there was even this highly theoretical project had to demonstrate that there could be some application in order to be financed. But vice versa I think you can also get some very interesting problems to solve from the applications we're involved with. Yes, definitely. So I think what was said before is the right thing so when something is born and people would come and try to exploit it is one way, the other way indeed is talking to industry in a point to point way and learning about difficult questions and working on them and I think it happens it used to happen in Europe I think it's vanishing because of this age 2020 nonsense let's call it by its name where you essentially push people to meet and act jointly I think it's a dead end and NSF doesn't go this way it goes to individual grounds to individuals who are taking risks and doing the right thing Europe is going the wrong way and we should say it clearly and I'm ready to say it anywhere anytime but I don't think Europe Commission is doing the right thing in terms of funding research so it should be done and other continents or countries do differently we should look at how NSF works which was properly and so I pretty much would prefer that we go by point to point interaction with industry asking questions, interacting with industry as many of us do and learning about problem thinking and slowly developing a way of thinking we'll give answers to industry needs other than going to things which have no content essentially by nature and prevent young people to take the right risks so point to point interaction with industry are the right way these collaborative things to me is I mean it's just not what should be done and it's not like that things work in the US and I say well better in the US than in Europe in terms of real interaction between industry and academia and I think the cause of that is the way H2020 and the spot assessors have been working putting the emphasis on what should not be the emphasis so I don't know, I mean it's easy to when there are good research groups to create point to point interaction by direct funding from industry to academia and that's how it should work and I think it's much more developed this is vanishing in Europe I used to work with Alcatel, we designed some architecture DSL architecture with them in the year 2000 and then there was a big European project, it said well why don't we make a big project 20 of us were their competitors and the thing collapsed so I think point to point is disappearing therefore this way of asking fundamental questions to the right academic is disappearing and these projects don't work correctly so young people who are bright are disappointed and they go elsewhere so that's as simple as that and so we should say it Thanks, I had another couple of questions but I think the time is running I'll just ask the people around if they have some questions to ask to the panel No questions? In Hawaii we're trying to do it point to point by the way just for information the scheme we're doing we're trying to identify the prestigious researchers around the globe at least for us it's in Europe and in France and trying to create a point to point link by inviting people and trying to produce a document of research together on our problems In any case I'm very happy, I'd like to thank you the two others went away and I hope also from Yahshua's point of view we'll have the opportunity to continue working together I know we identified somebody there that we're going to welcome to talk to us and give a talk at our place so we can maybe see how we can apply these new results you've been producing and I hope we're going to have a successful next year workshop the bus is waiting for you outside and thank you all for coming and staying until 4.30pm