And the basic, 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 down, 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 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 were the 20 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 this century. So it was, I was quite surprised to see that many of the, 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. 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 Poincare conjecture is still there. It has not changed. Then 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 still, is there, at least. There's also here, in integral zeros of a polynomial of one variable, which from my point of view, we haven't still found any urgent need, but maybe there will be in height bounds for the

often 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

equilibrium mechanics, which also was not a 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 could find some application of this conjecture. Introduction of dynamics in economic theory, which is also something

very important. It turns out that this result is of 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. Here you can see, is there a polynomial time algorithm over the real numbers, which decide the feasibility of the linear system of inequality Ax higher than b? I'm quite surprised that at least, then the closing lemma, I'll finish all this, one-dimensional dynamics generally

hyperbolic, centralizers of deoformism, Hilbert's 16 problems, still there, Lorenz attractor, Navier-Stokes equations, still there, not still there, but at least a big issue.

Solving polynomial equations, so wait a minute, the Jacobian conjecture, solving polynomial equation, you can see here, can you solve, can a zero of n complex

polynomial equations and unknowns be found approximatively or on the average in polynomial time with the 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 years. 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. 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. The aim of at least this first presentation here was just to show you that the fact that I would say that the community of mathematicians is

working on these problems is also a sign 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. This goes back to my questions that I'm going to ask the panelists because I'm not here to the presentation, is 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. I can start with you.

I know you're going in two minutes. Let's start with you if you have some ideas. I can make two comments on this list, which we just saw. 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 a better understanding of turbulence. For turbulence, for instance, it might be actually more important to understand the route toward 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. This general equilibrium theory of Arrow and de Broglie

I had some contacts with people in theoretical economics and 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, these subjects.

So I think partially some of these subjects very quickly date. I mean, some of the subjects which Smale put up there I think probably already completely dated. So one should be careful there. So do we have some new problems coming in by New Helbert? Were there any 20 problems posed?

There were seven problems by the Claire Knight Institute. Yes. Okay. There were seven problems. So one million dollar problems given by the Claire Knight Institute.

And so I think you had the Riemann hypothesis, you had the Poincare conjecture, you had Navier-Stokes, P versus NP, and then you had the Hodge conjecture, so it's more on algebraic geometry, and the Bertrand-Svenot and Dyer conjecture more on number theory. Maybe I'm missing some of them, but I think maybe that's all.

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, what are the new needs in terms of research in mathematics to solve the issues where you are? I mean, 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 2-sphere, which has to be generalized, obviously, for us,

for the n-sphere, and even Grassmann 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 diaphantine approximation.

Not exactly this problem, because in fact, this tool of diaphantine approximation appears in our area when we want 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 computed forward, which in fact is very,

I mean, its performance is very related to multi-dimensional diaphantine approximation. So there is also the problem of interference alignment but not the one that has been popularized 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 that,

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 Manuel this morning, for example, this problem of solutions for non-linear integrable equations,

not only solutions but especially some superposition principle, et cetera. Okay. Yeah, so the idea of here what I wanted to do is basically to come up, but I mean we're not going to 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 in the next years to help engineers. Because the ones which you said were pure maths,

what would be the applied problems or at least the applied mathematical tools we should solve or conjectures to move forward in our disciplines? Well, to the list I would add 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, information theory is the basis of the industry, a big share of the industrial revolution from wireless to devices to memories to everything. And the fact that

some of these problems are there. information theory is not very developed in math departments, generally speaking. And so, I think it deserves the multi-user version of it and a lot of problems

which we'll discuss a bit later. I mean, what deserves to be as important if not more. I mean, physics is changing, the world is changing, so 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 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 course in Shannon 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, this would be questions that really come from a very practical motivation of modern economy, right? And physics, I would say,

because this is the physics of things, physics of data, physics of communication and so physics goes beyond what we have learned about fluids and things like that. There are still all these problems but it seems this list is apart from the P versus NP story. I mean, makes as if the world

has not changed at all in the last 20 years or so. So, no, isn't it? Isn't it? Yeah, yeah. One comment following what you said about multi-user information theory. There is this also another problem

coming from group theory 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

let's say Ingleton-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. So I didn't want to quote it thoroughly but I mean large network and network mathematics, I don't know how to phrase it in terms of, I mean, is on 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, wireless internet all social interactions are in terms of these objects.

I mean, how is it that we didn't even see the world graph but understanding the mathematics of graphs in anything that goes beyond what has been done and there is some structure in there it starts showing up in the work of Aldous so this seems to be absolutely fundamental

and so I find this list very interesting but sort of oblivious of the fundamental evolution of our societies. The world has been changing in a very dramatic way and if mathematicians don't want to address this reality

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. Could you formulate a question in this area?

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. In the first set of questions

we quoted the relay channel and the interference channel and we can quote a few others which have been open for a while and for which there are some results The solution is mainly 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 relay and the interference then you can add it up.

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. I just would say things.

That would be my answer. I think also that these non-channel type inequalities these non-channel type inequalities for more than three variables is very I mean something very important because it's

it's not easy to understand where they are coming from and when the number of variables increase so when your network increases then there are many more and so Ok So another point

the question of the panels 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 H2020 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 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 economy and I know there's been some studies I don't know if it was done by Cedric 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 afterward you have to know that 15% so mathematics have an impact on the French economy of the order of on the GDP on 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 and 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 for mathematicians that can be translated into something. Do you think that we are overdoing it towards what we think of what mathematicians can bring us?

Or do you feel comfortable with that? As an academic do you feel comfortable with the situation? Do you see 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 the so

he's not maybe the best guy I should ask a guy from USUS to answer this For my own work I'm not able to answer because I'm really not more and more involved

with industry Yeah 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 defend 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 easily but yeah it's the kind of thing I have to

advocate You 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 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

We have just one small story It's not exactly about industry but it is the relation between let's say high theoretical mathematics and applications and some years ago maybe 6-7 years ago

I don't remember very well I was still in telecom at that time and I was contacted by Leila Schnapps who wanted to to renew I don't remember exactly what it was, it was a European project on

Galois theory La Grotendique something very theoretical and they asked her probably the one who was responsible of of let's say the financing

this project they asked her in fact to find some applications of it and so she went to me to try to ask me if I wanted to be part of this and finally I said yes but of course it was something that I didn't understand very well

so I didn't do anything but at that time there was I mean even these highly theoretical projects had to demonstrate that there could be some applications in order to be financed but vice versa I think

you can also get some very interesting problems to solve from the applications where you are involved yes yes definitely yeah 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 even if there is no substance yes yes I think it's a dead end and NSF doesn't go this way it goes till individual grounds to individuals who are taking risk and doing the right thing Europe is going the wrong

way and we should say it we should say it clearly and I'm ready to say it anywhere anytime but I don't think the European Commission is doing the right thing in terms of funding research it's not the way it should be done and other continents or countries do differently we should

look at how NSF works which works 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 problems thinking and slowly developing a way of thinking

that in the end we'll give answers to industrial needs rather 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 this collaborative thing 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 they work I must say 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 spread SSOs

have been working putting the emphasis on what should not be the emphasis so I don't know it's easy to when there are good research groups to create this 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 Al-Khatil we designed some architecture DSL architecture in the year 2000 and then when there was a big open project he said well why don't we make a big project 20 of us their competitors

and the thing collapsed 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?

okay, so I'd like to in Hawaii we're trying to do it point to point by the way just for information and that's the scheme 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 Yashua's point of view who will 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 also another successful next year workshop the bus is waiting for you outside and thank you all for coming and staying until 4.30 okay, bye