 So yesterday, over dinner, Thibault said that we shouldn't heap too much praise on him. But I just cannot avoid it. It's unavoidable. You can just measure the depth and breadth of his contribution. If you look at the conference program. But there's also another Thibault, and that's the Thibault I've learned. I've gotten over 20 years of collaboration and interaction, and it's a sort of, should I say, non-physics Thibault. And I've just put up three things here. This was already mentioned, this prize-winning comic book, which is really ingenious. It's probably the best population quantum mechanics that I know of and have ever seen. Then there's this. This was only touched upon in Pierre's introduction. At least I saw something. It looked like a picture of Einstein next to a picture of Proust. And Thibault, as far as I know, is also an expert on Marcel Proust. And actually, this is a copy of a book you gave me as a present some years ago, just to tickle my interest in Marcel Proust. And now this thing has been sitting on my diet table for many years. Because Proust, as you know, it's formidable. Yesterday we had the discussion of the meaning of the word formidable. But the situation has changed a little bit because at the beginning of this year on my local radio station in Berlin, they're now reading all of Proust from beginning to end, 50 sessions of 40 minutes each in German. Yes, of course. But the thing that really sort of got me really excited is to have a feature that comes every Monday, a quarter past seven. And it's called Lust und Frust mit Proust. And this is a young lady. She does it brilliantly. She picks out certain aspects of Proust and then dissects them in a way that really makes the thing much more accessible if you just try to read it. And this also reminded me of discussions we had about the word lust because you kept asking me in connection with Rielke's epitaph because also the word lust appears. And then, you know, we had discussions on how to best translate this into French. So anyway, so this is another inspiration I got from Thibault over the years. And of course there's also music. I just remember that you were practicing the Winterreise, I guess, with Hugo Moschela, Katra Ma, and, you know, that's another summit of Western culture. Okay, let's now pass to physics. So first I want to tell you how it brought us together. And what brought us together is actually a stunning insight by our chairman, Bernard Julia, who had this conjecture in connection with maximal supergravity where they discovered the emergence of exceptional symmetries in supergravity. And then he had this staring conjecture, this is almost 40 years ago, that there would emerge something called E10 symmetry in the reduction of this theory to one dimension. Now over the years we have learned a little bit more and the way I would propose it now is the question is, is this really the symmetry underlying M-theory, this hypothetical, mythical, unified, non-perturbative unification of string theory? In which case you would have an infinite increase in the hidden symmetries of these theories and the unification of matter and gravitational degrees of freedom into a single, you know, monstrously large duality symmetry. So actually I was hooked on this idea from the very beginning, but so I think it was about 2000, I came to Paris and gave a talk and Thibault was sitting in the audience and afterwards he got to me and said, why don't we start discussing this? And so we did and then together with Marc Enneau we started working on, well, something called E10 over K10 Sigma model, never mind what it is, not worrying too much about what E10 really is. And now after 20 years I have to say we still have no idea what it is and not as anyone else. And this was the other thing he said, not too much praise, let's have a look towards the future and you know the problems that are still open. Now he has a real problem, open problem, namely to understand what this is, as Igor Frenkel always emphasizes to me, he says this is deepest mathematics. So it may take another, you know, we have to calculate in decades, for progress. Anyway, over the years the investigation revealed a number of really nice things, among other things. And first of all a crucial link with the BKL analysis of cosmological singularities, actually a fascinating link. And with this Sigma model actually a concrete proposal of what M-theory really should be because these days you hear a lot of people talking about M-theory and they never tell you what it really is. Much less give us some kind of formula that would encapsulate M-theory. And then there were many, many further results than also together with Axel Geinschmidt who is also here. So I will just, because time is short, I will just touch on some aspects of this without being complete and the intention of my talk is less to emphasize finished work because there's hardly any finished work here. But rather to emphasize what we don't know, a huge level of ignorance that's still there. So I've already mentioned BKL. So this is one aspect I want to illuminate. And I guess that Marc will also talk about this Westmark. You will talk about it. So with some things I'll be a little, a part of it rather quickly. Anyway, the core of the BKL ansatz can be encapsulated in this Karstner metric because the statement as you approach, this is about the generic behavior of Einstein's solutions. The idea here is as you approach space-like singularities is the sort of generic behavior which was analyzed by, first by Berlinski-Khalaptikov-Liftschitz, one of the greatest coverage of mathematical cosmology of the last 50 years. And the statement is somehow the main action takes place in the diagonal phase factors, scale factors, which you see in this ansatz, which is the Karstner metric. And when you substitute this ansatz into the Einstein-Hilbert action, this is what you get. It's just a free relativistic particle in a massless particle in this beta space, not in real space, but in this beta space of logarithmic scale factors. When you translate it into Hamiltonian, it's just a statement that p squared equals zero. This is something well known from elementary quantum field theory. Now, the crucial thing or one crucial observation is that the metric that you get from this action is, first of all, Lorentzian. That's one of its crucial famous features. But later, and this is the connection with E10, it turns out that this David metric coincides with the Carton-Killing metric of some indefinite-cut smoothie algebra when you restrict it to the Carton subalgebra and then you identify the Carton subalgebra actually with these scale factors. Indefinite is important here because this follows from the indefiniteness of the David metric. So that's an essential feature which comes from gravity. Now, it has turned out to be convenient to describe the motion, this relativistic motion, which in beta space is just a straight motion on null lines in the forward light cone. It's convenient to project this motion onto the unit hyperboloid in the forward light cone by means of this projection here. So we have coordinates beta. I should say that most of my talk will be for E10. That means 11-dimensional supergravity, which means 10 of these scale factors. So we now project these 10 vectors onto a 9-dimensional unit hyperboloid. And then there's this row, which is simply the length of this vector beta and the omega parameterized unit hyperboloid. And it proves convenient to choose coordinates or the lapse functions in such a way that the distance between epsilon and T equals epsilon, T equals 0, the singularity is blown up to infinity. So this is why we call this zeno time. So the singularity in this coordination is reached for row to infinity. Now, the effect of the matter degrees of freedom is of course very complicated, but there's a crucial simplification, the BKL limit, which you can study by, well, this is the language of high energy physics, integrating out matter, off-diagonal curvature, and so on, degrees of freedom. At the end of the day, this free motion, relativistic motion, is modified by potential. And in the limit towards the singularity, these potentials become sharp walls so that as a result, this particle moves on light-like lines and bounces off the walls of this chamber, which is formed by these sharp walls. So that's a very nice description of the BKL situation. And it's sort of universal, applies to all dimensions and various types of metacouplings, supergravity, whatever you. But now we want to immediately go to the quantum theory. Now, quantization is simply replacement of momentum by d by dx, and the constraint is replaced by the, well, Klein-Gordan equation. So this is what this is. And then what you get is the so-called Wieler-Witt equation. It's really like the Klein-Gordan equation, except that this H operator now acts in this beta space of scale factors. And if you now decompose the coordinates in the way I've shown before, then this D'Alembert operator has this piece, and then he has a piece depending on the Laplace-Beltrami operator on the unit hyperboloid. So you see that rho is really this radial coordinate, it's really like a time. And this is a well-known mechanism in how to extract time out of the timeless Wieler-Witt equation. Wieler-Witt equation, constraint to start with, has no time. But this is how you get time out of it by taking one of the degrees of freedom. Okay, so this is work we did like 11 years ago with Axel and my PhD students, Michael Köhne. So let's assume that we diagonalize this, first of all we factorize, separate the wave equation, and then it's an elementary exercise to work out, once you know this eigenvalue, what the wave function looks like, or the radial part of the wave function, it looks like this. So it all depends on what these eigenvalues are. Now there's a nice trick to derive an inequality on this, which I don't go through in all detail. It involves directly boundary condition, it involves using the Cauchy-Schwarz inequality. And at the end of the calculation you get this inequality, where d is the number of spatial dimensions. So d would be 3 for Einstein gravity in four dimensions and 10 for a little dimensional supergravity. Now you see that this is just the combination that appears here. So what you see here, because d is greater than 2, you have the wave function decays and otherwise oscillates. And this inequality ensures that what's on the square root here is positive. So this is really, if it were negative then you might get another i and then it might destroy this behavior. But the universal behavior is here like this, it goes to zero and then it also oscillates an infinite number of times as you go towards the singularity as long as this is positive and depending on what the energy eigenvalue is. So this means that generically in this approach the wave function vanishes at the singularity. And there are already three physically important things that you can deduce from that. Because first of all this is David's original proposed mechanism for resolving classical singularities in quantum gravity. You had this idea that if you solve the wave function for the wave function of the universe it might have zero support on singular three geometries. So this is actually borne out by this example. Furthermore you're forced to a complex wave function. And this is another question. The Wheeler-DeWitt equation starts with a real equation. So where do you get complexity from? Here it's forced to be complex and in this way you create an error of time which was not there to start with. And finally, because the singularity, the black hole singularity is the time-reversed picture of the Big Bang singularity this could also be relevant for the resolution of the black hole information paradox. This is a recent paper of Malcolm Perry. Now it can be a little more specific about this wave function because we have the walls. But the walls that reflect in this case, the effect of this is that the wave function has to vanish at these walls. Furthermore, because the walls are identified or determined by the wild reflections of this underlying Katsmoody of the wild group associated with this underlying Katsmoody algebra so you just have to impose these boundary conditions with a minus sign if you want directly boundary conditions. Now for E10, that's M theory. So D minus 1 is equal to 9. So we have to coordinate the unit hyperbolic and we do it in such a way that this is like the upper half plane, U plus IV which is what would be in the case of Einstein gravity. But because we are in nine dimensions, this is now what we call an octonionic upper half plane. So this is an octonion and with this you can realize the wild group in a sort of modular way which is completely analogous to the modular realization of Einstein's theory in which case you just have the usual modular group. And an important fact here is that these are the units, simple roots in E8. Here we exploit the fact that the E8 root lattice can be endowed with the structure of a non-computative and non-associative ring and these integers, these are octonionic integers, quote unquote, called octavians and then the E8 roots are simply the units of this ring. And in this way you get a sort of completely modular realization of the wild group of E8 and which means that actually the wave function of the universe is a modular kind of modular form but now with respect to this generalized modular group. Now here's another problem for the future, a proper theory of nomophic forms and this kind of group remains to be developed. These results actually can be super-symmetrized. This is an extremely simple truncation of the theory but they're already with the fermions to get into trouble because the fermionary fox space has two to the 160 components even for this utterly simple truncation. There's also work related by Thibault and Spadelle which provides further evidence for these structures but in ordinary Einstein gravity. Now, okay, so BKL's story is very nice but actually one thinks about embedding gravity into this huge structure, you also have to worry about other degrees of freedom and this is actually taken, argument taken, a paper I wrote with Thibault back in 2005 which was really about order corrections in M-theory and how they related to E10. This thing is just an appendix but I don't think this may possibly be the more important insight in this paper. Namely, formally proceeding, we take this infinite dimensional coset manifold, we formulate the geodesic deviation equation which you can look up in Weinberg's book for example. XI describes some geodesic which you take to be in the Carton subalgebra so it's just straight line and then delta XI is the deviation and this deviation equation describes the divergence of two geodesics as a function of the Riemann tensor, this is what this equation says. So you evaluate this for this huge coset and then you have to label these directions of the indices of the Riemann tensor which of course there are infinitely many. So we pick just a particular one associated with some root, there are also root multiplicity, worry about but never mind. But first of all it's negative, okay, this is a hyperbolic manifold so that's okay but the important point is that this can be made as negative as you like and this has to do with the presence of so-called imaginary roots in E10 and one result about imaginary roots is that any multiple of such a root is again a root. So by simply multiplying the root by n, any integer you get another root which means you get a factor of n squared here which means that the geodesic equation has this n squared here and then it will just deviate exponentially and no matter how small you choose the initial delta xi to be 10 to the minus 100,000 or whatever you can always pick an n large enough so that this deviates. So this suggests that the analysis for the BKL analysis close to the singularity where you just restrict to diagonal degrees of freedom is sort of fails in this much bigger context which suggests that the analysis cannot be confined to diagonal matrix but then one must take into account all the off diagonal degrees of freedom. So now we go into the realm of speculation because then you ask yourself, okay, I showed you this Klein-Gordon equation for the diagonal scale factors which gave rise to these automorphic forms and yes, we have more on this and that's already in the early work because many years ago we showed that the Bosonic Hamiltonian of or at least part of it, the Levner-Welchel supergravity theory, alias M theory and the E10-Casemia agree up to level L equals 3. I can explain what level means but let's not stop here it's just a sort of way to probe into E10 and this is a very non-trivial agreement actually. It tells you that E10 somehow knows everything about maximal supergravity and then this simply suggests why don't we simply generalize the wave function, the equation in this way where now the box is something much, much, much bigger namely the box is the E10-Casemia. Now E10-Casemia exists. You can read it in Viktor Karp's book. It's a unique expression and it's written here. Here where you have the Carton sub-algebra, this is the Weylvector and here you have the sum of all the raising generators a normal order if you like because it's something of a positive roots. This is unique. There's a unique expression for this and now the point is that actually this realization that we have you can in principle at least represent these generators by differential operators that act on the degrees of freedom, off diagonal and so on. Now this is a calculation where we'll get stuck after a few steps because with each step it gets more complicated never stops getting more complicated. So these are, this is not hardly a Gedanken calculation but anyway you can do this at low levels. What you get is a kind of Weyler-DeWitt operator for 11 dimensional supergravity. His coincides with this. And furthermore, by this prescription it will give you a definite ordering prescription. This is another problem with the Weyler-DeWitt equation that people don't know. There's no unique ordering but here it is, it's there. So this is a nice feature. However, unfortunately this cannot be the whole story because first of all beyond the matching works perfectly up to this level three. It gives you all of 11 dimensional supergravity, Weyler-DeWitt, but beyond it just stops. It just disagrees. There's no way, we haven't found in 20 years a way to fix this. So that's already a first indication. Secondly, it indicates that this doesn't include the fermions. Fermions are a crucial part of the story. So that's another thing. And finally, you know, suppose it were true. This Casimir operator is well defined on highest weight or lowest weight representations. And there you can evaluate it. It's a finite number. But then again you know that if the Casimir is zero then you expect the representation to be trivial. So that's nothing left. So the other possibility is that it's not a highest weight representation. But then you can kind of convince ourselves that this expression no longer makes sense. So it's ill-defined. So here it's sort of between Schiller and Carriptis. We have to, you know, one or two bad options. So anyways, this is not a complete story. And this is something we've been thinking about for a long time. And we now think that one will have to better understand the fermions. Because this is something I've learned over many years of working in supergravity that no matter what the problem is it's always easier to analyze if you look at the fermions. Basically because fermions obey linear equations of motion whereas bosons quadratic. So for example, if you know this whole story about consistent truncations in Kaluzzak lines here it was finally only solved because we first did it for the fermionic sector. For the bosons it's usually much more complicated. So we've been looking at the fermions spending most of our time now understanding fermions. They don't transform under E10 but under its compact subgroup which is equally difficult to understand. But there's this curious feature that only so far we only know finite dimensional unfaithful representations. These are the ones that are inherited from supergravity. So we showed that they can actually implement it as representation of the full KE10. But this is as much as we know with Axel we found two more that are beyond supergravity. And recently actually there's been some progress at least with Axel, Ralph Köhl and Robin Lautenbacher. This is this paper here because we know at least for the affine case we know infinitely many such representations we've actually I think we've understood the result that for the existence of these unfaithful representations we've shown that they can be made arbitrarily big. And this I think is also important with regard to understanding E10 because the thing that mathematicians have tried look to me at least hopeless but this is a completely different way of thinking about it where you exhaust this group by larger and larger and larger groups where all the KE10 transformations are realized but unfaithful. So maybe there's a way to construct it as a limit. But there's another idea here which you know has been on my head for a long time and maybe we may need to bosonize the fermions because while there's some hint from old work by Goddard Nam Olive from 1984 where they show that fermions in the 128 can be bosonized into 120 bosons. It's a non-abelian bosonization. So if you take the multiple of maximal supergravity 128S plus 128C, the conjugate spinor representation replace one by the adjoint of the K of E8 which is SO16 then you get the adjoint of E8. So you might expect there's a similar mechanism for a KE10 at work and if this is true then something like what I showed on the previous transparency might still be true at the end of the day in a way that we, you know, at the moment have no idea how it could possibly work. So let me conclude. There are now many, many, I think also first of all there's the mathematics which is as Igor said, deepest mathematics but there's also physics questions here namely first of all can a sensible wheeler to width operator be defined for E10? Is, and this is an idea that was first put forward by Origano, is psi then a kind of generalized modular form but you know vastly, hugely generalized with respect to something like E10 over the integers, whatever. And this is also a question. I argue that the geodesics restricted to BKL are unstable and you have the same question. Does the wave function close to the singularity necessarily involve all off-diagonal degrees of freedom as you get closer to the singularity? In which case, you know, there's a diffusion quote-unquote in huge phase space as you approach the singularity and as I said this may have, you know, given an entirely new twist this whole plug, whole information problem as argued by Malcolm Perry because maybe information doesn't get lost if it doesn't get crushed in the singularity but you know if it diffuses into this huge phase space maybe it cannot be recovered. Also the question is does, and this is actually what we've been discussing with Axel does the wave function continue to vanish at the singularity with, and explain it, well there's a beautiful book by Axel and friends called Eisenstein's series in Automorphic Representations just for ordinary automorphic forms and there indeed it's true that the wave function decays exponentially in these off-diagonal directions. So here's the final question that I've written, published at CQG Note just a few months ago on this. The question is, is life at the singularity infinitely simple or infinitely complex? Much of current work on quantum cosmology makes this assumption that life is infinitely simple you just have a few variables, a scale factor spatially homogeneous scalar fields and then you'd some partial differential equation but what would also happen is precisely the opposite this is actually what I argue here that as you try to get closer to the singularity it becomes more and more and more complex. So to summarize there's a vast terror in Cognita out there wanting to be discovered and explored and you know we have had wonderful time together I still hope that we're not too old to sort of take up some of these questions in the not too distant future so let me conclude with a few more pictures. In 2005 my wife here and I went to Bern for the Einstein Centenary to meet with Thibault and then we went on this on what we call the Einstein pilgrimage visiting all the places that Einstein had been in to in Bern and here you see a sitting actually like schoolboys in Einstein's living room reconstructed of course but you see nice pictures of Miléva and the children and so on and this picture was taken three years ago in Waldemossa in Malacca you were at my ERC kick-off workshop and this is a nice monastery where Frédéric Chopin spent an unhappy winter it's a wonderful place it's like you wonder why he was unhappy well I guess because the heating didn't work something like this and finally here, undated in Thibault's office this is one of the countless notebooks in Thibault's office and of course here's Ulysse now Ulysse has already appeared in previous talk I would just add to this that so Ulysse was a gentle dog always lying there sleeping, dosing sometimes squinting at us but that was about it but at lunchtime he had this habit of getting up and moving between us and the blackboard to tell us now over lunchtime stop it at tea time anyway so with this note if Ulysse was the chairman I think he would now also move between me and the so happy birthday thank you thank you very much, I think we have time for one question just curiosity you mentioned this beautiful work of Bichet as a classic work of Ulysse how would you expect this Zeno approach to be modified by Dupato? well the Zeno, I mean it's just the choice of a time coordinate and it's I don't think it's a convenient choice because in the original picture you have the BKL oscillations in the chaotic case take place in infinite number of times so what this does it simply stretches this finite interval to infinite time and then you have a very nice straight description of these oscillations so I would say it's just a choice of time coordinate and you know the Wieler-Dewitt equation has no time to start with the genius thing about the Wieler-Dewitt equation because you can choose the time coordinate as you like but this is you know this just illustrates the mechanism that's often assumed to make this work is there another question? it's the same thing as you said I'm not a physicist, I'm just asking because I read this quantum gravity one paper of Dewitt and he used this as a global condition so my question is a bit kind of confused because this is a local time it has to do with the fact that this is where it connects up to Bernhard's conjecture because Bernhard said you should see this in the dimension reduction one dimension and the BKL is a sort of dynamical realisation of that because it says as you go towards the singularity the time dependence overwhelms everything else so it's de facto effectively like a dimensional reduction to one dimension so this one way to interpret BKL is to say that as you approach the space like singularity the system sort of decays into a continuous superposition of one dimensional systems and this is why I think you know you start seeing this huge symmetry and this air of time corresponding it is also local yeah sure I mean in quantum gravity you can pick time coordinate locally as you like but because here we sort of starting from compactification to one point another unsolved problem here is how to excite to get back the spatial dependence from within this hugely algebra we have some idea lots of ideas actually but nothing so far has really worked so as I said this there's a lot of Terrain Toki Koknida here so let's just thank him again