 Well, thank you very much and thank you for inviting me here. On the one hand, I feel greatly honoured to give a sort of closing lecture of this conference. And I should also say it's, as always, you know, immense pleasure to be here. And I really regret that I couldn't take part in previous part of this conference because even from what filtered through from the questions and illusions to previous talks, it sounded really intriguing. After some consultation with Thibault, I've decided to give a more general talk. So I apologise in advance to those of you who are really specialists. Certainly string series will not learn very much from this talk. As I said, it's going to be a brief survey. It will be rather pointillistic, I would say, in the sense that I will just touch on a few things and make a few remarks and they will also not cover everything, but it will certainly take the liberty to spend the last part of my talk on things that are very close to my heart when it comes to what a potential theory of quantum gravity could be. So I would like to start out by recalling the reasons why we should be interested in a theory of quantum gravity because at this time there's actually no pressing need from any experiment or observation to really proceed in this direction because the theories that we have at this time seem to be working extremely well. This applies both to general relativity but also to the standard model of particle physics which gets better and better with every measurement. So the reason to be interested in the theory of quantum gravity comes from basically the fact that the two theories that we have are incomplete and possibly even inconsistent in the ultimate consequence. On the general relativity side it's the issue of singularities which seem to be generic. This follows from the singularity theorems that were proven by Hawking and Penrose long time ago according to which generic initial conditions like the existence of a universe will inevitably lead to singularity formation. And of course there's the cosmological singularity, the Big Bang singularity at the beginning of our universe and this is the greatest puzzle. In some sense this is what we really would like to understand. I mean one of the amazing stories of modern physics that with the understanding we have we can get very close like 10 to the minus 30 seconds or so to next to the singularity but it's just this tiny little bit that causes all our headaches. And of course there's the question of structural space time at the smallest distances. On the other side in quantum field theory there as well we are faced with some lurking inconsistencies possibly. So as you all well know in perturbation theory we encounter ultraviolet divergences and Feynman diagrams for the standard model symmetries. These can be removed by infinite renormalizations, order by order. This is rather sophisticated procedure that has been developed over decades actually and has led to amazing agreement with experiment. If you think about things like G minus 2 or precision experiments they're not doing for the electroweak theory but nevertheless I think Kostas showed this or quoted Dirac on this who never accepted infinities. He said the right theory should give finite answers period. So we're not really sure what we somehow feel that this cannot be the end of the story and in some sense you can make it a little more precise because you can ask yourself if you want to go beyond perturbation theory, does the standard model which works so well in perturbation theory does it make sense beyond perturbation theory? And I think there's now some reason to believe that it will probably not exist in the rigorous sense of constructive or axiomatic a quantum field theory. And this is the reason why people are talking about the possible ultraviolet completion of the theory that is some other theory that contains the standard model as a limiting case. So the difficulties in both cases in some sense may have a common origin because in both general relativity and quantum field theory we treat spacetime as a continuum or differentiable manifold meaning that we can in principle should be able to go as small distance as we like. And secondly in Stanley quantum field theory so far we treat elementary particle as mathematical points exactly point like excitations and this is of course the origin of these ultraviolet divergences. But there's also a difficulty when you try to describe point particle mechanics in general relativity. So the conclusion of all this is at least majority of the community believes in this and this is independent of further string theories to loop quantum gravity. You expect something traumatic to happen at the scale at which we believe quantum gravity might be relevant which is the blank scale. So the question is really what happens? How are these theories modified? What happens to continuum spacetime? Is it sort of dissolved in a kind of discretum? Is the continuum replaced by a discretum? Or what is it that or maybe it's non-commutativity of spacetime coordinates but something is expected to happen at this scale. So this is the relevant scale that we're talking about. Hopefully something will be found before that scale otherwise. Well I'm not... And now with regard to once you accept this point of view that you can adopt two different kinds of attitudes and this also marks the dividing line between the different approaches to quantum gravity. One hypothesis that quantum gravity essentially is just a non-perturbative quantization of Einstein gravity in whatever formalism, metric, connection, loop, discrete. And once you solve this and treat it suitably you just have to complement this theory by the standard model of particle physics and that this together then correctly describes the physical degrees of freedom also at the very smallest distances. The opposite hypothesis is that actually general relativity is just an effective or low energy theory arising at large distances from a more fundamental Planck's scale theory and that we do not at this time know what the basic degrees of freedom of this theory are except that we suspect they're very different from either general relativity of quantum field theory as we know them. In this case general relativity and with it spacetime covariance, such notion of spacetime general covariance are assumed to be emergent much like macroscopic physics emerges from the quantum world of atoms and molecules. And from this point of view this first approach would seem as sensible as trying to find out about the microscopic world by say applying quantization techniques to the Navier-Stokes equation. I mean everyone would agree that it's very unlikely that you would find out about atoms and molecules in this way. Now a basic fact about quantum gravity that sets or quantized gravity that sets it apart from the matter interactions that characterize the standard model is the fact that gravity Einstein theory is perturbatively non-renormalizable. This means that at each order in order to get sensible results we have to introduce a new kind of counter term to render the theory predictable and finite. So here's the famous two-loop counter term whose coefficient here was calculated some 30 years ago. But I should like to emphasize not just the fact that you have divergences because you might think well let's subtract them and make the theory finite. It's the fact that you need to adjust an infinite number of parameters and this is believed to render this theory non-predictive. Because then if you have an infinite number of coupling constants there's no chance to pin down what the theory is. Again there are two possible conclusions in line with what I said before. The first conclusion is that somehow you have to have consistent quantization of gravity that requires that we need to introduce a radical modification of the theory at very short distances. In particular include matter, supersymmetric matter, fermions and maybe even something more radical like going from point to particles to extended objects like string or membrane theory and thereby to get rid of this divergence. The opposite point of view also expressed most colorfully I would say by Roger Penrose is that actually the ultraviolet diversions are simply an artifact of a bit of treatment of the theory because when we do perturbative quantum gravity we do some violence to the theory. We destroy somehow the geometrical structure, general covariance. This is no longer there in a nice and manifest form and therefore the hypothesis is here that these would disappear upon proper non-predictive quantization of the theory. However, and this is something I would, I mean whatever your attitude towards this is I think no approach to quantum gravity can claim complete success does not explain in full gory detail what happens with this divergence. Now this is easier for approaches that start from particle physics because that one somehow tries to get rid of the divergence right directly but this comment applies more to the non-predictive approaches because sometimes they say they don't even see the divergence. I just want to say if you claim you don't see the divergence then I think that you need to investigate your approach a little more closely because whatever your non-predictive approach is it should admit a semi-classical limit and then once you do this then this thing will reappear. So there's no way around anyone who claims to have a theory of quantum gravity and does not explain to me what happened with this divergence I will not accept this as an answer. Are you sure you want two negations? No, it's not. Sorry, it means the contrary point of view. Yes, they are two negations. Ah sorry, you mean double negative, sorry. Now there's also the thing about gravity and matter. This is what Einstein himself said about his equation. He said the left side is built from marble, beautiful, unique. The right hand side he said is made from timber, this is the energy momentum tensor of what, dust, whatever standard model the team he knew. So he spent a good part of his, later part of his life trying to see whether he could understand this in a geometrical way. So the wish would be to somehow right the left right side bring it to the left side and somehow understand it as some kind of... When you agree that it's the other way around actually. The left hand side is the height of dynamics and the right hand side. Timber and this is the marble. You mean the opposite of what Einstein said. It depends on this. I think on this I would side with Einstein. Anyway, this has led to an enormous effort over the past decades really. For example, Kalutza Klein's theory is an attempt in this direction when you take the theory in higher dimensions and explain the matter energy momentum tensor as coming from higher dimensional geometry. So in part sort of going in this direction. The other thing is that you can introduce supersymmetry which is like introducing fermionic coordinates and they're also enlarging a spacetime and then also endowing this right hand side with a kind of geometric interpretation. There's also the question of gravity versus not quantum field theory or quantum gravity but just standard quantum mechanics because we have this kind of schizophrenic Slava would probably say situation where this is treated completely classically and this should really be sort of as a quantum object or expectation value. And this has also led to a long debate. In particular the question is whether we need to change the basic rules of quantum mechanics as was suggested originally by Hawking. For example, should we really allow for the possibility that pure states can evolve into mixed states? And the other thing I would like to mention here is that because after all at least my favorite approach space and time are somehow thought to be emergent and so you should all start out with a theory that does not have space and time and then I would just like to emphasize that many of the paradoxes that we think occur in quantum mechanics appear in a completely new guise because most of these paradoxes have to do with non-locality and that you really need some notion of space or maybe even time in order to make sense of it. So the whole question I just want to raise it not answer it is if you're really in a sort of pre-geometrical framework, what's the... Is this a different question or how should we address it? Finally, the final point is the so-called hieric problem which is the fact that gravity is so much weaker than the other forces. In my popular lectures I illustrated by taking a pin and a little magnet and then the pin is drawn towards the magnet and then I say you know this little magnet beats the gravitational pull of the whole planet and this is a rather direct evidence of this hieric problem let me that the scales are so much different so in particle physics we talk about the masses we actually cover wide range so you already have something like a little hieric problem but all of this is tiny vis-à-vis the Planck scale of 10 to 19 GV which is the equivalent of 10 to the minus 33 centimetres. So a key challenge and perhaps the crucial issue in this whole game is for any proposed year of quantum gravity is that you should offer quantifiable criteria to confirm or falsify the theory and in particular you should be such that they allow to discriminate your answers from somebody else's answers. You know it's not much is gained if you know some theory that explains structure formation or whatever there will be many theories or things like hypothetical things like violations of Lorentz and variance if this was really seen there will probably be many and in fact conventional explanations of such facts so I think the challenge is here much more pronounced and of course you can always say my objects are quantized in units of the Planck length but this is something that we will never be able to measure so here some degree of sophistication is required so here is a list of the approaches not complete I would say that roughly speaking the community is divided into a majority part that does a string series supergravity and so on there's the more path integral is something in between there's canonical quantization loop quantum gravity out of which we develop the modern discrete approaches like spin form, scoop field theory and so on but apart from this we also have non-commutative geometry we have something called asymptotic safety and then you can go even more abstract in the sense that you could even look for a theory where everything is just information theory and I should say there's no agreement it's really that we have different communities in fact with very little communication and Jörg showed this different picture but this is the same event, historical event and as you know this event or this effort failed because at some point people could no longer communicate with each other and this is something I also see happening a little bit in the different communities working on towards quantum gravity and I think if you sort of zeroing in on the right theory we should see things falling into place and that should emerge some agreement so I want to now go through some of them but just very briefly now the first thing I would like to mention here just one slide is this idea of asymptotic safety I said that most workers believe that something has to happen on the blank scale but not everyone because there's the question of whether standard quantum field theory concepts are actually enough to quantize Einstein's theory so this approach in some sense is very close to conventional quantum field theories RG flows, RG group and so on and in fact does not require anything special to happen to the continuum space time somehow assume that you can go as much as you like below the blank scale so here one hypothesizes that there's a non-gaussian fixed point and this really means if true that the non-renormalizable quantum gravity is really renormalizable in disguise because there's some in this infinite dimensional space of couplings there's a hyper surface to which things can flow and so in fact you would just have to fix a finite number of parameters rather than infinite number of them so the aim is then one constructs a scale dependent effective action one can discuss in some detail what this means because in a generally covariant theory you know the measure of links etc. it's something that's not fixed a priori so this requires some rather sophisticated assumptions but this approach is essentially agnostic about the microscopic theory it's essentially just the normalization group equation and what you put in as initial condition and then the hope is that maybe these things fall in so what would be observable here? Well we don't see running Newton's coupling constant when we look outside the window so all of this is going to take place after the blank scale which is here thought to be analogous to lambda QCD so observable effects would be very hard to come by the only thing I can imagine is somehow that you know in this way you can predict that there should be a small logical constant at large distances but this remains a hope so let me now say a few words about canonical quantum gravity because this was the first serious attempt to tackle the problem and that was actually the first non-perturbative attempt which was also back right in the pen end which is a direct attack on Einstein's theory in terms of textbook methods of quantizing now what you need to do here is of course you have to give up manifest space-time covariance by a split of space-time into space and time so that space-time geometry can be viewed as an evolution of spatial geometry and time according to Einstein's equations geometric dynamics then is defined as follows the dynamical degrees of freedom are the spatial components of the metric the components with one with a time index are Lagrange multipliers that lead to constraints and they're associated canonical momenta and then the dynamics is defined by the constraints namely Hamiltonian constraint that goes with time reprimaritalization invariance and the spatial and the logarithms are associated with the spatial different morphism constraints now the general covariance of the theory in the classical context is in this constraint algebra which is more complicated but I haven't suppressed all indices but the difference between this and the standard gauge series that the dynamics is somehow encapsulated and constrained so what you would do in this approach is to replace this by a functional differential operator you express everything in terms of operators and then when you express the Hamiltonian constraint you get the famous wheeler-de-witt equation this equation has been around for well, almost 60 years nobody to this date has been able to make any, I mean because it's ill-defined even by the sloppy standards of physicists and in particular what you would have to satisfy you would have to say what becomes of this algebra as you replace classical variables by quantum operators and as we know from many examples they would expect anomalies just like in the case of two-dimensional different morphisms which becomes the Virasor algebra and in four dimensions you would expect even more trusting modifications and I'm just saying this because classical spacetime covariance in this theory would be replaced by quantum spacetime covariance just like one here, one has a very concrete example how this could happen but it's something that you have to prove and this has proved to be impossible but there's no solution that people agree on there have been proposals of course so this whole wheeler-de-witt program came to a halt because no one was really able to make any progress on this and it got a new impetus with Achtigal variables which is a canon switch to another set of canonical variables which are essentially the following quantum mechanics when you replace Q and P by creation and annihilation operators AQ plus Ip so here you have the spatial spin connection this is the extrinsic curvature this is essentially the canonical momentum with a parameter which was later called Barbera-Mitzu parameter and originally this was plus minus I and then he was able to show, Achtigal parameter and he was also able to show for gamma equals plus minus I then in this case the constraints canonical constraints take a simple and rather nice form in particular it turns out that everything that depends on this variable does so via this curvature tensor which we know very well, this is just the Young Mills field strength associated with SU2 so it's almost like the phase space of the SU2 Young Mills theory except that you have two more constraints these are diffeomorphism and Hamiltonian constraint what does the approximately equal to zero mean? a weakly zero, you can put it weakly zero in the Dirac this is the Dirac notation now I think originally it was plus minus I later people put this to a real constant and I should say in my opinion this is where this whole subject took a wrong turn because once you make this real then the nice form is destroyed you get some other ugly extra term that is very hard to handle anyway the modern version of this is loop quantum gravity where you don't work with a connection but with a path-ordered exponential along some edge and of course a conjugate variable is the flux defined like this in terms of this E variable and then your wave functionals depend on these olonomies a new feature is that the kinematical Hilbert space can be defined but it's non separable which is kind of weird but you should keep in mind that all new physics that is claimed to be associated with loop quantum cosmology which is a mini superspace version of loop quantum gravity analogous to the old mini superspace of Willa DeWitt all the new physics really comes from this non separability this is one of the reasons they don't see divergences because you know normally would put here delta function for this in quantum mechanics so when you square this then you have the usual difficulties but here this difficulty doesn't appear this form so this is one of the reasons they so far cannot see the divergences hence no anomalies and the other peculiar feature is that although you're working with connections there are no negative norm states and this has to do with this non separability non separability have they tried to construct Yang-Mills theory? no this program has not led to any kind of valuable new insights for Yang-Mills theory but it should be a simpler theory and we know so why is this going to work? well I will not say anything on this I just present the facts but it's very good okay now the crucial much of this is just kinematics defining state space it's scale up products it's just kinematics the core question canonical gravity is the Hamiltonian I mean first of all can you define it and then can you do anything with it and I will not describe this in detail because this is rather elaborate formalism was mainly developed by Thomas Tiemann and this is claimed to be a main success namely that the Hamiltonian can be defined rigorously as an operator on this kinematical Hilbert space by the way this is the ugly term that nobody talks about which would vanish if you put gamma equal to plus or minus i this is a little bit like the McDahl formulation of gauge theories where it just allow yourself to work with field strength now the proper definition is rather rather integrate and the one thing I would maybe raise here because this is a valid question the question is whether Hilbert space is really the right concept because what you find here is more you need to go to some kind of distribution space and it's also the question when you're talking about the wave function of the universe does it make sense when you calculate scale up product does this make any sense or is it such that you need to give up even such things that we take to be granted in ordinary quantum mechanics anyway there's also some argument that this Hamiltonian is not very physical because what it does that these pictures of the spin networks which I'm sure you've seen spin networks are graphs with vertices and edges and things like this so what the action of this Hamiltonian does in some sense it adds what we call spider webs like this so it doesn't really yeah question we know that the metric does make sense at least classically so we know that somehow the gravity is the background that is Mikovsky or some background which breaks the symmetry to this guy who works one of the difficulties here is that the metric is represented because it's like switching from q to creation and annihilation operator so the metric is represented by a singular functional differential operator like this xx so collided two points and so in order to define a classical state you would have to find psi such that this well there's ij I guess that this is something like this would be like a classical expectation value gij of x and of course there are all kinds of difficulties in making this well defined so this is one of the reasons like in the Wilson loop way of describing the gauge theory I mean we have this problem but in the case of gravity for sure what do they say that we know that there is a notion of actually no no they say eventually they will be able to define a semi-classical limit and but all I'm saying is this is why it's too difficult for them to find a semi-classical limit because the metric operator is represented by a singular functional differential operator so if you start with all this for my death are you able at least to quantize linearized gravitational waves you're asking no linearized gravitational waves condensation would require solving this problem first then what they are doing they don't really have metric how can they do fluctuations it's all recorded so let's continue you know I have no time to because this is just a pointillistic survey as I said so let me just summarize a little bit summary and critique of these you can ask all the the non-perturbative approaches this is LQG but something that developed out of spin forms I have no time to talk about and group field theory I put the main emphasis not on the things that we like in quantum fields here but the things that we like from general relativity which is background independence although very often what really is is only spatial background independence because space-time background independence would imply that you solve the wheel of the wheel equation Hamiltonian constraints so that's not there so this is a bit kinematical however this approach is so far they're not able to recover the semi-classical limit but they also do not incorporate essential insights and successes of standard quantum field theory which I think is a big shortcoming like anomalies you see this is one thing that's usually not emphasized very much there are tons of quantization ambiguities so matter couplings the restrictions that we know to hold in the standard model like anomaly cancellations for the quarks and leptons that's a very interesting and very subtle consistency condition is not seen in these approaches but as I said these issues will be hard to settle without a detailed understanding of how standard quantum field theory and the semi-classical limit would emerge from this framework so let me now move on COSTAS gave a very beautiful review of string theory so this is you know just you said poor man's rubber key so this is a poor man's version of COSTAS presentation so super world basic strategies now quite different coming from quantum field theory there one tackles this perturbative divergences directly by trying to modify gravity at short distances this is by introducing supersymmetric matter because fermions have a tendency to cancel bosonic divergences so this is realized in supersymmetric series it's also nice because in some sense it gives a raison d'etre for the existence of matter in the world and also there are such things as maximally I mean if you really want to push the notion of symmetry to the end maximally symmetric point field theories n equals 8 supergravity to which I will come back 11 dimensional supergravity and then also one can go to supersymmetric extended objects so I think that the point like singularities due to the point like interactions are dissolved by blowing up point like objects to extended objects so this has led to the super string but I would also like to remind that there is a theory called super membrane theory which you can think of as a non perturbative version of a string theory and from it follows the matrix model. Now this theory is much harder to deal with than string theory. String theory is basically free field theory on the world sheet but here on the world volume this remains an interacting theory and there has not been so much progress in understanding it. So as Kostas explained beautifully so I'll just go down the list of course this is a huge subject it's very much modeled on concepts from particle physics so there's no problem with the semi-classical limit. However there's a problem with say the formulation beyond perturbation theory. But we now understand that it's not simply a theory of one dimensional extended objects they're D-brains, they're M-brains, they can lose their client excitations or what have you. This is one of the big successes explaining giving a microscopic combination of black hole entropy including higher order corrections to this Beckenstein formula there's all these ideas about holography some people think that holography is the key to quantum gravity I don't quite share that. What would be a physical consequence that might be able to measure of this formula for the entropy? Oh well you know this formula has been proven for you know very ideal extremal black holes and first of all you would have to find an extremal black hole in the universe and then But suppose there were an entropy formula for a visible black hole what could I do with it? Not much really because visible black holes are well the other thing is that the derivation is done in a completely different domain. It's using D-brains the way I send derivation which is semi-classical and applied to even... No but I think there is a slight problem here because the derivation is done in a domain which is far far away from classical and then you extrapolate over I don't know 20 orders of magnitude and for this you have to invoke supersymmetry and BPS you know BPS is invoked a lot these days for doing all kinds of arguments and strings here but you should keep in mind BPS is light years away from the real world it's not you know so you know of course it's a real range of challenge to sort of derive this formula or understand it for real life black hole well anyway there's a lot of new ideas that are currently being tested low energy supersymmetry, large extra dimensions and so on and of course I will not say anything about the implications for mathematics because this is something everybody knows and appreciates certainly here at IHES so what are the open questions? Well I mean it's strings here it's been a massive intellectual collective effort for almost 40 years or so but we're still struggling to reproduce the standard model as is not some you know fancy extensions with tons of extra super multiplets and so on but just assume you know no super particles are founded at LHC then what are we going to do here this is a real challenge similarly it's a struggle to incorporate there's a way to get a positive cosmological constant supergravity superstrength here you love negative cosmological constants but you look outside the window and it turns out to be positive I will say a little more about maximally extended supergravity also in connection with finiteness because the question is what is the place of this theory here in this whole scheme so that's been impressive progress but I would say a convincing scenario for resolution of space-time singularities certainly not cosmological or space-like and the question that was posed initially namely what happens to space-time continuum at blunt length is as open as it's ever been so and the real question is really what is string theory in the sense that nobody thinks that at the end of the day it will be the way it's done nowadays you simply assume a classical manifold and then you have these blunt size extended objects and then you quantize them on this classical background we all think that this is not going to be the final answer so one thing I would like to emphasize and this is a little bit a problem with all approaches is the lack of there are too many ambiguities I'm a little bit polemical here because there is this well-known number of consistent vacuums which is far too much for what we need but this is as I said this is a bit hushed up but if you look at trace the ambiguities quantization ambiguities and so on you also find that there is similar non-uniqueness in loop quantum gravity in discrete quantum gravity and also asymptotic safety because I keep hearing these talks I've heard them now for quite a while and I always sit there and think this is working too well because no matter what they try they find a fixed point for example if you do gravity in 67 dimensions there is also a fixed point and I find it hard to believe that this should also work in 67 dimensions anyway so you know there's all this ambiguity which is precisely the opposite what I said you should try to force your theory to make some kind of statement that can be checked or verified or falsified so the question is really does nature pick the right whatever it is answer at random from a huge variety of possibilities or are there criteria to narrow down the number of possibilities so I think that in order to discriminate between a knowing number of diverging ideas better start looking for inconsistency or else and this is something Gary Gibbons said to me about certain approaches in quantum gravity said this is going to remain fantasy if we can't come up with a solution to this problem so the last part I would like to go a little more in the direction that I like namely the question of what is the role of n equals 8 supergravity in this now n equals 8 supergravity was in fashion 35 years ago and then it became out of fashion with the advent of string theory and obvious problems but it has recently turned out to be it's much more finite than was originally expected and in fact up to four loops at least it behave exactly like young mills maximal young mills theory if you continue like this then it would be just as finite as n equals 4 young mills theory not proven to be divergent in high ropes not proven young mills is divergent in 6th dimension well young mills is yes it's divergent above 4 dimensions there's a formula that tells you at which dimensions the divergences start and here the claim is that for n equals 8 supergravity up to 4 loops when you survey different dimensions it's exactly the same formula as in as in young mills theory so it could be it's not ruled out it's not excluded that it might be finite to all orders and you know this is one of the things I now start worrying about as I go older will I ever know whether it's finite or not I'm not sure anyway there's no effort towards 5 loops but as we heard yesterday this effort seems to be thoroughly stuck so it's not clear what's going to happen interestingly the difficulties here at 5 loops are somewhat maybe related to a similar phenomenon appearing strings here because there as well as 5 loops there appears a peculiar difficulty to do with the fact that supermodular space is no longer split which means you're not allowed to first integrate over the fermionic moduli and then over the bosonic this is at least the one sentence I understood from Witten's presentation two years ago I didn't understand much more but so this is not excluded here but even if it is finite of course you can raise the question what about nonperturbative quantum gravity what does it tell us about the resolution of singularities in relation to real physics and most people would answer this question in the negative but I would like to advertise some recent work which goes back to some observation of 30 years ago which could become relevant if they really continue not to find new spin 1.5 degrees of freedom at LHC and this is the following this is really very strange coincidence which is the following the same number of quarks and leptons three generations they are 48 and this number is the same that appears in supergravity when you take the 56 fermions spin 1.5 remove 8 golstinos to break all the supersymmetry it's the same number and furthermore the following strange coincidence if you break n equals 8 supergravity to SU3 cross U1 these are the representations you get for the SU3 and this was an idea of Gell-Mann's really strange you know I would never had such an idea but he said he described it as a last ditch effort to salvage n equals 8 supergravity so he took the standard color assignments for the quarks and leptons and all assigned the three these quarks and leptons to representations of a family SU3 so this is a new symmetry that acts horizontally among the generations in this peculiar way here and when you work it out it's exactly the representations that appear upon this making this decomposition of the n equals 8 fermions it's also a little bit curious I think it just indicates one should start looking in unusual directions because one thing you see here that this SU3 family would not commute with the SU2 weak because it puts the upper and lower components of the would be electric doublets in 3 bar and 3 of this family symmetry anyway there's this strange agreement and furthermore if you calculate the electric charges of these particles so they are given here and then you calculate what you get from supergravity it turns out that they are sort of systematically shifted away by 1 over 6 so it's 1 over 6 for anti for the 3 of family and it's minus 1 over 6 for the 3 bar so this has been known for a long time it's sort of you think is this a mirage or is it something real and in fact we found already long ago with Nick Warner that this scheme has actually realized that the SU3 goes to one stationary point of n equals 8 supergravity gauged n equals 8 supergravity now very recently we've figured out a way to fix up these charges by deforming the u1 and we've found it in a certain way that goes outside of n equals 8 supergravity so there's no extra motivation to look at even more symmetries and I think this is really the core of the subject namely the duality symmetries that were originally discovered by Kramer and Julia because we now believe that these symmetries are more important ultimately than spacetime symmetries and what was well realized already long ago is that these exceptional symmetries become bigger as you go down it as you do a dimensional reduction dimensional reduction means you drop the dependence on internal coordinates it's just like a Kalusa Klein theory but what also happens here is that by doing this you transmute spacetime former spacetime symmetries into duality symmetries and if you carry this process to the extreme all spacetime symmetry has disappeared and everything is duality symmetry and by doing this you follow this chain I'm sure you've seen this before so this is E7 that appears in 4 dimensional supergravity E8 in 3 dimensional maximal supergravity and below it becomes infinite dimensional so this is now what we've been working on for the last 12 years or so so this is the symmetry that was already conjectured to be in the dimensional reduction to one dimension now long ago I thought well who is interested in one dimensional reduction because the theory becomes essentially trivial you drop the dependence on all spatial coordinates but then came this other mysterious hint namely connection or link with work that was initiated or done by Belinsky, Halatnikov and Liftschitz long ago, this is BKL their characterization of the singularity so here's a singular big bang spacetime and what they hypothesized is that as you approach the singularity spatial points decouple this is just the horizon problem of inflationary cosmology and what you get is a continuous superposition of one dimensional systems in first approximation so this effectively means there's something like an effective reduction to one dimensions, one time dimensions as you go closer to the singularity or in other words the true symmetry of the theory becomes only apparent as you go towards the singularity so this is somewhat analogous to the high energy limit of gauge theories but with higher and higher energies you see more and more of the symmetries maybe I should not this is another just to explain the philosophy I said before that cosmological evolution can be viewed as a one dimensional motion and an infinite dimensional moduli space of three geometries, this is what Weeler called super space which is a rather complicated moduli space such that the cosmological evolution is really like the motion of a point particle in this infinite dimensional manifold this is what leads to the Weeler-DeWitt equation the wave function of the universe in Weeler-DeWitt this is the space it lives on but of course we believe that unification of spacetime matter and gravitation should tell us that there's something more simply something you just glue to this space but there should be more unified structure so the question is whether you can understand the full moduli space and simplify in some sense by embedding it into a group theoretical coset and this is the proposal that we have been following with Thibault, namely to somehow map this evolution of 11 dimensional supergravity onto a coset space which is of this type this is the hyperbolic Katsmoody group or Algebra E10 which is the maximally extended hyperbolic Katsmoody Algebra so this is work I've been doing over the last 12 years with Thibault, Marcheno and Axel Klein-Schmidt in particular so E10 I have no time to tell you what E10 is but nobody knows what it is and nobody knows what one of the nice challenges so here's the sort of way the mathematicians would define it so it's just Dinkin diagram, generators and relations a la chevalet serre but you know this of course works for finer dimensional E-algebras in a standard way, no big deal but it's just, you know you think just adding a note to the Dinkin diagram doesn't mean very much but the difference is an explosion of symmetry and in particular for these algebras which the Carton matrix is indefinite you have what's called exponential growth and there's just no way known so far how to have a sort of global description of this algebra, not to speak of the group that's even more complicated so we've approached the subject in a somewhat pedestrian manner I think this is something no mathematician would do this is just too simple minded in a way this is the level decomposition and the nice thing is when you do this with E10 you just pop up the bosonic fields of 11 dimensional supergravity, graviton 3 form and also their duals but the explosion you see this is first three levels if you go to 28 you already have more than 4 billion young tableaus of SLT and the monstrosity, the monstrous complexity of this early algebra because when you try to calculate like structure constants you get stuck at level 4 or 5 well my student Fischbauer, he's genius with a computer tried to go to level 6 but he also failed and what's worse is no matter how far you go you're sure that if you continue it gets even more complicated anyway as physicists we're not too worried about this we just go ahead and so we've really with this idea of mapping the evolution of the cosmological evolution onto such a coset space so this is exactly what we did just restricting to the low level things that we can identify with 11 dimensional supergravity and on this coset space so this is something like the maximum compact subgroup and then we found agreement on the certain conditions but it's certainly non-trivial between the supergravity equations of motion and the sigma model equations and the emergence of spacetime, he is conjectured to emerge as follows in the sense that information that was previously in the space dependence of the fields or the field theory now in some sense gets spread all over the E10 Lie algebra there's no more spacetime to start with and in it are all the degrees of freedom that you would like to see in the field theory but of course there's a lot more it's not just this because the spectrum here this level decomposition when you expand it it has about the same kind of exponential growth as the string spectrum so this is the thing I wanted to mention just because this I find this really surprising I told you that we had with Christoph Meisner we found a way to embed this or deform the u-wang to take care of the to get the correct electric charge assignments for the quarks and leptons and I said in order for this you have to go outside supergravity but we put a paper on the net just yesterday where we show that this deformation is contained in this compact maximal subgroup which you can think of as an infinite dimensional extension of the r symmetries that appear in supergravity SU8, SU16 and so on and first of all I find this rather amazing that in order it would really be striking my mind if in order to link n equals 8 supergravity in the world, quarks and leptons you would have to go to such a fancy infinite dimensional extension of the standard r symmetries and standard duality symmetries and the other thing I would like to emphasize here with this you know you somehow completely change the picture on how you would get standard model fermions out of a Planck scale theory and in particular this symmetry is also chiral, I think there's more than enough room in it to also accommodate the electroweak chiral-gauge interaction so this is something I'm quite excited about but you should keep in mind the moment they discover a new spin one half fermion at LHC were dead so at least this proposal satisfies the requirement it's clearly falsifiable ok so I'm almost finished so here's once again the philosophy that we sort of been following with Thibault, Marc and Axel so here's again this space-time diagram a cosmological space-time and of course BKL of course was just classical all the way down to the singularity but of course everyone knows that or we expect something happens at the Planck time so you can really cut off your classical part of your manifold and replace it by something else which was, Thibault always reminds me that this is the famous objection of Zeldowicz to BKL, he said it's all very nice but after a few oscillations you're already in the Planck regime and then you know can forget about your classical theory so the idea here is really to sort of take this as a guiding principle by looking for the theory that comes here below the Planck scale that replaces space-time by something else and here the hypothesis is that it's the E10 or K10 sigma model possibly well certainly quantized which is something we haven't really done and also of course taking into account the fermions which I didn't really say much about let me conclude my initial comment, the strongest argument for quantizing gravity is the incompleteness of the standard model and general relativity and the main question now is how do we resolve these short distance singularities and how can it and of course should be done in such a way that this resolution can be reconciled with the classical Einstein equations and continuum quantum field theory and so on and this may happen either by dissolving point like interactions like string theory, membranes and so on it could happen via the cancellation of ultraviolet infinities and it goes that supergravity is a candidate it could just be then this theory, you go down, there is no divergence they all cancel or is there some fundamental discreteness as something that LQG speculates about all these modern discreet models like spin form models and so on they are all models of discreet quantum gravity and they all have difficulties in recovering the continuum spacetime and that's the major challenge there or is it just the most conservative thing that quantum field theory basically remains valid but in a version with asymptotic safety so I would say just where do I I place my bets, I think symmetry based approach offers promising perspectives with n equals 8 supergravity e10 that's absolutely fascinating to me, I've been fascinated by this of doing all my professional life but there's still a long way to go so that's where we are, thank you thank you German, can you cover to the statement that gravity is equivalent to two copies of the unknown or it's supergravity which is equivalent? Well there are various versions that's the kind of obvious version of it in strings theory because you build closed string states out of left and right moving open string states and you put two spin one to make spin two so that has been obvious for 50 years now in field theory what these people in n equals 8, 3 bound and company are doing is this famous color kinematics duality where they actually for the Feynman integrands actually replace the Young Mills integrand by gravity integrand and the procedure is that for the kinematic part you try to find combinations that also satisfy like identities and then you simply replace the color factor by the kinematic factor that depends on momenta and polarizations but this is actually where they are now having difficulties at five loops so it's not clear whether this really works to all orders so what we had yesterday sounded more like eventually you have to give up on that idea or if it works you have to modify it in a somewhat substantial manner because that's in this picture of a billiard movement near a singularity so how should I think of it for a Schwarzschild singularity what happens? well that's of course the time reversed picture but the idea is that you know Schwarzschild singularity is also space like so as you fall into it it's a precise the mixer you will be stretched and squashed and I always like to emphasize I mean it's squashed to a point ultimately but this will happen in an extremely interesting way because before you get there classically at least you would be stretched and squashed in many different directions and of course then ultimately you know once you get to the Blank regime the classical theory breaks down and then you have to you know invoke whatever the quantum proper quantum but will I come out ever? not with a Schwarzschild you just of course what happens in the quantum theory I don't know but this is just not known there are some recent indications the work with Schwarzschild is that the first quadratic inference is a balanced behavior but this is so there is something quantum which is still to be explored and also physically do you have a question? no are there more questions? Schwarzschild is raising his hand yes even if n equal 8 supergravity is proven to be finite does it mean it's non ambiguous? what do you mean non ambiguous? well there are certain quantum field theories which are finite but nevertheless ambiguous in what sense? I mean there are more than one theory that has this property one example is John Simon's theory in three dimensions where you have to work hard to find divergences however the answer is not in the theory it's a long discussion debate an endless debate about should you find k or k plus g when you make your perturbation theory so is there something analogous in that? well there are certainly ambiguities here in the sense that the n equals 8 can appear in a gauged version there are different versions of the gauged supergravity although the SO8 is almost unique except for some recently found modification but what I would like to emphasize here and this was my last remark is that if there is any link to the real world it's not just n equals 8 supergravity even with this quark lepton thing you know you have to go beyond it in a certain controlled way that's I hope controlled by this fancy infinite dimensionality symmetries any more? Let us thank German