 Thank you and I want to thank the organizers for inviting me to this nice occasion to celebrate together with Boris and I'll try not to overshoot the time so let me start I will essentially talk about I changed a bit the title so this density resolved is almost disappearing but instead I added a little bit of quantum so I will talk about the spreading of nonlinear wave packets and finally time permitting come to two little puzzles with just two quantum interacting particles which nevertheless seem to be open issues so we had already some discussion about Anderson localization so let me start straight into the simple equation this is a discrete Schrodinger equation on a one-dimensional chain and psi is a complex color L is the latter side number and here's the disorder epsilon L which are random uncorrelated numbers drawn with equal probability from a an interval of widths W and here's the hopping with strengths one so the widths of the spectrum of the eigen values eigen frequencies eigen energies of this problem is capital delta four plus W now the surprising thing which was found first by Anderson long time ago is that all eigen states of this the corresponding eigenvalue problem which you see here are exponentially localized with a localization length which is a unique function of the eigenvalue so all these states roughly look like in this cartoon so they occupy a certain part in space which I call localization volume and sometimes most of the time I'll call it capital L sometimes also capital V and of course it's related to the localization length but by numbers it's not exactly the same thing so it's usually two or three times larger than the localization length I can also basically be very brief on that we already heard about successful experimental efforts to observe Anderson localization in a variety of physical settings in particular I want to focus your attention on the experiments done with light and with ultra cold atomic gases was Einstein condensates because in both cases you can tune the nonlinearity respectively the two body interaction strengths and get this system into a nonlinear wave equation or respectively a many body problem in the quantum case so therefore the motivation which actually goes back straight to discussions again initiated by Boris back in the time in Max Planck Institute in Dresden the idea then was to look at this nonlinear Schrodinger equation so this is the previous equation we have seen but now we add a nonlinear term here which is proportional to psi to the cube and there's a parameter beta here in front which controls the strengths of that term and then the question is what will happen to a spreading wave packet and you can generalize that to basically any wave equation problem where the linear wave equation has eigenstates which are all localized with a finite of a bound on the localization length and then you add some kind of short range nonlinearity like this one and you follow the spreading of an initially localized wave packet pretty much asking the same question as Anderson asked in his seminal work and then you ask whether it stays localized or whether it can be localized and there are two possible answers the first one is it will of course delocalize because when you excite a wave packet in an empty system you will always excite a finite number of eigenstates normal modes of the linear wave equation and due to the nonlinearity they will now interact with each other so you from a dynamical point of view you have a system with several degrees of freedom excited and the chance that this system is integrable is essentially seems to be zero and therefore you can expect that there will be chaotic dynamics inside this wave packet and chaotic dynamics will destroy face coherence but when you destroy face coherence you will destroy wave localization in the first place so therefore there should be some kind of incoherent excitation of the exterior of the wave packet which will lead to a spreading of the wave packet the other answer is no because there are conserved quantities at least there is typically an energy which is conserved but in this case of Schrodinger equation there's a particle number of the total norm which is also conserved and then if you look at the spreading if you assume that your wave packet does spread then in the course of spreading the densities inside the wave packet will become small and that means that this term here should become very small in the course of time during spreading and therefore it would be tempting to neglect this term at some sufficiently large time and if you do that you are back to Anderson localization and so your wave packet should actually get stuck and not spread okay so let's see what will indeed happen but before coming to that here let's look at these equations the same equations of motion in the normal mode space that is you take the equations in real space and you transform do a canonical transformation into normal mode space so the fine use are now the amplitudes the complex amplitudes of the new eigen mode of the linear wave equation and what you see is indeed when beta is the nonlinear parameter is zero then you deal with a set of non-interacting harmonic oscillators and of course you will enjoy still Anderson localization in such a setting however when beta is nonzero you get an interaction between these normal modes in forms of these triplets and there are some vertices here some overlap integrals I which are essentially given by sums over products of in general four different eigen vectors of the eigen states of the linear wave equation new counts the eigen modes you can count as you want you can count them in space as I'm doing here right now by their position roughly speaking but you can also of course sort the eigen modes with respect to their energy or some other parameter and so then what we will do is we will follow the dynamics of the wave packet and we will look at for instance the normalized distribution of the norm density of that wave packet and we will measure the second moment of this distribution which is sensitive to the location or to the distance of the two tails from the center of the wave packet so we will have a good idea about whether this wave packet stays localized or not a bit more on numbers so we already mentioned the total widths of the spectrum we talk about the lattice so the spectrum is bounded from both sides it has a finite width which is in our case here W plus four and in an example of disorder strength four which by the way means that the localization length is of order six this number ends up to be eight then the size or the localization volume of each eigen state which I call here L is of course also depending on the disorder it's the bigger the weaker the disorder and in our example it's about 15 to 20 let's say 18 so a typical state can occupy 18 sites in the real lattice now because of the nonlinearity this any given state will be coupled to a number of other eigen states of the linear wave equation and this number will be roughly equal to the size of the eigen state itself because there is a direct correspondence between the number of sites in our lattice and the number of eigen states of the eigen value problem so then if we take let's say these 18 other eigen states which are now through nonlinearity interacting with the given one then we can look at their eigen values sort them and look at the level spacing of this set of eigen states and arrive at this average frequency or eigen value eigen energy spacing inside the localization volume called little D and that's a new frequency scale or energy scale which is roughly given by capital delta over capital L and it is the smaller the weaker the disorder in our case it's of the order of 0.4 0.5 something like that and then there is another number which comes from the nonlinear terms because and that's the nonlinear frequency shift so nonlinearity in the first place as you know renormalizes frequencies or energies and like in any unharmonic oscillator and this renormalization very brutally speaking is proportional to beta times some norm or norm density and then you can compare now these numbers and end up with the possibility of three different regimes one when this frequency shift nonlinear frequency shift is much smaller than the level spacing little D then this regime we will call weak chaos for reasons to come and then when this frequency shift is larger than the level spacing little D then we can be in a regime of fully developed strong chaos when all normal eigen states are strongly renormalized and resonantly interact with each other and if this frequency shift is even bigger such that it roughly exceeds the widths of the full spectrum of the linear wave equation you enter the world of self trapping or discrete breathers which means that you renormalize parts of your system such strongly that their frequencies are completely tuned out of resonance with their neighborhood and therefore these states can stay for very long times again being localized but for a different reason or even be exact localized eigen states solutions of the non-linear wave equations okay so now let's see what happens when you switch on your computer this was first done by Mario Molina from Chile in 98 but it was a very preliminary study then there was a work by Shepel Jansky and Pikovsky by us and by many more and here's an example what happens when you start with your wave pack wave packet being localized on just a single site and we let it go and we integrate the equations of motion and we look at the second moment as a function of time on this lock lock in this lock lock plot and when beta the non-linear parameter is equal to zero we get this orange line a horizontal line the second moment doesn't change as it has to be this is Anderson localization the wave packet doesn't spread at all of course the single site spreads into a lingual excitation spreads into the single particle localization volume but in the frame of the normal mode representation none of the normal modes changes its norm and therefore the distribution of the norm in normal mode space is strictly constant now when you switch on some non-linearity let's say you look at this blue curve you find now that the second moment still stays around the old value but starts to fluctuate this is these fluctuations are entirely due to the non-linear interaction between the modes and then at some point you start to see a growth of the second moment and consequently a departure from Anderson localization and when you increase the non-linearity even further you basically push this this growth point to shorter times that you see from scratch a growth of the second moment and the loss of Anderson localization if you look at the way this happens these look like these are individual realization results here you see it more or less well can be approximated by a straight line this is a log log plot which means we have a power law and if you try to estimate the power and take for instance this dashed line the slope is 1 over 3 which means we are in a very deep sub diffusive regime it's a very slow process the slope equal 1 would correspond to normal diffusion slope 2 is ballistics but here instead we have 1 over 3 or something like that so in this cartoon you see how the wave packet is spread in one of the runs and if you now look at the end time of these simulations and you look at the distribution of the densities of the norm densities you find here this distribution as a function of the space parameter on a linear scale and down here exactly the same thing on a log scale so what you find is a broad distribution with exponential tails which are the remnants of Anderson localization and the further you integrate the system in time the further you kind of push these exponential tails away from the center of the wave packet and these little bars here indicate the size to which the wave packet is allowed to grow or to be within the linear equation so we see a clear destruction if you wish of Anderson localization and in practice we can reach an increase of the size of the wave packet which is of up to two orders of magnitude larger than the size in the linear wave equation we cannot go further because of time restrictions this takes quite some time to integrate this on a machine and we can go further we can stretch this thing to 10 to the 9 and even 10 to the 10 and we basically see the same features but we can't go further okay so as I already indicated one of the ideas is that yes this wave packet must destroy and Anderson localization start to spread because of chaos and because of defacing if so and if the defacing of the normal modes is strong enough then it shouldn't harm if we just do the same in parallel by hands which means we repeat these simulations but now instead of just running the computer and look what happens we will stop from time to time the simulation transform the field in real space into which is where we integrate the equations into normal mode space do a phase reshuffling in the normal mode space so randomized the phases in normal mode space transform back into real space and continue integration and then repeat the same process again after some suitable time lag of let's say a hundred or something like that and what we get are these lines they again follow more or less a straight line the slope is different and can be roughly estimated to be one half again sub diffusion but faster so which means that in these original data what we observe is a sub diffusive process which must be probably due to chaos but which is weak that is not all modes are strongly defacing in suitably short times because otherwise there shouldn't be any difference between the the computer results or the integration of the equations of motion or integrating plus defacing by hand so if you try to dig a bit and try to understand what's going on you can start playing games like this you write down the here again the equations of motion in normal mode space here the overlap integrals and then you say okay maybe we don't need chaos let's replace the whole problem by an integrable Hamiltonian you see which depends only on the actions but which carries the whole network of of connectivity the connectivity network between the normal modes of the original equation because here are the corresponding overlap integrals but it takes you one line to see that of course this Hamiltonian will not produce any new spreading it will follow Anderson localization as it happens for the linear wave equation the next thing you can do is you take these equations in normal mode space you do a local gauge on each of the normal modes such that you remove this term here this linear term here in front and basically arrive at oscillating terms in this nonlinear sum which now contain frequencies which are combinations of typically four different eigenvalues of the linear wave equation and then you could say okay let's average these oscillations out there's still something remaining of all these equations here of all these terms but if you solve that problem you find that it's first of all integrable and secondly that it again doesn't lead to any spreading of a wave packet so indeed there is a reason to assume that chaos is important in the in this process and also terms which we threw out by averaging over time are important so if you place them back and you look at one of those terms and it's clear that we have to focus on these quadruplets of on these frequencies which are these quadruplets here which are small numbers then you immediately see that you have to try to estimate whether some naive perturbation approach will break down or not and come up with an estimate for a probability of a resonance that is you now want to see whether in your wave packet some given normal mode can be treated using perturbation theory through its interaction with the other normal modes which surround or not and yes or no this depends essentially on a factor which involves this ratio of the quadruplet which you see up here and the overlap integral and then to cut a short a long story short if you work this thing through you end up with a probability that given let's say a wave packet which with a more or less equal norm density inside the wave packet then the probability that a randomly chosen normal mode in that wave packet will satisfy the resonance condition that is will not be you cannot treat it in perturbation theory is given by this expression 1 minus e to the power minus beta times the density and little n is the density of the normal mode or over the level spacing and then you see of course that when n becomes very small assume that your wave packet spreads a lot so the density becomes small you expand this and you see that the probability becomes small not zero but small linear in n and becomes the smaller the smaller n while in the other limit when n is big enough this probability is essentially expected to be 1 so then comes the next step which is to develop some kind of estimate for this for this spreading process which is which was called by others effective noise approach so basically you assume that you have a wave packet which is spreading which is chaotic inside and you want to know at which time a new mode which is sitting in the exterior of this wave packet with counting number mu will be excited to the level of the wave packet and become one with it for that you have to write down some stochastic differential equation this is the equation of motion for that exterior mode and what is important are the forces which come from the wave packet modes which we assume in time to be random so we have a Gaussian white noise function here f of t and then what is important is to estimate the prefactors of course proportional to beta is proportional to the cube of the amplitudes of the wave packet modes and then comes the purely phenomenological step which we had to do in order to save our lives namely to assume that it also is depending or proportional to the probability to have at least one resonance inside the wave packet because if there is no resonance then of course the wave packet should not spread so if you do that then you come up with an estimate for the diffusion rate and this diffusion rate as you see is a function now of the density so it's not a constant it's a it's a thing which depends on the density in this the smaller the smaller the density is and then you can happily write down the final result because you know that the second moment is essentially one over the density squared because the bigger the wave packet the smaller the density so distance immediately translates into densities or into inverse densities and then you can close the whole equations and come up with the following results that under these assumptions in the long-term limit in the asymptotic limit the second moment should grow like t to the one over three but there can be an intermediate regime when beta n over the level spacing is not less than one but larger than one then you indeed can obtain a sub diffusive regime where the second moment will grow as t to the one half now instead of bombarding you with lots of numerical data let me just try to summarize everything here in these few words so we do of course many computations we do averaging over thousands of disorder realizations etc etc and what do we find we can basically say that we can confirm a regime an asymptotic regime of weak chaos for spreading wave packets where the exponent can be estimated to be 0.33 plus minus 0.02 so which means one over three is well inside this result for the non-linear Schrodinger equation with disorder we did not yet fully observed a nice fully developed intermediate regime of strong chaos we see an onset of that but not a fully developed regime over several decades in time however what we can do is we can generalize this approach which i discussed here to higher dimensions of the lattice to different exponents which describe the non-linear term instead of cubic terms, quintic terms or whatever you want and then we can check the results and apply them not only to the non-linear Schrodinger equation but to a variety of other models as well for instance we observe the same weak chaos regime in the obriende potential so instead of instead of a disorder potential we choose a quasi-period potential which was discussed to the already by Zhora Schleppnikov and we find again exactly the same asymptotic spreading with exponent one over three we take other models of coupled oscillators called Klein-Gord models which essentially are different I think by not conserving the norm or anything similar to that but simply conserving the total energy we find the same weak chaos regime in these Klein-Gord models as well we go to dynamical localization in the quantum kick rotor we consider a non-linear version of this quantum kick rotor and again find spreading with the same exponent one over three in this case and we also checked that in the two-dimensional disordered non-linear Schrodinger equation we also again find the proper weak chaos spreading with a different exponent because you have to look at this outcome of these generalizations and we do observe strong chaos as an intermediate process in these Klein-Gordon models and also in the non-linear quantum kick rotor system and also maybe just briefly to mention that since we basically talk about a process which is non-linear diffusion then you can go back in time and look at the results which were found by Zildović, Kampanets and Barrenblatt on non-linear diffusion equations on self-similarity and scaling properties of their solutions and we checked that as far as we could and found very good agreement as well with the data of our spreading wave packets. Now the non-linear Schrodinger equation is however a bit different from some of the other models which I mentioned in the following that it does conserve two quantities the energy which you see here again and the total norm or the number of particles in the classical version. Then you can introduce densities right like I discussed already before but we have two densities we have the norm density a little x and the energy density a little y which we scale from the original densities by just multiplying them with this non-linear parameter beta and then we can remember that there was a paper already in 2000 where people studied this model for the ordered case so w equals zero so basically this term is not present they looked at that model and they asked what are the statistical properties of that model how did they address this issue they said okay let's take a partition function like that one it gives distribution with some chemical potential and the positive temperature and let's see which parts of this density space can be addressed by such a distribution so it's not a question about dynamics it's a pure kind of a statistical approach the answer is that there is a forbidden region which you see here this inaccessible region and then there is a zero temperature line which is this bottom line here which you can also find corrections for the case with disorder but maybe I shouldn't stress that too much here but we can talk about that later if you want so basically there's a zero temperature line and then there are states above that line which you can address using such a partition function and then there's a infinite temperature line which is this line which you see here and and that's the end that's the end which you can reach by a Gibbs distribution however that's not the end of the model because you can still address all those all all these all this white area up here which is called non-gibs so you can address this area in terms of densities but you cannot describe this using a Gibbs distribution with positive temperatures you can formally describe it with negative temperatures but you have to think what it means so what the upshot is actually what the outcome is of all this is if you go in there and you look at the dynamics of your system you will find that your system tends to segregate into two parts so basically your wave your field decomposes into two components one which is strongly self-trapped and sits on certain sides and another one which is floating around and presumably has an effective temperature t equal to infinity or something close to that but the other truth is also that if you actually go into this thermalized region and go here to very large densities you basically see the same thing the difference probably is in the lifetimes of this of this self-trapped component but other than that you find a lot of similarity also in the thermalized region because when you go to large densities you basically enter self-trapping as I discussed already before and therefore this region of course is not of interest when we look at a destruction of Anderson localization due to non-linearity but what is of interest is this area around zero now still if you are close to zero there is this Gibbs non-Gypps transition line which might be you know important or can be studied so what I can tell you right now is the following first of all if you look at the wave packet which spreads and you assume that at any time you can characterize this object by two densities by an energy density and a norm density in this assume I mean that it makes sense you can always do that but that it really makes sense then you get some point here in this phase space in this parameter space and as you spread the densities decrease and since they are linearly dependent on each other essentially you now follow a straight line which goes straight down to zero so that's the fate of your spreading wave packet if it will do so at infinitum and then in principle it can happen that you can start somewhere here and in the thermalized region and then you see if you have positive energy densities ultimately you'll have to hit this Gibbs non-Gypps line and enter the non-Gyppsian regime and the upshot so far is that we don't see any peculiar behavior in the spreading of the wave packet when we hit that line whatever the reasons and whatever the consequences of that fact are if you go too high here of course we see self-trapping we see non-Gypps we see everything but if we are close to if we are at small densities positive densities we don't see any significant effect coming from this crossing the other thing is that there's this little d the average spacing which is a scale which comes from the disorder problem and that little d can be directly now compared to x which if you remember is beta times the norm density so somewhere here is a marker which is shifted more and more down to zero the weaker the disorder is and and that's the interesting region where we observe our weak chaos spreading dynamics and we can actually see spreading wave packets down to density values x which are as low as 0.01 times the average spacing so we are really we can really go deep into this weak chaos regime and still observe happily spreading wave packets however at the same time exactly in this interesting region of weak chaos there will be ultimately another interesting phenomenon appearing which is going back to Kolmogorov Arnold and Moser which says that if you want to launch a wave packet at these small values of densities you will have a final probability that your wave packet will not spread and this is basically a one-to-one correspondence to the fact that you have a final probability that there will be no resonance inside the wave packet so then these wave packets will not spread but there is always a complementary probability that there will be something some resonances and then you apparently can spread so inside this this region of weak chaos we expect this KAM regime which was also discussed by Denise Bosko in some earlier papers and in that sense you can actually view this spreading wave packets as objects which you launch outside the KAM regime and then with probability one you have a chaotic trajectory it starts to spread but at large enough times you actually enter a KAM regime but you are still staying on a chaotic trajectory which might be quite interesting and this is maybe also an interesting regime where as now is very fashionable again thanks to Boris and Igor and the Denise Bosko's work at least initiated by that to look for non-negotic regimes with finite conductivities in the classical case but this might be work in progress so how am I doing in time Mr. Ciena? Mr. Ciena, Volodya, how am I doing in time? Five minutes okay so yeah so I will come then just very quickly to some recent results on a very very simple problem which is again a disordered one-dimensional shredding equation but now we just consider two interacting particles so the single particle localizes with localization lengths let's say xi one and the question is how big will be the localization lengths of two interacting particles and you of course perfectly know this is a story which goes more than 20 years back in time and was seemingly resolved and I want to basically tell you that it was not resolved at all and so the localization lengths xi two of course will everyone agrees at least on that will stay finite for finite values of disorder but how it will scale with xi one for big disorder is an open issue and one can however also ask how will the wave packet of two particles spread into this newly accessible volume which exists when you switch on interaction so here's the Hamiltonian basically both the Habat Hamiltonian but just with two particles and we solve this either by by diagonalization or by integrating the time-dependent shredding equation in time it's essentially equivalent to a one quantum particle on a square lattice you know if you look at the basis you can choose the basis where you put two particles on sides j and k and then you arrive at the shredding equation which is lives on a two-dimensional lattice with some on-site energies epsilon j k which are given by epsilon by the single particle on-site energies and or the on-site energies of the potential and of course the interaction and on the other side you can also transform it into normal mode space or if you wish Fox space so into the space of the eigenstates of the non-interacting two particle problem and if you go there what you find is a kind of an effective again a one-dimensional problem but a very multi-channel problem because many of these fox states are now coupled and they are again coupled exactly by the same overlap integrals which we already discussed and if you try to play here with perturbation theory again you find if you take one of these terms here you find that perturbation theory will work or will break correspondingly if something happens to this pre-factor which not surprisingly is again given by a ratio of the overlap integral and the difference of two two particle energies which essentially is a quadruplet of single particle energies right and then what can we say so uh the connectivity difference to the in nonlinear classical case is now not l but l squared the equations however are linear the differential equations are linear instead of nonlinear the width of the spectrum changes that's not so important there's a different average spacing you can try to come up with different games of estimating the localization lengths of two particles but the the problem is that you always have to say something intelligent about the overlap integrals as for the interaction itself if the interaction constant u is much bigger than the width of the spectrum similar to the classical case you have a kind of a quantum self trapping if you want you renormalize the states with two particles on one side out of the rest of the spectrum they form a narrow band with a very small localization lengths the rest of the spectrum corresponds to a spinless fermion problem with two particles so nothing interesting happening there when you have very small u you have of course weak effects so the best thing is to choose u of order one or two in this setting so then you can try to calculate numerically the localization lengths of two interacting particles the largest possible which you see here in this result you maybe don't see this is the ratio of psi two to psi one as a function of u and what you find is that surprisingly even if you go to disorder strengths down from disorder strength four to disorder strengths two when you change the localization lengths from six to about 24 you still observe that your two particle localization lengths is of the order of two or three maybe times a single particle localization it's not a big effect nothing really interesting seems to happen there and but things do happen but all the numerics which were done so far in some of on all of these papers which are listed here we're doing we're playing in this ballpark of strengths of disorder and then they try to find some some scaling laws and whatever fitting some exponents it's all garbage basically because these studies were not done in the right regime moreover in in some of these papers there were also wrong assumptions about about the statistics of overlap integrals and like for instance in these two first papers by shaperianski and imri but the truth is that when you go to very weak disorder you will see as you now see here in these figures you will see that something happens to the statistics of these overlap integrals and actually it's not surprising at all what happens so look at these results which we derived for w equal 0.5 we went to very weak disorder we cannot do any calculations of the two particle problem there but we can still solve the single particle problem and the localization length here is already 400 so it's a very big localization and then what we do is we find one single particle eigenstate we identify all other single particle eigenstates which live in its volume and then we have this set of eigenstates we sort them with respect to energy and then we plot the two indices we plot an index which counts them along the x and y axis and then a point in this two-dimensional plane is one foxtate of the non-interacting problem and then for instance if we take this foxtate to be here in the middle of this of this two-dimensional finite square if you want then we can ask what are the overlap integrals with all the other foxtates which are defined by the same single particle eigenstates which we found and what you see is at very weak at higher values of disorder I don't show you these pictures you get basically here a mess you get the mess which was assumed also in most of these papers which are cited here however when you go to very weak disorder what you find is that this mess clears out and most of the other foxtates are not connected to our reference foxtate but only those which are arranged along these two lines here are strongly connected and if you think about where these lines come from you immediately see that this is restoring of translation invariance because you go to very weak disorder you have to see some kind of momentum conservation restoring but you see lines which maybe you didn't expect here and this comes from the fact that your eigenstates are localized so you don't talk about periodic boundary conditions but about simply fixed boundary conditions so just do the game of momentum conservation in fixed boundary conditions and you'll see that you get such lines along which overlap integrals are non-zero while in the rest of the region they are zero okay so this is to say what that all these previous studies are doing basically wrong assumptions with respect to the overlap integrals but the next surprising thing is which probably is the last one where I stopped is that if you now take this this reference state here a foxtate in the center and now you ask yourself how many other foxtates are really connected in the sense of breakdown of perturbation theory so in the sense of of this prefect they become in larger than one and then what you find so you can count them you find that they are the positions of these of these other foxtates are indeed located along the lines of strong overlap integrals not along the diagonal line obviously because there you change the energy but along the enter diagonal where the energy is roughly the same and then if you now count this connectivity numerically and you do an average what you find is that is these numbers so you see that even at strengths of disorder two the connectivity is roughly one which means nothing you that's why you get the localization length increase of factor of two or three but if you go down to even smaller values of disorder 0.75 0.5 and 0.35 you start to see that the connectivity increases quite a bit now whether that will lead to an increase of the localization length of quite a bit I don't know and maybe there are experts in the audience who can tell how that can be interpreted but at least there is room for some expectations that there might be a proper whatever dependence of Xi two scaling or not on Xi one which is located in that regime of very weak disorder where so far all the numerics failed to to enter to and with that probably I should stop and I skip the last thing and I want to thank all the colleagues friends co-authors which you see here and of course him most importantly I want to thank Boris who discussed all these issues over many years and whose help was of course an is and hope also will be always very illuminating and and important that not only that also you see that I recently I moved to different places so this is Auckland and then more recently we started to set up something in South Korea and again I'm very happy that we have Boris who is actively helping us to do my science and with that I come to the most important and actually the only important part of this talk which is I want to wish Boris that you will always find time as you did before also for the future to do innovative work in any conditions even on the most beautiful and most remote islands on this world but at the same time that you will also always find time to share with friends and have nice food in any place on this world so happy birthday Boris and thank you very much for your attention