 Okay, I'm going to explain briefly our recent paper with Marcoli, which is in the archive, only it's not in mathematical section, but in physical section, so mathematicians usually cannot find it, Big Bang, Blow Up, and Modular Curves, algebraic geometry and cosmology. Well, I think everybody heard about Big Bang. Our universe seemingly started in a unique event. There was nothing before, then suddenly something blew up, and of course this verb blew up, suggest analogy with algebraic geometry immediately. Something blew up, and then somehow this universe started. And it's according to observational data and standard theories, it will end as a very, very large, maybe infinitely large and infinitely cold space without anything interesting going on. So recently, well, not very recently, but the last decade or so, Professor Penrose, Roger Penrose, who has a lot of mathematical and physical fantasy, suggested that, as a poet said in my end, is my beginning, namely that actually before the Big Bang, there was a previous aeon, previous universe that became very, very big and cold. But then something happened, and there was this Big Bang, and everything started again. Mathematically speaking, in order to invent a model of something like that, one should start with explaining what happens exactly at the moment zero and at the moment infinity. And geometrically, it means that we should define some interesting possibilities for future and past boundaries of space times. And if possible, the boundaries should be such that future boundaries of one aeon should somehow match the early boundary, zero boundary of the next aeon. That's one thing, and that will be the subject of the first part of my talk. Then I will say a few words about observational data, data, what is time in cosmology. And then I will discuss the moment that was central in my idea of how to cope mathematically with all this story. I asked the question, what happens with time at the boundary? And then I'll explain how modular curves enter the picture. This is a mathematical thing, and they match the so-called mixed master universe that was invented by physicists, and say a few words about boundaries and Roger Pendereau's crossovers. So, future and past boundaries. So, the general notion of a boundary is quite well known to mathematicians. If we have a manifold M and some subset in it, open subset, and then this open subset is embedded into something else, as an open subset. And it's embedded in such a way that its closure, its dense, and so its closure coincides with M bar zero. And then we say that this difference between M bar zero and M zero is a partial boundary of M. It's by no means uniquely defined. You need an external embedding for it. Boundaries are generally used for studying asymptotic behavior of dynamical systems, solutions to differential equations. So, in particular, when we are describing classical models of universe-satisfying Einstein equations, and we want to study their limiting behavior, we usually add some boundary, and then look what happens at this boundary. But I will start with two very elementary examples. The basic example one is the projective boundary of Minkowski spacetime. So, M4 is a fine four-dimensional space with a metric of Lorentz signature. And let's embed it into this affine space, into its projective space in an economical way. And then the boundary will be three-dimensional vector space. And basic example two is a blow-up where we replace a point in M4 by the projective space of origin directions. Again, boundary is P3 here. Here, there is one subtlety, whether we are imagining complex case, which is also quite interesting in the Sir Roger context, or whether we are imagining real space. We can imagine first complex space, and then real points of it if there is a natural real structure. But on the other hand, this boundary may have a non-trivial fundamental group. And it might happen that we should pass from real points of a complex picture to a cover of it. But this is related to possibility of non-trivial orientation in the real case. But these are particularities. So, here is the canonical linear model of universal non-gelativistic cosmic time. This is kind of synthesis of results of observations. So, we have this vague point of blow-up. Then, there is a very mysterious period of inflation. So, time here goes in the horizontal direction. There is a very mysterious period of inflation, during which the universe becomes very, very big. But the period itself is something like 10 power minus 32 or 34 seconds. The cosmologists invented this period in order to explain, of course, nothing is observable from this period, but they invented this period in order to explain some things that could not be explained otherwise. First stars appear about 400 million years after the Big Bang. And then there is a period till now, which takes about 14 billion years. And up to approximately this moment, the light did not come to our space section only later on. And during this time, so this is dark age, so to speak, then here development of galaxies, planets and so on. And here very essential dark energy becomes an accelerated expansion. Now, of course, I have absolutely no time and competence to discuss details of many of these strange happenings. But what I want to discuss a little bit is mathematical correlates of the idea that there is some kind of cosmic time, cosmic time. This is pretty strange from a mathematical viewpoint, and I will return to it a little bit later. But right now, using this picture, I will illustrate what two boundaries I suggest to add to any mathematical picture of this observational one. At the cold end of the universe, I presume that it is well approximated by flat Minkowski space, although there is matter, there are some curvatures and things like that, but globally it's well approximated by Minkowski space. And therefore, the future boundary can be approximated by the simplest boundary of Minkowski space, namely three-dimensional projective space. Now, it carries one, so time infinity, moment of time infinity corresponds to the section P3. But there is one additional structure there, namely, so to speak, infinite sky. The point is that if we take any moment of space time in the Minkowski space and then draw the whole cone of light cone, the whole light cone from this point, then at P3 they will have all one common base. Because, say, if you look at this light cone and this particular light ray and then this one and the parallel light ray, then at the end they have the common point. And this common point will be the point of this infinite sky. On the other hand, at the moment of zero I imagine that I have some classical geometric picture of this space. I take this point of Big Bang and I blow it up in the algebraic geometrical sense. So there it was my unfold with Lauren's signature metric. Here is this point. I blow it up and when I blow it up instead of a cone I get something like hyperboloid. And then I get the new P3 which takes place of this plane and this point, sorry, and its intersection, its base in the tangent space is again this additional structure in this P3. So we see that these two simple mathematical procedures give boundaries at the beginning and at the end of the same structure, P3 and a sky sphere. Okay. Now I want to discuss what is this mysterious cosmological time. As I said, this is an abstraction invented by cosmologists. The point is that of course in relativistic models of space time, time is a local notion. If you have a time like geodesic gamma, then differential of local time, time along this geodesics on gamma is the same as differential of the relevant Einstein metric restricted upon gamma. So I will always consider Lauren's metric of signature 1, 3, that is 1 plus time dt square minus 3 squares of space coordinates. So on space like geodesics then time becomes purely imaginary. We usually imagine that space like point are on a real distance, but with this convention the distance is purely imaginary. So if you wish it's the same time. Unlike light like geodesics, time stays still. Time is just zero along light like geodesic. So if you imagine different directions from one point of space time in various directions, so this light cone becomes a wall. The respective wall crossing in the space of geodesics produce the well-known to physicists the weak rotation of time from real axis to imaginary axis. Only of course physicists invented it in a totally different context. But nevertheless here it is very clear. Now cosmological time is a theoretical construct. It's not anything like this local times. It's a theoretical construct not unique and depending on the choice of model of the way physicists match observations with their mathematical model and so on. Here are two important examples that are bridging theory and observations. One example is the so called inverse temperature of the cosmic microwave background radiation. One divided by KT. It is accepted that the current value of this background radiation measures the global age of our universe starting from the time when we start seeing light. That's approximately as many years after the Big Bang. Before that it was a pack. Now this is one way to define so to speak cosmic time and another way is the red shift of stars in observable galaxies and then multiplied by Hubble's constant. This is also a very remarkable suggestion or discovery made in the 20s that the red shift in the usual way is proportional to the velocity with which the star goes away from us but that it is also proportional to the distance to the star. The coefficient is the so called Hubble constant. So we have then the observable stars and galaxies now are put to various cosmological time sections of our universe. So this is the Sloan Digital Scala survey. Unfortunately it's not so easy to see well the colors here. The point is so here we are this is Earth. Each point that you see here is a galaxy and the red date is the older Earth stars of this galaxy. The red date is the older Earth stars. Here is the outer circle. It's approximately the distance of 2 billion light years and white sectors contain unseen galaxies. They are obstructed by dust, the so called dust. Such a dusty matter in our own galaxy. So this kind of observations, radio telescopes and things like that, big computers of course, produce the basis of the reconstruction of the model of the universe. And when I produced this slide I found to my amazement that the scientific picture of observable universe appears on an uncanny resemblance to Marcel Duchamp classical of modernism. New Dessant d'en escalier. So the lady here is drawn at the various moments of her getting down the stairs. Of course for Duchamp the motivation was the new media movies and things like that but our universe behaves in a similar way. We see it in various moments of its existence. Now I will describe so I have described approximately to what we want to apply our mathematical models, how do we produce boundaries past and future of our models. And now I want to explain the idea that when time is on the boundary, then time becomes purely imaginary. But this weak rotation in two different models either happens instantly or else is mediated by the movement of time along the hyperbolic, along the hyperbolic geodesic in the complex half plane. So in the complex half plane we have real axis, imaginary axis and time can just the zero from imaginary become real or vice versa or this jump can be modelled by going along the geodesic. Here a little bit mathematics. So first of all one of the standard cosmological models of universe is the Friedman-Robertson-Walker one. Such a time is a product of axis of cosmological time t and three space of constant curvature k, one minus one or zero. That at the moment t is rescaled by a factor. To find this factor we should solve Einstein differential equations in various models. So we have the signature metric and depending on the curvature we have this value for coefficients s and r as I said depends on the chosen solution. So dynamics then is described by one real function r of t. It increases from zero in the big bank to infinity and this is the scale r equals to one now. Now if one writes the differential equations that constrain r of t, the following refers to the so-called cosmological constant three. Then in the strange way I was very much impressed when I've seen the equation of an elliptic curve. Physicist usually do not know what is an elliptic curve and cannot recognize its equation when they see it but for me it was a key moment, a decisive moment. I found out that what I thought about imaginary time somehow matches very well this appearance of this elliptic curve in the equations and moreover they then model the global time besides the proper time t from which we started and the scale factor r of t. They introduce a very natural again for a mathematician time, cosmological time, just the integral of the differential of the first kind along a real curve from zero to present moment r of t. Coefficient a and b here are characterizing matter and radiation sources so they also depend on time and in concrete situations they differ depending on what kind of conjectures you assume. But now the qualitative summary in the Friedman-Robertson-Walker universe the time evolution is described by a real curve on an algebraic surface so it's not universal but the universal surface having all elliptic curves on it. There exists of course a very well known to mathematicians the universal family of elliptic curves which is parametrized by the upper half plane. Just take a point tall, take and see the lattice generated by one and tall and take the quotient. This group acts upon the whole family in this way and quotient by subgroups are usually a finite index, are modular curves, they parametrize also other verse of family. Below I will show that as soon as we have this picture of cosmological time in our minds we can understand different picture of universe near Big Bang which is called the so-called mixed-max universe. So we will be imagining it as a statistical dynamics approximation to an unknown quantum field or string picture of the Big Bang. Okay so a new subject matter what is a mixed-max universe? As I wanted to say it is a certain classical dynamical system with chaotic behavior which approximates the behavior of a set of certain pretty simple solutions near but near the Big Bang. That is the point. So we will take the so-called Bianchi-9 space time which has a so symmetry of its space like sections and essentially its main difference from the Robertson-Wauker model is that now we had there say when we had a positive curve which is then we had essentially a sphere here only the radius has been changing but now we have an ellipsoid each axis can have a different coefficient of expansion A, B, T, B, C. They are called scale factors and there exist a fantastically simple family of the so-called Kassner solutions to Einstein equations. They are just powers of T or these three coefficients are just power of T and this Pi are points on the real algebraic curve which is of course just a circle. The metric becomes singular at t equal to zero which is the Big Bang moment. Okay now in the 70s, Belinsky, Halatnikov and Liefschitz argued that if we consider not specific Kassner solutions to this model, Bianchi-9 model but kind of typical solution and move not from the Big Bang to infinity but in the reverse direction when we look at the typical solution that goes backwards in time then this typical solution to this model can be described by an infinite sequence of Kassner solutions that is infinite sequence of points P1, P2, 3 on this real circle. The space of the sequences is endowed with symbolic dynamic that encodes the logarithmic time dynamic of the relevant Bianchi spaces and it was discovered quite early that formal is the same symbolic dynamic encodes geodesics with irrational real ends and complex upper half plane H and their projections to the modular curve. In this paper with Matilda are probably one of the main suggestions besides suggesting this relevance of the imaginary time during the transition periods. A certain suggestion how to explain this coincidence, the symbolic dynamics of Kassner solutions and the symbolic dynamics of geodesics. So we argue that the identification of these two dynamics reflects the picture of physical time undergoing as I said reverse recrotation mediated by the movement along hyperbolic geodesics. So now I have yet some time so I can give some details about these two different encodings. So one encoding is BKL encoding of Kassner errors. So we introduce here the reverse logarithmic time as I said there are many cosmological times so introduce this one. When omega goes to plus infinity, this means that you go backwards to the big bank and BKL argues that the typical solution determines a sequence of infinitely increasing moments omega 0, omega 1, omega n and the sequence of irrational real numbers un and during one semi-interval which is called nth Kassner era for a given trajectory gamma. The evolution of ABC is approximately described by several consecutive Kassner formulas. Time intervals where scaling powers are constant are called Kassner cycles. And the evolution in the nth era starts at a time omega n with a certain value un and this determines the value of u determines the respective scaling powers p1, p2, p3. Here they are. Then the next cycles inside the same era start with values u minus 1, u minus 2 and the respective scaling powers and after as many cycles inside the current era jump to the next era comes with this parameter. This means that the natural encoding of all un together is obtained by just considering an irrational number x together with continued fraction decomposition and that the time flow is modeled by the powers of this discrete shift. Just very simple. And xn is this. So now this is inside one era. So what happens at the next moment? We introduce an additional parameter delta n that translates omega n into new value m omega n plus 1 and then the information about both sequences together is actually modeled or encoded by powers of the shift of two sided sequences of natural numbers. When we didn't look at the omega then we got only one sided and here it's two sided and the rearrangement of coordinates like that in the increasing order induces generally a non identical permutation of coefficients and when u diminishes by one the old permutation is multiplied the old permutation is multiplied by this one and when we pass to the next era then this permutation occurs. Now I will show you that the geodesic flow on modular surfaces is very naturally encoded in exactly the same way. We consider geodesics in up a half plane that are intersecting the imaginary half line. If we take such two sided infinite sequences of case then we define using it the geodesic going from initial point x minus which is between minus one and zero to the end point which is in the interval between one and plus infinity and here are precise definitions of this and the relevant Poincare section for this symbolic dynamics on one of these curves is essentially the projection of the imaginary semi axis of h to m. Here are a few. Also on h we see tessellation by fundamental domains for PSL to Z and its projection its trace on the real axis is essentially a farry triangle. So we have three geodesics connecting three rational points or i infinity and two rational points here are farry neighbors. And encoding oriented geodesics with irrational ends by double sided sequences we do it in the following way. We go say from this point on imaginary axis in this direction. We intersect one triangle of the tessellation and look where is the point i infinity to the right or to the left. Here it is to the left of my geodesics. Then I pass another triangle where the respective point is here to the right then one more to the right and so on. So we have infinite sequences and in this context there is an explanation of coincidence of these two encodings of solutions of typical solutions of Einstein equations and sequences can get its qualitative explanation namely postulating that return of the cosmological time to its real values is mediated by a stretch of hyperbolic geodesics and we never choose any concrete geodesics we consider the whole dynamical system and in this way we get statistical classical statistical description which should be in principle the trace of a known for us quantum development. Well a small warning when we embed the real curve tool using invariant of elliptic curve that I have shown I choose the coordinate minus i2 and think about right complex of blunder than upper one but this doesn't matter much. Okay. Now I am returning to boundaries and crossovers. The remark I made at the very beginning about mathematical isomorphism of two boundaries that deal with their skies future boundary of asymptotically flat space time and past blow-up boundary modeling Big Bang and I am again here that it may be used to furnish a mathematical model of what Penrose calls conformal cyclic cosmology. So this is Penrose's picture of expanding universe and it is the same as was at the very beginning only very schematic and now times goes up instead of to the right. So we have Big Bang somewhere here is very very brief but enormous inflation period we are somewhere here space section can be finite or compact or infinite and here are the data that I have already explained as a provisional data in the initial picture just the picture taken from Penrose book and what he suggests is that somehow as the previous universe ends somehow the new one starts and his mathematics that he suggested concerns matching not actual geometry of space but the relevant Einstein matrix namely he essentially is saying that when we go to this boundary from the past that is from the future of previous eon or backwards to the past of current eon then the metrics match in the following way they become conformally equivalent but the conformal factor tends to zero in one case it tends to infinity in other case is his conformal rescading intuitively I think it agrees with the pictures that we suggest namely that our past and future boundary get blown together if one uses the Penrose picture of twisters so the metric can be then matched in a conformal way but generally our picture is this assume that we have two models of space time m minus for the previous eon m plus for the next eon and m plus must be endowed with a point big bank point of course there is nothing invariant about this point as soon as we are considering the four four dimensional space only when we introduce the cosmological time then we can say ah this is moment zero before that just any big back point we construct then a partial compactification of m minus by three dimensional projective space or a subdomain in it construct the algebraic geometric blow up m bar m plus and identify the boundary of m minus bar with the part of the divisor of tangent directions and this will be the crossover and identification will be through identification of skies of the respective skies this notion that future sky of previous eon somehow corresponds to the past sky of the next eon is very essential for any attempts to produce observational validation or or falsification of this picture so it's quite important that in our geometric construction it appears in an extremely natural way and I repeat that at the moment of the crossover t infinity previous eon t zero this one the cosmological time undergoes a recitation becomes purely imaginary and then undergoes a reverse recitation and the mixed-marked stochotic universe is accommodated here if the reverse recitation is mediated by a stretch of hyperbolic geodesic thank you so the way you include the geodesics in terms of words of left and right left and right and so on reminded me of how we can from the Christ making torus of a two-dimensional torus also by left and right angles is it just an accident producing no I mean this is the particular case of the general Poincare principle if you have some chaotic movement on a geometric manifold or whatever how do you encode the typical behavior you choose also to speak and then you count only moments where this geodesic or whatever it is intersects this wall this principle is so universal that it works almost everywhere but with different choices so do you see any connection from that in Torre or is this an accident? well what I do see is really this connection with real curves on universal families of elliptic curves maybe one and another themselves are connected somehow I do not know I do not understand why you you said that this is like a miraculous link between the BKM picture and well because you know, Mitzner and Chitra in the 70s understood that the BKM picture was a billiard on the hyperbolic space on the hyperbolic plane and then the shape of the billiard is the art in billiard which is half the modular billiard and this generalized to any dimension to the models coming from no what picture there that you did not mention which explains well I mean I think that the real explanation is even now not in place because we must have a precise mathematical bridge between solutions of Einstein equations and this picture up to now these two encodings exist so to speak juristically and independently our suggestion which is not yet made mathematically precise is that this bridge between two ways of looking at these encodings should be through considering the time curve as a curve on familiar elliptic curves this should be verified proved or whatever by looking at concrete solutions of Einstein equations those that were used by Misner and Bogojevlinski and Halatnikov and Belinsky, Lifschitz everybody but up to now the coincidence of these two encodings was only looked as a miraculous coincidence including the work of Misner I think well Do you see as it's kind of complex analytic continuation of space time around Big Ben is kind of nice analytic complexity continuation of space time is kind of nice around Big Ben Yes, in a way yes with one reservation namely this strange thing that when you do blow-ups in complex geometry and real geometry you get in real geometry this additional covering but otherwise yes complex analytic story is very natural like in Penrose twisters Because there was sometimes when we were coming by Neil Turok about making fundamental not around Big Ben but kind of contours around here doing the standard Yeah, we must find time on Turok I must look at it, I don't know Another question Let's thank Yuri Valovic