 Čakaj, imaš več, mnogo, za organizacij, da imaš zelo. Vsih, ki imam tudi tukaj, je... nekaj, da je tukaj od vsega matriča teori. Tukaj, da je tukaj vsega matriča teori, je zelo, da je vsega matriča teori, da je tukaj vsega matriča teori, da je tukaj vsega matriča teori, da je vsega matriča teori. I zelo, da sam jih je zelo, da sem daz sentagonalne skrana, na zelo, da sem jih ga predmelyljučna mat. Vsega, da sem jih ga predmelylčna matem, tudi je zač japovljanje sprem, eskud je poča, da se nisem, bolj saj. Aspečno, da sem jih ga predmelylčna matem, da sem jih ga predmelylčna matem. Protočno, da bi je je vneno, zelo s jednog branča in zelo so prikratili, da je najbolj spodobno, kako se vedno počkala. Taj počkala je ne pozdravila, kaj ste prikratili. Zelo počkala se, ko mi je povedila v kanve, na bajneri kvadratičke formu, je to, da je jedna situacija, kaj je tudi template, tudi konvej tapograf. Konvej inventat, tudi tapograf, imam tukaj, prvi, početno početno početno, kaj je bajneri kvadratičke formu, kaj je vse početno vse. Now you can view this as a quantum problem, you can see the Laplace-Biltrami on the flat torus, then corresponding spectrum is quadratic form, and then you ask in what is the spectrum, it means that what are the values of this quadratic form. So it's quite natural from integrable system, quantum integrable system point of view, it corresponds to the question of flat torus, which is clearly integrable in on sense, so this was the question of konvej, so I will start with explaining konvej tapograf and then I will talk about direction I can from and this is Markov equation and and then I will study some function which is describing the growth in the corresponding dynamics which is SL2Z, PSL2Z maybe better to say dynamics. Now why is SL2Z is here, because if you consider usual dynamical system deterministic one, you can view this as action of Z, simply you take a mapping and you iterate it, so it's just autology. Now you can take other group, infinite group and what I'm suggesting is to take as PSL2Z, modular group and there are many many reasons for this, I hope to explain maybe there are many interesting examples and one of them is konvej and the second is which was I think realized later was Markov equation and so I will describe some function and then I will explain how, because I think it's beautiful, the proof is based on links with hyperbolic geometry although at the end of the days and with bit of analysis of feather or grom of stable norm, so I realize that I have chosen criminal amount of material, so I will probably have to rush through sometimes, but I hope to get to explain the message eventually which is I think it's important. So this is the paper which is joined paper mainly with my PhD student Katie Spalding and with Alfonso Sorrentino, it's now published so the main material is there. So what was the idea of konvej? It's quite nice actually, so you can see the binary quadratic form and the point is actually it's not konvej was not the first, it goes back to 19th century and it was the student of Riemann I think it will come the name who used this idea. You should use not a basis but superbasis. Superbasis are three vectors such that e1 and e2 is a basis but you add also e3 which is such that the sum is superbasis and if you have that's one thing this idea was known konvej added one more thing which is conceptually it looks like trivial thing but actually it's very, very conceptually important step he said that one should consider not vectors but lax vectors mainly you have to consider vectors up to sin e1 and then every if you can see the so-called lax vectors then every basis gives rise to two superbasis and these are here for example so for example if you take e1, e2 then you have e1, e2 and e1 plus e2 or minus, minus it does matter remember that lax or you can consider e1 minus e2 so superbasis are living in the vertices of this of this binary graph so this is what is konvejt apograf konvejt apograf is just the all superbasis presented in a very clever way so that vertices so lax vectors are living in the complement to the binary t on the plane it's not just binary t ok so now this kind of tree of course was known before sometimes it's called fairy tree and fairy tree is is the representing all the fraction where at each point you have this is a median it's called if you add fraction naively 2 plus 1 3, 1 plus 1 is 2 I mean that's fairy addition but in fact of course it's nothing else but projectivization of pitch yes it doesn't matter sum is zero that's the rule so the rule is very simple sum is zero so superbasis is a kind of you have two and then one vector is so this is a kind of superbasis it's quite natural in a way ok or you can draw it like this if you want up to SL2Z are superbasis that's correct edges are basis and complement are lax vectors so this are basic so this is what convey topograph now you can use it in many many ways that's my point as soon as I realize that actually we have after that we can ask some question which I'm going to ask so once we have it so now let me look at this in a different way and this is very very important point of view let's look at the edge so you have one one here one to here let's put a matrix here one one one one one two it's a SL2 metric and if you take positive part for example this is a monoid SL2N it's a matrices which is dimension determinant one with positive integers so you can view this positive half of the tree as SL2NN means that non-negative only entries there are no elliptic elements of SL2Z so it's a kind of SL2N rather than this another thing so let's forget about all the labels let's look at this as a planner tool and graph what are the symmetry of this graph the symmetry of this graph is you can rotate by pi over middle of this by 2pi over 3 so the symmetry is Z2 free product with Z3 which is nothing else but PSL2Z so that's why it is actually and this convey did explain this beautifully so he clearly realized that this is a very very important object even if you forget about all this if you consider this as a planner graph of arithmetic because the symmetry of this graph is PSL2Z so basically now let me say a little bit about this so binary quadratic form you have 3k efficient so if you know the values on basis it's not enough to recover this but if you know the values on the super basis then you have 3 numbers and this is exactly the thing so you recover so C is the value on the E1 plus E2 so if you know A, B and C you know H of course and you know the whole thing so the action of SL2Z when you change the basis then it's changing the how the form look like but the values also of this satisfy this arithmetic progression rule which is nothing else but parallelogram rule sometimes it's called right so if you know U plus V then it's a kind of definition of quadratic form if you want you need homogeneity but that's all what you need to know and then it's quadratic form so then you have this conveying which is playing important role but growing it grows forever doesn't matter so maybe I'll skip this detail so this is an example so this is a kind of question goes to Gauss which integer can be represented as a sum of two squares and the usual thing is of course deep results you have to factorize look at the prime numbers 1 mod 4, 1 mod 3 and then you know this by Gauss but for this you have to know factorization and factorization is hard problem I mean if you know factorization then of course so convey saying basically well let's do it easily I mean let's do the following let me visualize them so let's start this and then apply this rule for example 2 plus 1 is 3 times 2 6 minus 1 is 5 is 3 just local rules only you grow the values and then you will have the tree which knows all the values well up to total squares but nevertheless that was the idea of convey and because of convey lem if you want to know up to the million numbers you stop when the number will be larger than million then you know everything about the values so it's totally different approach to the classical but anyway and so this is the value so one thing I'd like to mention this because it's really beautiful yes if quadratic form is indefinite it's not of course related to the spectral problem but nevertheless convey river it's beautiful object namely you have both positive and negative numbers and if you exclude 0 there are no zeros on this picture and river is dividing the convey topography into two parts in one is positive and another negative and he showed that it's periodic also convey is periodic it's related to periodic continued fraction expansion of quadratic rational and this is the thing so now going back to Markov it's daifantine analysis so you have it's a story of the most rational numbers so every number can be approximated by p over q with 1 over q square less than 1 over q square that's continued fraction theory question can we reduce this c one is always possible can we do it better the answer is yes and the possible values of c are known to be Lagrange spectrum sometimes called and reserving for Markov top of the spectrum is less than one thought so let me show you the most irrational number with largest possible c c is a function of alpha and it turned out that the spectrum is discrete top of the spectrum is discrete so golden ratio is a champion so you can always take c is less than 1 you can take it 1 over root 5 but then if you fix it 1 over root 5 it will be golden ratio but if you exclude golden ratio and equivalent equivalent means that the tail not consistent with 1 in continued fraction then it will be silver ratio which is root 2 and then bronze ratio if you never seen this story you would be surprised it's clear that there is something going on that's a bronze ratio that's the third most rational number and then there is something here so this is the thing if you never seen it you should of course think what's going on because here this is diagonal of pentagon this is diagonal of square there is no geometry here there is deep arithmetic which was discovered by Markov and this is a beautiful paper it's actually master thesis and this is a relation between this question and the following differential equation it's called Markov equation it's x squared plus y squared c x y z and then he showed that all a solution, integer solution 1 1 1 is obviously a solution and then you notice that this equation is quadratic in z so therefore if you know two roots if you fix x and y then with respect to z it's quadratic equation so if you know one root z then you know the second one using vieta 3 x y minus z for example if you have 1 1 2 in the end do permutation then 1 times 2 times 3 6 minus 1 5 so what I drawn here is actually I used convey topograph to represent it I think I don't know who was the first to realize that one can use convey topograph I mean it's I learned it from I think Volodya Falk first so it came from Taichmuller theory and from this but they didn't use convey topograph they just said that it's natural way to represent them now there is a unicity conjecture which is still open saying that every mark of number appears only in one mark of triple so for example 2 appear many times it appear in 1 1 2 and 1 2 5 but only once it appear as the largest one in 1 1 2 so you have as many triples as many numbers so this mark of numbers are the most but the conjecture is still open so now what was the discovery of this wonderful mark of is that mark of spectrum means that the values of c bigger than one thought top of the spectrum is discrete and that's exactly the values of mu and m are mark of numbers which are all the numbers which appear in mark of triples so this is the thing and this is if you want to know the explicit form then if you know xyz mark of triple then you can construct this rational number by the way this is 9z2 minus 4 9z2 minus 4 so just to show you again just second this list 9 times 5 squared minus 4 225 minus 4 that's the meaning of this so this is the third mark of number so this is the list of most rational number ranked by mark of constant what is the minimal the most rational number is the larger the largest possible c you have anyway so this is now story growth the growth of mark of numbers was studied by Don Zagir in 1982 and this is I think there is still some question remaining to this asymptotic behavior but this is basically what happens if you put them on the line so from this perspective they are growing on the tree and this is the tree and again to grow the tree you are using local rule so for example 2 times 5 times 3 is 30 minus 129 and then you start growing 3 times product minus 2 will be 433 so this is the thing so it's not natural yes if you order them in this order then this is m1, m2, m3, m4 just order them in increasing order so the fifth number will be 29 and this is how it's natural I should say that what Don Zagir is doing is quite natural because you want to know how they grow in the increasing but from dynamics point of view I mean from convey point of view it's natural to look at them at the tree so basically what Don Zagir and Markov essentially also doing put them all on the line order them and then you will have Markov spectrum now from the dynamical point of view they are growing on the tree so it's natural to see how they grow on the tree that's a good question I don't think so it's terminology this is the definition it's all possible c for given alpha you have c what are the values they used to call Markov spectrum it's not clear the top of the spectrum is discrete I mean the set we should say probably set it's discrete and it's explicit integrable part of the spectrum the bottom is presumably continuous it's feeling from zero to some number which is called Friedman number I think so it's continuous but in between it's counter like set it's very complicated I think it's still a lot of questions but usually people call the whole spectrum is Lagrange spectrum for some reason I don't understand why but Markov is reserved for the integrable part which is top discrete spectrum and it's kind of suggestive that maybe but I don't know it's a bit like Riemann zeros I don't know, speculate but no but it's common to call it Markov spectrum the set it's a set of these things now what is the idea of it was the idea of Don's idea and we will use it too so there is much better actually equation than this if you add four nines then it will be integrable and this surface is actually well known classical surface from 19th century it's called Kelly cubic it's a cubic with maximal number of conic singularity four and this is one of the version of them now what's special about cubic because it's a kind of rational degenerate you can parameterize it explicitly you can parameterize it explicitly by kosh and then you reduce this equation cannot be reduced to the linear form but if you do this change of variable you reduce one of them at least to the Euclid equation A plus B equal to C linearizable explicitly linearization okay so it's only but you have to spoil it by four nines of course you have to forget about arithmetic at that point but if you're interested in growth then it's fine and that was Don Zagir used but actually this observation I don't think he knew this but Mordell in the theory of Diphantine equation noticed this already in 1953 so people that actually it's much better because he used it normalization which is integer anyway so this is the kind of thing and this is what behind it that's all what is related to integrability it's close to integrable equation so to study growth you can use this and therefore you can reduce it to Euclid equation and Euclid equation means that so this is but Euclid equation is very simple if you have A and B here you put A plus B here 3, 5, 8 and so on and because koš reduces basically not for this equation but for modified for Mordell equation as we call it then it's just koš minus 1 explicit linearization so this is linear equation here so this is tricky Markov equation it's very related only asymptotically approximately but if you add four nines then it's and we want to use fairy tree of course fairy or convey topography if you want and then now what's the question what is the growth you should label first of all the path so to work on the tree you have always two way to go left or right and let's label the path by a continued fraction basically c0 is number of left tones if it's 1 then it's if it's 0 put 0 and then c1 it's how many right tones after that right, right, right left, left, left and so on this is the thing I mean you may say that why don't you put 0 and 1 left 0, right 1 then it will be kind of diadic this is another way and this is you will see to switch from one picture to another you need this Minkowski question mark function which I will mention at the end gosh it's fast so right right so as I said SL2Z PSL2Z is acting on the whole tree on the convey tree and the four it's acting on the solution of this and it's acting PSL2Z is Z2 product free product with Z3 Z3 is cyclic rotation Z2 is disinvolution Markov and the orbit of 111 is Markov orbit it's just one as PSL2Z orbit okay so we the whole story of Markov is about one PSL2Z orbit so SL2Z is acting on the tree and if you take one element then it's there is Kanta and Laray and Iwasaki Ohara they studied dynamics of action of SL2Z on this but for a given element, right we want to study the so basically they pick up A and then iterate it A square, a cube and so on usual dynamics so we want to study growth along the path and so this is actually historically for some reason we define we were interested in Markov dynamics so you have to take double logarithm in order to make it finite so now the thing is that you can it's equivalent so these are 3 equivalent definition this is from Markov and this is from Euclid Cn remember A plus B equal to C so you can use usual logarithm because of so this is integrable limit basically integrable modification you are using or you can use growth of spectral radius spectral radius of A so which is exactly so you may say that we looking at the growth of average of topological entropy on the Markov spectrum so it's the same and we using simply this observation as now coming back to this quadratic form so you can view these values of quadratic form along the path and then the answer is that then you will have twice the same function except with two exception exception are the ends of the river convey river which is so quadratic form you have hyperbolas and hyperbolas has asymptotes and asymptotes have they define corresponds to two points on absolute alpha and alpha alpha plus minus so river is flowing is a kind of you can view this as geodesics so yeah that's another thing convey topography you can view what model discrete of discrete hyperbolic plane and I think this was probably I've heard that this was an idea already of Gromov at the point of view so therefore it's interesting that absolute so you can view this disk as a punker model of hyperbolic plane now what is discrete so this is a kind of skeleton and this is a very clever skeleton because the absolute is the same you see because to go to infinity you need to choose path and path is labeled by continuous fraction continuous fraction real numbers you will have exactly the same absolute so it looks like much smaller set but it has the same absolute at infinity so this convey topography is actually very clever and this is a geodec so the river in this picture is just geodesics coming from one hyperbola end of hyperbola to another so this is the thing so if you start here then everywhere else you can go it's twice lambda except this when if you go along the river you have zero growth so this is anyway so I have to skip now so let me maybe go to the main point so the main point is that I would like to advocate that lambda of x is probably very important function with very interesting property that's the main message actually so first of all it's a pgl2z invariant so you may say that it's real modular function but people usually when say they modular you assume some analytic property and you will see that there are no analytic property it exists only in absolute and whether it's related to anything inside it's not clear so lambda of x is almost everywhere zero but host of dimension of its support is one actually it was pointed out by Michael Maghi to us so the Leppunov spectrum of this form it takes all the value from zero to log phi in spite of being zero almost everywhere it takes all the value from zero to log phi and and this log phi corresponds of course to the golden path and golden path so you remember this left, right, left, right path that's the maximum value and this is the most important thing because at some point we were lost so you have a function which is a cell to z invariant it means that it's zero at every rational number but on the other hand there are quadratic rational for example it's not zero I mean how would you ask about analytic property and the answer is the following choose a special set and if you restrict on a special set and the special set is of course the most rational numbers which Markov studied, right so they correspond to it's a cell to z invariant so therefore it's a kind of function if you want on the quotient of absolute by cell to z which is nasty space, right how would you work with them so the idea was carefully orbits and order them carefully then you will see some structure and that what we managed to see we realized that if you can see if you can look at at the most rational number in fair parameterization it's continuous and convex continuous and convex function continuous and convex and then result based on hyperbolic geometry now I think it's a beautiful piece but I think I have to skip but let me just flash something so this is uniformization theorem so what is the main object you take T2 and T2 of course the best metric is flat but if you make a puncture then the metric become incomplete by uniformization theorem there is constant curvature metric here unique which is conformally equivalent to this this is what you get is you will have hyperbolic torus punctured hyperbolic torus and it's important that this is equianharmonic means that this is rhombus and this is 60 degrees so you take flat metric make a puncture is not complete anymore introduce complete hyperbolic metric it's uniquely defined by this after that you can see the simple geodesics non-self-intersection geodesics and mark of numbers are cosh of their length this is beautiful observation due to Garshkov and Kohn so maybe I'll skip this in fifties this is how it proved and it goes back to really skip this this is another thing which we have used it's Harvey Kohn observed we actually introduced themselves we call it quantum mark of quantum Euclid 3 but it's actually when we realized that Kohn introduced it so you basically replace A plus B equal to C by AB equal to C in B are 2 by 2 matrices and if you start with a particular matrices like this then Kohn showed then you will have this Kohn tree a very beautiful actually object now they are related to usual mark of tree this one by taking one thought of trace one thought of trace for example 1 this is the thing and this is a really beautiful object it will play also some role and they actually generate a special subgroup of SL2z which is commutator commutator of SL2z is generated by these two matrices and this is actually explicit uniformization of these torus I was talking about so it's really beautiful piece of geometry happened in 1950s in hyperbolic geometry and then that's contribution of Volodyfok as I said I learned about this stuff he rediscovered probably this convey representation for Markov I don't know who was the first anyway he definitely knew this so he introduced this function psi of p over q and remember that we have labeling of Markov numbers by comparing by rational numbers 1 for example is function m of 0 2 is m of 0 1 half, 5 is 0 1 thought so now we are ordering them not according to the growth but according to the position on the tree and then you have as many Markov numbers as many rational numbers positive rational numbers and then you take m of p over q 3 half and this 3 half and cos minus 1 is coming from this integrable modification of course remember that was cos 3 half so this was fork observation that this function actually is quite nice m of p of q is tricky but if you take cos it's still bad and this is what I never understood this coefficient I knew about this fork's function but I couldn't understand this was a kind of contradiction this 1 over q it was clear that it's crucial but I couldn't understand why I mean it's anyway so if you add this 1 over q which was clever thing to do then the function is extended and continuous convex function so now this is our observation so it turned out that lambda our function lambda evaluated at the most irrational number corresponding to m Markov number m of p of q so this is m of p of q is integer and this is the most rational number this quadratic rational I have written explicit formula for sorry p of q I mean this is what I tried to explain here it's given by this so 1 is m of 0 1 on this ferry 3 and this is addition ferry addition it's just labelling by conveyed apograf if you want because these are conveyed apograf but I'm writing this in the usual way so this is different parameterization of course not like Zagir did and 1 and 2 and 3 in order it's a parameterization by p of q and if you add this 1 over q so this is that was the crucial observation that lambda if you evaluate on this most rational number then you have one half of fork function and that means that it can be extended if you choose carefully these orbits and if you order them properly and if you put them together it will be convex function so this is and moreover corresponding matrix on the ferry 3 because you have to climb is precisely the matrix from the con tree everything suddenly fit together marker results orbits results con observation yes the limit on absolute you have real number and you can go there between on this tree and this is the limit of course of this and we want to know what is the average growth of topological entropy along on this tree and the claim is that if you if you choose special mark of and numbers and then yes sorry and if you choose this special most rational number then lambda on them is nothing else and you need this q this q is crucial the q which fork put the whole thing works and we have found some generalization now I have to let me skip the federal grom of stable norm I'm afraid so basically this is another interpretation if you know this then the fork function is another interpretation it's half of the restriction of stable norm for on the first homology of this hyperbolic torus on the line and if you restrict norm on the line then of course you have convex continuous function so this is explanation it's restriction of the norm of certain norm and this is and as a result we prove something about fork function which was not known to fork namely that it's differentiable at all rational numbers so it's quite peculiar function psi even when we carefully chosen the representative we made it maximally regular it's still not differentiable at any rational number I mean it's quite peculiar and this is everything was prepared by Bangert and McShane and Trivin in 90s so just the last my message my message is that we are looking at special function as complex analytic function but I read actually it was talked by Hadamar in 1930s in the first congress of Soviet mathematician I think in Kharkov 1930s and he spoken about special real function it was a kind of a side remark he said that was surprised that he essentially mentioned exactly these two function which I knew and this is two example I want to point out so this is one Weierstrass introduced it in 1996 but he said that he heard that Riemann actually produced but it's it's debatable whether there is no records of Riemann considering this function but anyway it's usually called Weierstrass function and this is the thing it's signed by pi k squared x divided by pi k squared k squared so it's very so and that function was started he produced it to remember that in 1990s it's before I think he was produced the function which is differential nowhere started with more interesting example from my point of view and he considered this only because somebody said that Riemann actually considered this and then Kharkov did study this and then it's until now so the answer is that it's differentiable at every only rational points and very specific rational points for example at all rational point and irrational point is not differentiable so this is one example another example is due to Minkowski again we rediscovered this function because we need it you can label the path either by binary 01 or by continued fraction and the difference is exactly this Minkowski question mark function and I didn't know this actually this remarkable function I did not know this because he denoted this by question mark I don't know how would you know this but anyway so eventually we found the reference and it was Minkowski geometry I mean the famous book and then he mentioned this question mark function and this is exactly this homeomorphism 01 with 01 which was started by Zohar and Salem and it's very interesting it's continuous both ways it's homeomorphism of 01 in contrast to devil staircase which is not homeomorphism so it's strictly increasing but derivative is zero almost everywhere and basically my idea is that maybe maybe lambda and psi and fork function psi belong to the similar list because they have you see many natural interpretation and it's what we notice is probably a little bit of their remarkable property but they are it's really require presumably like in all these cases you see it require proper analysis from people who doing real analysis so it's a kind of special real valued function which has peculiar property Thanks