 Thank you very much for this invitation. It's a pleasure to be there. It's late for me. I learned that for some of you, it's very early. Probably for some of you, it's a long time, but in fact, doing mass the day evening is beautiful. It's a nice place to discuss. So I prepared something, which is, there will be some serums at the end, but the serums are old, and they are not fully written yet. I mean, there is a small version on archive until you see the date. It's 2012, and we are still revising the manuscript and every time we revise it, it grows. So probably it's an end of the task. We'll see. And so what I want is to reach this material, but before that, maybe spend more time explaining why we're interested in doing that. So here is the abstract and I skip it. But in the abstract, you'll see that I'm going to discuss heights, a record of geometry, non-archiving geometries, which will be motivated by questions of archival geometry. And then the end is that real forms and currency, non-archiving geometry, so mixing real numbers and non-archiving numbers, and then a bit about mid-tride line models and complex geometry. So first is about height. So the height is sometimes described as naive. The height of a point is just a way of measuring its complexity. And you should take a point, a rational point of the particular space with homogenous coordinates, x1, xm, and you can choose them co-prime and integer, and then the height is usually just logarithm of the supreme number of those numbers. And with an unfortunate typo here, it's xm and not xj. One very important feature is the Northcote property that, once you bound the degree and the height, and you get only finitely many numbers. And for when the degree is one, it's very, it's obvious on the definition because you bound the homogenous coordinates. And one, one standard, which is built into the machine of heights is good functional properties, which at this level is very elementary looking that you take a rational map from a prototype space to another one. And it's given by a homogenous polynomial, f not fm in n plus one coordinates. And it's not always well defined because if a point cancels all of those polynomials and you won't be able to evaluate this, evaluate the map at this point, but otherwise you can. The first inequality is that when you evaluate the map at such a point, you just multiply the height by some constant, by the constant d, which is a degree. And maybe you add some error term. And if you have close to variety, which does not meet the determination locus. And if the bf is empty, then you have an inequality in the other direction, so the height of the image is greater than d times the height of the initial points minus some constant. Just let me mention that in some applications, it could be important to know something about this constant cf and cx, but for the moment I will contain myself with this inequality. So then what Veid did almost 100 years ago is to transform this weak functionality properties into a height machine that becomes quite more sophisticated. It's not obtrusively related to varieties in some project space, but rather to varieties equipped with a line variable l. And usually if x inside some pn, l will be the line variable that gives you the so-called O of one line variable that gives you basically functions, homogeneous functions of a degree one on x. And then to do x and l, Veid defined a height functions hl on the algebraic points and for every morphism an equality of heights. So the height of f of x is the height of x with respect to the back line. Except that this is not true, because we had unspecified constants as cf and cx before. And so all these equalities do not really hold, they only hold up to a bounded function. And so these functional equalities up to a bounded error term. So for some applications, it's really innocuous, especially if you look at points of large height, it could be nice. But if you want to know what the height of some specific point is, then it's not sufficient. There were various ways in the second half of the 20th century to get rid of this indeterminacy. For example, John Tate and Joseph Silverman introduced a dynamical height so you assume that you are given on your variety x morphism f and you have the line model l is such that when you pull back l by f, you get l to some power q greater than two. And then there is a unique way of normalizing the height functions such that the equality for at least for this morphism is a precise equality. And in the case of dynamical systems, the interest of doing this normalization is that you get a characterization of the preparatory points, the point whose four orbit is finite, ultimately your cycle finitely many points, and then they are characterized by the vanishing of the height. So you should consider an abandoned variety and some morphisms such as multiplication by two, and assume that, for example, then what this function is, it's called a near and dead height. It's a positive quadratic form on the point of the abandoned variety tensile with r. And it's well known that this tool is crucial in half of the proof of the model Bayes theorem, for example. So the way where you see precise normalization is when you look at the mother measure of the point, so it's even more elementary than the previous case, you just take the point of p one. And you don't look at point of infinity, so you take a point of the former one with homogeneous coordinates which are one committee. The one that the height of this point is one divided by the degree of the minimal polynomial times the logarithm of the mother measure of the point. So you get an expression which is the other degree in the denominator and one logarithm of some integer leading coefficient. And the something which is the sum of all roots of a function which is essentially log of absolute value of the complex number if you are outside of the circle and zero if you are inside the circle. For example, on this formula, you see that from this formula, you can see that if the height is zero, then that imposes that the leading coefficient is one and then all the conjugates are inside the unit in circle and by the unit disk sorry. In this context, I should also recall the lemma conjecture, which makes sense because you have a precise normalization here, so the height of some point t should be at least some constant divided by the degree of the point of the degree of f. So it means that this should be universally bounded from below, unless the point is zero or with infinity. This conjecture is open but it's known that there are first of all a weak version due to the Rowalski up to some logarithmic additional term and there is also a recent version of an analogous inequality called a conjecture by Schindler and Zassenhaus and proved by Dimitrov. So, I missed the title. So, in this context what Fekete and Secker did is replace the disk by some compact set and replace this function log super number and one by the potential of the compact set. So this is a function on C which is zero on the compact, harmonic outside of the compact and goes roughly like an logarithm at infinity. And what it corresponds is that if you view, if you are inside the Riemann sphere and the north pole is a point at infinity, and then you have your compact set, what it means that if you are putting your your contact into some conducting materials of metal and the rest is supposed to be empty and you put a unit charge at infinity and then you get the capacity of this electric system. And from this function, what happens is that GK is very close to the function that appears into the definition of the matter measure so it would give another height function relative to the compact K and which would correspond to other geometric questions. For example, you could ask and Fekete and Secker asked consult whether you have infinity many points, all of those conjugates belong to the compact set K and which are also integer. And what Fekete and Secker proves is that if the capacity of the compact is too small, if it's more than one, then you're only are fine human points satisfying this condition. And if the compact has a great capacity greater than one when essentially you have infinity many points. It's another instance where having a precise normalization of the height allows to solve questions which makes number theory and geometry. Arachel of geometry was invented in the end of Arachel of this first work in 1774 and then it was in dimension one. Arachel was retaken by Fettings and Gillet and Soulet around 1985 to say 1995. And it furnishes a high tech machinery to specify the height in a fully functional way, but in a way which is also geometric. So, instead of looking at projective varieties and limelows, when we look at schemes over the integers limelows on that schemes and Hermitian matrix on the complex limelow over this meaningful. So the, the, the relation with the classical theory is that you could take X inside the projective space over Z. On this scheme you have the classical or one nine mobile. And X would just be defined by equations. Zero FN and the FG would be polynomial with integer coefficients. And on all of one, you also have the fubinister geometric, which is very classical object from complex algebraic geometry or analytic geometry. Which allows to measure the size of homogenous polynomial at a point X. So usually if you want to measure that, you could try to evaluate the homogenous polynomial at the point, but it doesn't make sense because when you multiply the homogenous coordinates you multiply this by the multiplication factor factor to the degree. So you just divide by say, say the norm of the point of the coordinates of the point divided by the degree of F. If you do something like that, you get, you have expressions which are well defined. And in some sense the classical theory of Bay Heights is just the particular case of molecular geometry with this, this functions except that the norm is not the precise norm. But it's a minor difference. This is essentially due to two things, that once you have such such shriples. So X and norms. There are functions which are absolutely well defined, and that apply to all close subskims of X. So you don't only have heights of points but you have the heights of everything lying on X. In a scheme of a Z, you have a classical definition that your scheme X over spec Z. Then you here you have the generic point of spec Z, which is spec Q. And then you will have a variety of a rational, but you also have a variety of two. And then you have varieties, which are in the language of the previous slide are just the variety of F2, F3, FB, etc, defined by the reduction module two or three, etc, of the polynomial that define the X. And then you have two types of tools. So you have two types of varieties, you have those which are fully vertical, such as a variety there. And then the height, which is defined in this matter is essentially the degree of the variety here. So you have a variety in some particular space of FB and that is the degree of respect to the corresponding language. And it's essentially the degree, you have just a multiplication factor, logarithm of one number. And in the other direction you have schemes which are this way, which are horizontal, they are subjective of a spec Z and they correspond to close the varieties of the generic fiber of a Q. And then what you get when you take the closure of something about the generic fiber, so something that will be horizontal. Then you get essentially the height, what we want to call the height of the generic fiber. And in some context, the height is normalized, so that's what you define by induction, something which looks like C1 hat of L bar to the power dimension of Z plus one, which is a notation like that, which is inspired by intersection theory. This would be this number, so essentially the height, and here is a degree of the variety in the generic fiber and this is just for normalization. So what perspective do we get during this, using this machinery, then we get that we transfer the question of having appropriate height functions into having appropriate models or appropriate metrics. And there is some hope that having a question of finding new models of varieties or good metrics and models is a question, which is of a more geometric nature and maybe can give a more precise answer. And in the context of a billion varieties, regarding the new haunted heights, the models are given by the number else and the metrics are classical that just a matrix whose first form is invited by translation and it's exactly the, the metrics that you can build on the line of all from the complex definition, complex point of view on a billion varieties and then I will not. For example, in the context of the appell, the bar theorem. Regarding the question of the KT as a really introduced a potential theory and cause and pay added capacities to have a question of conical type of a KT as a good type, but introducing compact sets at local places. And in algebraic dynamics. This business could be useful, but it's not so useful. Because it's very rare that you have some good reduction process that, for example, it would, it would be necessary that you really have a morphism over the integers and tending extending the morphism on the. Generic fiber and this is quite rare. And therefore, one still needs an approximation process of the same type as take, take, take process, and this is done by the theory of analytic metrics of chosen. And in some sense, this approximation process. It's, it's, it's very useful in practice, but still it may lack some geometric definition. For example, if you apply this geometry, this is the process on a billion varieties with bad reflection. It's very it's quite complicated to understand what to get. But if you do. If you look at the non-accommodian geometry on the non-accommodian uniformization of the other end varieties and you are supposed to see what happens in a cleaner way. Just to give for additional motivation of a record of heights. Statements which are impossible to, to give if you don't have those heights, except in particular contexts, such as dynamical dynamics. Is that you have inequalities that relate the height of the variety with the height of the points. So this is the height of points, small x, and this is the height of varieties. And this is a general equality. And essentially, essentially the, the, the height of the points of the variety are always greater than the height of the variety, except that they could be sub varieties, close sub varieties where the inequality doesn't, doesn't hold. So you remove them and you might be obliged to remove infinity when you write. So it means that this complicated to supine means that whenever you fix, you fix epsilon, there exists why such that for x in x minus y, the height of small x is greater than capital X minus epsilon. And whenever there is such an equality like this. You can use it as a variational principle, and Spiro, you know, Jean did that. And from this inequality proved liquid distribution theorem for for height of points, which are close to this law of bond. And, in turn, this could literally be a key distribution theorem, while the cornerstone of the proof by you more and John and the, of the Bookman of Conjecture, that says that that generalizes the monument for conjecture for sub varieties of varieties and roughly that says the monument for conjecture says that if you have a curve in its Jacobian, then you have only finitely many portion points on it. And one of the torsion points are exactly the points with neuronal height zero. And what the book of Conjecture says is that you have finitely many points. Whose height is what it says that away from those finitely many points, the height is bounded from below by a strictly positive constant, which is just the height of the curve. So, ending this this part with motivations for non-accommodian geometries. First of all, it could be just purely philosophical to put non-accommodian places exactly on the same footing as Archimedean ones. So this is what some people do, but in reverse, there is a lot of work trying to put the Archimedean place on the same footing as the non-accommodian ones and doing algebra, the skins of F1 and so on. So here I want to do exactly the opposite. I want to, I think that complex analysis and real analysis is much more developed than algebraic geometry. So one should be able to have tools from analysis to do algebra. It could be useful to do that when you have bad reductions and no good models. For example, one of my hosts is that it would be interesting to use non-accommodian analytic geometry on Shimura varieties to understand the heights instead of constructing in a difficult way models of Shimura varieties. So it could be interesting for, it's also interesting for distribution serums. And there are also other aspects of Archimedean geometry that lead naturally to non-accommodian geometry is that in some sense as you said, so for example, and Johnson observed, in some sense, the, the, the limits asymptotic limit of Archimedean geometry is non-accommodian geometry. So you could, one could understand the generations of, of Archimedean phenomena in a non-accommodian way. In some sense, this is what non-standard analysis claims from, from complex numbers you naturally build a field of non-standard complex numbers which is of non-accommodian nature. So let's discuss a bit non-accommodian geometry. There are building blocks. And I have two columns that explain the building blocks from the right geometry and those of non-accommodian geometry. For if algebra is usually algebraic geometry is about finding the general algebra of a field. And non-accommodian geometry looks at algebras which are questions offering of porous areas which converge with whose coefficients converge to zero. Algebraic geometry defines the space, the spectrum of an algebra and growth index, a set of all prime ideals with some Zaris ketopology and there is analogously an analytic spectrum of an affidavit of algebra. And which consists in, in multiplicative semi-norms on the algebra A. And another way to describe this, you can view prime ideals as equivalence classes of morphisms to fields. If you have two morphisms from two fields K and L, and if you can build a composite extension such that two morphisms from A to E coincide, then you claim that those morphisms are equivalent. And if you do that, you get precisely the prime ideals of A. If you do the same job by looking, with looking at morphisms to complete valued fields, because you are doing analytic geometry, so you need to work with complete fields, and then you will get the set of all multiplicative semi-norms. So what Berkowitz did is that defining the analytic spectrum, say all multiplicative semi-norms, so I just recall the definitions here to be clear. So it's just a function with positive values that the semi-norm applied in the sum is smaller than the sum of the semi-norms. It will become an Archimedean at the end, in some cases, and you will get a super-norm of P of F, P of G, but don't claim it is. The fact that it's supreme, the semi-norm is multiplicative, is that the semi-norm of the product is the product, and I think. And you also, during algebraic analytic geometry or K of some ground fields, so on the ground field, you ask that the semi-norm restricts to the absolute value that you're given. And on the set of multiplicative semi-norms, which is a spectrum of like in the Gelfand theorem, there is a natural topology, which is a topology of point-wise convergence. So you just ask that all evaluation maps from, so these are maps from whenever you have F in A, it's not F max, but P of F is P max to P of F. So for all F in A, you have an evaluation map from P, which maps a semi-norm P to the value of the semi-norm at F. And you ask that all of these semi-norms, these maps are continuous and it gives you some topology. And the point of view is to look at those semi-norms P, not as functions on the algebra, but really as points on space. And so one will view the value of the semi-norm P at the function F as the absolute value of the function F at the point P. So it is reversing the point of view, which will be important. So the, I skip a whole part of the theory, which will teach you how to glue analytics spectra together to get varieties which are not, which are more complicated than the varieties that analytics spaces built from affinate algebras, but it's not very relevant today with me. And I will also skip one important property is how to define new functions on your algebra, for example, analytic functions on the concept, but this will be enough for me today. So the interest of this, those spaces is that the topology is quite good. The spaces you get, we get, I will describe them in an example later, are locally contractible, they are locally compact. And if you try to do analytic geometry of the CP, for example, though the completion of the right closure of CP, then you get something which is definitely not locally compact and which is totally disconnected, so not locally contractible. We will see that there is an interesting interaction with the real numbers. And those spaces also have topology and also have a growth in the topology and growth in the topologies are is what happened when you try to glue compact sets. And when you want to glue open sets your topology. And it's useful to have to have both, both topologies. And indeed, there are many other theories. And so the naive theory has not enough local compactness in the. It's very good when you work over QP and do something which is relative to QP, but it's really insufficient here. The theory of state would, which only looks at CP points, but forgets about the topology and only looks about the growth in the topology has not enough points so you don't see really what happens. There is a theory by Renault built on formal schemes, but then one really lacks a space on which one can do analysis on geometry, the space exists, but it's not. It's not very visible. And there is also more recent theory by Huber, but this one has too many points for what to do. In some sense of the back of history which was developed roughly here is exactly what what we need. So just me let me give you an example. So we'll just the only example that people can do is they are fine. We want to look at some norms on the ring of polynomials in one variable. And so back of each claims that there are four sources of points. The first one are just given by you take a point in the, in the UK. And, and you look at some of them that takes a polynomial and takes the value of a and the absolute value. So you have the point zero, you have a point for one, and the other point for, I think I will pretend case QP bar. So we take point B here. Now what one can do is you can look at the point of the disk of center zero and register and you can look at the Gauss norm on this disk. And it will mean that you take P, Pd of F. In some sense it's just a supremum of f of a for all a there but you really need to take the supremum on the algebraic closure. And it's, if you expand this, the polynomial f, square efficient, Cn Tn, then it's just a supremum of absolute value of Cn all to the end. And it's not trivial that it is a, it's trivial that it is a semi norm, but it's an even norm is positive, but it's not trivial that it is a multiplicative and this is a content of the Gauss theorem. We have a lot of semi norms, and I will draw them. And we have a line of semi norms. And here I will write zero and one. And when one can go on with this line. In this line, I will use both rational disk and irrational disk according to R being rational, P to some rational or being really a real number. If you look at both points of both types, you get these segments, but you could also start from the point one and look at the disk from one, and then you have disks. You have disks at the point P, it's not, okay. And then you can grow the disk, but what happens is that when you, when you've reached the disk of center zero and radius one. It's not equal to the disk of center one and radius one because of non-accommodian property. And so you get that your line, which was supposed to go from one and meets the initial line there. And then draw the point P at some point. And when you start from a point P, it will do the same it will reach the initial line, but exactly at the disk of center zero and radius one of a P and then it will go. And so what you can also do that with this P squared, which I'm trying to do like this. So let's do P plus P squared will be there and P squared. And it will be disk of center zero and just one of a P squared. And you get something which is a kind of tree, and which is very branched because you can, you have a countably many branches, countably, countably down set of points. It's very, it's a tree, but it's very heavy. And here is a nice picture of this tree, when you draw it on the plane taken from the paper by Roschowski and Leuze and Poon. So here is the proof that this projective line, Bakovic, actually embeds in the plane and it's an object which is well known in topology. It's called the Roschowski dendrite. It's some kind of universal space which topologists like to study. Nevertheless, the point of view of Bakovic fits very well with Reynolds approach. For example, when we have a formal scheme or a scheme of a spec Z like before, but I will do it over the I think of the valuation ring of the few K. Then assume I have a generic fiber, which is an antique space, but there is also a special fiber and there is a map from the generic fiber to the special item to the special fiber. And there is also a very important process, which is which has been understood by Bakovic. Which is very important, especially when you have that reduction, because then you would have a complicated special fiber with many, with, with many components and many interest intersections. But what Bakovic proved is that for each of these components, there is a specific point in the space. So here in my picture there are two points. And related to this intersection point, there is a real segment. So it means that from this picture and then if you had three by three intersection, you would have to draw two dimensional simplices and so on. So just from the combinatorics of the special fiber of this stuff, one sees in a very concrete way inside the analytics space of Bakovic. Some, some, some things, some object that is usually called a skeleton, and which is a polyetron inside the analytics space. And this is the one reason for which I say that those spaces have a nice interaction with real numbers, because we, we see real objects from topology, which are, which are nice, which are skeleton and also Bakovic proved that there is a retraction from the generic fiber, this analytics space, and to the skeleton. So this is why those spaces are well behaved. They are very hairy trees, but you can contract the tree. So the time goes too fast. Other than that, I've prepared too much stuff. So I will. Okay, that's really nice. Okay, in 10 minutes. Don't pay attention to the number of pages below. So now I think I can try to say a word about how to define real forms and currents in an archiving geometry. So it was on the implication, maybe it was on the implicit, but let me just say that, for example, a regular geometry is full of currents and real forms and differential geometry. So it comes from tropical geometry. So we have an analytics space of a field which always be cold care and will be assumed to be a normal Canadian and complete. I will consider tropicalization maps which are morphisms from X to to the tourists, which is the analytic tourists, but it's not necessary to pay attention to that is just given by and functions on X which are homomorphic in the sense I did not describe and invertible. And from this F, what I can do is I can evaluate the functions f1fn at some points of the space and can take the absolute value. And because we prefer addition to multiplication, I take the logarithm. And so I get one gets a continuous map from the analytics space to to the real real space. So this is something which is which was studied by delphan Capranoff and Zalevinsky of as a complex number called Amoebas. And here we have none of us. And it was shown by Gary Groves in the 80s and in this context by Capranoff and Eisenhower and Lint around 2000. And so if X is compact, then the tropicalization of X is not a complicated space is not a numberable like people have in complex German analysis, but it's a compact political subspace of and its dimension is just a smaller than the dimension of the initial space. And so it's another way of having interaction between analytics spaces and real geometry is just by looking at these tropicalizations and very useful. Maybe you've seen the definition of like Capranoff and you see that sometimes people don't take the F drop of X but closure of F drop of X, and this is because they don't use back of this space. So if I claim that those tropicalization maps are important, which I do. I will also use this to define smooth functions on an intake spaces. And the idea is that that moments are nice maps and antique maps in non-accommodation geometry. I will take the C infinity functions on the real numbers. And if I compose a C infinity function on the real numbers with the tropicalization map, I will claim that I get a smooth function. So, and there are a lot of smooth functions on spaces or ones are technical assumption that the space is separated and it's locally it's reasonable it has locally not philanthropic functions. I will not forget about this. And then there is a stone by a structure and this the smooth functions are dense in the smooth functions in the continuous function for for the compact open topology uniform components on compacts. And it's the same if you look at C infinity functions with compact support. And if the space is para compact, then you have smooth partitions to even up those partitions of unity. The compactness assumption is a bit subtle in back of its geometry. If the base field K is something like CP, if it has a countable dense subsets, then almost all spaces are very compact. But if K is the field of Laurent power series over the complex numbers, then if you take P2 and you remove one one point. You get something which is not not anymore para compact. So, anyway, this is an important assumption and sometimes it's annoying. We need differential forms and those differential forms are provided by the Bayer-Ezin Lagardex supercalculus. And so it comes from an analogy that on complex numbers, we have prairie subharmonic functions on real numbers, we have convection chance. On complex analysis, there is a mojampere operator and there is a real mojampere operator, which is slightly different. So here you differentiate with both the complex and the holomorphic and the anti-holomorphic variable, and then you only have the real variable. So here you have two n variables and on the real, you have n. And so while on complex analysis, you can do differential calculus with holomorphic and anti-holomorphic variables. You also need to develop some supercalculus, which means that you totally formally take the potential product of two copies of the differential forms. So it means that really what you do is you take one copy of the differential forms and you write them with D prime, and you take another copy and you write them with two different forms. And then you can do differential geometry, as usual, you can take D prime of this, you will differentiate the functions, and you will add some D prime in front of that. You can also take D double prime, you will add some D double prime, but you still will do different selection with the same variables. So it's very nice that if you do that, and you do D prime, D double prime to the end, you get exactly the real non-jamper operator. So this totally algebraic business that might look strange and unreasonable, but in fact it gives you naturally the operators that complex geometry, real geometry. From now on, we just take differential forms, super forms on the tropicalization and we pull them back formally on the space, meaning that locally I start from objects which are an open sets, a moment, and a superforms on the real space. I define a formally f of a star alpha, and then it's more or less standard to glue these objects into shifts. And the first proposition that showed that what we get is reasonable. For example, if you have two moments which have nothing to do, but f-trop is the g-trop, so there are two sets of functions which takes the same absolute values, then when you pull back the forms, you get the same thing. Also, if you pull back one form and you get zero, then essentially what you have to pull back was zero, so we don't lose anything. And just formally from the lag of that business, we get differentiations. One important stuff is to be able to integrate forms. And the idea is just to try to integrate formally back on the tropical side, but then it's complicated because here you have two n variables. And so you cannot integrate with respect to two n variables on Rn, so you remove alpha to them and you try to integrate. The problem is that this will depend on the choice of coordinates. And so one needs to add an additional structure on the tropicalization, which we call calibration, and which corresponds to the weights of tropical geometry. And we also have a balancing condition. So, from there, the machinery of currents begin as in analysis. We define the currents as a dual of compactly supported forms. Just by duality, we have differential operators and push forward because we can pull back forms. And because we can integrate forms, we have an integration current. And because we can also integrate forms on boundaries on subspace of some forms which have dimension n minus one n can also be integrated. And so this gives you different kind of integration. And as a matter of fact, we have a stocks formula. And we have also a formula, for example, it makes sense to take, if F is a neuromorphic function, it makes sense to look at logarithm of F. This will naturally define a current. And if we take the D prime, double prime of this current, we get the integration current on the divisor of F. And another formula, which I like to show, is that the analog in complex analysis of the formula that D prime, double prime of the, of the, this is exactly the potential for the disk of center zero and radius r. And when you do Laplacian of this function, you get zero outside of the disk. You get zero inside of the disk because outside is harmonic inside constants. And you get something on the circle. And it's well known that it's a pretty real measure of the circle in complex analysis, integration current on the circle of producer. And in our context, you get, we get something different. In some sense, the boundary of the disk or this R is simply the Gauss point at the other one, which was related to the semi norm P, D zero R, and we get the integration on this space. And it's an analog of this function in works. So I've skipped my time over by more already two minutes. So, I will just tell you what I wanted to don't don't try to read this. So, just to show, I will show them but in from this, once you have forms and currents, you can define a matrix on nine walls, just by copying the definitions from a 19 geometry. In the next nine walls, we'll have coverage of forms. And because of the point career and formula, when you integrate products of coverage of forms. We get the degree from algebraic geometry, from algebraic geometry when the situation comes from algebraic geometry. The more interesting and more difficult is that when we take, when we start from a natural matrix format for metrics which are natural for from algebraic geometry but given by models like in a record of geometry, those metrics are not smooth. They are out corners are defined by some maximum functions. And so when we take the, the churn form we don't get a form but a current. And for for the same reasons that we can, we are able to define products of those currents and and to give formulas for for for the for the products of those currents and in fact, we recover exactly what is given in a record of geometry by formulas which are natural from from the point of view of geometry and which allowed me 15 years ago to define measures on on back of each faces. So here we don't in the end we don't have only measures that we have measures for matrices and goals that we have forms and of all types from for for these matrices and goals. Also, we are able to understand the positivity we are able to understand the intermediate currents and the positivity. So this is just the starting point of business. It promises to be useful and we hope it will be. What is remarkable for us is that it has been used in many works. Meanwhile, so our paper is not yet finished but people have done interesting stuff with it. So, so for example, gobbler and Kuhneman should gobbler and Kuhneman have shown that something is making a lot of noise is my my efforts. So gobbler and Kuhneman used it together with other ideas from tropical geometry to to define quite a full set of geometry in the moment setting you and Mikami have shown a relation of the natural geometry groups that can be defined by these forms with child groups and K theory. It also appears more recently in a work by Ducro Ruchensky and Leather, who showed that Archimedean integrals naturally converge to non Archimedean integrals in the sense that I described. So, I think it's a last slide. There are a few three additional slides that are empty in order to be able to write something. If I need it. So, thank you very much for the attention. I'm sorry to be slightly overturned.