Thank you very much for the invitation and for coming around. I want to talk about density of rational points close to manifolds. And I'm going to start with the question of rational points on algebraic varieties. That's where I came into this field from this point of view, in particular about the dimension gross conjecture and a brief sort of account of what this is about. Then I want to talk to this more general question of looking at a manifold and rational points close to the manifold and explain how this has shed light on the rational and the dimension gross conjecture in recent years. And in the last part, I will then outline applications of also some recent work, including collaborators Ratschler-Schribastava and Niklas Technow-Meinz to defend an approximation. So let's start at the very beginning. If I take a projective variety, say, given by our homogeneous polynomials with individual coefficients, let's start over the rational numbers Q for now, then I obtain a projective variety, say, V. So this is zero state of my R, homogeneous polynomials. Then some very basic questions that you may want to ask about your variety V is, for example, about the existence of rational points. Of course, a very deep and vast topic, which I'm not going to touch upon, but another question could be if there's maybe infinity maybe points, how about a distribution of rational points on V? In terms of distribution, if you have infinitely many points, you may want to count them up to a certain height. So let's first define some height function. Let's say, again, V is our projective variety. Let's assume for now it's irreducible under a certain degree, say, D. If you have a point in projective N space, say, X in homogeneous coordinates X0 to Xn, then we can always find a representative where all the coordinates are integers and co-prime. There's exactly two such representatives. On this plus or minus sign, you can change the whole vector. Then we define a height. It's a max of the absolute values of the coefficients. So this is well-defined. And this gives you what we call an naive height on there. So given we have the naive height, you may then ask the question, if I look at my variety, V, and the rational points, and then I can ask how can you maybe count the number of rational points of height bounded above, say, B, where B is some parameter which you may envision going to infinity. In particular, we're going to talk about a batch number of upper bounds for this function. A first trivial upper bound is given by the dimension B to the dimension plus one. You can prove that by induction, basically by slicing, if you look at the affine cone. And we're going to look now at one concrete example here. So this is, again, the definition of the height function. If you have a point in Pn, the height is given as the max of the absolute values of the coordinates if they are given by a set of co-prime integers. If we then count rational points on the variety Pn itself, then we want to count all such two both x0 to xn, that a GCD is equal to one, coordinates bounded by B for all i. And we have to divide by a factor of half because there's exactly two representatives, the vector, and minus its vector. They give you the same point in projective space. So this would be a concrete counting problem. And you already see, if you for now ignore the GCD for a moment, then for each of the xi's, we have roughly B to B plus one choices. So you should have something of odd of magnitude B to the power n plus one. It's actually true that GCD only changes the leading constant, the number of rational points of naive height bounded by B grows like B to the n plus one. This matches, again, our trivial bound with the dimension plus one. So the dimension of Pn is a projective variety. It's n, so we get it B to the dimension plus one back. But this is not a hard theorem. You can also do such things over a number of fields and proof asymptotics, where the shadow will. But for us, this is now enough. And I want to reformulate it now for a moment into a linear subspace. Well, it's basically the same thing. I just embed in a different way. If say V is the linear subspace in Pn given by one linear equation, it's not the general. Then the dimension of this linear subspace is n minus one. And by the same arguments, we get up to a constant B to the n points. So think about choosing x1 to xn freely, and then x0 is uniquely determined. Linear subspaces tend to have many rational points. And you may now ask the question, if we assume that we do look at irreducible varieties, which are not a linear subspace, can we maybe get some non-trivial, upper bound some improvements on that. Some improvements may be so wrong about some stronger upper bound. If you move from linear hypersurface to quadratic hypersurface, as all the AIs are non-zero, say a full rank might work here, then one can count rational points of bounded height. And you will obtain that there is constant times B to the n minus one points on there. I assume n is at least 4 for safety, because otherwise there may be at some point a logarithm popping up. This can, for example, be done by things like the circle method. And so heuristic of how to arrive, there could be that you say the xi's, they all have size at most B. They're all integers. So it has something like B to the n plus one choice is for your variables xi. And this is a quadratic equation. So if the values were all, say, randomly distributed in the image where they could land, which is an interval of lengths around B squared, because it may be a typical value of this quadratic polynomial but stick in variables of size B, maybe size B squared, then if they were distributed equally in this length of size B squared, it would give me the probability of B to the minus two, that this is zero, this heuristic would lead to the asymptotic B to the n minus one. You can now do the same thing for larger degrees. So if we say take here degree D in the diagonal hypersurface of degree D, you can look at the same equation. And by the heuristic, I would now still have B to the n plus one choice is for the coordinates but a degree D equation. So say the probability of this. So again, it will distribute it would be B to the minus D for one such sample choice to be a solution to the equation. One can actually prove such things for degree D hypersurface. This is true of roughly n is of size D squared, which uses a resolution of nucleotus mean value theorem. You can then we learn Gussner, general case of a really previously on a degree three and also what's general later on. I don't go and talk about the fine details here, just want to sort of point out that what it seems like if you look at this question from a naive point, if you had a higher degree, what are the last points you'd expect here? This is just an example, but one can phrase much more generally in terms of mining's conjecture for rational points and varieties. If you read here this n plus one minus D and you think from an algebraic geometry point of view, you may say like this n plus one minus D is if you look at a canonical sheath of the endless of minus n minus one, the canonical sheath of the hypersurface is O of minus n minus one plus D. So the anti-canonical sheath is exactly given by O of n plus one minus D. And this is exactly sort of this matching exponent. In other words, if you would use an anti-canonical embedding of this variety, you would somehow normalize this exponent away. Sort of that is exponent is somehow related into looking at an anti-canonical embedding that's part of what is formulated in mining's conjecture for rational points and projective varieties of a number of fields, but we're not going in this direction. Despite it seems that higher the degree is, the less points it has, this is all conditional on this varieties being nice enough. If we are given a variety and you don't know anything about it, maybe you don't know if it's smooth or anything, the worst things can happen. So this is a very simple example, but just to illustrate a point, if you don't make any assumption, say look at this example, a hypersurface again in PAN, take some polynomials, should be homogeneous of the same degree of zero F one, then look at equation X zero F zero minus X one F one zero, that contains many rational points because I could set X zero and X one equal to zero. So we produce a high dimensional rational subspace on there. If you choose X zero and X one equal to zero, other coordinates are free. So you should expect B to the N minus points on there, which is again B to the dimension. And you can make the degree as large as you like if you have this sort of linear subspace container, that kind of singularity which is reflected and you can't do better. Dimension close conjecture gives indeed an upper bound which works or which tries to cover all projective varieties or Q of a given degree at least two, no matter how singular they are. And the conjecture is that indeed, this example that we've seen on this singular thing and the same that thing that we have seen for the boardwalk is the worst situation that can happen. So the conjecture is that if you have an irreducible projective variety of degree at least two, then the number of rational points and there is bounded by B to the dimension plus epsilon. So that matches this example because N minus one is the dimension of your hypersurface and also the one for a quadrack. It's called weak dimension close conjecture. And you could also conjecture stronger things and also in certain situations, stronger results are known. There's for example, a plus epsilon, you could debate when does it actually necessary or when it should maybe disappear. I put here a V in the absolute, in the constant. You could also here ask maybe there's some uniformity in terms of the degree or dimension of the variety. Both of these, there's a meaningful, yeah, there's work results in this direction. So indeed, sharper versions exist in some situations in the weak versions already solved in the work of Browning, East Brown and Saltberg are using their determinant method. So that's a very, say, a arithmetic approach where you try to locate your rational points in some auxiliary sub varieties and then try to intersect those sort of by pursuit type styles so we'll just get our bounds in the end. And just let me mention here along the way, we see very nice work of Astrid, Klugeus, and Lien and Dittmann who managed the least degree, at least five to remove the epsilon and get even here a polynomial dependence in the degree. So not really a uniform result but even a very explicit one. So this was an introduction of one side where I think I got interested in this area. Instead of counting points on a algebraic right, you could also try to count them close to a variety or more generally you could take any manifold I started now just with a board we could just not define over Q. You may now ask here for integer points close to the rational points which are close to it. This example by open minds conjecture where we'd get arbitrarily close to this quadric. But so imagine like instead of you're lying now in the thing I want to have points close to being somewhat just in a neighborhood. We can formulate this as follows. If you have some bounded set manifolds here of dimension little m that's gonna now stated dimension for the rest of the talk here. Then we define a counting function which comes with two parameters. First capital Q and delta. Q measures the complexity of your rational point the size of the largest denominator. If I look at a rational point and say parameterizes with P over Q where P is an integer vector and Q is a natural number of size up to at most capital Q. So this is just a fraction here of an n tuple of rational numbers which are written with the same denominator. And you may ask now for points of this form which are close to M. M is something bounded so automatically then P will be also bounded by constant times Q and you may ask for the number of points which are close to M by distance at most delta over Q. So the delta measures how far you want to be away. Distance could be for example Euclidean distance. This Q is there for normalization purposes but in practice this doesn't hurt much. So that basically is like taking a shell around you manifold and asking for rational point which are close to there and that's closely related to the dimension close conjecture because if you take some algebraic variety and look at some alpha and patch and rational points on there then rational points on your manifold which say comes from the algebraic varieties exactly the same which you would count here for delta equal to zero. So you're kind of sticking the thing up and look at what's in here. That's also trivial bound. The first real upper bound you could get is of size Q to the M plus one. We may say about is a bit more later for example if R one, if you look in R one at an interval from zero to one then I would count here fractions P over Q one dimensional rational numbers. Fractions P over Q between zero and one and that's about Q square of them. This is what you see here. This would be a Q to the power one plus one. Sometimes there are that many points because you could for example choose as a manifold a piece of a rational linear subspace. Just the example we had takes some interval from zero to one in the line R one and then we will find actually Q square points on there. So this trivial upper bound of Q to the M plus one is sometimes realized for rational linear subspaces. This may be not a typical case. There's no alternative for realistic that I want to draw your attention to. So for a moment let's say M is the dimension of your manifold, R is the co-dimension of N is the dimension of the ambient space. Then you could try to understand this problem as counting lattice points. You could now imagine in this formula to first fix little Q that gives you capital Q choices. Now if I fix this little Q then I look at somehow a shell around M of well, sign up by delta over capital Q little Q is of size roughly capital Q for now for simplicity. This thing has co-dimension R so the volume that I get by sign up my co-dimension R manifold by delta over Q should roughly be delta over Q to the power co-dimension. So this delta over Q to the co-mine dimensions the volume in a shell I look at. And in this shell you then ask for rational points with the same denominator of Q. That's a lattice. So you look at if you fix that Q do you look at lattice and it says the determinant Q to the minus M. So if you naively think about this like taking the volume dividing by the determinant of your lattice you would expect in this shell delta over Q to the R times Q to the N points. And that's another factor of Q because we then in the end let vary Q. So if we calculate this we get the other to the co-dimension and Q to the dimension plus one. That is another characteristic that you may come up with. So maybe some point before we go ahead that rational linear subspace is containing many rational points is something special. If you take any manifold which is not a rational linear subspace similarly as in the dimension gross conjecture you may expect that there's far less rational points on there. So you might expect that if there's far less rational points on there then maybe if delta is somewhat large this contribution is the one that you could maybe see you in exhibit. But even then your manifold may still contain itself a reasonable number of rational points. There's just two examples maybe you can look at a second one for time reasons. If you take an N minus one sphere that's one equation say one squared plus XN squared equals to one. If you ask for rational points on there then by the homogenizing this is the same as asking solutions to the equation say Y1 squared up to YN squared is equal to say Z squared in integers. So again this is like the same question all I said we asked for a quadric value on and if I said your delta equals to zero and count the rational points on the sphere then you get back to constant again Q to the power N minus one points. So you expect on the sphere Q to the N minus one points and if you compare the data heuristic R would be equal to one in that case and M would be N minus one. So we would compare this delta times Q to the power N. Can you see that as long as delta times Q is larger than one this heuristic time should dominate the points which are on here. So we may expect that there's some lower bound for delta for which somehow in this shell around a manifold is large enough that say this heuristic expectation is the one that you actually count which is larger than the same the stuff that is going on the manifold itself. Nundi this is a conjecture which in this form has appeared in work of Yen-ni-huang I'm going to come to that in the next slide. If you take a bounded submanifold of our N boundary is proper curvature conditions this is actually regnet's supposed to be like that. So the question in the end under what conditions one can establish that conjecture then first of all one can conjecture that is there's a general upper bound which consists of this kind of heuristic how many things we generically expect plus a term which is the same that we've seen in the dimension cross conjecture. This is Q to the dimension plus epsilon is for the same reason here because if you think about an algebraic variety and I take an affine patch of this say projective variety and then take an affine patch and then could consider this as your manifold and then you may expect by the dimension cross conjecture maybe in the worst case Q to the n plus epsilon points on there. And if you then vary both the conjecture maybe that no matter what manifold which is somehow reasonably behaved has the same property that on the manifold you don't find more than these points and sort of yeah you could say like this should be maybe all the contributions you see the points on the manifold plus the thing that then adds up. You can of course then this compare those two terms so if you look at those two terms you see that as soon as delta to the r is larger than Q inverse or delta is larger than Q to the minus one over r then the first term is larger than the second term so you could conjecture that if delta has at least this size you may get this as an asymptotic delta to the r, Q to the m plus one if Q goes to infinity. For example if r is equal to one we get here again the bound delta is at least Q to the minus one which we've seen in the example on the previous slide. And there are examples like if a Markov that show that some curvature conditions are necessary even for every point like if you look at a Markov or D at least three then the point one zero zero one you have a vanishing Gaussian curvature and you get that many points which you could stick in delta close to Q to the minus one is gonna be larger than what would it be conjectured. This main conjecture has been proved in rather seminal work by one in 2019 for the case of hypersurface smooth compact hypersurfaces is the restriction that a Gaussian curvature is bounded away from theorem. Well, if we need some constraint here for example, yeah, rationally near subspaces we have we need to exclude for sure but maybe also other things like if a Markov and this is one on theorem that we now have in this generality. There was lots of previous work in this direction for curves Huxley was obtaining upper and lower bounds which were almost sharp then there was for C3 curves or C2 curves hence if you talk about upper or lower bounds works of Berezniewicz, Dickinson and Villani and one in the line upper bound first ones for the lower bound and also then a work of one for curves. So there's been a lot of building up in this direction but this was really a very milestone result and some very interesting consequences. That's why I started a talk in discussing the dimension curves conjecture that you can reproof the dimension curves conjecture for the varieties for which the corresponding hypersurfaces have a non-vanishing Gaussian curvature. This is because if I stick in in this conjecture delta equal to zero you get as an upper bound exactly due to the M plus epsilon if you rephrase this language from the algebraic varieties side to the manifold side this is exactly the same thing that you talk here about. This is very interesting because one's work is purely analytical. You throw away all the algebraic structure and you only remember that there's a manifold that you work with. And so in contrast to the earlier proofs using the determinant method this is very interesting that you somehow don't need any arithmetic structure at least for a certain class of varieties. Something that I want to touch upon for a moment is work in higher code dimension that a few years ago I was interested in together with Shintari Yamagishi which needs us to more general results. So a serious restriction is the cyber surfaces and we want to point out that one can do for certain classes more. If you say parametrize your manifold by r functions so say I take a vector x this is a vector say x1 to xm and then I take r functions this parameterizes a code dimension r manifold in r to the m plus r of dimension m maybe consider this thing locally in some ball then first step is you want to formulate this is a counting problem we are going to throw in smooth weights to make these things better behaved this may look like that. We look at this counting function here we count vectors a and set to the m and q up to capital Q so this a over q is the first m coordinates that you look in here for this x variable. And if I stick in for this x and rational point a over q I want to somehow that all my f i's evaluated at a over q are also close to rational point that you can formulate by asking for q times f i of a over q being close to an integer. This notation up above alpha for real numbers the minimum distance to the next integer. So if qf1 of a over q is less than delta for example it means that q times the first function f1 minus some integer b1 is an absolute value less than equal to delta on other words f1 minus b1 over q is less than equal to delta over q. That's exactly what we were looking for because this sort of says that the coordinate here is also close to a rational number with the same denominator little few. So is this really formulated and asking for vectors a over q which of these functions get well approximated. And here again we see it back another agenda heuristic but we expect how this may grow in a generic case. If I would drop the conditions of these functions f1 to fr and if w is some smooth weight function maybe think about just an indicator function of an interval for a moment then we would have the q to the n plus one points that we can't say little q runs up to size capital U and the others a run up to size q to as well. So get q to the n plus one and each of these integer parts lands between zero and a half so the probability that you hit delta is something like two delta and if these events are independent we should expect that say a proportion of delta that are often points actually satisfy these inequalities and that's the same heuristic that we had early on. In the case of a hypersurface say if r is equal to one and your parameterized by one function f1 then non-zero Gaussian curvature is in a given point x0 is the same thing as saying that a Hessian matrix does not vanish at that point. The class of many faults of one can extend this work of one which is more like a cloud. It's an interesting class with one example where one can do something is asking for the Hessian in the whole pencil of functions that parameterize your manifold never to vanish. It's a rather strong condition. One point out that he asked this in a real pencil in a complex or complex numbers it was not be possible so he asked in a real setting about this. So think about your Hessian matrix and take the linear combination of those then if you compute a determinant you end up for example with a power of a positive definite quadratic form and d1 to dr that would be one example that is would be realized. Such examples do exist for example from results of determinants of representations you can construct those but the nice thing is that one can prove again similar asymptotics and other bounds and the reason why I think this is a nice class on interesting class for one reason is that this conditions actually not as completely maybe random unless you may sing in first place also an analysis of one of my co-authors Wachler-Chevastava then later explained to me that also over there this is an assumption that sometimes you may like to use but a nice thing from an arithmetic point if you is nice that if you look at this class for which well there exist examples then we actually can count the number of points in a much thinner shell than we were conjecturing to. So the results that you get are actually stronger than what you would conjecture. So you conjecture that only up to q to the minus one over r you can count stuff but actually see we get a smaller at least if the co-dimensions, yeah, certain ranges. Well, even if there's something that is for example if the dimension m goes close very large and you actually approach delta as large and q to the minus one which would be the more the same for hypersurfaces which I think is kind of interesting because very often in a dimension gross conjecture you just project things down to the hypersurfaces there may be also sometimes certain conditions more going on in higher code dimension which would be suggesting by this example. So if we the reason why we can count in thinner shells is that we can show there's less points on the manifold that's why we can zoom in more because there's just less that could be disturbed and may encounter and display here the concrete upper bound it's maybe not, yeah, I can talk about what this is the trivial one, not a trivial so this would be the conjecture q to the m and we save some power. And again, if we think about the dimension m is being very large you actually say from r minus one here so for each code dimension you seem to save maybe you could save close to a power but I'm not gonna, yeah, in the detail here too much I just want to also maybe point out along the way of that it's not so easy to get away from the curvature condition that the corrosion curvature is not equal to a zero. So in higher code dimension even if this someone make this much stronger there's one very nice example by Rajschlaj Shrivastava Nick Klaas-Techno where they can remove for one point this corrosion curvature condition essentially spirit they look at hypersurfaces it's a kind of this shape which are given by a homogeneous function so they have function f such that f of lambda x is lambda due to some degree times f of x you could think about something like an L2 norm to a power of D as a concrete example and they manage yeah, so this by construction if D is larger than two it's gonna have a vanishing hasheon in the origin so you kind of construct examples that way that the hasheon vanishes exactly this one point in the origin that's the situation that they study that hasheon otherwise is non-zero and they need to obtain here upper and lower bounds of the sharp for the range delta if you inverse or the thing where you would want to study and something very interesting here is that there's another term that pops up in fact the result also holds for capital D being larger than n minus one and I would have to write it up slightly differently so this is one say sample result something interesting about their verse that they actually showed that if delta gets somewhat small and if the degree D if your homogenous polynomial is large then there is actually some other contribution that you're going to see from some kind of what you could call a cap set to think about a thing but just being very flat your hyper surface such that if you're zooming into the origin that somehow you just behave sort of almost that linear that has somehow many points in there that also shows that you can't just conjecture everything to go through without any further assumptions on curvature for example so I really like this work a lot because it brings a lot of interesting tools from the analytic side and sort of bootstrapping arguments which are very nice to have here but of course you may also want to go further and ask what happens now for even more general manifolds that these are all very nice examples that do something higher code dimension or for one point being maybe exceptional but what if you want to drop now basically all conditions in a bay maybe you don't want to look at a rational linear subspace we understand them but maybe one of the biggest classes that you could consider are what is called L non degenerate manifolds would say that if you have some manifold which again is parametrized here say by our functions and X now again an m dimensional vector so this m dimensional vector and our functions f one to f r you would say it's L non degenerate if the partial derivatives this vector of order up to L span or n in a point and say it's L non degenerate if this happens thing for all the points if you manifold completely this just means that your manifold is not contained in some linear subspace if you think for a moment about a hyper surface so one function f then being non degenerate means that there's just some derivative somewhere which is not going to be zero in every point that's really the biggest thing oh sorry derivative of all at least two that's not going to be equal to zero so in a way it's somewhat of a very very big notion that you could ask for and something surprising that together with Ratschler-Schwesterer and Niklas Teichler and we figured out is that we can actually do something about that class formally there's some counting function you would associate it we will stick in even more smooth rates even smoothen out the nominator and everything else but so sing about this counting function here as the same counting function that we have seen previously so this counts the rational points close to this manifold where the nominator is bounded by capital Q and you look in a shell of size delta divided by capital Q and under this very weak assumption of L non degeneracy we do manage to get an asymptotic with the constant the one that you would predict under that smoothing and delta to the RQ to the M plus one with some power saving in Q as soon as delta is not too small that we need a lower bound the delta is certainly expected this is most likely not a correct lower bound but it's some lower bound that's already very nice because I hope that I tried to illustrate a bit with the previous exam it's not so easy to just get rid of somehow the curvature conditions so in this context it's to our knowledge the first bound that gives an asymptotic in that generality yeah again sort of disbound while it gets bigger the larger dimension co-dimension or dimension and embedding dimension and also the non degeneracy parameters we can get lower bounds which are stronger for the lower bounds we actually get here the same constant so this constant I phrase it now here there exists a constant but what I really mean is we can produce a lower bound with the exact thing that you expect up to an error term which again is power saving in Q as soon as delta is at least as large as Q to the minus three over two and minus one there's been in 2012 rather groundbreaking work of Viktor Boris Nevich in publishing the animals in this case where he produced similar lower bounds for non-degenerate analytic manifolds so we could relax that a bit too smooth instead of analytic for delta larger than Q to the minus one over R so with that for the case that our depends on the dimension and co-dimension we actually get a stronger bound with three over two and minus one that's very nice because we can produce in some sense now even more lower bounds and also with different techniques we can do that there's also upper bounds you can stick all of these things together that we've seen before you get a uniform upper bound which incorporates the term that you expect and some well maybe not so nice looking term but what you get in the end if you balance these out and think about what can be the worst case being you obtain a bound for the number of points on the manifolds so that comes basically from saying that if I make delta smaller then also get less points so that there's a certain point for delta where at both times your balance and then this is an upper bound also for delta equal to zero because there's only getting more points if I think this up so there's a bound of the Jp plus one oh, okay minus one over two Ld times n plus one and here I made now here a yeah, I'm sorry about that I will correct that in the things to be uploaded this is the dimension plus one this is an n plus one minus something so we don't hear the dimension because conjecture is dead but we get some very general upper bound if you would think about projective varieties there's been some very nice work of a Schulte animoto where he gets for projective varieties some upper bounds though then you need to know something about the geometry even to compute the exponents and that's a nice aspect here that we get something of free now I'm going to say just a few words about a proof if you follow some or part of our proof isn't filed by a work of there is Nevischen Yang which I'm going to get to in a moment but the new part is using pro-assamation and introducing more free analysis into this thing we use pro-assamation we want to use things to studying oscillatory integrals the main strategy and using here already the language that there is Nevischen Yang introduce of generic and special parts in a way of previous work you try to some look at your manifold and there was some or there was there was Nevischen Yang introduced there were some special parts which are hard to handle which you have to somehow cut out and don't know what to do about this much very well and some generic part where you can somehow try to get up a balance we also do something similar but analytically we construct carefully a rate function to do this and then so for the generic parts we use even rapid decay estimates a bit more and we have to be a bit careful because things depend more parameters than maybe yeah your standard application maybe used to but with careful bookkeeping this works out and for the special part we use work of Bernick Kleinberg and Margules which already appeared in the work of Paris Nevischen Yang with quantitative non divergence which basically tells us that a set of the manifold where you have a point for which there exists some shift vector on a power saw and some which makes this integral somehow in a way that we can't control that this volume is small so basically tells you that there's not many points for which there exist shift vectors which are bad and this work of Bernick Kleinberg and Margules has already been also used in many contexts but for example in this work of Bernick Nevischen Yang so for us the new part is that we combine it with the free analytic side so for the last 10 minutes I want to discuss applications in diaphragm and approximation and see where also the work of that dimension of Paris Nevischen Yang which has been motivated from was not enough in the very beginning here again but directly lemma you can approximate a real number theta with a fraction a over q by at least one over q squared and multi-dimensionally if you have a vector theta one to theta n if you want to approximate it with a fraction a i over q simultaneously you can do that up to q to the minus one minus one over n you can also formulate that in terms of approximation functions if you're given a function psi and a point in Rn we say that it's psi approximable if there's an equality that a vector y minus a vector a over q in the infinity term norm is less than psi of q over q for infinitely many such choices that this infinity is supposed to be the infinity norm so we want that each coordinate y i minus a i over q is an absolute value less than psi of q over q for all i so for example by Dirichlet's theorem if you choose what psi of q function q to the minus one over n then you get here on the right-hand side minus one minus one over n that's exactly sort of what you come from this multi-dimensional Dirichlet lemma and Dirichlet's theorem would tell us that everything has infinitely many such approximations we can approximate every point that well you may now look at other approximation functions a very classical theorem in this area is Keynesian theorem which basically says if I look at some monotonic approximation function then there's a break of the pending whether the series of psi of q to the n is convergent or divergent so psi of q to the n is convergent which means some upsides rather quickly decaying so I want a rather very good approximation then u n is the the back measure of the psi approximate of the points in our n is zero so if you want to approximate your point so well that is psi of q to the n is convergent so then you make this very small here yeah then this is only for the back measure zero point at most possible on the other hand in the divergence case the complement has the back measure zero so almost all points are that well approximate if you would take Keynesian theorem and stick in again the approximation function q to the minus one over n then we are in a divergence case and so you would get that a complement has the back measure zero so almost all points are that well approximate while we even know better by Dirichlet's theorem but just want to show you that this is the normalization which we can which pops up here Keynesian theorem is basic application of Borel-Cantelli lemma which I just want to call quickly here so we are in a case that this approximation function has this convergence and if you then say look at a cube and of the n dimensional cube in our n and the psi approximately points in there you could ask the other question if I fix the denominator q which real numbers are close to a rational part with denominator q in this box zero one to the n so if I look at vectors a i over q and if you think this up by psi of q over q then you would look at such kind of it's a little boxes in each direction you're sitting up by psi of q over q and you take the union over all the a's this would be then everything that would be set of real numbers which are somehow close to one point with the denominator the little cube the measure of this thing is psi of q to the n because each little box has measured psi of q to the n over q to the n and you have q to the n such rational points denominator q but the sum of these measures of the eq is convergent and by Boyle Cantelli that means that for almost all points you can at most slightly and finitely many of those so the measure of those where you are contained in infinity many of these e sub q's must be 0 and that's exactly what we wanted to show we wanted to say that for most points there can't be infinitely many approximations if you're in a convergence case so I'm showing this argument to you because I want to point out that what we use here is basically we are able to count rational points of denominator equal to q in a box and if you now want to do approximation not in rn but close to a manifold that's where then the results of counting rational points close to a manifold come in very much for that reason a question has been phrased in different variations different works that I named here it's been a following that if I take a sub manifold in rn and again a monotonic approximation function for which I assume this series to convert to diverge then you could ask if also restricted to the manifold almost all or no points are psi-approximable so imagine you take an approximation function for which this thing converges then you know that in rn the lag measure is zero of the points that you can very well approximate but it could be that maybe they are all somehow contained or that in your manifold it could be that your manifold behaves exceptionally that despite an rn you can approximate that well typically you can on m still also the other way around and so this question basically asks that if I restrict to a manifold does this still behave somewhat the same way does a typical point on rn behave like a typical point on a man-vehicle the divergence case has been essentially proved for smooth manifolds from the general by the work of barris-nevich that I mentioned earlier on on lower bounds and rational bounds and close to manifolds and very recently the convergence case has been proved by barris-nevich in young exactly using other bounds for rational points close to manifolds and the reason why you need here other bounds for rational points close to manifolds is exactly the argument here by Borel-Cantemini more or less if in total they are not for each denominator Q or h of denominators 2 if they are in total not too many rational points then there can't be also very large measure of real points which are close to these rational points so this is a direct application and with our work we can actually re-produce that theorem so our upper bounds that we obtain for rational points close to manifolds are stronger what is produced in there but maybe because we've managed to kind of not in the free analytic techniques on that side that is not visible in re-proving that theorem but it's visible if you ask finer questions in this direction if you say take some subsend to find dimension as a host of dimension and write this what host of measure HS then you could ask if I look at some approximation function Q to the minus tau and for example for tau is equal to minus 1 over n this is what's covered by Dirichlet's theorem host of dimension should be n everything Rn is that well approximate but if I make tau a dull large so if I want to approximate better then I know that a Lebesgue measure is going to be equal to 0 of the things that I can approximate well but maybe the host of measure is somehow still non-zero you could ask about a host of dimension of the things that are somehow even better approximable and in Rn one can solve this work of Janik Bezikowicz for large tau's for just Rn which again is sort of a very similar concept as to what you've seen in the Kimchins type theorem and you can now ask a similar question for a manifold so if you take a manifold and look at the points which are very well approximable as n of psi you somehow want to approximate very well can you still find the dimension the host of dimension and there's a question or conjecture I will move to this slide this appears in Barysimic in Yang as well so they say that maybe by a similar heuristic of volume computations so this is a very natural thing here one can compute a dimension the host of dimension of these very well approximable points for certain values of tau and there's a certain limit where this may not work anyone has also good reasons for this and one question is now how large can we make this tau so to what extent can we still understand the host of dimension so the larger tau gets the better we want to approximate the points and so it's a challenge to ask like how much large can I make the tau how well can I approximate and still understand at least the dimension of the things that are that well approximate and given that we get stronger upper bounds by our techniques we can improve on that aspect by well that's something neither do the work of Barysimic in Yang give the conjecture now of course do we but a range of say these values of tau such that we can find a dimension that gets larger there's different inequalities and we're not so illuminating but I just want to say that sort of information that we have on rational points close to many false gifts sort of here of information on about a fan and approximation so I think now we are maybe just in time so thank you very much


So today we're going to talk about Gitbomb. Now, originally, this talk was going to be given by Ed Warnocky and Ava Black. But Ed Warnocky got iced in in Austin. And Ava Black came down with an illness. And so we went down through the Rolodex of various other omnibore people who are present. And I was top of the list. So please excuse me. The slide deck is not one that I wrote, but I will do my best to give the spirit of the talk. I did write a significant portion of two of the six implementations of omnibore. So I can definitely answer questions on the technical side. Please raise your hand and interrupt me if you have questions along the way, because it's better to get the questions out as we go rather than let them sit and linger. Because there'll be other people with the same set of questions. But it's up to you on how you want to deal with that. So first, let's talk about timeline. So we had a colonial pipeline hack, which occurred in May 2021. Pretty sure everyone here is familiar with it. And that ended up causing the cybersecurity executive order to be put out by the Biden administration. But if we go back in time, we actually can see that much of the work that had been done that led up to omnibore was the creation of SPDX, which was launched in February 2010. And before that, where some of that work came from was from initially an industrialized SBOM that would have been done at Cisco, where Cisco had a requirement to know exactly where all their infrastructure or all of their stuff was coming from. So they can work out where did the hardware come from, what software patches and so on within their switches and routers. And so, however, mistakes were made. But at the same time, much was learned. So again, back to the timeline. Colonial pipeline hack, May 6. Take a look at the timeline. Between May 6 to May 12 was the time between when they, and actually what this means is that they were actually working on the release, but they were just waiting for the opportune time to actually release it out. And this was huge. So the timing was perfect. They launched it, got the support. The first omnibore talk, well, at the time it was called Gidbomb, but now the first omnibore talk occurred May 28. We actually were doing a lot of work before this as well to lead up to it. But we ended up talking about amongst each other and having various private conversations between people. We actually did the community launch in February of last year. So it's slightly under a year old. When we announced it, this was on the day of the reach that we had. So we had some pretty good outreach. We were pretty happy with that. And the interest has definitely kept up. So however, one of the things that we did, and especially Ava Black, was absolutely fantastic with this for six months, went around and spoke with various people. So she would listen to what people had to say about the space, usually over a cup of tea or coffee or something similar, hence the baby Yoda. And found that there were issues around how S-bombs were done, how do we actually correlate them to the things that we're working on. And reality is, in the small scale, it looked OK. But in the large scale, when you actually start to scale it out, it actually looked a little bit more like this, like trying to trace where everything was. And the general response to that was horror or frustration or a mixture of both. And this is all to answer a simple question. Am I safe? Is what I am doing safe? But in order to answer the question whether you're safe or not, you also have to look at the simplicity of the thing you're doing, because you cannot answer that question if things are too complex.  And so we found many of us on the project have a significant distributed systems experience. And so we aim towards things that are more simple to try to keep reliable, try to keep it performant, and also so you can actually reason about the security. I want to make a point here. There's actually great talk by, I think it was about 15 years ago, maybe, a strange lube by a guy named Rich Hickey, who created the Clojure language. The talk was not about Clojure. The talk was about simple versus easy. And just because it's something as simple doesn't mean it's easy, just because easy doesn't mean it's simple. And very often, you can get lulled into the idea that something is easy, and then work out later on that when something breaks, you now have this massive rat's nest that you have to deal with. So we wanted to avoid that. Like, yes, there may be some areas where the simplicity may be at odds with the ease of use of it, but we want to make sure that what we were doing is something to be reasoned about. And there's different ways, as well, when you look at the complexity of something. So on here, we have a very simple example of a line graph. And you think, OK, well, I can actually use two points x and y in order to identify where something is. If you change your perspective, you actually only need one number because you could say, well, let's angle ourselves in such a way that we only need one number in order to identify where things are actually at. So part of what we're looking at is, well, how do we change our perspective a little bit in the software supply chain so that we could see if there's some way we can simplify the problem a little bit further? And that was part of what we were asking the question on with our neighbor. We actually had something that was a little bit more complex, we worked out if we make it a little bit smaller, a little bit smaller. What actually gets to the heart of what we want to do? So there's three simple questions that we came across. The first one is, what is the identity of something? By identity, I don't mean this is the URI or URL of the package, or this is the human name we give it. What is the canonical identity of what a thing is? Dependency is, what is in the artifact? Metadata is, things that are not in the artifact, there are not the identity of it, but are something extra that we've learned about it. So we have to have the ability to attach metadata to whatever it is that we're doing. A really good example of metadata could be something like the license of a project. Could be who compiled it is metadata. What did your image scanner say about something as an initial metadata? So jumping more into identity, we ask what is a software artifact? So an artifact in this scenario is any software object that is of interest. One thing that all software artifacts up to now have in common, and who knows, maybe in the far future this will change, but something I'll have in common right now is that they're represented as a byte of arrays. It doesn't matter whether you're using source code or object files, JAR files, Python or Python C, Debian RPMs, OCI images, they're all represented as an array of bytes. So we say that two artifacts are equivalent, if and only if the byte array of one is equal to the byte array of the second. So with this, this gives us a sense as to what a unique artifact ID. Of course, it'd be terrible to stick the whole application in an S-bomb, so we have to do something a little smarter. But in general, there's three properties. It's canonical. Independent parties presented with equivalent artifacts can derive the same identity. That they are unique in that two non-equivalent artifacts have a separate and distinct identity, and that it is immutable, that once you have the identity of something, it's not going to shift into something else. So these are the three properties we're looking for. Some identity non-solutions that we ran into was, you'll often see many things, this is actually prevalent in many S-bombs, when people think of S-bombs, is like, well, the file name, the file name is the identity of the thing. Well, file name is actually metadata. It's not actually the identity of it, because your file name can change. I can change foo.c to be bar.c, or I can actually change what's in foo.c to something else, and so now the identity should change. If I move it to a different directory, as I changed. Or what if I take that file and just stick the entire contents into a database somewhere where there's no file name? So file name's not quite there. They're good for locating things, which is not good for defining the identity of something. The same problem is with URLs. PURL's a little bit better than just the URL, but it still has the same set of problems. You're looking at the location of where something is, which is a hugely valuable thing, but is not the identity of an artifact. The minimum elements of an S-bomb, which I am actually a fan of, I know a lot of people rail against them, but the reason I'm a fan of them is because they're very simple. There's something that companies can do today that move the needle, but also sets them up so if they can do this, and then they can do a little bit more next time, and then they can do a little bit more of this year, next year, and so on. So it actually gets them in a baseline. We can increase that baseline over time. But that being said, it's not an identity. It doesn't actually provide you. And especially when you look at the kernel versions, like I have kernel version 5.17.3. I compile it. You compile the same version. We're actually coming up with two different pieces of software because in general, there may be around 3,000 files that actually get used, and depending on how I configure it and how you configure it, you're gonna get two different outcomes. And that's assuming that the builds are also deterministic as well.  So it turns out that we already have an interesting solution for identity. So if you look at Git, Git actually computes an object ID to be the identity. And you have this algorithm called a Gidoid. And so in terms of a Gidoid, it takes the contents of that and it generates a 20-byte hash or I should say 160-bit hash, which is a size of a shot one. There is work to turn it into shot two. I know that violates the immutability property that we were talking about before. We actually have a solution for that that we've included in the spec, but we'll get into that later. In short, it has a solution towards that, which is basically that you have a Gidoid, that Gidoid ends up in a file system, which happens to be the Git object repository itself. And everything inside of that is identified by the Gidoid. And so what we did with OmniBoard to really dive into this is we set up the identity and you have an artifact. And that artifact is you have an OmniBoard document and that the OmniBoard document itself, you say like a specific file, is identified by its Gidoid, went into the creation of a particular object. So you have identity of that object and the identity of the things that went into that object. So that's the example of the lowest one right here. You have a Gidoid of each file. Those all get wrapped up into whatever those things came together to produce ends up with its own Gidoid that is unique to itself. And that's then represented as part of the dependency graph. The same thing happens when you go further up, where you have an OmniBoard document that is consumed by another OmniBoard document as you get more hierarchy. You say that the particular blob of a particular item is represented by an OmniBoard Gidoid of that particular thing. It gives us a graph. You can think of it like a, I often call the BOR a bag of receipts, of cryptographic receipts, because at that point what you're doing is you have a set of identity. You have a tree of identities or a DAG of identity of things that you use in order to build that particular system. So Gid is a Merkle tree masquerading as a VCS, as a version control system. So does everyone know what a Merkle tree is? Is there any people who don't know what a Merkle tree is? Okay, so a Merkle tree is a special kind of tree that is defined as, if you have a hash of all the subtrees, you have a tree that has several subtrees underneath of it and you take the hash for those subtrees and those subtrees themselves have leaf elements and those leaf elements also have their own set of hash. So you're looking at hashes in such a way that if you change a file, it only changes that file. All of the parents of that particular file going up to the root. So that way you don't change most of the DAG. You only see in the representation the changes of the leaf element that was changed and the parents moving up to the root. So like minimum amount of changes you can make. Generally from a growth perspective, the growth of changes over time, each change might change maybe, it's logarithmic based upon the quantity of data you have in it. So it's very efficient. It's the reason why I used to get up in the morning, I'd go to work, I would hit SVN update, I'd go have breakfast, come back, watch it finish.  So Merkle trees is the magic behind it. And so Git uses that, it uses the GitOid as the identifier. So now that we have this dependency graph, we wanna ask the question, what is in the artifact? So in this scenario, we have the identity of a C file, maybe some headers, and those go through a compiler, it releases out, it generates an object. So we wanna capture that information and the same with the other sites. We don't have two object files and then those get linked together to create an executable. It's actually more complex than that because you also have shared objects that may already be present on the system. So when you run an application, it's not just the executable image, it's also all of the additional dependencies that were in the environment that get pulled into it through your operating system, which also have their own set of things. So each of these can be represented by a Omnibor and that Omnibor at the top, you could then generate something. So it's not just for things that are static at rest, but also you could use it to represent things that when you run, I give you something that gives you a little bit more information. Also works in other environments like in Java, everything is linked in dynamically. There's no concept of statically compiling things once you cross class divides, the class files. So the same pattern works for things like Java or other systems. So we take this and what we did is we generalized it and we said we have this executable goes back to what you saw before with Omnibor. So we generalized that particular path. So in short, we're able to use those getaways as artifact IDs to represent the whole set of dependencies that come along with it.  So you see a blob, you have the getaway that's attached to it. And then you have an artifact with artifact three has its own Omnibor that's attached to it as well in lexical order. So that way you get a canonicalized view of it. So you're not, you're not guessing us to like, what order did I see things? Everything is explicitly ordered by lexical. So this brings us to the question of metadata. What is known about it? So there was a great comment that was put together by Jeff who's another Omnibor community member and contributor. He said an SBOM is a format for organizing metadata that describes the makeup of software artifacts. So in other words, Omnibor is specifically about the identity of things. It's not about how do you store the metadata, but we still have this metadata we have to look at. So we have dependencies, all the stuff in orange and stuff that is within Omnibor. And you have all of the purple stuff on the side that is metadata. So what we are looking at as a pattern is that the metadata can use the gettoids in the orange boxes to decorate the tree. So you might, you'll have a separate database where one of them might be, I ran an image scanner. Another one might be, I ran something that checks licenses. The third one might be something like who compiled it and gave it to me. So you have this metadata that sits to the side. And that metadata actually can be dynamic because I ran an image scanner today. I ran it again six months from now. It's probably gonna change unless it's a very simple program that's, yeah, it'll change. So in short, we wanna make sure we can keep this separate but use the Omnibor graph as the identifier in order to work out like what does the metadata apply to. So getting back to the example from Jeff, Omnibor stuff on the left. The stuff on the S-bomb is on the right and we actually have been working with the various S-bomb vendors. So for example in, I believe, is SPDX 2.3. They've added support for gettoids which then gives the ability to reference the Omnibor graph. And that was the direct collaboration we have with them. So in that scenario, it's also, we're looking at what is known from a metadata but it's not enough to know what is known. It's also important to know how it is known. And more specifically, once, if you can understand how it's known, the question then is can you trust that information that's on there? So this particular section was written by Ava and Ava is a major voice in terms of trust. And I'll try to paraphrase the best because this was her part of the slides. And to paraphrase, when you look at something that's trust, trust is, you have to look at what is it that you're trusting? Are you trusting the source code that's going into it or what something says? Are you trusting it because I gave it to you? Are you trusting it because my signature is on it? What is it really, where we root that trust in? And signatures are a great example because something could be signed, like how do you know what was behind that, what was signed, what was in it? Like signatures give you an important piece of information but they're also very limited in terms of the total quantity of information they can give you, which then brings us back to our sad current. By the way, the first time this image was used, the contrast was way off in a previous presentation and it was like filmed war version of Kermit. It was fantastic. So when you ask a question of like what is trust, because this can bring us towards actually helping answer that question, there was a person, Dorothy Denning, who wrote about the Orange Book back in the early 90s where the Orange Book said, you can build a system that you can categorically trust as like, you can trust it because it's a property of how it was built. And Dorothy came forward and said, trust is not a property, but rather is an assessment based on experience, a declaration made by an observer, not a property of what is being observed. And this is super important because, just because I give you something and I say, oh, I used all these secure development practices in order to, so you can trust it. You can't trust it simply because I followed a set of patterns. I said, what do you have to do is observe what's around it. So part of the idea is like, how do we give you that metadata? How do we make sure we can link everything together and pull that information from the places that have the ability to actually pull, to give you the best information. And trust is also, Mike Roussell wrote a book on trusting in computer system, of which Eva was a editor on. Trust is time dependent as in, I trust my image scanner today, but am I gonna trust the output a year from now? Absolutely not. It's asymmetrical. And asymmetrical, the best way I can describe this is the relationship you have with your bank is not the same relationship your bank has with you. And so it's also contextual. Like you go into your bank, you're gonna have a different set of services that you trust them for than compared to like maybe you go into a doctor for surgery or something similar. So it's contextual in that response. The same goes with computer systems. It's like it's time dependent, asymmetrical and contextual. And so people often put their trust into the build tools. And so, but what we're looking at is, if you look at what a build tool actually does, a build tool will transform your inputs to some output. The problem is that you can scan the inputs. And so this is where you scan the source code. You say, okay, well, that doesn't necessarily explain the output. It helps give you a direction of what the output can be. It could definitely constrain that, but you don't know exactly what it is. How about scanning the output? So how many people here can say what kind of a pie this is? Is this an apple pie or is this a, is this pears? I mean, for me, if it has coconut in it, I'm probably gonna have a bad flavor that I'm not gonna like. I got a friend who, if it's an apple pie, it'll need an EpiPen. So knowing what's inside of it is super important. The same goes with software. You have to know what's inside of it.  So what you wanna try to do is you wanna try to drive that to something that's a little bit more accurate in terms of what we can trust. And we can actually trust the build tools. So we look at the build tools. The build tools will give us the information. Have the information. They have what was going into it. By build tools, I mean the compiler itself, not Jenkins, not Maven, or the make file. I specifically mean the build tool that is actually doing the transformation. And so we can trust the build tools to give us more accurate information, not to say that it's 100% perfect, but it is the best place where we can find that information and at least get some form of attestation out of it in terms of like what C files went or what Go files, what headers, what was it, what were the environment variables, intermediates, and so on. And so the goal of Omnibore is to be able to take that information from the build tools themselves and to be able to create that artifact, that DAG, so that we're able to then decorate that DAG with information. And we build that DAG and bed an ID for everything that's there in a language heterogeneous environment, regardless of the packing formats, regardless of, and the key here is it has to get to a point where there's no developer effort.  Like this is stuff that should just work, it should just give you that information and you can start decorating the trees. And to get to that answer is to am I safe? And part of it is that when you look at what actually went into it, now that you have this information, let's say deep down like several dependencies in, you have this log for J that was injected into your system. You have that information available because you have the metadata that was attached to it. And you can say these getaways are known to have that vulnerability based upon where, when the patch was in, based upon when the version, the version it was first found. So you can look and see whether or not those files were included as part of the build. And also in the future, let's say six months or a year from now, another vulnerability comes out. The ability to tag those particular getaways as being unsafe is something that would be of huge use to infrastructure and consumers. So in short, we have a construction of multiple systems. Actually, this is an older deck. We actually have two others that we have. So we have Go, Rust, we have work going on in LOBM, BOMSH. There's also work going on in GCC as well. So we've been working with the compiler communities in order to get this stuff in with the expectation that if you're building tools, you'll be able to use those annotations to generate the data. I'll make sure when this gets published, I'll publish the new version of it. It has the additional two links on there. And with that, you're also welcome to join and get involved. So first, well, I want to thank you all for your time and I don't know how much time we have left for questions. So does anyone know if we have time for questions? Seven minutes. Cool, so are there any questions that we have? Okay, that's a good question. So non-compiled, we can take the Omnibor or the getaways of the files themselves and put them together in terms of saying what a package is and also analyze. So we still get some of the benefits of, like I have a file that has an own vulnerability that I maybe pulled from another project and then be able to tell that that was an, or maybe a dependency when I compile in. But things like Python and Ruby and similar are a little bit more difficult and then you have to work them into that. We've not worked with those particular communities just yet, because we've been focusing primarily on the compiled ones, but we definitely see ways that we can tie into it. Like maybe when you load in a Python file or load in a Ruby file, you could check the getaway and check against a list and see whether or not you want to accept that or not. That is work that has not been done yet. So that's an area of interest. We definitely would appreciate some help there. The question is whether we intend to capture the getaways of the build tools themselves. So we're having some discussion on this because if you capture the build tools that does affect the output and whether you want to do effect or not is sort of up in the air. There is integration that we've been looking at with a project called Intodo, which captures those build tools are. And I have an implementation of Intodo that I've been working with to capture the build tools as getaways so we can get that information out. So the question is how do we establish the trust of the information that we captured there? So the first one is in terms of the inputs and outputs. So again, this is not designed to solve all aspects of it. So we're trying to solve a very specific problem within it, but we need help from other tools in order to capture that. So I mentioned the Intodo one specifically because many of the Intodo systems are designed so that you can capture not only what process was ran and in other words like I ran this transformer or preprocessor, I ran this compiler, I ran this test and this is the group that built it. That gets signed, gets stuck into maybe a six door or something similar. But there's also the question as to whether you can trust the tool or not. And part of, there's a couple things towards this. So one of them is when you're building out your CICD system itself, first you do want to try to keep that under control so that the limited people have access to it. But simultaneously, some of the Intodo tools, one of the ones that I work on is a project called Witness and have the ability to hook in with analyzing what files that the compiler open and what to capture metadata on some of that stuff as well or maybe there are other system calls like make a connection out to a network somewhere to be able to capture some of that. So there's, there are things there that help with some of that observability. But at the end of the day, if it's about should you run a particular piece of software or not, you still have to root that trust into something and that something is tied to do you trust the agent that built it and do you trust the agent to try to capture that information effectively. So you still have that issue in terms of build trust. So it's not designed to solve all the issues up and down the space, but it's designed to help give you additional information so you can make better decisions. So is there any other, any other questions? Fantastic. Well, thank you all for your time and I'll be around. So if you have any questions you want to ask me in person while I'm here, I'd be more than happy to answer.


Welcome to All Hands Update, I'm Petty Officer Andrew Johnson. The Invictus Games ended with a closing concert in London September 14th. The U.S. team won gold medals in several events including the 100 meter sprint, the 200 meter sprint and the discus and shot put competitions. The Invictus Games is an international tournament that brings together wounded, injured and ill service members in the spirit of friendly athletic competition. The athletes participating in the games competed in several sports including cycling, rowing, track and field and wheelchair basketball. The games use the power of sport to inspire recovery, support rehabilitation and generate a wider understanding and respect of those who serve their country. Check Navy.mil for more updates and results from the Invictus Games.


Stakeholders in Iqiti State have called for transparency and accountability on issues of public procurement in this state. The two-day workshop, which was organized by the Public-Private Centre, focuses on ways to improve transparency, accountability and to ensure that the interest of the masses is protected. Speaking with Newsman, the Senior Communications Officer Nina Eze and other stakeholders said the program is to ensure sensitization of citizens to their rights and encourage them to ask the right questions. The first is empowering citizens and ensuring that citizens get value for money, they get better service delivery. Because at the end of the day, it's when citizens know how to actively and effectively hold government accountable, then they can speak out, then they can demand that this school in my community is not working. The toilets are not good, the fences are not good, the structures, there are no tables and decks.  So we are implementing this program in Iqiti State, in Adamawa State and in Kaduna State. Just basically bringing together stakeholders, all the relevant stakeholders, so government... Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


can't leave out houston yeah i was waiting for you to say it yes that's it you can end the video i was waiting for him to say it he didn't what's good y'all is your boy ross back again with another video so i'm gonna check out 10 cities with the hottest wrestling crowds uh definitely been looking forward to checking this particular video uh video out uh wrestling crowds can make or break a show upset that plenty of times now some people would disagree with that but wrestling in front of a hot electric crowd it just adds something to the atmosphere to the product especially those that are watching at home seeing the crowd reacting ooze and awe to the moves and people being invested in the storyline itself it just it's that extra layer of the complete package of just wrestling as a whole we saw what it was like wrestling in front of no crowds you know during the pandemic era and yeah people were putting on good matches but it just doesn't have that same feel that's why they brought out the thunderedome to try to replicate real people being there and even though that was cool it still doesn't have that same feel as live audience a hot crowd cheering on some of their favorite matches booing the the heels and cheering the baby faces and just saying this is awesome when there's something awesome happening it just it makes it that much better man so we're gonna check out some of these uh cities that had a hot wrestling crowd appreciate all love and support let's get right into this one man great wrestling city is characterized by its boisterous crowds and storied past for this video we're covering the most famous wrestling towns and countries will detail a bit of history and look at the iconic moments that occurred in each location nuclear heat will also highlight some memorable crowd moments from these particular places listen to that as we showcase 10 of the hottest wrestling cities before we begin let's give an honorable mention to boston massachusetts green'sboro north carolina up on the top st louis missouri that was a cool moment too that was such a cool moment now let's start this chicago is a premier wrestling city with a building that many talent considered to be the best to wrestle in the all-state arena formerly known as the rose month horizon this venue and others in chi town have given us a ton of fantastic moments that featured a terrific crowd on hand to make things even better it was the land yeah great moment haha boost bump and doosan classic oh date man this is peak daniel bryan yes movement oh my god the nuclear boos oh the authority was getting odd this was so good so good second city saint c m punk is the most patriotic wrestler towards the windy city his legendary homecomings are among some of the great wrestling moments back this is cool too oh boost bump and doosan that seems so lit philadelphia was a wrestling institution cursey of extreme this is why wrestling in philly this year whoo i can't wait it's gonna be great it's gonna be a great one boy it's gonna be a great one championship wrestling this was a promotion that had a relationship with his fans comparable to that of the sports team of which the ECW arena in south philly was its home ground there you would be hard pressed to find a more impassioned and loyal group of wrestling fanatics boys just throwing chairs look at this oh my god oh my god sure the people could be crazy this look at this at the end of the day they just wanted to be entertained they came up with some of the most creative chance that made for a truly unique viewing experience philadelphia has always been one of the wwe's best markets as well so good so good each year it's always a treat when the wwe or aw run shows in the united kingdom the fans in britain absolutely love their wrestling and it shows they have some of the best crowds and chants filled with british humor this is why man uuk crowds man you guys y'all are just so fantastic when it comes to like just chance and the energy y'all bring to the shows that's why i am definitely one of the uh i'm i'm sure it's not just a few of us out there but i am definitely one of those people that would love to see a wrestle mania in the uk i'm so dead serious about that if there's another place they have a wrestle mania it should definitely be in the uk bro i think that she would just be fucking magical so i'm i'm all for that deadass haha tribal wanker it's the type of stuff you'd expect to hear shouting on a match day inside a football ground this all makes for a really fun party atmosphere they were so old love it to do the same to body rude and lose those slipped down inside the ring we're seeing that right now the last is a dual star athlete just imagine i love it this was such a man that was man clash at the castle fantastic moment bro fantastic but again how do you expect to be treated Roman Reigns is treated no one see memorable moments go down during uk events yeah look at that bro there should be listen to that reaction right one of the best monday night raw matches of all time john sena hbk arguably this match is better than the wrestle mania match this shit was great they went out there for a whole hour and gave you a five star classic it's literally one of the best monday night raw matches ever dead ass just fantastic bro this was cool too and they see the wwe champion canada and wrestling go together like pancake such a good moment x and maple syrup from the dates of stampeding calgary to the incredible pay-per-view event that's name paid tribute to the territory where a number of future stars honed their craft meanwhile montreal and toronto continued to be two of the best c's for wrestling they have some of the best fans that never miss when it comes to tremendous reactions classic g-tap instantly it's staining chance and supporting their favorite wrestlers this is so cool bro sammy right before elimination chamber so cool bro such a great moment so good great match to great atmosphere so good lucha libre is integral to mexico's cultural heritage across numerous promotions and venues such as arena mexico which is known as the cathedral of lucha libre wrestling in mexico is all about la familia from the fans that come to see the matches to the resting tradition that runs through the famous families at the guerreros as well as the mendosas better known as the villanos the fans most beloved wrestler el santo was a movie star cultural icon and revered folk hero who never lost his mask in the ring and someone whose legacy was passed down to his son in this modern era fans of idolized stars like mistico and rey mysterio much like mexico in japan wrestling is taken very seriously with so much history ranging from the legend of rikido zan to the four pillars the japanese fans have been spoiled with so much great wrestling over the years with even annual visits from the wwe to look forward to dallas is a city steeped in wrestling tradition that continues to hold big shows to this day who could forget the days of world-class championship wrestling in the sportatorium where the von erics piled their trade and got over huge as excited fans came in their droves each week we loved our fans and they knew we loved them and they could tell we loved them look at the reaction is the big d hosted two huge russell manias in 32 and 38 that was cool deep in the heart of texas love it we can't leave out houston yeah i was waiting for you to say it yes that's it you can in the video i was waiting for him to say it he did it he did it city which held what many consider the best russell mania of the wall in 2001 wrestle mania returned to houston in 2009 it was notable for paying tribute to some great texas legends the night before at the hall of fame which is followed up by one of wrestling's greatest matches featuring two texas icons literally it's the match that always gives me goosebumps and it gives me that extra pride of knowing it was in houston i got to watch this live 25th anniversary of wrestle mania this is still one of the best russell mania matches of all time simple simple fans couldn't get enough of memphis wrestling during its heyday where crowds packed out the mid-south coliseum to see the likes of jackie fargo and jerry lawlor while people tuned in on television to hear the legendary lance russell call the action memphis loves to welcome their legends back when they return that's awesome man memphis is a place with a long resting tradition where great moments continue to be made that was a cool moment i didn't know this i forgot this was in memphis i forgot well memphis been having some great moments we just saw it with the rock the rock concert that was in memphis that's crazy memphis has had some great moments wrestling wise the beautiful island of puerto rico is renowned having the craziest group of fans this deserves to be here people throw rocks at you they throw piss on you they throw urine on you they throw feces on you they want to see somebody get hurt they love the baby faces and boy did they despise the heels has sido mas un padre para ella puerto rico is still a resting hotspot where the audience remain a focal point and highlight of the show new york was the wwe's original home hey bro Puerto Rico y'all did y'all thing from back last y'all honestly made that one of the best baby views from last year bro it's because of their reaction because of their energy they made everything feel so important and you so good bro y'all showed out man from base daying back to the days of bruno san martino and his countless sellouts of madison square garden while holding the wwf championship for two thousand eight hundred and three days insanity the garden continued to be a hot bed for wrestling with rabid fans that were treated to so many classic moments of the world's most famous arena iconic moments so many great moments the manhattan center has housed some of the best resting crowds under one roof staying back to the early days of monday night rule manhattan center also featured the famed hammerstein ballroom that paid host to events in the original ecw as well as for wwe's revival of the brand in 2005 and 2006 there we saw many of the most passionate and rowdy wrestling fans that made for an unforgettable audience bro that crowd we'll we'll never see nothing like that in wwe ever again i want you to know that because guess what they're gonna bleep out everything they're just gonna bleep out all noises yeah in more modern times the wwe has used the barclay center in brooklyn as its main hub for events in new york where a good crowd is always guaranteed underneath all of that it's king listen to the this is such a great moment too it's the ball now if you enjoyed this video be sure to check out a similar video of the top 10 oh man this is great this was great this was great this was great this is what i like to see man right here man just going back down memory lane and in watching these crowds you know be electric and bring that energy it adds to goosebumps this is why i i think it's important to you know recognize how good a crowd can save a show not even save a show can make it better when you go back and watch some of these fake like your favorite wrestling clips a lot of times you're watching it because you want to relive that feeling you had when you're initially watched it and you heard the crowd pop you want to relive that i don't know how many times i went back and watched kofi uh defeat daniel bryan to win the wwe championship or when daniel bryan won in the triple threat match and he made batista tap out and you heard michael called tap out batista tap out batista and the crowd went crazy when he actually won like it's just one of those type of things like you just go back and you relive those moments because you're you're watching it you're watching it but you're also listening to the crowd and the energy and just seeing that on video and hearing that on video it brings back the nostalgia even more so comment down below let me know the city of your favorite wrestling crowds whether it's an aw or wwe or wherever if you watch the pay review or watch the show and they were in that particular city every time and they always brought the energy let me know what city that uh that was but that's your favorite of course but i appreciate all love and sport wrote 250k and i'm still gonna unspeak it i can't even talk man y'all know you know i'm not even gonna do the outro man y'all y'all know y'all know the outro i'm out peace