I'll be speaking about Laplace eigenfunctions, so this talk falls in the Riemann spectral geometry, but I'm not going to be talking about eigenvalues though. So this is not a talk about spectral rigidity, it's more a talk about how eigenfunctions of the Laplacian behave. So I'm going to start introducing the setting. So we are going to work on a compact Riemannian manifold, we are going to assume it has no boundary and throughout the talk I'm going to write little n for the dimension of the manifold. Now the manifold is compact, so you have the Laplace operator acting on it and the spectrum of the Laplacian is discrete. So if this is the standard Laplacian, I put a minus sign in front to turn it into a positive definite operator. We are going to denote by philand.j the eigenfunctions, all the eigenvalues are positive and I'm going to be writing philand, lambda j squared for the eigenvalues. And because again the manifold is compact you can form an orthonormal basis of l2 of the manifold when you and l2 with the Riemannian volume form by this using these eigenfunctions that throughout the talk I'm going to take to be normalized. So the reason why I'm interesting studying eigenfunctions is because they encode all the dynamic and geometric properties of the underlying manifold. From a quantum mechanics point of view if you want to understand what's the probability that a quantum particle belongs to this region A in space, what you do is you grab this square of the modulus of the eigenfunction and you integrate it over A and then you get the probability of your particle being in that region of space. So eigenfunctions they carry all this information about the underlying manifold and they reflect a lot of what's happening with the dynamics of the geodesic flow. So just to illustrate that point I have this picture here you have a disc and a cardioid and in red what you see is the trajectory of a geodesic in each of these two surfaces. And well there is a big difference here this one looks quite chaotic and in these four plots you have the density plots of four different eigenfunctions. The eigenvalue is growing in this direction and what you're seeing here is the plot of the modulus square of the eigenfunction. So you're seeing this function plotted. Black means that the eigenfunction has a that the modulus is high while white means that you're getting a zero there. So for example in these pictures here it looks like the probability of my particle being near the center it's zero or very small. For example in these two my particle is going to be concentrated near the boundary. While here where the geodesic flow is highly chaotic it looks like the region is becoming evenly grayish which would mean that the probability of finding the quantum particle in any region A in this cardioid would be comparable to the area of the region. So this is one way in which eigenfunctions keep track of what's happening with the dynamics of the underlying manifold and this talk is going to focus on two aspects of eigenfunctions the critical points and the zero sets of the eigenfunctions. You can think of critical points if you think of maximums and minimums as the places where this modulus square is the greatest so those are the most likely places for your quantum particles to be found that while zero sets are going to be the least likely places for this quantity quantum particles to be at. And what I wanted to do before starting to talk about which kind of questions we are going to ask was to give you some pictures of how zero sets look like and the standard thing is to show you video of the cladney plates experiment. So here in this video what you're going to see is a metal membrane that's put its place on top of a speaker. This speaker is connected here to a frequency generator and what you're going to see is that for different frequencies this membrane is going to start vibrating and what they just did was to put very thin grains of sand on top of a plate. So the frequencies that are appearing here in this corner you'll see and what's happening is for each different frequency these frequencies are associated to standing waves so they actually correspond to eigenfunctions so the wave function with which this membrane is vibrating it's an eigenfunction whose value lambda is this one here on the corner and what happens is the membrane is vibrating and the places where it doesn't vibrate at all those places are attracting the grains of sand so those places are the zeroes of these eigenfunctions. So what you're seeing is just different zero set configurations of eigenfunctions for these different values of lambda and you can see that as the frequency gets larger and larger the configurations become much more complicated. This was an experiment done first in I think it's 1680 by Hooke he did it at that time with a metal plane and a violin bow and then it was replicated a hundred years later by Kladni and since then known as the Kladni plates experiment he was the first one to record at least a hundred configurations of the zero sets and what we are going to do throughout this talk is well when we are in two dimensions is to try to understand what the structure of this zero set is we are going to talk about how these components are going to be nested within each other if you look at the complement of the components those are called novel domains we are going to try to understand what the connectivity of these components are and what I want you to keep in mind is that we are going to be working in this limit where the frequency lambda is going to infinity so just as a reminder so lambda j square is the energy lambda j is what I refer refer to as the frequency okay so these are pictures of zero sets on different surfaces so that you don't only have the picture of what happens with a square this is a quarter stadium this is a biturist this is a square biturist and here you have the round sphere the lines up here are the zero sets of a high frequency again function and in the two bottom pictures what you see is the complement so in black you see where the egg and functions take positive values and in white you see where the egg and functions take negative values so the zero sets in these bottom pictures is just the lines divided in black from white and what I want to do first is to tell you what is it what do we know about these zero sets and I'm going to do this in the realm of surfaces which is what we know the most so for zero sets you can prove that they are rectifiable so you can actually measure their length and there is this construction by STR that says that the measure of the zero set should grow like a constant times lambda when lambda gets large so this construction was open for a long time and very recently low now prove this lower bound here the construction says that there should be also a constant up here but we don't we are nowhere close to getting that constant at the moment um what we also know about the zero set is how it's pressed across the surface so we know that there exists this constant c here such that if you grab a ball of radius c over lambda no matter where you place it on your surface it will always intersect the zero set so what that is saying is that if I start here at a point x in my surface and I walk a distance of c over lambda then I'm always going to see a sign change in my function because this way functions they are oscillating at frequencies that are comparable to one over lambda what we also know is if you take the complement of the zero set we know that the inner radius of the complement is bounded in between two constants over lambda so there exists this constant here time is c such that if in every nodal domain you can fit a ball of radius c over lambda so this is giving you a notion of how thick the complement of the zero set is and finally what we also are interested in is in understanding the number of components of the zero set in all the pictures that I've been showing you it looks like the number of components grows to infinity as the frequency grows we have no proof of that at the moment in a very in very specific cases we can show that the number of components goes to infinity but it's actually quite hard to do and we believe that it should grow like lambda square if you are on a surface or like lambda to the end if you are in a compact manifold but the only thing that we know and this is a standard result is quran's nodal domain theorem that tells you that at most you will have a constant times lambda square number of nodal domains okay so this is what we know on surfaces on general manifolds we know even less and what I'm going to do in my next slide is to tell you what are the questions around which this talk is centered so we are going to discuss number of critical points of the eigen functions when you divide this by lambda to the end we think that this quotient should remain in most cases bounded above and below by two constants so the number of critical points should grow like lambda to the end and being the dimension of the manifold but there are no results point that prove anything like that at the moment the measure of the zero set as I was saying before this question should remain bounded above and below by two constants the number of components of a zero set and so this is going to be the first part of the talk and then it's going to get a little bit crazier and we are going to talk about the different morphism types of the components of a zero set and also about the nesting configurations of these components okay so the first thing that you have to do if you're trying to attack these questions is to realize that very little is known so one of the things that you can do is to try to randomize the problem and instead of answering what happens with these quotients or these quantities for an actual eigen function you can ask what happens for a random eigen function so um suppose you're working on the n sphere or on the n torus okay where multiplicities are high and now fix an eigenvalue and what you will do is to consider a linear combination of eigen functions whose frequencies are equal to the frequency that you started with okay so what this is is just an eigen function with eigenvalue lambda square that's what you just did but if you pick these coefficients at random so we are going to allow these coefficients to be standard Gaussians independent then what this becomes is a random eigen function with eigenvalue lambda square um there it is okay so here in the slide I have a normalizing constant here one over squared n lambda and lambda being the number of frequencies equal to lambda so the multiplicity of the eigenvalue this is just a normalizing constant so don't pay attention to the point is that you have now this random eigen functions and um on the on the round sphere they are called random spherical harmonics on the on the torus on the flat torus they are called random arithmetic waves and the idea is to try to answer these questions but the these questions but now for this random eigen functions so um on the case of the n sphere and on the n torus eigen functions can be computed explicitly we know how they look like we know their eigen values so you can actually say a lot and there has been a lot of work done in this direction so for example on the two sphere uh there is a series of works by Nicolaescu, Camarota, Maninucci and Bigman that prove that the number of critical points will converge to a constant in probability so what this means is that the mean is going to a constant and they can actually show that the variance goes to zero and they actually in this last paper they get a nice rate of decay for the variance so for the number of critical points indeed these quotients are converging to a constant it's not only that you get the get it bounded above and below by two constants but you actually get convergence in this random realm for the measure of the zero set on the two sphere there are computed the expected value of this quotient and then you highest and Bigman control the variance so they can also show that the variance not only goes to zero but they get actually really good rates of decay and you get that the probability of so you get convergence in probability of these quotients to a constant so yeah this construction is holding in this random realm so these two first questions are what we call local quantities because you can if you want to understand the number of critical points or the size of a zero set you can start with your manifold and you can chop it up in neighborhoods that have sizes comparable to one over lambda and then compute the number of critical points in these tiny walls and then just add them up however you cannot do this with the number of components of the zero set because if I have two tiny walls I may have components that are going from one ball to the other and you don't want to over count and actually these components will definitely go from one ball to the other so this is a much harder quantity to study and there is the work of Nassarov and Soling that they first did it on the two sphere and then they were able to generalize this where they show that in mean the number of components of the zero set is converging to a constant it getting the variance is much harder exactly because this is not a local quantity so they only get convergence in mean and the idea of what I want to do now is to start working on these questions but on a general compact Riemannian manifold where you don't have formulas for the eigenfunctions or the eigenvalues so that's where we are held the problem is there that if you want to work on a general manifold if you fix the base manifold and you look at the space of all Riemannian metrics that you can put on it generically all the eigenvalues are going to be simple so each eigenspace only has one eigenfunction so you can not do linear combinations of eigenfunctions within an eigenspace so you will have to really like change this definition this doesn't make sense anymore you don't get anything random here so one thing that you can do is to fix an epsilon and instead of working with frequencies that are exactly equal to lambda you can work with frequencies that are in a window from lambda to lambda plus epsilon no so what you're doing is you're incorporating more eigenfunctions to your random linear combination mix by working now with a window from lambda to lambda plus epsilon so what you just when what you get when you do something like this when you're working with this random linear combinations it's a function that has a frequency that's very that's concentrating concentrated near the lambda that you pick but it's no longer an eigenfunction right i'm mixing different eigenspaces i'm mixing frequencies from lambda to lambda plus epsilon so that's one of the things that you have to keep in mind from now on however we do believe that they should behave like eigenfunctions that these are an honest model for what eigenfunctions look like and this is the content of the random wave construction by by very that what it says is that if you're working on a manifold where the geodesic flow is chaotic enough what happens is that the statistics of these waves that are called monochromatic random waves by the way should be the same as that of actual eigenfunctions whose eigenvalues are in these windows whose frequencies are in these windows so what you have here for example is just a histogram for the value distribution of an actual eigenfunction whose frequency is 500 on an arithmetic surface and so you can see that actually so that those are the points that are plotted and you can see that it actually adjusts to a Gaussian distribution and there is some numerical evidence towards this construction but nothing like this has been proved but just keep in mind that we do believe that these are a good model for how eigenfunctions should behave and so okay suppose we want to answer these questions for these random waves and now the question that you need to ask yourself is what do you need to study the number of critical points or the size of the series sets so these waves that we are considering here since they have Gaussian coefficients they are called so they are Gaussian fields on the manifold and there is this theorem by Kolmogorov that says that Gaussian fields are completely characterized by the two-point correlation function so if you know the two-point correlation function for your field you completely understand how the field behaves you can compute any quantity related to the field so the two-point correlation function is exactly this so what you do is you fix two points x and y on your manifold and you compute the expected value of the product of the value of your wave at x times the value of your wave at y so you are trying to understand how these two values are correlated to each other depending on what x and y are and because the Gaussians the variables that we are considering are have mean zero variance one and are independent it turns out that this is exactly what you get as your correlation function so you get the sum of the cross products of the eigenfunctions whose frequencies are in this window so if you're familiar with Val's law this is an object that appears a lot to compute number of eigenvalues say for example in a window from lambda to lambda plus epsilon only that what you do is you deal with this object on diagonal and you integrate over the manifold with respect to x but here if you want to understand these waves you cannot evaluate on diagonal and you cannot integrate x out you actually have to deal with this sums of cross products what this is is the kernel of the projection operator from l2 of the manifold onto the direct sum of eigenspaces whose frequencies are in these windows from lambda to lambda plus epsilon so this is just the kernel of that operator so if you want to say yes are you thinking of epsilon as small as a fixed small number okay how many eigenvalues do you expect to find in that window excellent so now it really depends on on the geometry of the manifold that you're working with from now on i'm going to work on there an assumption that says that at least there has to be one point in your manifold for which the measure of geodesic loops that start at that point and close that is added has measures zero so that under that assumption you have for that epsilon fix epsilon lambda to the n minus one eigenfunctions roughly in that window so there are a lot of them and what you need to do as i was saying if you want to understand this two point correlation function is to understand this spectral projection operators and if you want to understand actually what happens with the zero set and the number of critical points you need to understand this two-point correlation function but at one over lambda scales because that's where the eigenfunction is how the eigenfunction oscillates so the largest correlations are going to happen in those scales so what you need to do is to be able to so if this is your manifold and this is a point x that you fix on the manifold so if we identify it with its tangent space be an exponential map what we need to do is to work with vectors here that have that are of the form u over lambda b over lambda and map them here be an exponential map so i need to be able to evaluate my two-point correlation function at these points so one over lambda close to a fixed point x so what we really need to control is the two-point correlation function evaluated at the image of u over lambda and the image of b over lambda this is the quantity that i need to control as lambda goes to infinity um are there any questions so far so now as i was saying to make sure that i have eigenfunctions and to actually be able to prove any result i need to work under the following conditions so fix a point x on your manifold and then look at the space of all initial velocities that generate closed geodesic loops so it doesn't have to close smoothly just closed okay and you look at the set of all possible initial velocities so you're working in sx star m okay and here you equip this with a liberal measure on what you're the condition that we are going to be working with is that the measure of the space of initial velocities that generate geodesic loops has to be zero okay and under that assumption what we proved together with Boris honey is that we have a limiting function for this uh scale core two-point correlation uh for psi lambda so the assumption that we have to work under is that uh if you fix the point x the measure of geodesic loops that close at x has to be zero and under that assumption we can control that lambda to the n minus one shouldn't be there cross it out the limit of this rescale covariance function is the function uh that's also going to be the two-point correlation function for a field so what i'm going to do now is to explain what this infinity field is so what you get is that this is converging to epsilon times the two-point correlation function for a Gaussian random field in r n so this infinity field is what's called a superposition of random planar waves what they said what it satisfies is that it's an eigenfunction for the inclusion laplacian with eigenvalue one okay and uh since it's a Gaussian field it's completely characterized by the two-point correlation function you can actually define it in terms of the two-point correlation function and the two-point correlation function is exactly this thing that we have here on the right hand side so you integrate over the n minus one sphere e to the i u minus b and and w so what you have here on the right hand side is actually um the Fourier transform of the spherical measure so this evaluated at u b is simply the Fourier transform of the spherical measure evaluated at u minus b so what we are getting is that no matter what the geometry or the topology of the manifold that you start with is you when you rescale the two-point correlation function like this you get always the same limit and this limit only depends on the dimension of the manifold you're integrating over the s and minus one sphere and that's the only thing that you remember about the starting manifold so this result is true as long as you have this hypothesis however I strongly think that it should always be true to prove this result we use microlocal analysis and so we have to work with the wave operator with the wave kernel and we have to provide the singularities along geodesics and that's why we need this condition in the set of geodesic loops but I really think it's a problem of our proof that we have to enforce this condition I think it should really be true on any manifold it is true on the sphere and it's true on the torus and the fact that it's true is what allowed all these people to get these results on the n sphere and on the n torus for the number of critical points and the size of the zero step so this this convergence holds in this infinity topology so you can take as many derivatives in u and b on both sides as you want and you still get the limit and it holds uniformly for u and b inside of all here in the tangent space of constant radius r another way of reading this result is that if you start with your wave and you rescale it so you fix the point x and you rescale it at one of our lambda ranges about x so now you see you think of this as a function in rn in rn so this is a function of u okay so as a random variable in rn it converges in distribution to this fields this random planar waves in rn that's what this result says because you have convergence of the two point correlation function so all the moments converge so these random waves really behave like this limiting guys that we have in rn and just so that you have an idea of what the heuristics are behind this statement this is not a proof and actually it has nothing to do with the proof but what happens is that if you grab an eigenfunction an actual eigenfunction of the Laplacian and if you fix your point x and you rescale this function about x then when if you hit this with the Euclidean Laplacian plus some lower order differential operators that I do not want to define then what you recover back is the eigenfunction itself where it's scaled so what's happening is that to leading order in lambda terms the rescale eigenfunctions behave like eigenfunctions of the Euclidean Laplacian with eigenvalue one which is the property that this limiting guys have so that's that's what's happening behind the scenes and this is why and that statement holds on any manifold so that's why we are getting this universal limit that forgets the metric g or the topology of the manifold okay so are there any questions about this statement what I'll do now is to show you how you can apply this result uh so for example if you try to count the number of critical points or measure the size of the zero set we can primarily mean they converge to a constant this constants an and b and only depend on the dimension so they are the same for every manual manifold compact manual manifold of dimension n as long as you work under this assumption that what you need is that for almost every point on your manifold the measure of closed genetic loops has to be zero so uh yeah that's assumption and if you want to control the variance uh you actually need to ask for something else because when you when you control the variance you need to control your two-point correlation function in both that are in points that are very far apart for example picture the sphere like the values of your function on the north pole should be super correlated to the values of the function on the south pole so it's not always true that only the one over lambda ranges matter you have to be careful of those things uh so uh if we work under the assumption that if you for almost every pair of points x and y on your manifold if you look at the set of arcs geodesic arcs that are joining these two points that the set of initial velocities will have measured zero for almost every pair of points then we can control the variance and not only we have that it decays to zero but we can also control the rate so we have convergence in probability of these two of these quotients to this constants a n and b n okay so now uh for the second half of the talk i'm going to start talking about uh the components of the serial set and their different number there are different morphism types and how they are nested so this is uh the serial set for actually it's not for psilamide it's actually for this guy here psinfinity in r3 where we just plot it to a box this picture is by alex barnett and uh so when you're working in dimension three uh your serial set has dimension two so it's going to be a surface or actually a collection of connected surfaces and the limiting guy the serial sets looks like this so you can really not tell what's going on there um the first question was can i count the number of components of the serial set and this question was answered by nazarevan solving and in min if you are working under the assumptions that boris and iphone you can show that the number of components will converge to uh will grow like a constant times lambda to the n and um what uh you can ask after that is well what happens if instead of just counting the number of components what i want to do is to count the number of components with a given difium morphism type can i say that 90 percent of my components are always going to be spheres that's the type of question that we are asking um so let me tell you how we are going to go about thinking of this problem so suppose you have a realization of c lambda and these are the components of your serial set so the manifold has dimension three the components are a collection of uh surfaces and you are going to organize this surface according to their genus so here i have ten components in total i five of them are spheres three of them are tori nothing of genus two and two components of genus three so what you are going to do is to collect that information into a measure into a probability measure what this probability measure tells you is the frequency with which each difium morphism type is appearing so five of those ten times i get a sphere three out of ten times i get a torus and the question is as lambda grows is there going to be a universal distribution for my difium morphism types is there going to be a law that tells me for lambda large enough 90 percent of the components will be always uh will always be spheres five percent are going to be tori and so on yes so i mean the random sort of this random random kind of linear combination zero is a regular value again oh yeah with probability one it's a regular value so these are going to be smooth manuals but the i mean the collection of difium morphism types of course depends on the choice of the ij aj yes definitely yes so what these are these are probability measures but they are random probability measures for so for each realization of cylinder you have a different distribution of the difium morphism types so let me actually define these measures so they are probability measures so they give you back a number between zero and one and the domain of these measures is the space of difium morphism types so if you grab a component of your zero set it's going to be a compact manifold it will have dimension n minus one it will have no boundary with probability one you can show that it's going to be smooth and with high probability you can show that it can be embedded in our n so this is the collection of components that we are looking at and we are going to question this by the space of difium morphism types so that's the domain of the measure to each difium morphism type i associate the frequency with which it appears among the components of a zero set so in this example i have one over ten ten being the total number of components and then i'm putting a delta mass every time a difium morphism type is hit that's how the measure looks like in this example so i have five times a delta mass of genus zero three times a delta mass of genus one and so on and that's in general how you construct the measure so if i call c capital c sub c lambda the collection of all the components in my zero set then the measure looks like one over the total number of components and then a sum of delta masses where i'm going to add a delta mass every time the difium morphism type of that delta mass is hit by a component in my zero set okay so the question is again in the limit as lambda grows do i have a limiting probability that will encode how these difium morphism types look like and the answer is yes this is a result by peter sarnak and jor bigman so what they show is the existence of this limiting guide mu infinity to which the mu epsilon does are converging and in the the way in which they converges the space of difium morphism types is discrete so to measure the difference between these probability measures you simply evaluate them on finite subsets of their domain and and you can compute the difference so what this statement says is that for any epsilon fixed okay any small epsilon that you pick the probability that the difference between the two measures be bigger than this epsilon is going to go to zero as lambda grows to infinity so in that sense these measures are the limiting guides and the assumption under which you need to be working is that the for almost every point on your manifold the measure of closed geodesic loops has to be zero simply because you need this convergence of the two-point correlation function to prove any result like this okay so now there is a limiting distribution i have to say two things though so from the proof of sarnak and bigman you cannot track how this measure looks like this is an existential proof they prove the existence of the limiting measure but you cannot keep track of how this measure is built so you cannot say that 90 percent of the components are going to be spheres you cannot hope for anything like that the second problem is the following we have no clue what the domains look like for high dimensions so if the manifold has dimension three the zero set is a collection of surfaces so you can definitely organize them by their genus so in that case you do know the domain of the measure but if your manifold has dimension four or higher then the zero set is a collection of components of dimension three or higher and we really don't know what the space of the morphism types of such manifolds look like so we really don't understand the domain of these measures so that's part of the problem that one has but despite of that what we did with bitter sarnak is to show that the support of the measure is the entire space so even though we have no clue how this looks like if you give me a morphism type with a strictly positive probability that the morphism type will occur in the zero set of your random ways for lambda large enough that's what's happening you're observing all the possible morphism types once the frequency is large enough okay so what I'm going to do next is to discuss the similar problem of the nesting of components so are there any questions about this statement yeah like how well I guess super hard but how computable who would be to find the limit from an example from an example no at the moment we really don't know how to find any lower bounds on this probability yeah yes yeah yes we have Alex Barnett running numerical experiments what he's observing is that there is going to be a giant component and then tiny components around it that will have these different diffeomorphism types but actually in his numerical experiments he's only being able to see spheres so it's very likely that the probability of seeing all the diffeomorphism types are super super tiny so I mean at this moment in time the computers cannot give us any information yeah yes yeah so all these statements are about in the end so if you want to understand this limiting distribution what you need to understand is the limiting distribution uh it's it's this gosh of fields infinity you need to understand the diffeomorphism types of this guy and what happens with this guy is that it satisfies this equation so it's a it's it's quite rigid it's an eigenfunction for the Laplacian with eigenvalue one so uh it has a lot of structure into it okay so now for the nesting of the components if these are the components of your zero set the way in which we are going to record the nesting since it's using a finite rooted tree so for this tree the root of the tree corresponds to the big nodal domain each of the nodes is a nodal domain so it splits into three pieces so there's no domain here that one and that one and then for example this one splits in three pieces further this this and that okay so each nodal domain is one of the nodes in my tree and you put an edge joining the nodes every time that you have a component of the zero set separating the nodal domains so you can actually record the nest things in your zero set components using finite rooted trees and the way in which we are going to record the proportions of the different nest things within your zero set is the following so to each component of the zero set so suppose this yellow one you look at the edge that's associated to it in the tree and once you remove that edge from the tree it's going to split the tree into two pieces and you just grab the smallest one so we are we are defining this map that to this yellow component of the zero set it associates this small sub tree here or for example to this blue component here that's associated to this edge you associate the leaf of the tree and the way in which you build the probability measure that's associated to the different nest things is simply on the space of finding through the trees you put a delta mass every time one of these sub tree configurations was hit among the components of your zero set so it's exactly the same construction as before only that now what you're recording instead of a few morphism types are these nesting configurations and the question again is in the limit as lambda grows to infinity will they have a universal distribution of the nesting configurations and the answer in this case again is yes there is in the same paper peter sunnag and your big man prove the existence of this limiting guy new infinity to which the new c lambas will converge so there is a universal distribution of the nesting configurations of the zero set once lambda is large enough there will always be a proportion a fixed proportion of components that are going to be isolated and then a fixed proportion of components that will be above inside another bubble and so on and what we prove would be their sunnag is that the support of the measure is the entire space of trees so if you give me any nesting configuration we can show that for lambda large enough that nesting configuration is going to occur with a strictly positive probability that's what we were able to show and what these statements this the proofs of these statements what they reduce to is actually to working in the n and what you have to do is to find solutions to this equation whose zero set contains a collection of components that are nested according to any tree that you make a map with or a solution to this equation whose zero set components has a component with the defymorphism type that you like so that's how actually the proofs of these statements on the supports of the measure what they really are about is about working in our n on finding solutions to this equation for which you can make sure that the zero set will have at least one component with the defymorphism type that you want or at least a collection of components with the nesting configuration that you want okay and to finish the talk in my last slide what I wanted to do is to show you the only setting in which we actually understand what the limiting probabilities look like because of numerical experiments so this is when you work in dimension two in dimension two the defymorphism types of the components of the zero set are boring it's because it's just you can prove that all of them are going to be embedded circles the components of the zero set so what you can do instead which is way more interesting is looking at the complement of the zero set so you have all these neural domains each of them with a different color and you can study the connectivities of these components so you can count the number of holes that these components have so for example this green component here has a has another one inside so one hole for it that one this violet mass here it looks like it has at least two holes and so on so you count the number of holes in each component and you ask is there a universal distribution for the number of holes the answer is yes and alex but not computed the limiting distribution in this case so this is the only setup in which we know what the limiting measure looks like what he's getting is that 91 percent of the components will have no holes about five percent of the components will have one hole one person will have two holes and then they the the the number of components having higher and higher number of connectivity starts decreasing super rapidly that's what he's observing his table actually goes all the way up to connectivity 20 and the error if I remember correctly was in the fifth decimal place so you don't even see it here and this is really the only case in which we understand what these limiting probabilities are in the other settings the only thing we know is that the support is the full space and that's it that's the limit of our knowledge that's it that's all I wanted to say thank you very much this function sir infinity does it depend on the manifold or what no no it's the same one for all the manifolds what same thing it is is a Gaussian random field in our n that's defined by this two point correlation function so Gaussian fields are completely defined by a two point correlation function so this Gaussian field is the one that has this two point correlation function so you just take the Fourier transform of the spherical measure and you can also I mean physicists like to think of it as a superposition of random planar waves they are equivalent and the only information that you have is that they solve this equation but it's it's just a function in our n and it has nothing to do with the manifold it's always the same limiting guy but yeah but it's not connected to the manifold it forgot it yeah yes when you build your measure mu counting the types of different morphisms would it be meaningful to ponder it by the average measure of each component the size of the component the size yes definitely that's actually the right question to be looking at sir i'll expand what he's observing in his numerical computations is that he's getting in dimension three or higher dimension two is completely different but in dimension three or higher he's getting a massive component of the zero set always so it looks like there is this percolating component of the zero set no matter how many experiments he runs that's eating up the whole space and then he has like this small isolated components that are then reflecting all the morphism types that we are saying but the main guy it's like this huge component that it's taking most of the volume of the zero set most of the house of measure of the zero set so yes definitely that would be the right question and the answer would be probably there is this guy that it's actually leaving the behavior of the zero set but so far we cannot show that there is a component that's large actually in dimension two dimension two is different because this is connected to percolation so the probability of being able to cross from the bottom side of the square say to the top is the same as crossing it this way so you it's unlikely that you will have percolating components for the zero set in dimension three it's different because to close that I mean to go from one side to the other you would need to block it with something of dimension two so it's much harder so yeah you you do have this percolating component but we have no proof of that like we are far away from a proof of that yeah and by the way I should say similar things can be done when you instead of working with this again functions when you work with a problem that's slightly easier so you do the sum from zero all the way up to lambda so you're mimicking polynomials on the manifold and you look at the zero sets and the proofs there are much easier and a lot more can be done so you can bound from below the probabilities and stuff this problem is much more rigid because you have an elliptic that needs to be satisfied throughout both your experiments on your explanations suggest that the number of critical points in a small region is kind of proportional to the volume and and lambda to the power n of this small region can you show some kind of equidistribution so in small scales yeah you mean well like you yeah does the the repetition of critical points goes to remaining volume or something that's yeah so for the critical points it's hard for the zeroes so let me explain one thing so for the zero set we can show that if I okay let me say it in words so fix a ball of radius c over lambda okay fix a c and look at balls of radius c over lambda and look at the zero set inside that ball so now think of the that zero set and the remaining measure that it's induced on it okay let's call it d sigma lambda what we can prove is that this d sigma lambda will converge in distribution to a d sigma infinity so in small scales you have conversions of this measure and one what's crucial to get these conversions in distribution is that we know that in our n the zero sets of this equation if you restrict to a bounded ball it will have bounded measure so the moments are going to be bounded so we get conversions of all the moments for the number of critical points if you try to do that we don't really know that the moments are going to be finite we actually come check sure that at some point actually for high enough moment it will be infinite so we don't we cannot get this convergence but we don't really know what happens with the moments yeah but that's a really nice question yes I think maybe related to that so for a lot of the local things there's a common limit distribution that doesn't depend on the manifold is that just because everything is almost everything starting to happen in really really small balls that are almost like euclidean space so the distribution is the one so eigenfunctions they are oscillating like crazy as lambda goes to infinity so really what's happening is that the larger picture really looks like a student and yes that's what's happening there was a question


Is there anyone wishing to speak to the board of finance at the public forum? Looking online, so you have several attendees, please raise your hand if you'd like to be recognized. That will close the public forum and move to the consent agenda. It's like welcome motion to adopt the consent agenda. Take the action indicated, which is essentially adopted. It is adopted and submitted to the public forum. So vote. Thank you. Is there a second? Second. Second by, that's a high tower discussion of the consent agenda. Seeing none, we'll go to a vote. All those in favor, motion please say aye. Aye. Are there any opposed? Consent agenda is adopted unanimously. Okay, we have three delivered items. First one is 4.01 amendment. Assignment of grounds to reserve formal housing at 300 Lake Street. So two item and let's see what we got. Todd, I hear you come back, share, let's give us a quick summary. Good evening all. So by way of background in 2003, the city leased an acre and a half of city land to the Burlington community land trust to construct 40 units and permanently affordable housing on the waterfront. That building waterfront housing is just per reference located near where the skate park is currently located. And it is to my knowledge, the only permanently affordable housing located on our waterfront. Since it was completed 20 years ago, this successfully operated serving primarily folks at 15% to 60% of area median income and better rent, which is affordable to those households. So recently due to an undetected leak in their wastewater system, the building experience very substantial damage, which will require approximately $3.1 million in repairs to the building and made a number of the units inhabitable. So given that the insurance will not cover all of these costs, the most feasible way that we understand to fund these repairs is to bring new low income housing tax credits into the project, which will infuse a substantial amount of capital to fund those repairs. So CHT is proposing to do exactly this and has two requests which are necessary bring this funding into the project. The first is to allow the city ground lease to be assumed by this new low income housing tax credit entity. And the second is to modify the terms of the lease to maintain the annual lease payment at its current rate of $4,700 a year. So after CEDO has examined the financial performance of the projects to determine whether or not the property could continue to be viable with an increased rent, we determined that it could not based on information we perceive. And so consequently, we are recommending the approval and we CEDO supports both of these requests from CHT. Great, thank you, Ted. Lawrence open for questions or motion? Councilor Hightower. Yeah, just curious as to if we know what CHT did for or with any of the residents of the units that had to be vacated and... Yeah, my understanding is that the folks who were displaced have found temporary homes in different locations within CHT's Port Collea. Unfortunately, we're also joined by Amy Dimetrowitz of CHT and she may be able to give you more detail on that. Sure, what are they? Yes, we were able to... The leak was discovered over time and so we were able to relocate all of the residents to other homes. Some are choosing to stay long-term and others, everyone has been offered the ability to come back once they're completed. But some are deciding to stay where they were relocated to. Great, thank you. Further discussion? President Paul. Unless others want to, I'm happy to make the motion to take the action as recommended before this. Thank you. Is there a second? Seconded by Councilor McGee. If you know further discussion, we will open the vote. All those in favor of motion, please say aye. Aye. Any opposed? Motion carries. Thank you. Thank you. Okay, one other item for approval and recommendation of the city council and its BEV property, machinery, insurance and all. Welcome, Paul. Let me get us a quick discussion. Sorry, I'm over here. Absolutely, so this is our annual profit-boiling machinery, insurance and all. The biggest line by far is all of what we've come with. Every year, the name when it's been on the board for a while, it's been probably too many times, but we've had significant increases in the last three or four years. Three or four years ago, about 225,000 we jumped to 660 degree when we're paying now. Good news this year. There are four different insurance carriers that underwrite this. The big one, the station is particularly unique. So we have AIO, which is huge, Zurich, StarTech, and just good news this year for the first time in four and a half years. You have a slight decrease predicted not to exceed 3%. That's what we put in our budget this year. So we're here in front of you, looking for that approval to not exceed the good news. The thing probably to pick up in Boardman, which is the insurance, I hope, provides a quick city hope to come in after a little bit less than that, about 670 to 670,000. So it's a 3% increase, we're looking at probably 1.3%. The timing is such that that policy doesn't renew until November 20th, and it's been November 1st, 20 days in the insurance world, like you. So I'm reluctant to walk into the last minute, so I'll look at the staff that are working extremely hard. Guaranteed to find coverage not to exceed 3%. I guess the other good news, in addition to the decreased costs or not to exceed, is that AIG has actually wanted to write what we call capacity reading. Instead of 25% of the appeal, they wanted to write a monthly vote, 50%. And that's significant because their rates are less than half of some of the current numbers that are writing that policy. So the good news, they raised it to 35%. That's why we're seeing a slight increase. Fingers crossed, we're hoping to, as early as next year, if you have AIG, right, just kind of have capacity and then we'd see even more significant savings. So that's it in Nutshell. Here to answer any questions. Great, look at Bob. Floor's open for discussion, promotion. Because of Bob. So you've never been tired of seeing people. I do remember when you were here. The only reason I remember is because if you say that you're only here for this one, I remember when you were here. So it must be this one. And I do remember when it went up so much, probably because I believe Paul Plunkett was sitting here and I was like, wow, that's a lot. So this is actually much better news. And thank you very much for the, thank you very much for the presentation. It's interesting, at least for me. It's interesting to know why, you know, the why going into the why and that's where, with all of why we're honest. It's not as high a difference. It's very interesting. I'm happy to make a motion to take the action for the board to us. President Paul, is there a second for the motion? Second by Councilor Hightower. Discussion of the motion. Seeing none, we will go to a vote. All those in favor, they go to the chair. Aye. Aye. Very opposed to the motion. Thank you, Paul. Thank you very much. See you next year. That's right. Paul, the risk guy shares that sooner is a problem. Good night. Thank you. Okay, final items for tonight is for board approval. Only is a request to a contract with PC construction company for roof structure stabilization and the board auditorium. This is the first piece of work in the structural stabilization at Memorial Auditorium that we discussed with several times with the bond funding that was allocated to make sure we keep this building standing for the next few years when we continue to determine the best path forward for the building. So again, this piece was the first one to get underway that structural engineer had let us know in the report over the winter that there was a risk of roof collapse due to damage to or just time age of some of the roof permoids in heavy snows. We're trying to work hard to get this work done before we have heavy snow and we're excited to get a few bids and moving forward. So having an answer and a question. While we're here, just a note and give it a piece of story not long ago about how to use it. Sure, this is a parallel process to that obviously the stabilization of the building. RFP, which we call that are similar, is sort of out on the street with proposals due on December 2. So we'll be working in the next few weeks to put together that committee to review proposals, hopefully, as they come in, but have a second. Great. Good. Both processes moving. It's a large open for discussion or motion as indicated. Thank you, Councillor James. Seconded by President Paul. Any further discussion? Seeing none, we will go vote. While we prepare for the motion, please say aye. All right, opposed? All areas unanimously. And we have completed our businesses for finance. Seeing none for the business for objection, we will adjourn at 5.18 p.m. for finance. We're actually at, well, how to schedule. So I know it's just a little bit of a result poll. Maybe I'll hand it over to you to share with folks how it works. So to just give us the opportunity to celebrate the huge news today, like the big news in Burlington, the roundabout is open. The roundabout is opening because there was just a capital to a reality that was major big news. News that you were one of the great champions of President Paul for many years.  But it's very exciting. It's today was the day. And I wasn't there at 10 o'clock in the morning. I wish I could have been, but they opened it, apparently, and took all the stuff down at 10 a.m. this morning. And there's no lines yet. They haven't painted the lines. But you can literally go around. And there's actually people standing, waiting to be able to cross, so they can do it so much more safely than with the students. So it's, and they're ahead of schedule. Like six months ahead of where Ireland said they would be. So it was supposed to be in June of next year. So how often can you say that? I'd get that in the schedule. We got another project coming in the way. I'm going to carry it here. Yeah, we can wait, but we can certainly grab it and say, what is it, 521? We're going to have to wait. So we're going to be here. So Joe and Soraya, just so you know, we're going to be in this room. So we're not going to have to go anywhere. So we'll just sit here. I assume others will know.


Here's some details of section 3 with the soft kick. So we're in this position and we figured it. Now we're on the Haomang back stance, 80-20. Our fists are to our chest and lift up. So the difference, like I mentioned, is that this one lifts up. It'll be a little bit higher so that we can stomp out and down with your weight and your hips. The first time we did an upward action with our toe or with our heel. This time we're going to do a downward action. So you lift your knee higher and use a downward action to stomp down. The target should be your heel. So from this angle, lift it up. Your toe shouldn't be straight like this. It should open up like this. You can have it be 45 degrees. You can get it slightly like this and we stomp it down like that. Stomp it down like that. From here, when you drop, use that momentum to bring your body forward. So, time, stomp. When it drops, use that momentum to bring your body forward. From this side, this direction, knee up, stomp down. Use that momentum. Use that momentum.  Some more time, knee up, stomp downwards, but not down here. When you stomp as far out as possible, at the same time, extend as far as possible that way. But use your weight to stomp it downwards at the same time. So if you can't lift it that high, you can start and try to stomp it down. Lower, if I feel it, you want to lift your knee a little bit higher, stomp it down as high as possible.  So that's some more details on the stomp kick and the shuffle.


My name is Max Geigel and I interned at the Rhode Island Office of Innovation and did a project at the Rhode Island's Division of Information Technology. I was fortunate enough to connect with a number of people around the United States who all pointed to Rhode Island as a great place to be doing design and innovation in government. And shortly enough, I was talking to the people in the Office of Innovation, so I connected with them and ended up working in both of those offices. The state government has a number of websites that are not governed by an overarching web structure. And the specific project I was working on is to help design the rules of engagement of how someone might design a new website underneath an information architecture for the state websites. Entering grad school, I was very interested in design and innovation in government. And the experience only stoked my interest in working in innovation. I had some really hardworking, passionate people working on the ground in Rhode Island. And I was very fortunate to learn from them and learned what it actually means to work innovation into government practice. Some advice for the incoming student. The internship is a great experience and one of the best experiences to apply practically what you're going to learn in grad school and where you might get opportunities after grad school. So what I would suggest for anybody coming in, if you're looking to design your own internship, figure out what you like the most out of your year. Figure out what you want to do, what you want to explore. Put those down and start networking as soon as possible. Reach out to the people here at Ford School, your fellow students. Reach out to your professional network, make lists and work your way through them and talk about exactly how they can give perspective to you in your second year as well as your professional endeavors outside of it. Eventually you'll start talking to people that have internships and you can go from there to see if those might be of interest to you for the summer in between your two years.


This 10th year of Daily Tech News show is made possible by its listeners. Thanks. That's you. That's you listening right now. Maybe you're Dale Mulcahy or Matt Zaglin or Kelly Cook or our brand new patron, Ari, whoever you are. Thank you and welcome on this episode of DTNS. So, uh, where are y'all getting your social media news these days? Is this the year of VR or is it still early days? And Google uses some AI to shorten your wait time at a stoplight? Finally, a good news. This is the Daily Tech News for Tuesday, October the 10th, 2023 in Los Angeles. I'm Tom Merritt. And from Studio Accidental Pecan Milk. It's a long story. I'm Sarah Lane. And I'm the show's producer, Roger Chang. Uh, patrons, stick around. We'll try to get the story of the Accidental Pecan Milk. Oh, I'm ready to tell it. Yeah. Uh, meanwhile, let's start with the quick hits. That's the wrong one. There. At its annual conference called Max, Adobe announced the Firefly Image 2 model meant to be better at rendering humans like facial features, skin, body, and the previously dreaded AI hands that always end up looking real weird. Adobe said that Firefly has now generated three billion images since launching earlier this year with one billion generated in September alone. So a lot of uptake and usage. 90% of Firefly users are apparently also new to Adobe in general, likely why the company decided to convert a demo site for Firefly into its own creative cloud service. Sony announced a new PS5 will begin shipping in November. And it's an interesting configuration. There is a detachable disk drive. If you want the Blu-ray drive, it's detachable. You also get a terabyte of internal storage, and it's the traditional slim form factor, weighs 24% less than the existing model, takes up 30% less space. It also has two USB-C ports on the front instead of the USB-A and one USB-C. You can get it in the US starting in November for 500 bucks with the disk drive or because it's removable, you can buy a version without the disk drive. That'll cost you $450. So you save 50 bucks. And if you change your mind later, you can actually buy the disk drive for $80 and slide it in. The PS5 is the new version of the PS5 will roll out worldwide over the following months. Somebody should make a song called detachable disk drive. They will now. Just in case, you know, you wake up one morning and your disk drive is missing. Okay. Google has announced that it is making pass keys that's a phishing resistant alternative to passwords that allow users to sign into accounts using the same biometrics or pins that they use to unlock their devices. The default sign-in method for all Google account holders. Quote, this means next time you sign into your Google account, you'll start seeing prompts to create and use pass keys, simplifying your future sign-ins. That was Google Product Manager Christian Brand and Serum Cara explaining how it works. Our goal is the same as always been, giving you technology that is secured by default so that you have the strongest security, but without the burden. Here we go. Passwordless future, still future. Bring it on. By this Friday, the UK Competition and Markets Authority is expected to conclude its collection of feedback on its provisional approval of Microsoft's acquisition of Activision Blizzard, at which point the deal is expected to close. Now, who knows, right? With this thing, but it's like 98% certain that the deal is finally going to close this Friday. And yes, the US FTC is still pursuing a case against the acquisition, but that's not going to be what gets in its way. US courts ruled against the FTC being able to block the merger in anticipation of the end of this long-running Sega. Activision Blizzard posted that Activision Blizzard Games will not come to Microsoft's Game Pass subscription service until next year. Everybody's settled down because some folks are like, so I'll be able to play Diablo on Game Pass on Saturday, right? And Blizzard's like, no, it doesn't work like that. We hope to have a bunch of stuff coming, but not until 2024. Meanwhile, Microsoft is expected to announce that it is developing its own AI chip when it does its developers conference. That would put them right alongside Apple, Google and Amazon. Well, Unity CEO John Ricatello has resigned following the company's pricing, let's call it a kerfuffle, that angered many game developers and that is putting it lightly. In a press release, Unity said James M. Whitehurst has been appointed interim CEO and president of the company. Unity also said its previous guidance for its fiscal third quarter financial results will stand. Those are going to be reported on November 9th. So at least at this point, Unity is feeling pretty confident financially. If you can't take the heat, Ricatello, you resigned. Okay, never mind. Let's talk about VR. Let's do it. Yes, I love VR. Bloomberg's Mark Gurman reports his sources say that Apple's working on a lighter version of the Vision Pro. Now, a lot of people haven't tried it out, but supposedly the coming first version, which weighs one pound, has gotten reports by some testers as heavy enough to cause some next train, not very comfy. The next-gen Vision Pro might also build prescription lenses into manufacturing. The first-gen Vision Pro hardware has been ready to go for months again reportedly, but Apple is not launching it until next year because it's hoping to perfect some of the software around it. Yeah, get developers to make apps and all that. Meanwhile, Tech Crunch's Brian Heater reviewed the MetaQuest 3 out now, October 10th. It's shipping for $500, which is still a bargain next to Apple's $3,500 Vision Pro, I guess. Meta also dominating the market already, even before the new Quest 3 came out. 50.2% of the market share. That's what the aging Quest 2, so that's probably going to go back up now. PSVR 2 from Sony holds the rest of it, 27.2%. So you've got a somewhat casual, but very mobile headset with a lot of games in the Quest. You have a more sophisticated gaming headset, but tethered with the PSVR 2. And then shortly after the first of the year, we're going to have a fully mobile headset better at AR than the other two, carrying the Apple Mystique, but also the Apple price tag. Heater suggests in his review of the Quest 3 that all of this is the makings of 2023 being the year of VR, or at least maybe setting it up for next year to be the year of VR. Sarah, are the early days finally over? Gosh, I don't know. I mean, I would like to say that I really, really care if everyone else gets into VR because that just means that there will be more apps for me as a VR enthusiast. But there was a time, not that long ago, that VR was something that we covered on DTNS and TND before that.  And then once I tried it, and I got that killer app for me, Supernatural Exercise app, I was like, oh, this is fun. This is awesome. I can't do this another way anywhere else I'm in. And from there, I still feel like I can count all the apps that I ever launched in VR on one hand, but it's worth it for me. It's worth it. Even if you're, I don't know, paying more money for a subscription service, which I do for Supernatural, for example, which Meta now owns. But yeah, I feel like the Apple Vision Pro stuff is going to change the landscape a lot. You're going to get enough development that there are going to be even, again, can them on one hand, a few apps that just, you know, cream of the crop, rise to the top.  Because I cannot tell you how many people in my life, friends, and these are tech people who are just like, I'm just not into it. I'm not interested. You know, I don't need you to convince me that VR is cool. I just, I don't want to do it. And I get that. But I also feel like I was that person at one point, and I can see where it really does change the landscape. And mixed reality is a conversation that I actually don't really have a lot of experience with. I kind of understand how it works. I'm still into VR itself. Like, let's just go away and have some fun for a little while. But yeah, I think with the Vision Pro, the Quest 3 being the, I guess, consumer-wise, the best option that you have, Quest-wise, they have gotten better with each iteration. You know, why not? And even the tethered people, the tethered people who say, I'm not jumping around on wearing a VR headset. Like being tethered doesn't matter to me. I want better quality. That's also an option and has been for a long time. Yeah, I think that is a fairly specialist case where you're in the Sony universe and you want really high quality gaming. And I think it'll always have its place and probably won't always be tethered. But I think where Quest 3 is going, even with its weird mixed reality thing that's still a little low res, even if it's got better colors than the Quest 2, and Apple, with its very good, apparently, mixed reality where you can see the world around you very clearly. I think next year's make or break. Apple's either going to, with all of its developer work that it's doing right now, find a couple of those apps like you're talking about that just make this sing and make everybody go, well, I don't want it for $3,500, but for a cheaper version or a lighter weight and cheaper version, I'll do that. Then we're on the road. That's what Apple does. They have very expensive first gen stuff that doesn't do all the things, but it makes people want the next version. If that happens, then we're on the road and VR will take off the way tablets and phones and everything else has. But if it doesn't, if people are like, yeah, I still don't see it. I still don't see what I'd want it for at any price, even if it was cheaper. I don't see that coming out of the Quest 3. Then we're still waiting for somebody to come in and do that. I think a lot of people are just banking that Apple's going to do that. They certainly have a good track record of that sort of thing, but nothing's guaranteed. It's all so much. As much as I say like, oh, VR is so cool. You just have to find your killer app and then you'll never want to take the thing off, which is true. That was true for me. There's also a lot of junk. There's a lot of social like, oh, let's all sit in the theater and talk to each other as avatar stuff that, I don't know. Some people might get a lot out of it. It's never made a lot of sense for me. If you're not a big gamer, a lot of fun games that are a little bit more full body than city in front of a monitor, that offers you, but I don't really play a lot of that stuff either. I found my niche and I feel like it is a game changer for me. It's not going to be for everybody, but I, but yeah, I really, I want to have this conversation in six months when some vision pro stuff starts coming out because you know, Apple is already working with certain developers very, very closely about making something that's going to wow the rest of us. We just haven't heard about it yet. Two weeks after the vision pro ships, two weeks after people get their, them on their heads, almost said in their hands, but they'll be on their heads. We'll know. We'll know, we'll know whether that thing is in existence and what it is. Yeah. On their heads, but in their hearts. We'll find out. That's free. Just doing some marketing stuff for everybody. All right. Let's talk about something that Google's doing. Google launched something called Project Greenlight two years ago. The idea was to reduce the amount of time that cars were stuck waiting at red lights. So stoplight stuff, green light. In theory, this would also help reduce emissions because the cars would be idling less often unless they're EVs of course, which is a whole other thing. Google trains machine learning models on traffic data from its maps product. Then the model can make recommendations of changes to traffic like timing to reduce the time spent waiting at those lights. It's been tested in a dozen cities worldwide. And on Tuesday, Google felt pretty confident announcing results and expansions of the project. Google VP of Geo sustainability, Yale, McGuire said early numbers indicate a potential for us to see a 30% reduction in stops in the city of Manchester, England specifically. It saw air quality improved by as much as 18% says Google. So they're all signs pointing that this is a good thing. Yeah. So if you're wondering exactly how this works, participating city gets access to a dashboard, to a website interface. And the Google Maps data will be trained with a little bot. And the bot will then tell the city manager, hey, if you adjust the timing at the corner of first and main to be this pattern, so maybe it's equal during commute times, but it's longer for main street during the 7am to 8am period, et cetera, et cetera. They change the timing, which cities can already do. They don't have to put in any new infrastructure to use this. They just need to use the data. Then supposedly if they follow that traffic flows better and you'll get to a stoplight and not be sitting there going, why is this red when there's nobody coming? At least that's the idea, right? And it's working on the same principle as Waze to say the machine knows what the patterns of traffic are. And in this case, instead of telling you what route to go, it's telling the stoplights, oh, you should be green more this direction than that direction right now to keep traffic flowing. Yeah. I was just thinking about this over the weekend. There is some construction going on in my area. Basically, if you drive in any direction, you're going to get stuck in it on some level. And for that reason, the stoplights have been kind of weird, but they're trying to route us the best they can. And I've often wondered, am I just not close enough to the crosswalk and the light to know that I'm there? I've been sitting here for a while. Is something wrong? We've all been in that situation. Sometimes there is something wrong, but often it's just traffic pattern stuff that was programmed way before you ever got to that intersection. Anything that can make this make more sense I'm down for. I have to assume that the lights themselves, because there are sensors at a variety of intersections, they probably have a lot of this information anyway, but Google, for example, and other companies too, but Google especially has so much more information of how many cars aren't there yet, but will be there soon enough. And maybe. And historical patterns too, particularly like, oh, every day at 8 PM, it backs up at this light, et cetera, that kind of right. Or like for, for whatever reason today at 8 AM, it's not backing up. Is there an accident, you know, somewhere else where we can then, you know, help the information with everybody. And honestly, I don't know how adaptable the system is, because this is saying, we will train the machine to tell you like, oh, this is what your optimal timing is, but it's not real time as far as I can tell. It's not responding to patterns in real time. It's only telling you what timing is optimal versus what you based on typical patterns. And that's, that's a question I have, which is, and this would require infrastructure. I get why they want to do it this way, because this means I don't have to put anything new into my system, because they're, they already can program the lights, right? But isn't it, wouldn't it be even better if you had real time sensors combined with real time data, which like you just said, Google has to say, oh, there's nobody coming that direction for a long time, even whether it's the traffic pattern or not, let's make it green this way, because there's a car coming, like just, just have them adapt in real time. I feel like that would be ultimately the way to go. And this is a step on that direction, I guess. Hey, for those of us who spent a lot of time in traffic, I'm actually, when I said those of us, like, it's not really me, I work from home, you know, I'm pretty lucky. But, but yeah, I, I think, depending on where you are, what metro area or even suburban area that you might be in with something like this, companies are trying to help your commute be that much less of a headache. I'm well, if you are a Google Maps user on Android, in particular, you're going to want to check out Android Faithful. If you haven't already, host Ron Richards and Huantui Dao bring you all of the Android news and information. And they've got flow. If you know flow from previous Android shows, she'll be joining them this week as well. Android Faithful is a podcast devoted exclusively to Android. Catch it Tuesdays at 8 p.m. East Coast, 5 p.m. Pacific Live, or subscribe to it right now at at www dot, you know, the www part Android Faithful dot com. Casey Newton's latest issue of his platformer newsletter discussed how the war in Israel has highlighted how news has moved off of Twitter now X but hasn't really found a permanent new platform. The short version is when it was Twitter during breaking news events like the outbreak of a war, for instance, you would find verified users in affected areas, posting reports, videos, images, etc. Sometimes they would be journalists. Sometimes they'd just be people on the ground, but they were verified so you knew their reports were reliable. However, with X having a reduced safety team and a change in how they verified users, it's made it harder to sift through some of the false posts. Verified users are paid, so it's difficult to tell if someone is a legitimate or not. Not so much because of the paid ones, but because a lot of people who were verified before are not now, and they choose not to pay to be verified. So it's a different landscape of who's verified and who isn't. Fewer people are also posting on X than used to. So there's just not the concentration of folks posting. I've run into that with less significant news events in my area where I'm like, oh, I'll go to X and I'll look it up and nobody's posting about it. Whereas a year ago there would have been at least a dozen in my neighborhood. Casey Newton notes that a lot of people were posting on threads this weekend about the events in Israel, and he discussed meta slash Instagram's unwillingness to embrace threads as a news platform. Instagram's head Adam Osseri has described threads as particularly focused on creators and sort of not saying never put news here, but saying we're really not trying to cater to that. Newton describes it as TikTok, but for text, which is interesting since Bloomberg has an article with the headline, TikTok footage becomes world's window into Israel conflict, and the BBC covered this too, which is all about people posting their videos of what's actually happening on the ground to TikTok. And of course, TikTok and Instagram have pretty big trust and safety teams. Doesn't mean stuff doesn't slip through, but they are on it and trying to remove it as fast as possible. Sarah, I'm curious. I know you followed the news this weekend. Where were you going to find out this sort of instant on the ground from people involved kind of reporting? Well, it kind of reminds me the landscape has changed. Well, I mean, gosh, that's almost a strange comment in light of this. But this reminds me of a conversation that we had, I believe, with Justin Robert Young when the Russia-Ukraine war broke out. And we all kind of talked about, okay, where do we feel like reliability is? Where's the best place for that? And I had said, you know, I'm reverting to cable news. Now cable news, depending on the channel that you might be watching, whether you're watching it to either agree or disagree with the information that you're getting, in general, I feel is, okay, the network itself, and I'm talking about legacy networks, obviously, you know, believes in its reporting. And there are a variety of journalists that I very much trust and, you know, want to, you know, encourage to give me good news. And that is sort of, I found myself for the first time kind of going, you know, I'm going to turn on the news and just kind of watch it for a while or watch a special that I know is coming on later type thing, because the internet is a little bit too much of a landmine. Now this was like still very much Twitter days, but it started to already feel a little crumbly to me where it was like, I felt like I was spending too much time either trying to figure out if a piece of information was correct or not on my own or try to school people that were not doing that for themselves that I thought should know better. And both of those things fell not worth my time.  And over the weekend, I couldn't find anything of value. And I'm talking people I follow who mostly just complained that the social network sucks now. Trending topics that was just a landmine of misinformation, you know, and everything in between threads. I'm still, I feel like you still have a lot of people on threads being like, see, this is better. Why isn't everyone here yet? And I, I'd love to get past that. So it just becomes better instead of talking about it being better.  There are other options, obviously, quite a few of them. But, but yeah, that's, that's kind of where I am. Yeah, I think you don't even need cable to get the experience of live news. CBS news is available for free, ABC news available for free. I think NBC news is still available for free, but also a version in Peacock. Max just launched a version of CNN's 24 seven coverage called CNN max. So if you have max, you've already got access to that. But the difference was always that on Twitter, you could get the things that those slower moving news organizations didn't get to the person who was actually there video making video that those cable news networks would then take and put on the air for them when they verified that person. And they were still doing that with acts. It was still happening. But it's just not happening as much as it used to. And yeah, I feel like threads was the place that a lot of people wanted to turn to for that. But because threads isn't really encouraging that, then a lot of people feel that threads is failing. But the fact of the matter is there's just, there's just, there's just not a place that everyone has agreed to go to the way they had agreed to go to Twitter in the past. I also, I mean, listen, say what you will about Facebook, but I can kind of see, you know, some people high up at medicine, let's not get into that big old news thing on threads. Let's make this a fun creator thing. It's Instagram and it's easier. Yeah, controversial way easier. You know, just make it like a good place. Canada passing a law against you. Right. Right. So, so there's that. That's what's going on. And yeah, you're right. When I say cable news, I think I use the word cable now to say like non streaming modern stuff. So yes, it doesn't have to be actual cable news, but and you know, a lot of, a lot of these outlets will let you stream, you know, directly from their website, maybe not all day, but you know, at least a short period of time. You can do CBS news and ABC news all day 24 seven for free. They'll have ads, but you can do it from the website. So there's plenty of that kind of news, but I know what you're saying. Like that's, that's TV news. That's slower moving. It's going to be more reliable, but it doesn't give you that up to the minute like, Oh, this is a person actually there right this second. That's what it used to do. Yeah. Yeah. Yeah. It simply isn't the same. And for anybody who's sort of like, I never use Twitter for news anyway. Maybe you don't see a big difference. There is one though. There really is. And that says a lot about the people that you follow and the people that you interact with. So, you know, nobody's going to have the exact same experience. I think threads should step up and I do too and work with the, the actual news platforms to provide this, because that would then encourage the people who are involved to be like, Oh, that's where the news is. On the other hand, maybe I don't want them to maybe I don't want that to be Metta's choice. Maybe it would be better if Mastodon was the place, which is what BBC has chose as their primary place to share news. Maybe it doesn't have to be anybody. Maybe maybe we can just see where people go and end up. Let us know your thoughts. Where do you think this should happen? Again, not news publishing, not big cable TV style news. But, but when someone is involved and says, I need to tell the world this, I don't know, they used to be able to go to Twitter. Where should they go now? Well, Tom, when's the last time you flipped a coin to see who had to take the dog out? Yeah, certainly, I just take the dog out so that there's no coin flipping there. But, but yes, I, I don't know. It's probably it feels like if we don't have a better idea, this is at least a 50 50 way to do this, right? Yeah, right. Flip a coin heads or tails. It's random. You pick one. You hope you're right. And, you know, you have a 50% chance of being right. Researchers at the University of Amsterdam say, Oh, not so fast, everybody. They said after flipping coins, over 350,000 times in the largest study of its kind, they found coins have the ever so slight tendency to land on the same side that they started on. So if I'm holding heads up and I do a flip, I have 51% chance of landing back on heads than tails, almost 50 50, but not exactly. The researchers used coins of 46 different currencies. So it wasn't just a penny, wasn't just a quarter, which were flipped by hand, caught in the palms of 48 student participants, and they videotape all of this for proof. The same side bias has previously been predicted by scientist Percy Deaconis. The theory being that the physics of coin flipping with the wobbling in motion of the coin makes it slightly more likely to land on the same side that it started on. So heads to heads, tails to tails. So if you're like, I already knew this, great. Good for you. I didn't. Now it's been proven by another study. So that's, that is interesting. And if you want to try to bias the results in a coin flip, now you know it's only 51%. It's not like it's going to be guaranteed every time. If you want to guarantee it every time, see Brian Brushwood's Modern Rogue. He did a video once to show you how to, how to, well, it's cheating a little bit, but how to make sure your coin flip lands the way you want every time. I've also also been like, if someone's a card counter, like, why are they penalized? They're good at this. They should win.  All right, patrons, stick around. We have got the mystery of the almond milk. Also, Samsung joining Google and shaming Apple over the RCS messaging protocol. I don't know that this is helping anybody, but we're going to talk about why they're doing it and whether they think they should stop. We certainly are. Just a reminder, DTNS is live Monday through Friday. We were off for a U.S. holiday yesterday, but Monday through Friday, 4 p.m. Eastern, 200 UTC is when we record the show live and we'd love to have you join us if you can. Find out more at dailytechnewshow.com slash live. We're back tomorrow with Scott Johnson joining us. Talk to you then. This show is part of the Frog Pants Network. 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