 So, it's my great pleasure to introduce our today's speaker, Professor Sylvester Ericsson-Bick from the University of Yves Gle. So Sylvester obtained his PhD in the current Institute of New York University in 2017, and then he was an NSF post-doc in UCLA till 2020, and later he received a highly competitive Finnish Academy research fellow and moved back to Finland. So Sylvester was first a system professor in the University of Oulu, and then moved to Yves Gle in 2022. Okay, so Sylvester is a pure mathematician, but working in foundational and interdisciplinary questions with applications in both theoretical math and computer science. So magic embeddings won such questions which attract great attention and interest from both math and computer science community and has generated a vast range of applications in machine learning, in computer vision, etc. So we are very happy to have Sylvester here as a visiting scholar, and today Sylvester is going to tell us about embeddings as a problem-solving technique, so please. Thank you, Sheridan, and thanks for having me. It's very nice to be here at OIST and to have this opportunity for a longer visit, so thank you to the TSVB program for making that possible, and thanks everybody for coming here. So this is going to be a very general talk, so I wanted to first just give a little my perspective on my general overview of my work. So yeah, I'm from Finland, and this here is representing a word cloud of the research papers that I've written over the past around ten years. And so maybe some things to highlight here is that metric spaces is very large here, so a lot of my work concerns metric spaces, different questions on them, such as embeddings. There are also other words here that you might notice like next, thus and therefore, so you can see all of my bad writing habits where I use the same word over and over again, but and so these problems have applications, but I will admit that I'm a pure mathematician at heart, and most this work that I will talk today and present here is a part, one aspect of my work, so metric embeddings. Other things that I also do are related to fractals, causes of metric embeddings, and then the tools that are used in their study, which is often called analysis on metric spaces. So but today I will just focus on the metric embedding part, the embeddings part, and for me what is an embedding? An embedding is something that takes an object, a task or data of some form and converts it to vectors or points in a Euclidean space, RN. And so that's already some narrowing down. I will talk about more specifically what I mean by an embedding later, but I want to at first have this very general view. The only restriction I will do is that the target here is some sort of vector space, Euclidean space. So there I'm aware, for example here at OIS that there's been some interest in other targets such as hyperbolic space, there the questions are pretty different, but I will focus here today on this Euclidean question. And I will tell first about generally why one would consider embedding problems, what one can do with them. And this might be a bit simplistic, so for especially for people who have used these before, this might not be so deep and profound, but I want to give a sense of where these embedding problems come from, what they can do for you and start from something very, very simple. So there was a Finnish mathematician, Lassi Paivaen, who worked in inverse problems and he wrote once a nice little article about mathematics and he said in Finnish mathematics grants you new eyes. And similarly metric embedding or these embeddings, what they can do is that they can take your problems and they can convert them to a space with more structure where we have more tools to study them. So this is an extremely simple example of such a setting. So suppose you want to add the first n numbers, 1, 2, 3, 4, 5, etc up to n. Well we can take this problem that's a problem about numbers, some kind of one dimensional problem and we can represent it as a two-dimensional object. So the representation here takes all of these numbers and converts them to a rectangle or a column of rectangles where we have number of squares is equal to the number. So 1, 2, 3, 4, and 5 to this case, so n is equal to 5 in this picture. But now that we have represented this as some geometric figure in Euclidean space in the plane, in fact, what we can use is the additional structure that that gives us, namely in this case symmetries. We can take two copies of this gadget which have the same area and we can form a rectangle out of them. Now this area of this figure is the same as the sum if the side length here is 1. The area is the same as the sum that we're looking for. So now we have two of these, we can form a rectangle whose area we can simply compute as n times n plus 1 and the original thing was a half of that. So what happened here was that we had a problem that was just about numbers that was just one dimensional, but by adding in a dimension or representing it in different space, we got more structure, we were able to utilize that to solve this problem in a very conceptual and clear way. So this is a, even though I would say that this is not the type of embeddings that I really consider, I'm not looking at numbers, I'm looking at spaces and embedding them, this is the type of phenomenon that occurs in practice. So this adding dimensions or representing the vectors in the Euclidean space and then using that structure gives you something more. Well, another thing that was also mentioned by Xiaodan in this introduction that we can do with embeddings is visualization of data, extracting something out of it that we would not see otherwise. So I should say already, what do I mean by data? That can be a very broad concept. So the first approximation of what I mean by data, and we'll give a more precise version of it a bit later, is objects and relationships. And for me as a mathematician, what that means is that we have objects which are correspond to some vertices, x, y, z, any set of points, abstract points. They don't lie in a space at first. These could be images, websites, etc. And then we have relationships between them, some connections between different objects. What this looks like is something in a nice setting, something like here on the left. So the objects here are these points, and the relationships here are represented by the edges. But what I want to show with this figure is that the visualization of this matters a lot. So these two things are actually the same set of points, same set of relationships, but drawn in a very different way. So one on the left gives you a very clear picture of what is happening one on the right. You don't really even see that they are the same. I mean, this connects to some ideas about distortion and about choosing a good visualization that I will get to a bit later. But this also connects to another aspect that I won't emphasize so much, but I want to mention because it's pretty cool in and of itself. So this is an example of a planar graph. A graph that can be drawn into the plane in such a way that the edges do not intersect. It can also be drawn in a way that they do intersect. But among all the drawings, we can choose a straight line drawing where the edges do not intersect. So it turns out that this property of being planar, you can characterize by embeddings. But whereas here I'm thinking of the graph as my object that is being embedded into the plane, the characterization becomes something embedded into the graph. So there's a characterization of planar graphs called Kuratovsky's theorem that states the graph is planar if and only if there are two things that you cannot find when it is, that you cannot embed inside it. A complete bipartite graph with six vertices or the complete five graph. That are depicted in the figure here. So why I mentioned this, even though this will not be a focus of the talk is that sometimes thinking of our space not as the domain of the embedding, but more as the target, we can study the target by which things embed inside it. So in this case, the planarity of the graph can be studied by which graphs embed inside the graph. Also other graph properties can be studied in the same way. But this idea goes much deeper and much deeper than I can touch upon on this talk. Namely, you can study properties of Banach spaces or vector spaces based on which metric spaces embed inside them. So you can characterize some properties of them in terms of that. We actually see one example where you might recognize this later. But there's a whole program in functional analysis called the ReBit program, which is basically hinges on this fact that we can study spaces based on what things embed into them. But as nice as drawing good pictures are, eventually we want to give some hard actual computations. We want to solve algorithmic problems involving the data. So where embeddings really show their strength is in reducing complexity of representations in both in space and in time, potentially. So here I want to previously, my data was just objects and relationships. But in practice, what we also are interested in not just whether there is a relationship, but how strong the relationship is. And that can be one way of expression that is through the notion of a metric space where we have a distance between our objects in the space. And this is this object xd. So we have some function that for every pair of points gives a similarity measure, say, between them. And that satisfies the natural properties that probably you've seen. And this was also mentioned in Yanob Yarn's talk a couple weeks back. If you, in case you were there. But there is a fundamental problem when studying objects as metric spaces. First of all, this metric D can be very difficult to compute in some cases. Say if it's similarity measure between two images can be quite challenging to compute. The other thing is that we certainly can't, while the data we can store, this metric tensor metric distance we cannot store. And this is related to the fact that linear algorithms are something that we can actually do, but quadratic algorithms are often intractable in practice. Even just quadratic algorithms. This maybe comes as a surprise if you haven't done computer science involving very large data sets. So I took one simple example. The number of websites is on the order of one billion. I mean, that's a large number, but that's something that you can store in a computer. You can store a name or some identifier that's unique for every single website that exists in the world. That's not a challenge. I mean, it's about one gigabyte of data. But what you cannot do is store any kind of reasonable distance between them. That's just impossible. You know, you cannot buy a computer that is capable of even storing one bit of information and push on to get a pairwise distance because 10 to the 18 is significantly bigger than 10 to the nine. So if we want to compute something on the order of that involves the communication between all people in the world, so on the order of billions of operations, we cannot do quadratic things. Those are out of the question. They will take too long time and we are unable to store that information. So if we want to do computations here, only thing that's allowed are linear computations. And there are, there's an ample amount of research about on this topic. And I chose this one example that's maybe the simplest example of this is computations of diameters of a metric space. I mean, this is just an extremely simple task. You have your data. What is the furthest apart that two elements can fit can be? What is the maximum distance x to y? And there's an extremely simple algorithm to this. We compute all the pairwise distances, but it's intractable if your data is on the order of billions. You cannot compute this. But if your data happens to be represented in a very nice way, you can compute it in roughly linear time. Of course, this is a very special representation that is that and computing this representation is its own task. But suppose your set is actually sitting inside some vector space rk. And moreover, suppose that the distance in that space is given by not the Euclidean distance, but the maximum of the coordinate-wise distances. In each coordinate, we just look at the maximum, the coordinate difference, and then we maximize that over the coordinate direction. So the L infinity metric, so to speak. In this space, we don't have to use n-squared operations to compute the diameter. It suffices to do n times k operations. Why? We have k coordinates. In each of the k coordinates, we compute a minimum and a maximum. And a minimum and a maximum you can compute just with a single for loop and linear time. So if k is a lot less than n, say in some settings it can be done in logarithmic in n, then this is a way faster than the n-squared algorithm that is actually tractable. Of course, what I'm hiding here is defining this representation and this say if your data was websites, you'd have to first choose such a representation that that might be highly non-trivial. But there are other examples of this, and this is just one simple example that I can represent without too much background. So also clustering, this has been applied to clustering algorithms, nearest neighbor searches, hierarchical decompositions of data, etc., the lots of applications of this idea. So okay, that was some complexity, but as much as I like applications and probably most of the many people here prefer applications, I still feel obliged since I'm a mathematician to tell a little bit about what is my mathematical motivation for these objects. And there's one setting and it's a little abstract where this can't be explained. And some of you might have seen this. It's about how we define spaces or surfaces or what in mathematics are called manifolds in space. So actually the original definition of a manifold, if you go back to say the work of Gauss and some of the older books at the beginning of the 20th century, how they define a manifold is always in a way that is some smooth subsurface of Euclidean space. So it's some say two-dimensional sphere in three-dimensional space or some three-dimensional surface in four-dimensional space, etc. So you could think that this is an extrinsic view on spaces or surfaces representing the most subsets of other things. But then around sometimes early 20th century emerged also a different perspective of representing space that is not necessarily embedded but which is represented via the notion of some charts. So here we have the same sphere but we can give coordinates. So it's sort of like a book of maps of the space. And in this case I just have two coordinates, one for the north pole, one for the south pole. Each of them represents just a normal disk and these cover the space. Of course there's some area where they overlap. The picture doesn't show it very clearly. And there we have some rules of transformation between them. On physics this would correspond that you have some space that is given different coordinate systems. We have some quantities in these coordinates that change in a given fashion. So one of these quantities is how we measure distance, namely the metric tensor. So we have quantities like distance, coordinates, and then we have rules of transformation between them. But here there is no embedding given. Now what is the relationship between these two? Well first it was observed that even though we have these embedded surfaces and our definitions might depend on how they are embedded, then some notions actually are independent of the way that the surface is represented. So in other words they are intrinsic and do not depend on the representation of the surface. And Gauss's great theorem is the most famous example of this, that curvature which can be defined by looking at how the surface bends in the ambient space can be actually represented as an intrinsic quantity. So we take, I mean how it is computed is you look at some slices of the surface, you look at how the curve, how the surface bends in that slice, you get a single curve, you compute its acceleration, or how fast it's turning. This gives you a bunch of numbers. And in the two-dimensional case it's at the maximum and minimum value of those numbers you take their product and that's the Gaussian curvature. Now this is something that depends heavily on the ambient structure, we have to slice the surface, we have to compute these accelerations, but Gauss's theorem says that no it actually does not depend on how you draw, drew it, even though the computation entirely depends on it. So and then later, okay so this already shed some idea that there is an intrinsic theory that you could study, express this curvature say just by computing computations and charts. But then later there was also work, it's a real work on showing that these two approaches to defining these manifolds, one this intrinsic way of giving charts and coordinates and quantities in them, and the other of representing the surface as something embedded inside our N, these are essentially equivalent. So first there was Whitney's theorem that says that any manifold can be smoothly embedded, so without introducing any corners. And then later there was Mash's seminal result that said that any manifold can be actually isometrically embedded. So and at this point I should say very importantly that there will be two notions of isometric. The Riemannian isometric is something that preserves distances at infinitesimally small scales. So small distances are preserved but global distances can be distorted in arbitrary large amounts. This would be a point that we return back to in a little bit. And what you see in this picture is one famous example, actually concrete computation involving Nash's theorem, Nash's theorem is actually rather explicit, you can compute the embeddings that come from it. But this wasn't done until around 10 or so years ago. This is an embedding of a flat torus in R3. And it gets this kind of fascinating fractal like structure. If you just google flat torus and Euclidean space you find more examples of this picture. The reason that it's a bit fractal is that you cannot actually embed the flat torus in a smooth way in three dimensional Euclidean space. But you need four dimensions for that but you can do it in less regularity in three dimensions. Okay so it tells us something about the relationship between extrinsic and intrinsic notions. It gives us more tools to study a space because we can first embed it and use the ambient structure to define notions. Now there's a final one that in addition to this complexity theory perspective has probably been the most, at least in the last decade or so, the most talked about aspect about embeddings. At least from my narrow perspective. This one is a little bit different than the one before. This is also related to how embeddings help solve algorithmic problems. But previously in the setting it was about let's take our data, let's go to some Euclidean vector space, let's solve the problem there and let's go back. Here the approach is different. Here we just have graphs, we will solve some optimization problem on the graph, but embeddings will tell us how good that approximation algorithms. So embeddings will not furnish you directly an algorithm to solve your problem, but they will tell us about how good certain algorithms are. They also inform a little bit how you can design the algorithm, but it's less direct. And the reason again why you do these approximation algorithms is related to the same problem that I mentioned before that there are many computational tasks where the exact solution is either intractable because say it's quadratic time or it's even say an MP complete problem where the exact solution is just extremely intractable. So the approximations may be our only way of getting reasonable answers. But how good are those approximation algorithms? And that's where actually embeddings come into play. Now we'll tell about one problem where this is the case, which is a bit surprising story. There are others like this. But this is something you learn in basic, the first one is something you learn in basic algorithms or basic computer science courses. So let's study, we have now just a graph. Each edge has some capacity. So you could think of this as say a road network or some data network where we have connections, but each of them has a limited capacity of data that it can transport. And we want to study how much can we push through information, commodities, whatever, through that network from a single source to a single target. So here's a network. I haven't written the capacities for simplicity. Let's say each of them has unit capacity. So how much can I flow information here from material from S to T? And well, one way of doing it is the flowing everything along this blue path. So take a unit flow that flows along this blue path. And that unit flow will be the largest flow. We see that because it sort of exhausts this one edge. I cannot pass anything more than this one edge. And this is connected to the fact that we have a cut, some set of edges that separates S from T. In this case, it's that single edge. And any flow that goes from S to T has to go through this edge or go through this cut. The cut could have more than one edge, but not just drawn one. And in this, the blue flow has exactly flow one, the minimum cut. So the smallest cut possible will have exactly one edge. So these coincide, this is an upper bound, this is a lower bound, so these are equal. But in general, the max flow min cut theorem says that the largest amount of flow you can take from S to T is equal to the minimum cut between them. This fact is fairly intuitive. It's not super trivial to prove maybe, but say it follows from linear duality. And there are many other ways of doing it. And this is also algorithmically tractable problems. Let's make the problem a little bit more complicated. And this is where embeddings come in. Namely, multi-commodity flow or where you have multiple sources and targets. Not just a single source, a single target, but multiple sources, multiple targets. You could have even different commodities here, but I'll just say that all commodities are the same. And what I will now do is I want to see how much can I flow from these sources to these targets. Again, representing these conditions that I use every edge at most, it's given capacity. But now we have to make a choice about what we are maximizing because we have different demands. And this alters the problem a little bit, but one normalization that is used in these embedding problems is that we look at the proportion of the demands that we can satisfy. What percentage of these demands? So lambda is here the percentage of demands and say we want to satisfy 50% of the demands from all of them. What is the largest that we can do about subject to the capacity? Of course, lambda can be bigger than one here in this formalism. But you should think of it as the proportion of the demands that we can satisfy substitute the constraints. And we would want to maximize that proportion. In this case, it's clear, it's quite clear that we can do these cut arguments. We can say separate these blue vertices from the red vertices. We can count the value of that cut. But it's far less clear that there should be any connection other than a simple inequality between the size of the cut and how much you can flow from the sources and targets. And indeed, you don't get equality between them. But there is some constant that controls this ratio, the ratio of the quality ratio between the minimum cut and the maximum flow. And this quantity is given by a distortion. And what is this distortion? So we have some metric space X that will be associated to the flow. We have the Euclidean space. And then for normalization purposes, I will say that I will look at mappings that are expanding. So the distances in the target are always bigger than distances in the source. This is just a normalization factor. I can multiply my mapping with a constant to obtain this. And I assume that I'm looking at mappings that have some bound from below, so they're expanding, but they have also a bound from above by some constant L. So they stretch all the distance, but not too much. And this distortion is how small we can make this L constant here. What is the largest stretching that we have had to incur? So this is the distortion of a bi-lipschitz map. And it turns out that here, minimum cut always controls maximum flow from above by the same argument as in the single source and target setting. But the minimum cut is only a constant amount bigger than the maximum flow. And what is the constant there? The constant is actually something that comes from the sizes, the number of sources and targets that we have. And how do you prove this? You need to realize that this constant here that comes from the relationship between the maximum flow and the minimum cut is actually equal to how well you can embed a certain metric space in Euclidean space. So it's a distortion of some embedding. Do you actually have the distortion that you have? Is it bigger or greater? Or is it in any way related to the Johnson-Lender-Strauss bound? This one. So then Johnson-Lender-Strauss, you have a dimension reduction from Euclidean space. So there you have 1 minus epsilon and 1 plus epsilon usually. So this distortion would be the ratio of 1 plus epsilon to 1 minus epsilon. In the Johnson-Lender-Strauss case, we're interested in having embeddings that are very close to isometry. So the measurement is a little different in that setting. Well, the simplest relationship would be that df is equal to 1 plus epsilon divided by 1 minus epsilon in that case. So the Johnson-Lender-Strauss lemma you bound from the distretting is bound from above by 1 plus epsilon and from below 1 minus epsilon. So you get the same setting by dividing the mapping by 1 minus epsilon. Or if you want to have different constants, l and b here, it would be the ratio of l and b. Did that answer it? So yeah, dimension reduction is of course a very important aspect of this of embeddings. But unfortunately, I didn't have a chance to fit that into this talk too much. But thank you. Thank you for your question. It's often discussed in relationships to embeddings. Also for the reason that the embeddings that you construct might be very high dimensional. But again, to do computations, you have to lower the dimension. So with say Johnson-Lender-Strauss, you can post compose your embedding with a projection onto a lower dimensional space and then hopefully solve it better. Yeah. Okay. So the quality of or the relationship between this min cut max flow is connected to the distortion of some embedding. And this gives you, there are other examples of this that have had profound implications. Now a little summary of the applications that I've discussed. So what embeddings can do for you, they can transform problems to simpler settings where you have more structure. They can help you visualize data in various ways and hope the good ways. They can help reduce complexity and at least in some settings. Mathematically, they give you some insight on the space and way of using the extrinsic structure vector spaces say to study your space of question. And then there's this more complicated relationship that is like with the max flow and then cut. And in this connection, I will not return to this point. You'll have to come to my mini course that we'll save a little more to hear about more about this. But there is a very famous case of this 0.5 from the last, and that's about, I guess, already over 10 years. But Chieger, Kleiner and Naur, they solved using purely mathematical techniques, a conjecture in theoretical computer science called the German's linear conjecture. And how it was basically similar to this five. You showed that a certain approximation algorithm, the performance of that algorithm was related to how good an embedding you had. And they showed that you couldn't get a better embedding than something. Okay. So I don't want to give just a talk with motivations. I want to tell you something that I know about embedding, something simple. You'll first come to this course to hear more. So the mathematical question that I'm interested in, okay, those were the motivations why I consider it. But what I'm interested in, you have a metric space x with some distance functions d, how well can we represent an inequality in space? How small can we make this constant l here? Another distortion of the function f. And the first thing that one has to notice, and this is somewhat trivial, but it's not so trivial if you have, I mean, you can figure it out by writing a piece of paper and piece of paper thinking about it. But this l cannot always be one. Metric spaces, you cannot always represent isometrically in Euclidean space. So isometry would correspond to l equals one. Okay. I mean, this is a fun problem to think about, but I'll spoil it for you a little bit. And I'll tell you the simple reason why this is not possible. And this is an extremely simple phenomenon about embeddings, but actually I'll show you that a lot of the results that we have about embeddings, positive and negative, hinge on this simple geometric fact. So it's not usually expressed in this way, but this is underlying a lot of the stuff. So when we're looking at embeddings, representing a metric space into Euclidean space, what we're really looking at is how can we represent the geometry of X inside Euclidean space? And the distortion means that there are some differences between the geometry on a general metric space and the geometry of our Euclidean space. And what's a simple feature of Euclidean space? It's a takeful, it's pretty simple to explain, namely that every pair of points, say X and Y in Euclidean space has a unique midpoint. If there exists a midpoint and it is unique. Moreover, it has another property that if you take three points, then the midpoint between X and Y has to be different from X and Z, of Y and Z. So every pair has a unique midpoint and two pairs that share an endpoint, they cannot have a shared midpoint. But both of these facts are something that you can break in metric spaces. You can come up with simple spaces where you have non-unique midpoints and you can come up with simple spaces where two pairs share a midpoint. And here on the right, you see one example. This is a simple space that does not embed. And before I say that, I should just mention that if you have a metric space of three points, a triangle, so to speak, any triangle isometrically embeds. So three points don't give you examples, but four points already give you examples. And this is one of the simplest examples where you can see immediately why, by this fact, why it doesn't embed isometrically. Consider three points. And these are just the points. I've drawn the edges here, but the edges are just representing the distances. So these three points, their distance, pairwise distance, is two. And then I have one additional point. I've drawn it in the plane, but you have to pretend it's not in the plane. Its distance to these endpoints is one. Now, if you do this computation in the plane, this is not an isometric embedding because if you compute this distance, it's some square root three nonsense. It's not equal to one. You can give that the simple calculation. But now let's ask, okay, so if you have an embedding of this picture, it looks exactly like this. These three points map to some points in the space. But what is this center red point trying to do? If it was mapped isometrically, it would be a midpoint of this pair and that pair, this pair and that pair, and this pair and that pair, those three pairs. So it's a midpoint. So if you're trying to have small distortion, or even isometry, what it has to do is has to be close to the midpoint between these two. So it's trying to be there. It's trying to be there. And it's trying to be there, but it has to make a choice into different dimension. So in this case, the projection doesn't help. You'd have to change the norm of the space at least. And there are examples where even that doesn't help. So here, adding dimensions or reducing dimensions will not save you from this issue. I mean, you can argue it by this kind of midpoint argument, or you can create a geometric inequality that you witness. This gives you the sharp distortion. But inherently, the phenomenon really is that this point here is trying to lie along the midpoint of these different places, but it has to make a choice. And whatever choice it makes, it will incur some distortion. And you might think that, well, this is silly. I'm just explaining something extremely trivial. But this is actually what underlies a lot of the discoveries. So I'll give you another theorem that you will not probably have heard this theorem in this way. But in the course, maybe I'll explain how this works. So there's Oswad's theorem. And the way I will say Oswad's theorem is that if your metric space is not too big, I'll say something about that if it's doubling. And if it does not have any midpoints, then it bilimpsilates embeds. So if you remove that obstacle, if your metric space lacks any midpoints, so any pair of points acts, there is not a midpoint in a quantitative way, because you have to make it quantitative because the quantitative statement, then your space automatically embeds. So midpoints are really your enemy in considering embeddings. Because really, if once you remove it, you can construct embeddings. Okay, there is another embedding and another enemy, and that's this doubling. And what is doubling? It's the simple question of how many balls can you fit inside a box? And I hate to disappoint you, that's not me in the picture. I don't know who that is. That's just a picture from Wikipedia. It has no relation to me. But doubling is, you have an area, how many balls of the given size can you fit inside it? If that has a uniform bound in the space, we say that the space is doubling. And Euclidean space has this property, as the picture shows. But of course, in the metric space, similarly, like with the midpoints, we can construct examples where that doesn't tell. So this already shows, although this is not the way Oswad is usually presented, this already shows that there is a connection between midpoints and embeddability. Now, there are some other general results that I should mention. So Borg-Antz, the arm says, and this is what all the algorithmic results use, if you have a finite metric space of n points, or a log of absolute value of size of x points, then you can always embed it into a vector space of dimension square of the logarithm in distortion logarithm of the size. This is still, this is a sharp result. This is what you can actually, a lot of the mileage you can get. In specific settings, specific metric spaces you can better, but this is the only general and sharp problem on the general mind. So one case where you can get better, again, this is actually connected to midpoints in some funny way, is metric spaces that have a very simple structure. So Gupta, Croft, Gamer, and Lee showed that if you have a geodesic and doubling tree, the doubling is the same property that we had before, that the space is not too big. And geodesic, it's just that we have a shortest path between every pair of points, which is the graph, it's automatic usually, but virtue of definition. And tree means something like this our Alcárea from Chile and Southern Brazil, my favorite tree. And what fact of trees I'm referring to here is that trees do not have cycles. So in particular, if you have any pair of points, there's a unique path that goes between them. Hence, there's only one midpoint. I mean, this is not actually how the logic of the argument goes that you have an embedding, but it certainly has something is related, connected to this aspect. So in very, it's okay, we have some general results like Burgan, but then specific settings where we can control the structure of the space in a strong topological quantitative way, like trees, we can get results. So these are general results. I want to say something about my results, of course. So my first result about embeddings was about embedding manifolds. If you have a manifold with bounded curvature, or even a flat manifold, for that case, if you want extremely simple case, the flat manifolds are always quotients of Euclidean space by the action of some group. Actually, I recently learned by talking to some people and giving a talk at Codex seminar, it's also on YouTube, curious that these problems, embedding such spaces has some applications to data that has symmetry. So I showed that in these settings, you can construct biologics embeddings with distortion that is controlled by the dimension of your space and the curvature bounds that you have. This is different from Nash, because Nash says something about preserving infinitesimal distances, whereas here we are controlling distances at all scales of the space, and that's where all the difficulty lies. This is in that way different than Nash's result. Another setting that we can do, we can generalize this tree setting a little bit, so-called bounded turning trees, and there are trees that have a little bit of more fractal like structure to them, so something that looks like this. But again, there are still trees, so we can utilize these facts between every pair of points, there's a unique path, and it's not too many bend points. And there's also some ongoing work that I still work on these problems with my PhD student Neil Jotzenlahti. We study sharp embeddings of approximations of certain fractal spaces, and here you have a slit carpet. I'll say a little bit how this is constructed, because I'll return to this. You take a piece of square, piece of paper, and then you use scissors, you cut in the middle a slice away. And when you do that, this slice starts to have two sides, so there's really a point on this side and point on that side, and the shortest path, what the metric is, you put a shortest path metric on the space. So the only way to connect this point to this point is by going around, so the distance between them is roughly the size of that slit. So we study this type of embedding question, that the reason is that they can tell us something about extending sharp embedding results to different settings. So this is the positive setting. What about the negative setting? So the earliest negative example is that of the hamming cube that arises in error correcting codes. So here you have a bit, your elements are bit strings, so here are four bits of length four, but they could be of length N, and you connect them with an edge or you make the distance one, whenever the bits differ by only one index. So this corresponds to bound edges of a cube or corners of a high n-dimensional cube, and then the path metric where we are only allowed to travel along the edges. So this cube naturally sits inside the Euclidean space, we map the corners to corners, but the metric is different. In Euclidean space, the distance between the diagonal is say square root two, but here it's two because you have to go around the corner. This is a metric that arises in study of error correcting codes, naturally arises in functional analysis, etc., and one can show that the natural embedding is actually the one that minimizes distortion. You cannot do better than square root logarithm of the size of your space. So here there's two to the n points, so this is square root of n distortion. So this is one example, and where do we have the midpoints? Along every diagonal here we have a midpoint. So these are blue points and their midpoints are the red ones. We have non-unique midpoints that already tell us that they do not isometrically embed, but you can convert this to an effective estimate that gives you the distortion. Another example that was famous in mathematics that I do is a loxodiamond space that's depicted here. It's some sort of self-similar fractal space that is constructed from the interval by iteratively replacing it by gadgets that look like this. So imagine first you replace the interval by its diamond object, and then every edge is again replaced by the diamond object. But then you equipped it just like with the hamming. This is not the metric on the space. You have to distort the metric. You take a rescaled path metric on the space, and this does not embed very effectively. Actually the infinite space does not embed, and you can give effective estimates on the distortion. And why again? Actually this is a loxodiamond's original argument. If you look at his paper and also along the influx argument he was using exactly this, you have many midpoints. So these blue pairs have two midpoints, these orange ones, the orange and the blue one have two midpoints that are red ones. And this is actually how the proof works. So if you look at the proof, you look at what is the essential estimate, it's precisely this fact that gives you the sharp bound. You don't need anything else. And this space too also has midpoints. Whenever you cut a slit by scissors, there are the endpoints of the slit, and then two midpoints on the opposite side that I alluded to before. You have a lot of these midpoints, the non-unique midpoint configurations, and these are what cause that the infinite example of this does not embed. Actually we're still studying the precise implications for the quantitative non-embedibility of this example. That turns out to be more subtle than existing methods seem to be. But in all of these examples, both positive and negative, somehow midpoints at least play a role. And there are ways of expressing this idea of having midpoints. And some of these, since I'm again a mathematician, have deep connections to geometry of the space. And there was a funny theorem that we showed, and it's still a little bit puzzled by its consequences, and there might be some ways of further seeing what this theorem exactly means. With Guy C. David, we showed that if the space has many curves, which is another way of saying that you have many midpoints, because along every curve you have a midpoint, if you have many curves, and if you also bi-elliptions embed with finite distortion into Euclidean space, I'm thinking here of some infinite examples, then it has to look like a product, the space. There should be some quantitative version of this, but I won't conjecture on that in precisely the precise form of that here now. So if you have a lot of these midpoint configurations, if you have a bi-elliptions embedding, then the only way to do that is actually that the space looks like a product. And you can use that to prove that some spaces do not embed. This was a strengthening of existing tools that existed in analysis of metric space, and it showed that, so for the specialist mathematicians in the audience, I can say, this strengthens, say, results of Jeff Chieger and Pierre Ponsoud, that you didn't need any Carnot group structure, you didn't need any Poincaré inequality. All you needed was positive modulus of curves to conclude that you don't embed. That is the precise statement of this non-embeddability. And well, this was an interesting geometric theorem. Actually, this looked like a product, turns out to be a big headache, but maybe you'll believe me that this picture does not look like a product, but actually showing that it's not a product is a different story. So half of our paper was actually figuring out ways of showing that something is not a product, which is a task that should be simple, but actually turns out to be surprisingly hard. So summary of this, what I wanted you to get out of this task about embeddings is that, first of all, isometric embeddings are not always possible, and why? Because of the different behaviors of midpoints, and that these midpoints are connected to interesting geometric properties of the space. I've worked on some positive and negative results here. There are many more results, many of these results that I mentioned, like Morgan and Aswad. I hope to present in detail in this mini-course that is starting tomorrow, in fact. So if you are interested and have time to hear more, then feel free to join the mini-course that will then start the QR code takes you to the website. Thank you. Thank you very much for this very informative and nice talk. So we'll open the floor for questions and comments. So if you have any questions, comments, please go ahead. Thank you for the talk, Sylvester. A very enlightening one. And you pose yourself as a mathematician. And mathematicians always love to show existence and also to make the accountability of how many possible, in some sense, inequivalent. Do you have any comment on counting embeddings? Counting the number. So in this setting, so often, yeah, we don't look at very much the issue of uniqueness. So one thing is that, of course, it's not unique at least up to rigid motions of the ambient euclidean space. But I mean, there usually is the methods that we have are not, so for the positive results, if we have that something embeds, they usually not very sharp. They're used based on some pretty weak constructions using the distance function and distance embeddings. So there, I would not expect that most of the time it would, we could perturb the construction a little bit and there would be a whole family, uncountable family, of course. So where you could get uniqueness in some cases, and I haven't studied that question, say in this case of the hamming cube, whether these kind of, if you look at the minimizing maps that minimize the distortion, whether they are unique. I haven't studied that question of that seems, at least for particular spaces, like say this one, you could perhaps try to tackle that problem, but I haven't looked at it. This is a stupid question, but what are the black dots in this hamming cube? Are these black dots? Yeah. Oh, these are the pervertices of the cube. So this is a, No, but I mean, you have gray and black. Oh, that's the distinction between the black. That's just a picture I found on Google. So there's no significance within the color. The black and the gray are just a different change of gray for me. Yeah. Yeah. Yeah, the microphone. Thanks for the talk here, just about this image. So when you talk about these midpoints and so on, now, does this mean that you cannot define some kind of continuum theory on, on this, in the sense, differentiability and so on are not well defined notions if you lose, if you have this multiple midpoints and so on? What I mean is on some finite dimensional Euclidean, you want some continuum theory for this. So, well, I can, I'll interpret the question in a way that I can answer it. But so one thing is here, this is a discrete space. So I have just the corners in this space. I mean, you could add the edges, but that will then change the setting quite a bit. So this differentiation actually is one way of studying embeddings. So, but what, so I'd say maybe this is a better case for the differentiability because we have some continuum to work with. So, I mean, I said that this embedding that you have, that you cannot construct this embedding with a, because of these midpoints, but there's a different way that uses differentiation that leads to this argument. This is actually something that hasn't been written up anywhere, but it's a sort of a folklore argument. So what you can show is if you have an embedding of this, that's something in this space, what you can do is you can differentiate that embedding. And when you differentiate the embedding, what happens is that it becomes linear. I mean, you'll ask what do I mean by linear? Linear in this x-coordinate, but then you see that that cannot hold. If it's linear in the x-coordinate, then blue is mapped to say one, this one is mapped to one, this is mapped to three, and these two maps, because it's linear, they're both in the middle, they have to map to the midpoint of the blue values. So if the mapping is linear, if you could differentiate, then this mapping could not exist because these have to map to the same value, but they have to be separated. So actually, these differentiation arguments and quantitative analogs of them play a big role also in this theory, but you need some continuous structure or at least some approximate continuous structure. In the cube, you don't have that, the argument goes a little bit differently, but it also uses, there's some connection because here you can prove concurrent inequalities or some kind of analytic inequalities and use them. But so I'm not sure if I answered your question, but I said it. Yeah, that was sort of answered it. Thank you for the nice talk. I was just wondering very naively if the notion of midpoint, which is currently defined without any orientation of the connection between two points, has any counterpart when you introduce orientation? For example, if you look at your cube example, currently the red points can be viewed if I'm not mistaken as the midpoints of the blue points. This is the case provided that you assume the connection from blue to blue either ways is possible. Whereas if you introduce orientation, then some of these are eliminated. Is there any counterpart in that setting? I haven't thought of that. So if you just measure distances in an oriented graph. The only thing I could say off the top of my head is that potentially by reducing the number of midpoints in that case, it would make embedding easier, just potentially. But you may encounter some problems. Of course, then you could still make, you would want to make sure that your space is still connected, that the distance is not infinity between some pairs of points. Yeah, exactly. Some of the analysis you could do, but I haven't looked at that particular case in more detail to see. Yeah, so the question was about in this Bourguin's theorem, I stated the distortion here on the slide, but does it also give the embedding or the dimension of the embedding? The proof of the result also gives a bound for the size of the embedding. So it's a square of the logarithm that you get from Bourguin's proof. So I think in Bourguin's case, it's not optimal because you could do dimension reduction to get the logarithm. You can use Johnson-Lindus Johnson to get the logarithmic dimension, at least when people's two. So in people's two, you can have a logarithmic size and logarithmic dimension. But Bourguin's argument needs a logarithmic square dimension for by Lipschitz, in fact, for by Lipschitz. Yeah, so that's actually a good point. Yeah, so I mean, I showed this, my point precludes the or causes some distortion. Now, what I didn't maybe explain very much is that in these examples, okay, if you have any embedding, there has to be, in this configuration, there has to be some distortion, but that distortion gets confounded. It adds up because, okay, some edge here is distorted, but that edge is part of another square with another pair of midpoints. So you don't just get the distortion once, but you get it many times. And here it's maybe clear. So the fact by looking at these points, we know that one of these diamonds here, one of these four diamond configurations, they have to be stretched more. You focus on that one, say this one. And within it, you get some more stretching. And every time the distortion grows a little bit. Now, if you have infinitely many steps that goes to infinity, if you have n steps, you get exactly like some kind of square root of the logarithmic step. So in these cases, you actually prove that there's no by-elections embeddings because there are so many midpoint configurations that they add up the effect, the multiply the effect. So a single pair of midpoints doesn't really, it's finite distortion, but when you have many of them somehow nested or interacting with each other, that's when you get problems. Yeah, two by-elections embeddability. Thank you.