 Thank you very much for the organizers for making this possible. It's really my pleasure to talk to you a little bit about what I like to call the memory of persistence. So this is a retrospective, but also a perspective. And since I like puns, I mean, some of you might have recognized this title already. It's, of course, a play on Salvador Dali's work, The Persistence of Memory, where you have this marvelous melted clock drawn in surrealist style. I tried AI to imitate this type of art, and I gave it a few prompts where I said, show me a torus that is melting in surrealist style. Show me some polyhedra that are melting. The results are, well, I think they're interesting. We're not quite there yet, but potentially maybe the next time I'll be giving this talk, I can have AI draw up all the illustrations for this talk. And in any case, really glad to be here, really glad to start this off. Now, I want to start, of course, at the beginning and talk a little bit about what our field has come from, the roots, and also the things that make it a very interesting field to be in. And let me start at the very beginning. I hope that this is of interest to some of you and that you're not all already into the topology crowd, because otherwise this might be a little bit more of a historical refresher, but in any case. So I like to refer to algebraic topology as the field that does counts and some kinds of calculations. When you ask the question, what is algebraic topology, you find different definitions. Some formal definition might say that we develop invariants that classify topological spaces up to homeomorphism. Others might say that we use tools from algebra to study topological spaces. But the one that I like the best is that essentially we are interested in understanding shapes through calculations. The emphasis lies on the calculation side here, because we really want to have some derived value here. We really want to be able to implement these things and calculate them in practice and make claims about the spaces or the things that we're dealing with, the data that we're dealing with. Now, let me give you a really first taste. And maybe you're familiar with this. I think you're familiar if you have followed Michael Bronstein's great blog posts on topology and graph neural networks, but I think this is a great example so it bears some kind of repetition here. The first example of topology that I always think of is the seven bridges of Koenig's back problem, which was, of course, as almost anything in mathematics first studied by Euler. Euler is your kind of joker in almost all the quizzes and almost all the trivia games. So when someone asks who gave the first proof of this, it's always Euler, or Kaz, potentially. Anyway, Euler was asking the following question, is there a walk through this beautiful city of Koenig's back that crosses every bridge exactly once? So this seems to be a geometrical problem at first glance, right? So you have a city, we have a city map, we know where those bridges are, and then we try to calculate something with this. Now, the ingenious insight of Euler, of course, was that this problem can be transformed into a graph. So we can treat every bridge as a vertex of that graph, as a note in this graph, and then derive some connectivity from this. And then we can ask ourselves whether there's a walk through that graph. So essentially we're disregarding, and you can see this here, we're disregarding a lot of the geometry, not all geometry, mind you, but some of the geometry to make our problem a little bit more concise and to allow for calculations. Now, I don't have to tell you, of course, that no such walk can exist. There are steep reasons for this, Euler showed this, there are more than two vertices with odd degrees, so this is a general statement. Some of you might have seen a similar graph, potentially in what is known as the utility problem, where you have a utility company that tries to connect houses with electricity and so on and so forth. But this already shows that by changing your perspective by doing some kind of fundamental paradigm shift, you often end up being able to answer questions that could not be answered in other ways. Now, we can go further, and of course we stay with Euler because he's the MVP probably. The Euler characteristic of a polyhedron is defined as just the sum with alternating signs of its number of vertices, the number of edges, and the number of faces. And this is already also a very nice invariance, so to speak, because it doesn't change under certain transformations and it permits us to do certain classifications. So there is a deep theorem that says the Euler characteristic of every platonic solid is exactly two. Interestingly, if you know this, then of course you can also show why there cannot be more than five platonic solids and so on and so forth. So this leads to all kinds of interesting things down the line. Then we briefly show you that this is true, this is a very nice example that I like to show quite often. If we take the tetrahedron, for instance, we have four vertices, six edges, and four faces. So that gives us two if we calculate this with alternating signs for the hexahedron or the cube in layman's terms. We have eight, 12, six, which also makes two, and so on and so forth for the octahedron, the dodecahedron, and the icosahedron. So this is already a very nice way to classify these faces up to Homeromorphism. And this also demonstrates the power of the calculations that are underlying the topological concepts. Now, let's go a little bit deeper. Another invariant that is not due to Euler this time, but to someone else, Enrico Betti, is the Betti number. The Betti number counts the number of d-dimensional holes in a space, and this also can be used to distinguish between spaces. Now, I always say in these cases, when I show this slide or similar one, I always remind people that at least in low dimensions, the Betti number have some kind of intuitive explanation attached to them. So the Betti number in dimension zero is the number of connected components, the Betti number in dimension one is the number of cycles, and the Betti number in dimension two is the number of voids. So we can see that this can be used to classify some spaces. So point has almost nothing. It has only a single connected component, nothing else. If we go to a cube, we get a little bit more. We get a void that is enclosed by the cube, but nothing in dimension one. The sphere is similar to this, which also I think might start to explain some jokes that people have about topologists not being able to recognize certain everyday objects. And then of course we have the torus, who for reasons that I will not go into detail here, has two loops in dimension. One, making it different from the sphere. So there you also have one of the first interesting results, I would say, of this type of shape analysis of topology. Namely, we are able to tell a sphere apart from a torus, and we can do this in a very precise manner. Now it's all nice and good, but where do we go from there? Where does computation topology come in? So computation topology is still a very recent field in mathematical terms that aims to bring these sort of insights into the data analysis space. And I like to think about this as kind of a point cloud atlas, like this really great book by David Mitchell. And it can be formalized or it can be motivated by the insight that reality is often messy. So when we measure data, what we get is maybe something like this on the left-hand side here. So just like a point cloud that might look like a vague torus shape or a vague donut shape, whatever, but it's not a platonic torus in this sense. So what we would like to have is some way to say that this point cloud on the left-hand side is somehow more like this torus, this idealized version on the right-hand side. And to make this happen, we need to represent these spaces somehow. We need to bring the points into a shape that make them representable and amenable to algorithms and to calculations in the first place. And the simplest thing that people are doing, have been doing and will be doing for quite some time, is a triangulation. So you can't see it here because the resolution is not high enough, but just wait a second. This is the Statue of Winged Liberty and it's what computer graphics people would call a mesh. So it consists of triangles and they're kind of glued together. And when you zoom in, you might see some of those triangles. So that is a triangulation. And through this process, you have turned your point cloud into something, into a shape, into something that you can work with and that you can apply your topological concepts to. And mathematically, this even makes sense. I really want to stress this, that this is not just something that people are making up and saying, oh, this is a very nice approximation of reality, but no, no, there's some deep theorems underlying this. There is a theorem proven by Cairns and Whitehead independently from different perspectives that every smooth manifold can indeed be triangulated. The actual formulation of this theorem is a little bit more complicated. There is, it admits essentially a unique triangulation, unique up to certain transformations, of course. So the bottom line for this is that this works and this makes sense and it's connected to some kind of continuous assumption of the manifold or the space giving rise to a data set. Now, let's take a look how that, how we could leverage these sort of insights in practice. And persistent homology is, if I may be lyrical for a moment, is a way to make sure that the points cross the scales like clouds cross the sky. So when we are given a point cloud, we can approximate this point cloud at different scales. So we can calculate these triangulations at different scales and observe how topological features appear and disappear. And why would we be doing this? Well, we would be doing this because a priori we often don't know what a good scale might be for our data. And maybe that does not exist a good scale because of noise or other ways. So to capture the overall shape and the overall phenomena that are going on in a data set, such a multi-scale perspective is required. And let's just go through this and see what I mean. So in the beginning, not a lot of connections are happening. So we have to increase our scale and start connecting more and more points and make those grow. And as soon as we have reached a certain critical threshold, if you will, we suddenly see that out of this chaos of points, a cycle emerges. Now, mathematically speaking, this cycle is of course not there because it's just a bunch of disconnected points, right? But through this process, we suddenly can say, okay, at this scale, we have something that might be cycle, something that might be the loop. And this loop persists to already give you an indication of why this is called persistent homology until for a certain scale, it is closed because everything is connected to everything else. And this is, I would say, persistent homology in a very, very, very, very, very small, very tiny nutshell without going into all the spectacular details that have been behind this. But again, we are here in the retrospective part of this talk. So I want to wax a little bit on about where these concepts came from and what they actually mean. And the concept of persistent homology one of the first formalizations, and they're with me for a second, was given in the paper, Topological Persistence and Simplification in 2002. And of course, I mean, I picked this paper because this also means that we're now two decades into the persistent homology revolution. So this is really nice. And in this paper, Edelsbrunner Lecher and Zomerodian summarized their technique as follows. We formalize a notion of topological simplification within the framework of a filtration, which is the history of the growing complex. So this is what I showed you earlier. Now going on, we classify topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility. So this I would say kicked off the dawn of persistent homology and potentially also the dawn of computation topology. But of course, it would be unfair historically speaking to not mention some other formulations here. So I'll try to pronounce this correctly. And then I hope I don't get too much hate meal. So Victor Hugo famously said, on résiste à l'invention des armées, on ne résiste pas à l'invention des idées, which means something like nothing is more powerful than an idea whose time has come or a more honest translation could also be you can resist the invasion of armies, but you can't resist the invasion of ideas. And indeed, when you look back into the history now with the benefit of hindsight, I should say, with the benefit of hindsight, we can see some similar formulations to persistent homology, some vestigial formulations, if you will, or some precursors to that in different works. One is the great work by Forsyden from 1990, which is a distance for similarity classes of sub-moneyfolds of Euclidean space. This also give rise to what is known as shape theory or size functions. Then we have a work by Sergei Baranikov, which is called the Framed Morse Complex and its invariance from 1994, which also discusses things like the persistence of features, but gives a different names. And then we have size functions from a categorical viewpoint by Kalyari and colleagues from 2001, which is a little bit more recent in this term. So the point of this is not to take a side in any precedence dispute here and say that, no, no, this person invented first, but just to showcase that there is a confluence of ideas of multi-scale analysis made possible by the arrival of the digital computer, I would say, making it possible to bring those methods now slowly and steadily into practice. Now, where are we now with this? If you follow the current pipeline, I would say, of persistent homology, you have roughly the following steps. You start with a point cloud, you use persistent homology, which I like to view as kind of a coffee grinder, so it changes your data in something that you can taste better and something that you can palette potentially. You obtain a topological descriptor or a set of topological descriptors from those, the so-called persistence diagrams. Of course, they also come in different forms and variations. I don't want to go into too many details here that there's a bunch of marvelous work done by a lot of people, some of which I will be able to recognize in this talk, some of which I won't, unfortunately, but just for the sake of this presentation, let's just remain with these persistence diagrams that tell us something. And then later on, you put them through your machine learning algorithm. And it is this mapping from persistence diagrams to machine learning that I find most fascinating and that I have been spending some time on in the past few years. Particularly to give you a taste of why I find this so fascinating is the overall process of calculating topological features is still somewhat discrete. So you count things, you see where is the cycle, how long does it persist and so on and so forth. But machine learning, on the other hand, is a very continuous, a very fluid thing. You have gradients in there, you want to back propagate through your neural network and you want to make sure that all of this fits together. Luckily though, the last few years have shown that this is theoretically possible. So there's work by Poulinard and colleagues, there's work by Michael Moore and myself, there's work by Carrier and colleagues and all of those looked at this red arrow, essentially, this mapping from persistence diagrams to machine learning from different perspectives. And essentially what they all came up with is saying that under certain conditions, the mapping from the persistence diagram to the machine learning can be made differentiable. So it's possible to change things in the filtration and to back propagate and change even things in your dataset. So for instance, in the first paper, topological function optimization for continuous shape matching, the authors learn a function that best represents a shape in order to match it to other shapes later on. So they actually change the filtration values. In our paper from 2020 on topological autoencoders, we look at training an autoencoder that is a neural network that gives you embeddings of a dataset, for instance, to give us a latent space that best represents topological features in the original dataset. And with Carrier and colleagues, they were finally showing, and I would say this is a seminal work that you can optimize all sorts of persistent homology-based functions. So functions like the total persistence, functions like as such line distance, and so on and so forth. So there's more though, I should say. The current pipeline only incorporates point clouds, but this is not the only thing that persistent homology has been used for or can be used for. One particular interesting instance, of course, since they are already topological spaces or topological objects in general is graphs. But we could use persistent homology easily for the analysis of graphs because they're almost anyway a triangulation or a simple shell complex. The other thing that is now increasingly coming up, and I think a few of you, a bunch of the organizers here have been doing great work in this direction, is the analysis of time series. So by using something like a Tarkin's embedding or Taken's embedding, let's not fight about the pronunciation here, you can take your time series and you can turn it into a point cloud. And then suddenly your time series becomes amenable to all this persistent homology pipeline and you can actually link the holes in your data to circular behavior or to periodic behavior in your data. So this is great. And I think this highlights that persistent homology overall provides us with a new paradigm for thinking about data, for thinking about data at different scales and at different levels. And to paraphrase Gunnar Carlson, who is also one of the founders of founding figures, maybe I should say of persistent homology, data is shape, shape has meaning and persistent homology helps us extract it. Now, it would be of course remiss of me to not talk a little bit about some more whimsical themes here. One of the things that I observed and that I'm really glad to see is that there is a sort of, well, let's call it Cambrian explosion with a big question mark here of papers that contain persistent homology and that deal with persistent homology. So as you can see, this has been steadily growing from the 2000s on. This is data taken from open Alex, a conceptual working space for linking concepts, authors and other academic works, kind of a spiritual ancestor of the Microsoft academic graph. Now, when we mark a certain time in this chart here, namely 2012, which is widely considered as the start of the deep learning revolution, then we can see that we're not too far off here. So at least we can see that there has been a steady growth of works dealing with persistent homology and of works in particular dealing with persistent homology in data science and machine learning as this workshop so amply demonstrates. And I think that there's much more to become, but we'll get into this now. Now, in 2019 when I was a fresh faced postdoc and very naive, I'm still naive, but in a different form, I was able to talk to a few people about challenges of persistent homology. And this was more of a get together in a smaller venue. And I outlined three different challenges that I figured out would be relevant for the next few years. The first one is improving performance. The second one I called is caping flatland because I've only seen persistent homology being used for relatively low dimensional topological features. So almost always dimension zero and dimension one, at least the machine learning. And the third one was the first class architecture. So the idea of having architectures that work well and that integrate topological features directly. By the way, these pictures are used with kind permission from Professor Fomenko and they are found in his marvelous book, Homotopic Topology, which is one of the books that got me interested in algebraic topology in the first place. But back when I wasn't even more naive, a young undergraduate student that was roughly at the time when the dinosaurs roamed the earth. So where are we now? What did time do with these challenges? I mean, it's three years, right? It's not a long time, but there's some time has passed, right? And in machine learning terms, a lot has happened so far. So first of all concerning the performance. With respect to the performance and in general the shape analysis of data, something very nice has happened. There has been the discussion of the persistent homology transform and various of its ancestors and of its updates. The idea of this homology transform is that you can capture shape without having to calculate multiple filtrations because for mathematically deep reasons, the multi-filtration setting where you try to filtrate your object or your space in two or more directions at once is still not super feasible, I would say. Maybe we'll learn something about this in this workshop, but it's not as efficient as persistent homology and even persistent homology still has some performance issues but we'll come to this, of course. The great thing about the persistent homology transform is that it calculates a filtration of some shape in a d-dimensional space by shooting rays into this shape and just checking the inner product of this ray with respect to the points of the shape. So you just need to pick a direction of the unit sphere of your choice. This gives you a nice ray and then you just calculate the inner product of your points in the shape with respect to this direction. And you can see if I do this for torus here, I pick one direction, I pick another direction, I pick a third direction. You can see that the resulting filtration, so the resulting color schemes, they all highlight different things in the torus and now if you imagine that you put more of those together, if you choose a bunch of interesting directions, you can actually show that you're able to quantify and assess the shape even in an injective manner. So this is work by Turner and colleagues from 2014 and they show that this persistent homology transform can effectively be considered an injective mapping. This is great because it gives us a way to capture this type of information without having to use a multi-filtration. But it goes even, it goes even on. There is in the original paper on the persistent homology transform, the authors also described that you can evaluate things like the Euler characteristic or some other function even alongside a filtration, thus leading to a set of characteristic curves of the dataset. And this is something that we recently exploited in work with Leslie and Karsten which was presented at KDD. It's on filtration curves for graph representation where we use something similar. We pick a very simple function of our graph and we pick a simple filtration and evaluating this function along the filtration gives us a very simple way to obtain a graph descriptor. And again, to connect to what Tegan said at the beginning of this workshop, we don't beat all the graph neural networks with this technique. This would be folly. This would be also probably not work in any case. But what we have here is a very simple baseline method that often performs quite as well and that doesn't need as many parameter choices and that also scales quite nicely because picking a filtration is as easy as sorting the graph. If you can sort the graph you can still calculate something with it. And if you pick a function that is not too complicated here you also end up not losing too much time in the calculations for these curves. And moreover, you can calculate distances between those curves and you can calculate mean between those curves. So this perspective yields a lot of interesting insights and if you recall what I said earlier on this is again this paradigm of persistent homology. So we just go from a single perspective from a single scale to this multi-scale perspective by choosing a filtration. That is really great this really addresses some of the performance concerns. Other performance concerns are recently been addressed by Solomon and colleagues who showed that it's possible to infer a lot of topological information already from random samples of the data. So instead of picking the big data set and calculating all the topological features that we can find they just suggest that well you pick a subset and you repeat the subset procedure and you repeat the subsampling procedure and then you calculate your topological features and this can actually be used to derive a lot of value a lot of information from the data and as they show it can even be used to perform topology-driven optimization. So this is just an example I took from their recent paper inverse theorems for distributed persistence where you can increase the loopiness of a data set you start with something like this and then over time the points arrange themselves quite nicely. So this again addresses some of the performance concerns that we had earlier because well we can always subsample we can always do this and calculate these features for smaller cases. It doesn't solve all the issues of course. I'm very, very, I want to be very, very clear about this. Now the second challenge is caping flat land. I can just say that well sort of so I mean in my case I'm not aware of a lot of work that uses very high dimensional topological features. The highest I personally went in that sense is 3D. So we had a very interesting project in which the idea was to use 2D inputs so 2D microscopy images and reconstruct 3D cells from this 3D shapes. And first of all it turns out that topological loss terms or topological information is actually crucial to make the model learn something. So before the introduction of the topological loss term they had really some trouble fitting this model and making it work, making it give good predictions but as soon as the topological loss was added things converged much more nicely and we obtained much more high quality reconstructions which is of course great. It's also one of my personal success stories I should say this work with Dominic Weibel and a few others from Helmholtz Munich and it's currently in press. Now this is the highest I've gone so 3D features I hope that we can go higher. I hope that there is something higher actually to aspire to here. So in my personal work I've seen that already including information from dimension zero and dimension one seems to be relatively good in order to increase performance. So I do wonder whether we can escape flatland even more and whether we can for example, for instance show that in certain cases and certain data sets, eight dimensional features might be the most important ones. I'm not sure I leave this as a challenge to the room and the challenge to all the workshop participants. Now I'm happy to say though that my other challenge the first class architectures one this has definitely been solved and tackled now by the community and this is really great and it's amazing to see the progress that has been made here. So I want to point out a few works here that kick this off. So there's work by Christoph Hofer and colleagues on deep learning with topological signatures which is a New Europe's paper from 2017. This presented a layer that makes it possible to take any persistence diagram and use it with respect to a neural network use it for different classification tasks. There is also work by Carrier and colleagues on PERS-LAY a neural network layer for persistence diagrams and new graph topological signatures. There's also hints at some of the things that I consider important here but this is also a very great work that subsumes certain representations including if you're familiar with this term the persistence landscape by Peter Bubinig. Finally, there's also work by Kim and colleagues on PLLA the efficient topological layer based on persistence landscape. So it all goes together and this also shows that integrations into the deep learning ecosystem and thus also into the machine learning ecosystem are possible. So lots of progress is being made lots of progress has been made. And I hope that we can now tackle some of the performance issues which still plague us. I mean, I've outlined two ways here but they are not the only ways out. For instance, one thing that we should definitely look into is the implementation of GPU algorithms that can leverage the same improvements and the same speed ups that one typically gets for neural networks. Others might include some smarter calculations of persistent homology that we are not aware of. For instance, there's this great package by Ulrich Bauer called RIPSA which you might have heard of that is the fastest way to calculate topological features of a Revitorius RIPPS filtration on the CPU. There has been a GPU implementation but I'm not privy about all the details here. So these are still things that we need to tackle but overall I would say that in these three years since I last talked a little bit about the challenges in our field a lot of progress has been made. And as we come a little bit towards the end I'm not over yet there with me for a while. I want to briefly take this time to talk a little bit about the actual beauty of our field because that is what captivated me, what evangelized me or what lured me into the field if you will. First of all, I mean my personal journey into topology began when I figured out that there was this weird class at university where you could actually draw your proofs and you could just look at them a little bit and you could say, oh, well, this makes sense and I have a homotopy between two curves here and I just draw my proof and I know that this will visually work out. At the same time, this apparent lack of rigor was then counteracted by doing a lots of diagram chasing like illustrated in this cartoon here where the wizard is incredulous about the field of topology but then finds out soon enough that topologists have some kind of magic powers on their own because a hat is of course topologically equivalent to a symbol so we can change the head on the wizard without being turned into a toad because a toad is of course the same as a human being at least in terms of topology. So this is what lured me in this balance between being very visual on the one side and very rigorous doing diagram chases on the other side, that is what captivated me. Of course, it also helped that there were these really weird drawings that just were magnificent to look at something like this Alexander Horne sphere which you can find in a paper by Daverman and Venomar called Embeddings in Manifolds. I just love staring at this and thinking about what this means and how this all is entangled up in higher dimensions. And of course, this continues that does seem to be a whimsical streak in topologists or at least I try to tell myself that I detect some kind of whimsical streak in topologists because for instance, look at this great picture from Robert Grice book, Elementary Applied Topology which by the way, and I'm going on record saying this I think might be a little bit of a missile over here. I mean, it's certainly applied but whether it's elementary or not this depends on what you consider to be elementary. I still think it's a very, very great text but I would definitely recommend to pair it up with a few other of the resources that I will give in the subsequent slides here. But just take a look at this picture let it sink in for a while. It's one of the pictures that is drawn for the chapter of introducing some pleasure complexes. And now look at this. The picture is called The Ancient of Days by William Blake. So there is a certain kind of whimsiness I would say some playfulness and joyfulness in the field of topology and in the field of computational topology that I really appreciate that I really did not see in so many other fields. So that is what lured me in and I hope that it can also lure you in. Now, what are the next 20 years now? What should we do in the next two decades? I mean, the first paper by Adels Brunand colleagues came out in 2002. What should we do in these next 20 years? Well, what we need is we need more data of course. We need our own data sets I think. We need to make sure that we understand what is going on in this data set and we need to also connect this data set to existing data sets in machine learning. Primarily to say that these data sets should be interesting from the topological point of view and they should also be kind of hard to assess by non-topological methods. If we have this, if we find our MNIST where we can say this is what you should use to test caseatopological algorithm. If it doesn't work there, it won't work anywhere. This would be great. We also need harmonized frameworks and reporting. We need to make sure that when we calculate something we need to state, is this extended persistence? How do we handle infinite features? How did we choose the filtration and so on? So basically all the changes in hyperparameters, all the discussions of hyperparameter tuning that are going on in the machine learning world, all the talk about reproducibility, we also need to follow this discussion. We also need to make sure that we talk about in more details what we are doing because fundamentally computational topology is about computations and we need to be able to stand behind those computations and say, ah, this is what we are capturing. But last and certainly not least, we also need users. We need people to use this stuff. We need people to apply to a certain problem and tell us, well, this didn't work. We need also people to tell us, oh, I threw this at a problem and it really stuck. This really helped. We need these type of things. We need these success stories and we need to celebrate them a little bit more. Maybe that's the anathema to mathematicians to talk about these successes and celebrate them. But we should do this. We should look into this. We should also avoid certain things. And this is, I would say the ugly part of this coin. What I like to call this is the witches cauldron of persistent homology. And I've seen this in quite a few papers and I'm, of course, guilty of it myself. And let me try to be a little bit poetic here. This is a line that I stole shamelessly from Macbeth from the three witches here. Roundabout the cauldron go in the persistent entrails throw diagram that with many a pair makes the network look less bare. Double, double, toil and trouble, GPU burn and cauldron bubble. So we should make sure that when we use topological features they should be justified and assessed carefully. Because of course I am heavily biased. I'm not going to lie to you about this. I want to use topological methods. I want to save the world with topology and I want to analyze data sets with topology. At the same time, it might not always be the best method. And I think we should be honest about what we're doing. And I think we should make sure that we don't use persistence as a way to decorate an otherwise bare method or an otherwise bare neural network with something that really doesn't help in practice. And this is also a good practice for machine learning in general, making sure that whatever you do you can do an ablation and you can find out what is going on later on. Now at the risk of tooting my own horn too much but this is a keynote. So I hope I won't be hated too much on here. I think we did an okay job about this in a recent paper on topological graph neural networks where we actually showed that topological features are crucial for high predictive performance in graph learning problems or at least in certain graph learning problems. So we carefully tried to dissect the neural network and see where the performance gains are coming from. And we wanted to make sure that when we add some persistence or some persistent homology concepts into the network we actually gain something. And this was appreciated by some reviewers not by others, but of course that doesn't mean that it shouldn't be done. So it should always be our gold standard or our North Star to make sure that we justify our choices and we justify the things we're doing quite carefully. Now this brings me to a more joyful subject because I don't want to whack the finger here too much. There is also success stories. There's a lot of success stories. They might just be a little bit hidden at times. And maybe we could be more open about this and start a community here which is also what I'm getting into. There was a recent post by Michael Bronstein which discusses a new computational fabric for graph neural networks. And it spoke favorably of graph neural networks and of algebraic topology in particular with the idea that without citing here everything the field of algebraic topology which offers multiple theoretical and computational advantages. So this is a great start. This is a great way of having our foot in the door. And we should use this. We should use this momentum and keep on pushing a little bit to make sure that we have more of these success stories and that we can capture people, we can captivate users and we can get people interested in this thing. Now we need to make sure that we highlight the value of a topological perspective to connect this to my point about the cauldron. We need to make sure that it's clear what we gain from using topology. And then I think we're on a good track to be picked up by other fields and to be useful overall for other fields as well because again, computational topology is fundamentally about computations. If we do the wrong computations that don't help anyone then this is not good for us and our community. Now, what might be the way forward? Well, we could try to build bridges to machine learning topics that are currently going on such as explainable ML generative models and so on. We definitely need to dismantle obstacles to learning by investing in good explanations. And we should focus on intuition and good visualizations. That is what drew me in and I hope that it can draw other people in. Certainly the ML community is open towards this if we are careful. So focusing on these things will be helpful. And I would also stress that when we write papers we should make sure that geometry and topology are not opposed to each other. So for instance, Bubenek and colleagues they showed that persistent homology detects curvature. So that's great. It incorporates some geometrical aspect. So we don't need to have a fight where we say that geometry and topology are at opposing ends of the spectrum. And of course, theory without practice would be empty. So both TDA and the life sciences they care about one thing. They care about the thing that shape really matters. So understanding the shape of things is really, really relevant and really, really interesting to bring a field forward. And I wanna stress this as one of the best examples and one of the best ways of getting people interested in TDA or topological machining or whatever you want to call it. This is going into the life sciences, going to the users, looking at the real world. This is, they have marvelous data sets. They have marvelous problems and they really care about the shape aspects, the shape analysis aspects and the multi-scale aspects of our work. So there's work by Amaskita and colleagues. I think that by Elizabeth Munch on the shape of things to come, topological data analysis and biology from molecules to organism. They have these beautiful figures. They call his endless forms the most beautiful and I have to agree. So we need to pick our applications and we need to make sure that we pick the right applications. And I believe that the life sciences are definitely a source of good joy in this regard. So not only good data but also domain experts that are really happy about collaborations and that really can help us unlock the mysteries of our universe in a certain sense. Which is not to say that the other sciences don't have this appeal, right? It's just, this is what I know and this is what I've observed in the field. That people are really, really happy about these contributions and that our unique perspective is also appreciated here. Now, as I close, I wanna give you a few parting thoughts and this is something that is dear to my heart and I have to get really serious for a minute. The meme not withstanding. So this is my own experience. I recently got funding for a PhD position. I interviewed an awesome set of diverse candidates, a lot of people, I made them offers and they came back to me and they told me they think TDA is too complicated. So this was really something that stunned me for a second. I really had trouble getting people to work with me because they still see TDA as kind of a closed off field from the outside, which is totally opposite to my own experience. So we need to build a community here. We need to make sure that people are getting into the field and that they don't feel that we are closing them out or shutting them out because for whatever reasons that might be. So chat toppers has written a great blog post on this or great resource on this, getting started with topological data analysis. In the interest of time, I'm not going to summarize all of this, but he writes this very nicely. He said, I'm overwhelmed and don't know how to get started. Well, do I have a surprise for you? And then he goes on and explains his own journey into topological data analysis and the things you need, but also the things you don't need. And I think we should focus on this. There's a bunch of resources that can help us with this. There's the applied algebraic topology research network. There's the geometry and topology machine learning slack. There's also wind comp tops, the women in computational topology. So let's build a diverse community together. I really, this would be my parting thought for you. Let's build a diverse community together, but also let us know what we can do better as PIs, as the people that are trying to organize these things. Let me know, let us know. I'm saying this also particularly as one of the co-directors of AATR and we really care about your feedback. We really care about your voices. We want to make sure that we can capture people such that the situation of this hiring doesn't repeat itself such that people feel safe and feel that they can tackle topological data analysis. Now, this is the end of the Mobius Strip for me. I thank you very much for your attention and I give you a few personal remarks here. So 10 years ago, my first paper on TDA was published. So this means that I'm getting old. Almost two years ago, I organized a TDA workshop in Europe's 2020. I saw a bunch of you there. I will hope I see many of you again in other venues. And now I'm here. So I'm really thankful for this great opportunity. I'm really thankful for all the things that have worked out and for all the people that supported me, for all the support that I was able to give. This is really an awesome community and awesome endeavor. And yeah, thank you so much for having me. Okay, well, thank you, Bastion, for a fantastic talk. We have about 12 or 13 minutes for questions. So if you have a question, raise your hand, and Tim will come over with a microphone and you can ask it. Anyone to start off? Hey, Bastion, thank you. That was a wonderful talk. One of the things that I was curious about and as somebody who works in the space as well, you often see challenges from kind of drawing the parallels to machine learning where a lot of the times you try to force a fit to use something like a convolutional neural network. So maybe sometimes of the time series you'll convert to a spectrogram or something like that and then you can feed it into a CNN and sort of that deep learning hammer. How do you think we mitigate that in TDA, especially as we try to bring in additional users from other spaces so that we don't sort of promote the mentality of put it as a point cloud, compute persistence, sort of the blankets approach as opposed to the critical concepts for how do we actually pick the right scenario? That's an awesome question. And I wish I could give an easy answer here. So I think this might be connected a little bit to what I suggested with the data sets. So if we as a community, if we as let's say the experts come up with data sets where we say, ah, this has some interesting topological structure. So this is where it's very appropriate. And this might already also give an indication to users what might work and what might not work. The other side of the coin is, of course, if we make people use that and if we find that the application doesn't really need topological features, then we should also maybe say this. I mean, in some sense, okay, I don't want to take too long here, but this is a little bit of the issue with negative results, right? So how do we speak about negative results? We say, yeah, we slap persistent homology on that and it totally did not work. So I have no easy answer here. Maybe we should try to create a forum where we can discuss such negative results because otherwise, of course, people will fall into the same trap. We tried something similar in some work on MRI data analysis and not fMRI, but structural MRI where we were able to show that the addition of topological features doesn't really help, but these sort of publications are, I would say, they are rare because it's hard to get them through and it's not easy to tell people what they should and shouldn't use, but it's okay. So maybe to wrap this up, maybe we as experts should come up with some policies here and some good examples, some flagship projects where we say, hey, here it works spectacularly because of the following reasons or the following shape, if you will. And maybe it might work, maybe it might not work, but yeah, I see this as a critical problem moving forward. I'm sorry, this was not a very concise thing. I hope it helped a little bit. Yeah, great. No, no, great answer. I saw there was another question over here. Do you want to still ask something or? Hi, Bastian. So what I wanted to ask is you find out we shouldn't deceive ourselves that some things do not, or yeah, just sprinkling some TDA into our papers because it looks nice and is interesting and fun, might not actually work, but you also encourage building these toy data sets, so to say, that we can evaluate how well our methods capture TDA, yeah, how well our methods capture topological features, but you also point out many real-world data sets where apparently topological stuff already helped. So why not ignore the first part and make sure that we don't evaluate so much on our toy data sets to make our methods look nice and instead go with the real-world data sets where you already know that topology seems to be important. Yeah, thank you. So let me maybe clarify. I was not talking about creating toy data sets because I mean, this is what we are doing often, right? We're saying that, okay, we have to see that it works somehow. And I mean, we know that a synthetic data set can be useful for this. I'm thinking more along the lines of MNIST or of the graph benchmark data sets which have been collected in the wild by some process and created and curated and made sure to be available to the public. And this is, I think, what we should also look into. In particular, maybe data sets where we can say, oh, this has a, let's say this has an underlying manifold and we can leverage this. I do agree that there is a, that of course, if we stay too much with contrived data sets, whatever they might be, we do not end up convincing people, right? This is always the problem. But this is also the problem of the graph learning community when you use a small scale graph data set then reviewers are often bound to point the finger at this. But yeah, to come back to the second part of your question. Yes, I'm definitely opposed to sprinkling anything on the paper and just so that it looks nice, whether that's TDA or deep learning or some other things, right? That is something that we must fight. I see this in the machine learning community sometimes that people are not ablating their work correctly. So they are, they compare, for instance, one model with thousands of parameters to a model with millions of parameters. And then it's no surprise that the bigger one can often do better. We should avoid this, of course. I mean, we should make sure that we can characterize what is actually helpful about the TDA perspective. So your data set building suggestion is about aggregating these real-world data sets. For instance, yeah, yeah, they can and should include real-world data. They can and could also include synthetic data if we can, depending on your definition of this, right? So what I'm seeing, for instance, to make this very concrete, I'm seeing some TDA papers, they use very interesting things where they create something from chaotic attractor systems like a Lorentzian attractor, for instance. I'm not sure whether I would classify that as synthetic. For me, that is also a real-world example because, I mean, these dynamics, they exist, right? They can simulate them and we can make sure of them. So why not put something in there as well and create something like a benchmark for saying that, ah, this is how good a topological approach is, this is how good a non-topological approach is. Great. So we have time, I think, for one or two more questions if anyone has one. Looks like there's a question over here. So, Abastien, thanks for a great talk. So I'm wondering how do you think TDA can help with data visualization, for example, compared to TSNE and UMAP? Ah, that's a great question. Yes, yes, definitely. So overall, I should say that I think the potential of preserving certain kinds of information has not yet been exploited fully by all these techniques. I mean, all, of course, have shown good results and they all go in certain directions. They all have different objective functions. But I think we haven't reached the highest level here yet. So we could think about different topological properties and, of course, geometrical properties in the data that we want to preserve and use those as inductive biases for visualizations. Now, in particular, for the case that you were mentioning for TSNE and UMAP, there is always a very heated debate and I do what you could maybe call pulling a Switzerland here and I refuse to take a side there, but there is a heated debate in computational biology whether these plots should be shown or how they should be shown and so on and so forth. And I think there's great potential for us as a community to go into that space and say, okay, these are the following things that you can preserve in your data. If you go down too many dimensions, then you will not be able to do so. So we can at least tell you when something's going wrong. Yeah, I definitely think that there's a lot of potential for TDA in visualization research still. I wanted to ask if you're familiar with any techniques for creating synthetic datasets that will have interesting higher dimensional topological properties, but that you can't predict ahead of time what sort of topological properties they have because it seems like a lot of these things would be kind of, it's easier to test if something is useful if you can try and use it to discover something, but a lot of the real world datasets that I've used very rarely have kind of interesting eighth dimensional homology and the synthetic ones that I use either they're arranged to have eighth dimensional interesting homology or not and yeah, that's what I want to ask. Yeah, I'm sorry, I'm not sure I got the question there. So I should say that I'm not familiar with a lot of good generation algorithms if that's what you're looking for. Yeah, if you're aware of any way that we can generate synthetic datasets to kind of explore these questions that don't kind of, you know, bake the question. Yeah, no, okay, this I can say so yeah, honestly, this is very close to my heart. I was wondering whether it would be interesting to get the diffusion generative model crowd interested in this or the generative, or the I should say the overall general generative model crowd because maybe it would be interesting to see what when we start with the dataset that we already have maybe we should try to synthesize some new types of data from this. Such as, I don't know, taking MNIST for example and then trying to synthesize different datasets like this and seeing what we end up getting. Maybe this is also linked, maybe this issue of what we are seeing or not seeing in real-world data is also linked to our performance issues. Maybe if we were able to do approximations of higher order topology more quickly, maybe then we could also say that, hey, well, there's nothing there or maybe there should be something there. Okay, don't want to take too long but the last point about this, there's great work on statistical BT numbers for instance which shows that you get a kind of decrease in the overall topological expressivity. So who knows maybe in real-world data even if it's high dimensional maybe there's nothing there because the ambient or intrinsic dimensionality is too low for this. I can only speculate on this but I would be happy to chat with anyone and to have a longer discussion on this. Okay, thank you. Looks like we have time for one more quick question. Hi, so I had played with topological data analysis like five or six years ago before I was doing a PhD. So as a non-topologist, how do you gain intuition for higher dimensional holes because back then most of the papers I was reading were either that, oh, this is all just connected components or like basically things you can get from just like standard computer science approach on a graph versus I guess I found one paper or viral evolution which I talked about something interesting in terms of higher dimensional holes. So could you just shed some light on like for a non-topologist? How would you have intuition about this? Yes. So I think for this goes back to the education approach. So I think for me, the illuminating moment was when I sat down and did some of these calculations and I convinced myself that we could calculate connectivity of higher dimensional spaces and still get something out of that. I should stress that my intuition only also reaches so far as like three or four or something like this. So I don't really know what I would an eight dimensional hole to look like, but I can sort of convince myself that these structures are relevant. And for instance, in the graph domain, one could think about having cliques that might actually be useful. So having fully connected subgraphs there. So this is my go-to analogy at least when I try to convince myself, but it's certainly also a topic that we should explore a little bit more. I mean, as I said, we have some education to do. We have to get more people into this field. Okay, well, I think that wraps up our questions. Let's thank our speaker one more time.