Please see the schedule below for the time and place of each talk.

Some seminars will take place on the Lausanne campus and others at Campus Biotech.

### Program

Date and place | Title | Speaker |
---|---|---|

01.10.19, 16:15 PH H3 33 |
Equivariant simplicial reconstruction | Naya Yerolemou University of Oxford |

09.10.19, 14:00 Biotech |
Complexes of Tournaments in Directed Networks | Ran Levi University of Aberdeen |

15.10.19, 16:15 PH H3 33 |
General framework for testing Poisson-Voronoi |
Martina Vittorietti TU Delft |

22.10.19, 16:15 PH H3 33 |
Parametrised metrics on persistence modules and non-linear hierarchical clustering | Oliver Gäfvert KTH |

06.11.19, 14:00 Biotech |
Categorising Attractor Dynamics in Neural Data | Abel Sagodi University of Amsterdam |

26.11.19, 9:00 MA B1 504 |
Extracting persistence features with the stable rank | Martina Scolamiero KTH |

04.12.19, 14:15 MA B1 504 |
Audio identification with persistent homology fingerprints | Wojciech Reise EPFL |

17.12.19, 16:15 PH H3 33 |
Stratified Space Learning – Reconstructing Embedded Graphs | Yossi Bokor ANU |

14.01.20, 16:15 MA B2 485 |
Simplicial Mixture Models – Fitting topology to data | James Griffin Coventry University |

29.01.20 10:15 MA B2 485 |
Directed Topology Two Ways | Nicole Sanderson LBNL |

11.02.20 10:15 MA B2 485 |
Geometric and topological shape description using discrete curvature | Jean-Luc Mari Aix-Marseille University |

18.02.20 16:15 MA B2 485 |
Configurations spaces of particles: homological solid, liquid, and gas | Matthew Khale Ohio State |

25.02.20 16:15 MA B2 485 |
Signal processing and random walks on simplicial complexes | Michael Schaub University of Oxford |

03.03.20 16:15 MA B2 485 |
Topological Fingerprint for Periodic Crystals | Teresa Heiss IST Austria |

10.03.20 16:15 MA B2 485 |
Differential calculus on Persistence barcodes | Jacob Leygonie University of Oxford |

28.05.20 14:15 Online |
Disordered proteins construct order from randomness in health and disease: a computational model | Julian Shillcock EPFL |

18.06.20 14:15 Online |
Sheaves as computable and stable topological invariants fordatasets: from level-sets persistence and beyond | Nicolas Berkouk INRIA Saclay |

02.07.20 14:15 Online |
Topology-Based Representation Learning | Bastian Rieck ETHZ |

### Abstracts**:**

**N. Yerolemou: **We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction, see through an example how to answer our initial question and briefly discuss some applications.

**R. Levi****: **Clique graphs whose edges are oriented are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial sets, that we refer to as “tournaplexes”, whose simplices are tournaments. In particular, given a directed graph G, we associate with it a “flag tournaplex” which is a tournaplex containing the directed flag complex of G, but also the geometric realisation of cliques that are not directed. We define several types of filtration on tournaplexes, and exploting persistent homology, we observe that filtered flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of those ideas by applying them to graph data arising from the Blue Brain Project’s digital reconstruction of a rat’s neorcortex. Time permitting we explore connection to another graph invariant.

**M. Vitorietti:** Modeling microstructures is an interesting problem not just in Materials Science but also in Mathematics and Statistics. The most basic model for steel microstructure is the Poisson-Voronoi diagram. It has mathematically attractive properties and it has been used in the approximation of single phase steel microstructures. The aim of this paper is to develop methods that can be used to test whether a real steel mi- crostructure can be approximated by such a model. Therefore, a general framework for testing the Poisson-Voronoi assumption based on images of 2D sections of real metals is set out. Following two different approaches, according to the use or not of periodic boundary conditions, three different model tests are proposed. The first two are based on the coefficient of variation and the cumulative distribution function of the cells area. The third exploits tools from to Topological Data Analysis, such as persistence landscapes.

**O. Gäfvert: **In this talk I will show how to construct parametrised classes of metrics on persistence modules facilitated by the introduction of persistence contours. Using persistence contours we can compute the stable rank, an invariant which could previously not be computed, and show that it’s in general NP-hard to compute. Moreover, persistence contours can be used to put persistence modules in a machine learning context by providing a class of metrics that can be optimised over. This could for instance be used to solve classification problems in a metric learning sense, called contour learning.

**A. Sagodi: **Attractor dynamics are thought to play a central role in the representation of information and computation in the brain. While many observed phenomena are consistent with their existence, direct proof in the brain remains elusive. If detection were possible, a further difficulty would be that of describing the attractors states to determine, for example, if they form discrete points, lie on a continuum of states, or are have some other features, such as forming a limit cycle. In this work, pairwise maximum entropy models are fit to both real and simulated neural activity. Maximum entropy models provide the simplest surrogate model, where features such as attractors can be detected and described, while avoiding the biological complexity of the original system. Two methods to analyse the attractor dynamics are considered in this talk: one involving entropy curves and another topological data analysis, thus permitting inspection of both the local energy landscape in the neighbourhood of the attractors as well as the overall topological features of the landscape. To our knowledge, this work is the first to apply methods from topological data analysis in order to investigate the energy landscape of a maximum entropy model. The example networks presented here consist of data from neuroscientific experiments, simulated data of the head direction network and Hopfield networks. Results include support for a topologically nontrivial one-dimensional ring attractor underlying the mammalian head direction circuit. Furthermore, preliminary results are shown which may help distinguish discrete attractor network dynamics from continuous dynamics by investigating the entropy curves of the attractor states.

**M. Scolamiero: **In this talk I will present the hierarchical stabilisation framework which can be used to produce invariants for multi-parameter persistence. The key element of our approach is defining metrics induced by so called ‘noise systems’. Such metrics generalize interleaving distance. At the same time, in the one parameter case, they allow to overcome the usual dichotomy interpreting short bars in a barcode as noise and long bars as relevant information. I will then focus on one of the proposed invariants, the stable rank and address challenges associated to its computation in the multi-parameter case. Finally I will illustrate some statistical properties of our stable invariants with respect to subsampling of the data.

**W. Reise:** A fingerprint of an audio signal is a descriptive summary that encodes relevant information for a particular, signal-processing task. We focus on audio identification, which consists of detecting when pairs of tracks are similar, for example, one is a cover or an obfuscated version of the other. In this talk, I will review industry-standard approaches, introduce fingerprints based on persistent homology features of spectral representations of songs and propose an algorithm for audio identification. Finally, I will comment on the characteristics of the proposed fingerprints based on the performance of the identification algorithm on a set of obfuscated songs.

**Y. Bokor: ** Given a point cloud sample X of an embedded graph G, we describe and algorithm and prove its correctness to obtain the coarsest abstract graph and model the embedding from which X was sampled.

**J. Griffin: ** Lines and planes can be fitted to data by minimising the sum of squared distances from the data to the geometric object. But what about fitting objects from topology such as simplicial complexes? I will present a method of fitting topological objects to data using a maximum likelihood approach, generalising the sum of squared distances. A simplicial mixture model (SMM) is specified by a set of vertex positions and a weighted set of simplices between them. The fitting process uses the expectation-maximisation (EM) algorithm to iteratively improve the parameters. Remarkably, if we allow degenerate simplices then any distribution in Euclidean space can be approximated arbitrarily closely using a SMM with only a small number of vertices. This theorem is proved using a form of kernel density estimation on the n-simplex.

**N. Sanderson:** Topology is a field of mathematics created to study dynamic phenomena that obey the “arrow of time”. Yet the algebraic topological tools developed historically ignore this directionality. We present two models that reincorporate direction into the topological objects of study – one with application to classifying the behavior of neurons in the brain and one with application to verifying concurrent programs.

**J-L. Mari:** We present two approaches to extract geometric features on meshes, based on the use of discrete curvature estimators in order to define specific areas of interest before processing these meshes. The first part deals with the computation of feature lines using a homotopic thinning algorithm transposed on specific regions of a mesh. To do this, the notion of neighborhood is transposed from classical thinning algorithms (where the adjacency is constant) to the mesh domain where the neighborhood is variable due to the adjacency of each vertex. In the second part, we present a new shape descriptor for 3D meshes called SCG for ”Shape-Curvature-Graph”. It aims at representing an arbitrary triangular polyhedron using a graph. A discrete curvature map is computed and an adjacency graph is constructed with a node for each patch. We show that this specific shape-graph can be used to perform self-similarity detection, or more generally to extract patterns within a shape.

**M. Kahle: **Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the phase space or energy landscape for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions.

We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of “homological solid, liquid, and gas”. This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Kelly Spendlove, and Robert MacPherson.

**M. Schaub: ** In many applications, we are confronted with with signals defined on the nodes of a graph. Think for instance of a sensor network measuring temperature; or a social network, in which each person has an opinion about a specific issue. Graph signal processing (GSP) tries to device appropriate tools to process such data by generalizing classical methods from signal processing of time-series and images — such as smoothing, filtering and interpolation — to signals defined on graphs. Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian.

In applications such as traffic network analysis, however, the signals of interest are naturally defined on the edges of a graph, rather than on the nodes. After a brief introduction to the ideas of GSP, we examine why the standard tools from GSP may not be suitable for the analysis of such edge signals. More specifically, we discuss how the underlying notion of a ‘smooth signal’ inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode a notion of flow. To overcome this limitation we devise signal processing tools based on the Hodge 1-Laplacian and discrete Hodge Theory. We first discuss how these tools can be applied for signal smoothing, semi-supervised and active learning for edge-signals on graphs. We then explore connections of these signal processing tools to random walks on graphs and simplicial complexes, and give alternative interpretations of our previously derived methods in terms of diffusion processes on simplicial complexes.

**T. Heiss:** As the atoms in periodic crystals are arranged periodically, such a crystal can be modeled by a periodic point set, i.e. by the union of several translates of a lattice. Two periodic point sets are considered equivalent if there is a rigid motion from one to the other. A periodic point set can be represented by a finite cutout s.t. copying this cutout infinitely often in all directions yields the periodic point set. The fact that these cutouts are not unique creates problems when working with them. Therefore, material scientists would like to work with a complete, continuous invariant instead. We conjecture that a tool from topological data analysis, namely the sequence of k-fold persistence diagrams for all positive integers k, is such a complete, continuous invariant of periodic crystals.

**J. Leygonie:** In this talk I will define notions of differentiability for maps from and to the space of persistence barcodes. Inspired from diffeology theory, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. I will illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in the TDA literature.

**J. Shillcock:** Many proteins adopt unique, three-dimensional folded structures to function, but it is now recognized that a substantial minority do not. Intrinsically disordered proteins (IDP) comprise almost half of the human proteome and are typically unstructured in solution. They are enriched in cellular signaling pathways and regulation, and implicated in many degenerative diseases including Alzheimer’s disease, Parkinson’s disease and diabetes. IDPs spontaneously phase separate into fluid droplets, or membraneless organelles, that concentrate enzymes and other proteins to execute cellular functions. Phase separation is a common phenomenon in polymer chemistry and everyday life, where oil and vinegar separate into droplets. However, the condensed phase of IDPs is highly structured, and this structure appears necessary for their biological function. Experiments show that the interior of membraneless organelles is a fluid network in which the IDPs reversibly bind to each other creating voids or cavities within the condensed phase. The proteins diffuse through the droplet, although much slower than the free proteins in solution, and exchange with the surrounding dilute phase. I will present the results of simulations of a model membraneless organelle composed of a single species of semi-flexible polymer with *sticky* end-caps. The polymers phase separate into a porous, three-dimensional network in which their end-caps reversibly bind to each other at *junctions*. The distance between connected junctions is predicted to scale with the polymer length as a self-avoiding random walk. Also, the average number of polymers that meet at the junctions depends strongly on the end-cap affinity but only weakly on the polymer length. The structured porosity of the condensed phase suggests a mechanism for cells to regulate membraneless organelles. Protein binding sites may be turned on or off to modulate the porosity and therefore the diffusion and interaction of additional proteins that are recruited to perform cellular functions.

**N. Berkouk:** Persistent homology has been recently studied with the tools of sheaf theory in the derived set-ting by Kashiwara and Schapira after J. Curry has made the first links between persistent homology and sheaves.The theory of constructible sheaves over a normed vector space admits a lot of similarities with the theory of persistence modules. In this (derived) setting, Kashiwara-Schapira have introduced the convolution distance, inspired by the interleaving distance introduced. The convolution distance satisfies a stability property with respect to the L∞-norm in the same fashion as the interleaving distance. Moreover if the normed vector space is one dimensional, constructible sheaves admit a graded-barcode (alike barcodes for one-parameter persistence modules).

After having quickly introduced this setting and these results, we will explicit the connections between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over R, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of continuous functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a pseudo-isometric equivalence of categories between derived constructible sheaves with the convolution or (derived) bottleneck distance and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems. Ultimately, our results provide a functorial equivalence between level-sets persistence and derived pushforward of sheaves for continuous real-valued functions.Altogether, these results show that we can use constructible sheaves as a continuous generalization of level-sets persistence in a computer-friendly way.

**B. Rieck:** Modern machine learning techniques are based on the concept of having dynamic representations of a data set that can be updated continuously during training. This is in stark contrast to the ideas of topological data analysis (TDA), which are, for the most part, of a fundamentally discrete nature. Fortunately, this gap started to be bridged recently! In this talk, I will discuss recent advances in TDA that enable the development of topology-based representation learning techniques. I will provide a brief theoretical introduction to this problem and showcase two applications: one for learning suitable graph filtrations in an end-to-end manner for classification tasks, the other one for regularising the latent representation of autoencoders, thus taking the topology of the input space into account.