Channel Title: Applied Algebraic Topology Network Channel Description: This is the YouTube channel for the Applied Algebraic Topology Research Network. Instead of watching recorded videos, you can join us online for the live presentations! For details on how to join the network and how to watch the talks live, please see the links to the AATRN homepage and to the AATRN seminar below. The directors of AATRN are Henry Adams at Colorado State University, Hana Dal Poz Kouřimská at IST Austria, Teresa Heiss at IST Austria, Sara Kališnik at ETH Zurich, Bastian Rieck at the Institute of AI for Health at Helmholtz Munich, and Elchanan Solomon at Duke University. <video> Paweł Dłotko (12/11/24): Feature Extraction and Visualization Methods in TDA Description: Abstract: Topological Data Analysis (TDA) is widely applied in the applied sciences across two main areas: (1) providing features of multimodal data for statistical and machine learning analyses, and (2) offering visualization tools to explore and understand the structure of high-dimensional datasets. The most commonly used tools in TDA are persistent homology and the Mapper algorithm. However, each comes with its challenges. Persistent homology faces computational complexity and lacks a natural generalization to multiple parameters. Mapper, on the other hand, depends on numerous parameters and the sensitivity of its output to specific parameter choices. </video> <video> Nursultan Kuanyshov (11/14/24): On sequential topological complexity of group homomorphisms Description: We define and develop a homotopy-invariant notion for the sequential topological complexity of a map f:X→Y, TC_{r}(f), that interacts with TC_{r}(X) and TC_{r}(Y) in the same way Jamie Scott's topological complexity map TC(f) interacts with TC(X) and TC(Y). Furthermore, we apply TC_{r}(f) to studying group homomorphisms ϕ:Γ→Λ. We present some computations of group homomorphisms for specific groups, including free groups, surface groups, abelian groups, nilpotent groups, and almost nilpotent groups. Additionally, we provide a characterization of the cohomological dimension of group homomorphisms. Time permitting, we will conclude the talk with a discussion of several open problems. </video> <video> Monique Teillaud (12/04/24): Triangulations, CGAL, and hyperbolic surfaces. Description: The talk will survey incremental algorithms to compute Delaunay triangulations in various contexts, as well as their implementation in CGAL (Computational Geometry Algorithms Library, www.cgal.org). Then it will present work in progress and open questions for the case of hyperbolic surfaces. </video> <video> Ekansh Jauhari (10/24/24): On some distributional and intertwining invariants of manifolds Description: Recently, the distributional versions of LS-category and sequential topological complexity were defined and studied as lower bounds to the classical versions of LS-category and sequential topological complexity. More recently, the corresponding intertwining versions were introduced, which bound their respective distributional counterparts from below. In general, the theories of these three versions differ from each other in various interesting ways. In this talk, we will first obtain some sufficient conditions for the distributional LS-category (dcat) of a closed manifold to be maximum, i.e., equal to its classical LS-category. This will give us many new computations of dcat, especially for some essential manifolds, connected sums, and 3-manifolds. Then we will look at the intertwining invariants and interpret the notion of intertwining complexity in terms of motion planning. Using cohomological lower bounds, we will also determine the intertwining category and complexity of some closed manifolds. Finally, we will end this talk by discussing a few open problems. </video> <video> Maxime Lucas (12/3/2024): Revealing patterns in brain activity with persistent homology Description: Abstract: The brain is a notoriously complex system and its activity can exhibit a wide range of complex dynamics. We will discuss how persistent homology can help discriminate and describe these dynamics and associated biological conditions that may induce them. In particular, we will talk about two cases: brain recordings (1) in human hypnosis experiments and (2) in epileptic fish. We first apply time-delay embedding to the time series to reconstruct the associated dynamical attractor, and then apply persistent homology. In both cases, persistent homology and topological indicators associated to it allow us to unveil how the brain dynamics is affected by biological factors (e.g. hypnotic susceptibility in the first case, or fishline and genetic mutation in the second). In turn, these topological indicators may have the potential to serve as markers for these biological factors and be useful to practitioners </video> <video> David Loiseaux (10/16/24): Multiparameter Persistence for Machine Learning Description: The main tool of Topological Data Analysis is Persistent Homology, which captures the persistence of topological features as a filtration parameter changes---typically, a geometrical scale. For many applications however, it is beneficial to simultaneously vary multiple filtration parameters, such as, scales, the dataset's sampling density, or simply take into account other intrinsic properties of the data. To that end, a generalization of this construction, called Multiparameter Persistent Homology, enables the use of multiple filters at once. In this talk, I will present the recent advancements in multiparameter persistence aiming at making it practical for machine learning. To that end, I will introduce several descriptors; one is obtained by gluing one-dimensional slices, and takes the form of an interval decomposable module called an MMA decomposition, and the other is computed from the Möbius inversion of classical invariants called the signed barcode. Finally, I will discuss corresponding vectorization for these descriptors, and their differentiability properties. </video> <video> Elena Wang (10/30/2024): Measuring Graph Distance using the Labeled Merge Tree Interleaving Distance Description: Title: Measuring Graph Distance using the Labeled Merge Tree Interleaving Distance Abstract: Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets. </video> <video> Nikola Milićević (11/06/2024): Homotopy and Singular Homology Groups of Finite Digraphs Description: Abstract: Computing algebraic invariants of topological spaces such as singular homology is extremely difficult due to the infinite dimensionality of the associated chain groups, even when the spaces themselves are finite. The same is true for higher homotopy groups. This complexity is a barrier not only in pure mathematics but also in applied settings such as data analysis and network science, where finite structures like graphs and digraphs are common. In this work, we extend McCord's classical theorem, which established a weak homotopy equivalence between finite topological spaces and finite simplicial complexes, to the broader context of finite digraphs. This extension is non-trivial because finite digraphs are not topological spaces, making the classical methods not applicable. This result allows us to compute higher homotopy groups, and singular homology groups, of finite digraphs by replacing them with weakly homotopy equivalent simplicial complexes. </video> <video> Alexander McCleary (10/02/24): Persistent Homology for Infinite Complexes Description: Abstract: Persistent homology is a powerful tool for understanding the evolution of homological features in a filtered CW complex. Traditionally, its theory has been limited to finite complexes. In this talk, we will present a theoretical framework that extends persistent homology to infinite CW complexes. We will also demonstrate that the two cornerstone theorems of persistent homology—the Bottleneck Stability Theorem and the Algebraic Decomposition Theorem—hold without the need for finiteness assumptions. This extension opens new avenues for applying persistent homology to a broader class of problems in topological data analysis. </video> <video> Radmila Sazdanovic (10/09/2024): The shape of relations: knots and other stories Description: Title: The shape of relations: knots and other stories Topological Data Analysis provides tools for discovering relevant features of data by analyzing the shape of a point cloud. In this context we develop tools for visualizing maps between high dimensional spaces with the goal of discovering relations between data sets with expected correlations. Additionally we are adapting TDA tools to analyzing infinite data sets where representative sampling is impossible or impractical and using them in synergy with ML techniques. Most of the examples will focus on analyzing relations between knot invariants with additional examples in game theory and cancer genomics. This work is joint with P. Dlotko and D. Gurnari. </video> <video> Petar Pavešić (12-12-24): Covering type Description: </video> <video> Marco Nurisso (10/1/2024): Interactions and topological synchronization in simplicial Kuramoto model Description: Simplicial Kuramoto models have emerged as a diverse and intriguing class of models that capture the dynamics of interacting oscillators placed on the simplices of a simplicial complex. Being formalized with the tools of discrete differential geometry, these models reveal interesting relationships between topology, geometry and dynamics. We leverage their mathematical structure to give a microscopic interpretation of the interaction terms which, we see, include effectively both higher-order and self interactions. This naturally leads us to establish an equivalence between the simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of the underlying simplicial complex being a pseudomanifold. Then, we describe the notion of simplicial synchronization, its relation to simplicial homology, and derive bounds on the oscillators’ coupling strength necessary or sufficient for achieving it. </video> <video> Caroline Moosmüller (11/13/2024): Trajectory inference in Wasserstein space Description: Abstract: Capturing data from dynamic processes through cross-sectional measurements is seen in many fields such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this talk, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is instrinsic to the Wasserstein geometry. Combining subdivision schemes with optimal transport-based geodesics, our methods carry out trajectory inference at a chosen level of precision and smoothness and can automatically handle scenarios where particles undergo division over time. We prove linear convergence rates and rigorously evaluate our method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like supercells, comparing its performance against state-of-the-art trajectory inference and interpolation methods. This is joint work with Amartya Banerjee, Harlin Lee, and Nir Sharon. </video> <video> Sara Veneziale (09/18/24): Machine learning detects terminal singularities Description: Title: Machine learning detects terminal singularities Abstract: In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk. </video> <video> Corentin Lunel (11/30/24):Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect Description: Title: Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect Abstract: While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order. </video>

Description

This is the YouTube channel for the Applied Algebraic Topology Research Network. Instead of watching recorded videos, you can join us online for the live presentations! For details on how to join the network and how to watch the talks live, please see the links to the AATRN homepage and to the AATRN seminar below.

The directors of AATRN are Henry Adams at Colorado State University, Hana Dal Poz Kouřimská at IST Austria, Teresa Heiss at IST Austria, Sara Kališnik at ETH Zurich, Bastian Rieck at the Institute of AI for Health at Helmholtz Munich, and Elchanan Solomon at Duke University.

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