APATG - Past Seminars

Past Seminars

The slides and movies of some of the past seminars are available from:
Slide link
Movie link

#47 Date: 2025/3/21

Speaker: Siddarth Pritam (Chennai Mathematical Institute)

Title: Classification of Temporal Graphs using Persistent Homology

Abstract: Temporal graphs effectively model dynamic systems by representing interactions as time-stamped edges. However, analytical tools for temporal graphs are limited compared to static graphs. We propose a novel method for analyzing temporal graphs using Persistent Homology. Our approach leverages δ-temporal motifs (recurrent subgraphs) to capture temporal dynamics without aggregation. By evolving these motifs, we define the extit{average filtration} and compute PH on the associated clique complex. This method captures both local and global temporal structures and is stable with respect to reference models. We demonstrate the applicability of our approach to the temporal graph classification task. Experiments verify the effectiveness of our approach, achieving over 92% accuracy, with some cases reaching 100%. Unlike existing methods that require node classes, our approach is node class free, offering flexibility for a wide range of temporal graph analysis.

#46 Date: 2025/2/21

Speaker: Takaaki Aoki (Shiga University)

Title: Potential landscape of human flow in cities

Abstract: People are moving from one location to another in their daily lives, for commuting, shopping, entertainment, schools, etc. This human flow provides vital information for unfolding the actual shapes of cities based on lively human behavior by place-to-place interactions from origin to destination. However, it is not easy to handle massive data on human flows as it is because, for example, when there are 1,000 locations on a map, the flow dataset is depicted by a million links from the origin to the destination. In this study, we identified the potential of human flow directly from a given origin-destination matrix. By using a metaphor for water flowing from a higher place to a lower place, the potential landscape visualizes an intuitive perspective of the human flow and determines the map of urban structure behind the massive movements of people. From the map, we can easily identify the sinks (attractive places) and the sources of human flow, not just populated places. The detected attractive places provide beneficial information for location decision making for commercial or public buildings, optimization of transportation systems, urban planning by policy makers, and measures for movement restrictions under a pandemic.

[1] Urban spatial structures from human flow by Hodge-Kodaira decomposition, Takaaki Aoki, Shota Fujishima & Naoya Fujiwara, Scientific Reports, vol. 12, 11258 (2022).

[2] Identifying sinks and sources of human flows: A new approach to characterizing urban structures, Takaaki Aoki, Shota Fujishima & Naoya Fujiwara, Environment and Planning B: Urban Analytics and City Science, vol. 51(2), 419-437 (2024).

[3] Learning the liveability of cities from migrants: Combinatiorial-Hodge-theory approach, Takaaki Aoki, Kohei Nagamachi & Tetsuya Shimane, arXiv:2405.11166 [physics.soc-ph]

#45 Date: 2025/1/17

Speaker: Seongjin Choi (POSTECH)

Title: Sheaf Laplacians on hypergraphs via symmetric simplicial sets

Abstract: Sheaf cochain complexes and sheaf Laplacians on regular cell complexes, introduced by Curry in 2014, have emerged as powerful tools in topological data analysis. Hansen and Gebhart applied this framework in 2020 to develop sheaf convolutional networks (SCNs), extending graph convolutional networks (GCNs). In this talk, we generalize SCNs to hypergraphs by constructing symmetric simplicial sets, which categorically generalize simplicial complexes. We show that (1) sheaf Laplacians are well-defined on symmetric simplicial sets (2) two functors exist from the category of hypergraphs to the category of symmetric simplicial sets. We present and prove the advantages and limitations of each functor.

This is joint work with Junyeong Park, Gahee Kim.

#44 Date: 2024/12/20

Speaker: Woojin Kim (KAIST)

Title: Super-Polynomial Growth of the Generalized Persistence Diagram

Abstract: The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the Möbius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in Rd with d≥2. This computational intractability suggests seeking alternative approaches to computing the GPD.

In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with n simplices, its persistence diagram contains at most n points. This observation leads to the question: 'Given a d-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration?' This is the case for d=1, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for d≥2, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI.

We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of d-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations.

This is joint work with Donghan Kim and Wonjun Lee.

#43 Date: 2024/11/22

Speaker: Ángel Javier Alonso (TU Graz)

Title: A sparse multicover bifiltration of linear size

Abstract: The k-cover of an Euclidean point cloud X at radius r is the set of all those points within distance r of at least k points of X. By varying the order k and radius r we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to its sensitivity to density and its robustness

to outliers. However, it is hard to compute: previous exact or (1+epsilon)-approximation methods are of polynomial size or have an exponential dependency on k. In this talk, I will show how to obtain a (1+epsilon)-approximation of the multicover that has linear size, for a fixed dimension and error parameter epsilon.

#42 Date: 2024/10/ 18

Speaker: Gugan Thoppe (Indian Institute of Science, Bangalore, India)

Title: The Shadow Knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes

Abstract: In 1985, Frieze discovered that the expected sum of the edge weights in the minimum spanning tree (MST) of a uniformly weighted graph approaches the constant $\zeta(3)$. More recently, Hino and Kanazawa extended this idea to higher dimensions, replacing the MST with a structure called the Minimum Spanning Acycle (MSA) and the graph with a uniformly weighted simplicial complex.

In this talk, we will go beyond and explore the distribution of all the face weights in such random MSTs and MSAs. We will see that this distribution follows a specific pattern based on a concept called the shadow. In graphs, the shadow refers to the set of all the missing transitive edges; in simplicial complexes, it is a related topological generalization. We will also discuss similar results for the death times in the persistence diagram corresponding to the above-weighted graphs and complexes, a result of interest in applied topology.

This is joint work with Nicolas Fraiman and Sayan Mukherjee.

Paper Link: https://discreteanalysisjournal.com/article/73323-the-shadow-knows-empirical-distributions-of-minimal-spanning-acycles-and-persistence-diagrams-of-random-complexes

#41 Date: 2024/9/20

Speaker: Shunsuke Tada (Kobe University)

Title: Ａ Computation of Bipath Persistent Homology and Bipath Persistence Diagrams

Abstract: In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this talk, we extend this setting, and propose the use of bipath persistent homology, which can be used to study the persistence of topological features across a pair of filtrations connected at their ends, to compare the two filtrations. In this setting, interval-decomposability is guaranteed, and we provide an algorithm for computing persistence diagrams for bipath persistent homology and discuss the interpretation of bipath persistence diagrams. This is joint work with Toshitaka Aoki and Emerson G. Escolar.

#40 Date: 2024/8/30

Speaker: Herbert Edelsbrunner (IST Austria)

Title: Merge Trees of Periodic Filtrations

Abstract: Crystalline materials are examples of atomic arrangements that can be modeled as periodic sets in 3-dimensional Euclidean space. The corresponding merge tree reflects how the connected components of the sublevel set of the Euclidean distance function merge until there is only one component that covers the entire space remains. We show how the necessarily infinite merge tree of a periodic set can be compressed to a finite tree with annotations that express how often components repeat. These annotations are monomials, whose degrees and coefficients encode the growth rate and the density inside a progressively larger sphere.

Acknowledgements. This is joint work with Teresa Heiss at IST Austria.

#39 Date: 2024/5/10

Speaker: Hubert Wagner (University of Florida)

Title: Computing persistent homology of 3D images with billions of voxels using Cubicle

Abstract: Persistent homology is becoming increasingly used in practice. In particular, this is true in the context of (medical) image analysis and computer vision. However, specialized algorithms are necessary to efficiently handle large multidimensional image/voxel/cubical data. I will describe some mathematical and algorithmic techniques I found useful in my 15-year-old collaborative journey towards making such computations efficient. I will also overview available packages, including GUDHI, Cubical Ripser, TTK as well as my software Cubicle.

Main reference: "Slice, Simplify and Stitch: Topology-Preserving Simplification Scheme for Massive Voxel Data" (SoCG 2023).

#38 Date: 2024/4/19

Speaker: Areejit Samal (The Institute of Mathematical Sciences, Chennai)

Title: Geometry-inspired measures with diverse applications in network science

Abstract:

In this talk, I will present our work on the development of geometry-inspired measures for the edge-based characterization of real-world complex networks. In particular, we were first to introduce a discretization of the classical Ricci curvature proposed by R. Forman to the domain of real-world complex networks. Forman-Ricci curvature is an attractive tool in network science due to the following reasons. Firstly, most traditional graph-theoretic measures such as degree and clustering coefficient are vertex-specific, while Forman-Ricci curvature is edge-specific. Secondly, the mathematical formula of the Forman-Ricci curvature elegantly allows for the analysis of weighted and unweighted graphs. Thirdly, we have also extended the definition of Forman-Ricci curvature to the realm of directed graphs. Fourthly, an important distinguishing feature of the Forman-Ricci curvature, in contrast to the other well-known discretization, namely, Ollivier-Ricci curvature, is its simplicity and suitability from a computational perspective for analysis of very large networks. Fifthly, we have developed an augmented version of the Forman-Ricci curvature which is suitable for analysis of higher-order networks. In this talk, I will mainly focus on the successful applications of Forman-Ricci curvature to real-world networks across different domains including life science and finance. Specifically, I will present application of discrete Ricci curvatures to: (a) brain functional connectivity networks constructed from resting-state fMRI data, and (b) time-series of financial networks.

#37 Date: 2024/3/22

Speaker: Yulia Gel (University of Texas at Dallas)

Title: Coupling Time-Aware Multipersistence Knowledge Representation with Graph Convolutional Networks for Time Series Forecasting

Abstract: Graph Neural Networks (GNNs) are proven to be a powerful machinery for learning complex dependencies in multivariate spatio-temporal processes. However, most existing GNNs have inherently static architectures, and as a result, do not explicitly account for time dependencies of the encoded knowledge and are limited in their ability to simultaneously infer latent time-conditioned relations among entities. We postulate that such hidden time-conditioned properties may be captured by the tools of multipersistence, i.e., an emerging machinery in topological data analysis which allows us to quantify dynamics of the data shape along multiple geometric dimensions. We propose to summarize inherent time-conditioned topological properties of the data as time-aware multipersistence Euler-Poincar\'e surface and prove its stability. We then construct a supragraph convolution module which simultaneously accounts for the extracted intra- and inter-dependencies in the data. We illustrate the utility of the proposed approach in application to forecasting highway traffic flow, blockchain Ethereum token prices, and COVID-19 hospitalizations.

#36 Date: 2024/2/9
Speaker: Agnese Barbensi (University of Queensland)

Title: Topological Optimal transport

Abstract: Topological data analysis is a powerful tool for describing topological signatures in real-life data, and to extract complex patterns arising in natural systems. An important challenge in this area is matching significant topological signals across distinct systems. In geometry and probability theory, optimal transport formalises notions of distance and matchings between distributions and structured objects. Here we propose a way of combining the two approaches, to construct a mathematical formulation for a topological optimal transport theory. By building on recent advances in the domains of persistent homology and optimal transport theory for hypernetworks, we develop a transport-based methodology for topological data processing. We define measure topological networks, generalising persistent homology hypergraphs in a measure space context. We introduce a distance on measure topological networks and we study its metric properties. Our Topological Optimal Transport (TpOT) provides a unified framework for a transport model on point clouds minimising topological distortion, while simultaneously yielding a geometrically informed matching between persistent homology cycles.

#35 Date: 2023/12/08

Speaker: Xiaochuan Yang (Brunel University London)

Title: Boundary effects in some coverage and connectivity problems

Abstract: Coverage and connectivity of Boolean models are classical problems in stochastic geometry. In this talk, I will present some interesting findings concerning connectivity threshold and coverage threshold in the large n limit, where n is the number of balls in the Boolean model. In most cases, boundary effects play a dominant role in deriving limit theorems for these thresholds. This talk is based on joint work with Frankie Higgs and Matthew Penrose.

#34 Date: 2023/11/17

Speaker : Sebastiano Cultrera di Montesano (Institute of Science and Technology Austria)

Title: Chromatic Alpha Complexes

Abstract: Alpha complexes are widely used to quantitatively describe the spatial configuration of a point set. Motivated by a biomedical question that I will outline at the beginning of the talk, we introduce a more general version, a chromatic alpha complex, tailored to the setting where the input points come with an additional label (a color). In this talk, I will define the complex and illustrate some of its structural properties.

This is ongoing work with O. Draganov, H. Edelsbrunner and M. Saghafian.

#33 Date: 2023/10/27

Speaker : Bei Wang (University of Utah, USA)

Title: Hypergraph Co-Optimal Transport.

Abstract: Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. We develop theoretical foundations for studying the space of hypergraphs using ingredients from optimal transport. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps between the space of hypergraphs and the space of graphs, and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our framework through various examples. This is a joint work with Samir Chowdhury, Tom Needham, Ethan Semrad and Youjia Zhou. It is published in the Journal of Applied and Computational Topology (DOI: 10.1007/s41468-023-00142-9).

#32 Date: 2023/02/10

Speaker: Erika Roldan Roa (Max Planck Institute for Mathematics in the Sciences & Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) at Universität Leipzig)

Title: Topology and Geometry of Extremal and Random Cubical Complexes

Abstract: In this talk, we explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can have a look at some of these random structures here (https://skfb.ly/6VINC) and start making some guesses about their topological behavior.

#31 Date: 2022/12/2

Speaker: Jae-Hun Jung (POSTECH Department of Mathematics and POSTECH MINDS)

Title: Topological data analysis of time-series data

Abstract: Interpretation and use of persistent shape statistics, e.g. barcodes, often hinges on a many-faceted inverse problem: selection of "good" cycle representatives. Much is known about this problem, both theoretically and computationally, but numerous questions of practical significance remain largely open: How often are optimal cycles unique? How close to optimal are the cycle representatives returned by persistent homology solvers, in general? How do different optimization techniques vary, in terms of computational cost? How often do optimal cycle representatives take coefficients in the integers? As with many questions in TDA, the complexity of real-world data renders purely theoretical approaches to these problems highly challenging. We will present the results of an empirical study which offers surprisingly conclusive answers, for certain types of scientific data.

#30 Date: 2022/10/14

Speaker: Gregory Henselman (Oxford)

Title: Practical insights on the selection of informative cycle representatives in persistent homology

Poster Session joint with AATRN!

Date: 2022/9/20

Time: 12:30 (Bangalore, India), 15:00 (Philippines), 16:00 (Kyoto, Japan), 17:00 (Canberra, Australia)

Joint Poster Session Posters: https://sites.google.com/view/aatrn-poster-session/prior-poster-sessions/september-2022-posters

AATRN Poster Session Webpage: https://sites.google.com/view/aatrn-poster-session/home

#29 Date: 2022/5/27

Speaker: Tam Le (AIP center, RIKEN)

Title: Geometric Approaches for Persistence Diagrams in Topological Data Analysis

#28 Date: 2022/4/22

Speaker: Cédric Ho Thanh (NII, Japan)

Title: Recurrence theorems for topological Markov chains

#27 Date: 2022/3/4

Speaker : Shizuo Kaji (Kyushu University)

Title: Geometry and Topology of Mobius Kaleidocycles

#26 Date: 2022/2/4

Speaker : Koushik Ramachandran (TIFR-CAM, Bangalore)

Title: On the geometry and topology of random lemniscates

#25 Date: 2022/1/21

Speaker : Emerson G. Escolar (Graduate School of Human Development and Environment, Kobe University )

Title: Interval Decomposability/Approximation of Persistence Modules, and their Computation

#24 Date: 2021/12/03

Speaker : Ryan Armstrong (University of New South Wales, Sydney)

Title: A universal multiscale descriptor for fluid wetting behaviour in porous media using integral geometry

#23 Date: 2021/11/05

Speaker : Tomoki Uda (Tohoku University)

Title: On Interleaving Distance between Reeb Trees as 𝑹-Pospaces