Topological Data Analysis in Visualization: Theory, Applications, Software
Topological Data Analysis (TDA) has become an established framework for the extraction and analysis of hidden structural features in complex data with applications to a wide variety of scientific domains including quantum chemistry, astrophysics, fluid dynamics, combustion, material sciences, biology, and data science. The genericity, efficiency, and robustness of topological methods have made them particularly well suited for the multiscale, interactive analysis and visualization of the structural information of data.
With the aim to broaden the participants’ knowledge in TDA, its application to visual data analysis, and exposure to available software tools, we are organizing a summer school at Norrköping, Sweden during Aug 2125, 2023. The primary target group of this summer school is Ph.D. students, post docs, and master’s students working in this field. For the program, we plan to have lectures by invited speakers in the morning, a short session of presentations from participants after lunch, and handson sessions with different software packages (including TTK, Paraview, Inviwo) in the afternoon.
The registration is free of cost, but the participants will need to arrange and pay for travel and accommodation themselves. Note that we have limited number of seats available.
This event is inspired by IndoSwedish joint network project and sponsered by Swedish eScience Research Centre (SeRC).
Topics
The tentative list of topics we aim to cover during this summer school include:
 Mathematical foundations: manifolds, Morse functions, complexes, persistent homology, …
 Topological analysis of scientific data: scalar, vector, tensor, and multifield topology
 Featurebased visualization with topological descriptors: comparison, tracking, uncertainty, …
 Visual representations of topological descriptors
 Algorithms in computational topology
 Applications and case studies
 Software packages for TDA and visualization
Invited Speakers
Here is a list of confirmed speakers:
 Christoph Garth, Technische Universität Kaiserslautern.
 Jonas Lukasczyk, Technische Universität Kaiserslautern.
 Paul Rosen, SCI Institute, University of Utah.
 Martina Scolamiero, KTH, Stockholm.
 Raghavendra Sridharamurthy, SCI Institute, University of Utah.
Program
Legend  
Lectures  "Hands on" sessions  Talks by participants 
Please note:
 The venue for the lectures and handson sessions is room KO23 located on the second floor and easily accessible from Entrance 14 of Kopparhammaren 2 at Campus Norrköping.
 All times are specified in CEST (UTC+2:00).
 You can click on individual talk and lectures to expand and view their abstracts.
Monday, Aug 21
08:45 – 12:20  Morning session 
08:45 – 09:00  Welcome address Ingrid Hotz (Linköping University) 
09:00 – 10:30  Mathematical Foundations I  Algebraic Topology Basics Vijay Natarajan (Indian Institute of Science, Bangalore) Abstract: Topology is the mathematical study of the connectivity of space. This study is often based on topological invariants, which help establish equivalence between spaces or help distinguish between them. This lecture will provide a gentle introduction to topology with a focus on topological invariants that are amenable to computation. After a quick introduction to homoemorphism and manifolds, we will study invariants such as the Euler Characteristic, Betti numbers, and homology groups. Next, we will study two applications to illustrate the use of the invariants — the surface classification theorem and critical point classification via lower link. Biography: Vijay Natarajan is a Professor in the Department of Computer Science and Automation at Indian Institute of Science, Bangalore. He received his Ph.D. in computer science from Duke University and holds the B.E. degree in computer science and M.Sc. degree in mathematics from BITS Pilani. His research interests include scientific visualization, computational geometry, and computational topology. In current work, he is developing topological methods for timevarying and multifield data visualization, and studying applications in biology, material science, and climate science. 
Coffee break  
10:50 – 12:20  Mathematical Foundations II  Discrete Topology Martina Scolamiero (KTH Royal Institute of Technology, Stockholm) Abstract: During this lecture we will introduce elements of discrete topology such as simplicial complexes and illustrate how they can be used to give a geometric representation of point cloud data or metric spaces. Persistent homology, a fundamental tool in Topological Data Analysis (TDA) will then be defined. Persistent homology encodes homological properties of a sequence of simplicial complexes, revealing shape characteristics of the data. Several ways have been proposed in TDA to summarise persistent homology in a way which can then be used for data analysis; we will review some of them (persistence diagrams, barcodes, landscapes, stable ranks) with an emphasis on their robustness to perturbations of the data. Biography: Martina Scolamiero is an Assistant Professor in Mathematics with specialization in Geometry and Mathematical Statistics in Artificial Intelligence. Her research interests are in the areas of applied and computational topology. She works on defining topological invariants that are suitable for data analysis, understanding their statistical properties, and their applicability in machine learning. She is also interested in applications of topological methods to neuroscience and psychiatry. 
12:30 – 13:30  Lunch at Visualization center restaurant. 
13:40 – 17:00  Afternoon session 
13:40 – 15:10  Discussion and problem solving session Led by Martina Scolamiero (KTH Royal Institute of Technology, Stockholm) and Vijay Natarajan (Indian Institute of Science, Bangalore) 
Coffee break  
15:30 – 16:00  Participants' introduction session Led by Talha Bin Masood (Linköping University) 
16:00 – 16:15  Understanding materials using Topological Data Analysis Jan Felix Senge (University of Bremen, Germany) Abstract: Topological Data Analysis provides new approaches for the interpretation of (high dimensional) data exploiting underlying structures and relationships to be used as proxies for understanding qualitative similarities and differences. Suitable vectorization of proxies like the persistence diagram can enhance Machine Learning (ML) algorithms with an intermediate layer improving performance as well as giving (other) interpretations of the data. In this talk, we will briefly highlight some of the design choices and tools of TDA when introducing pipelines for persistent homology to understand the 2dimensional structure of processed surfaces [1] as well as the 3dimensional structures of pores [2]. Then we will focus on how different visualizations help practitioners to understand the structure.

16:15 – 16:30  Calculation of exchange interaction and emerging DMI in magnetic material under straingradient Mathias Augustin (KTH Royal Institute of Technology, Stockholm) Abstract: DzyaloshinskiiMoriya interactions (DMI) are an important component in most topological properties of magnetic materials. The existence of the DMI in a material is closely related to the crystal symmetries and breaking them may allow the emergence of DMI in materials where they are otherwise forbidden. One way to break the crystal symmetry that recently gathered interest is to apply a strain gradient (e.g., curvature) on the material. However, performing first principles calculations on such systems requires a big supercell to account for a realistic strain gradient. This leads to costly, if doable at all, simulations to compute the parameters. Here I will present a method to compute the parameters in these systems by treating the straingradient effect as a perturbation, circumventing the need for large supercells. 
16:30 – 16:45  Displacementsensitive charge density waves in NbSe_{2} bilayers Dhani Nafday (KTH Royal Institute of Technology, Stockholm) Abstract: We have modelled various periodic lattice distortions associated with charge density waves in bilayer NbSe_{2} by means of abinitio techniques. Our main finding is that inplane displacement affects the energy hierarchy. Although a natural consequence of the enhanced degrees of freedom in lowdimensional systems, this finding is nontrivial because the directspace structural details of the inner layers were not explored yet. We propose two independent techniques to observe our predictions, giving simulated scanning tunnelling microscopy (STM) images and diffraction scattering. Neat differences are noticed in the simulated STM images, as well as in the diffraction patterns. 
16:45 – 16:55  Reconstructing geometry with topological priors Federico Sichetti (University of Genoa, Italy) Abstract: I will briefly illustrate how TDA techniques can be applied to surface reconstruction, a classical geometry processing problem. 
17:30 – 19:00  A tour of Norrköpings Visualiseringscenter followed by a dome show hosted by Prof. Anders Ynnerman. 
Tuesday, Aug 22
09:00 – 12:20  Morning session 
09:00 – 10:30  VietorisRips, Čech, and alpha complexes Talha Bin Masood (Linköping University) Abstract: In this lecture, I will discuss practical aspects of how topology is used for understanding the shape of data represented as a point cloud in some space. Some key ideas such as thickening, filtrations, and simplicial complexes will be described. I will then introduce three types of complexes that can be constructed as a result of this thickening, namely, VietorisRips, Čech, and alpha complexes. We will learn about the relationship between them and their advantages and disadvantages. In the end, we will also take up some concrete simple examples of point cloud data and discuss how we can compute it's topological inavriants through persistent homology. Biography: Talha Bin Masood is an Assistant Professor at Linköping University in Sweden. He received his Ph.D. in Computer Science from the Indian Institute of Science, Bangalore. After that, he worked as post doctoral researcher at Linköping University from 2018 to 2022. His research interests include scientific visualization, computational geometry, computational topology, and their applications to various scientific domains. 
Coffee break  
10:50 – 12:20  Scalar Field Topology I: Critical points, merge trees, Reeb graphs, and applications Jonas Lukasczyk (TU Kaiserslautern, Germany) Abstract: Scalar field topology (SFT) deals with scalar functions defined on topological spaces, commonly referred to as scalar fields. Since scalar fields are very common data representations, SFT has successfully been applied in many research areas to characterize features. To this end, SFT provides several data abstractions, such as critical points, contours, merge/contour trees, Reeb graphs, ascending/descending manifolds, MorseSmale complexes, fibers, and so forth. These abstractions capture the inherent structure of the input data, and due to their generality, they can describe a plethora of semantic features, including pressure minima in climate data, boundaries between mixing fluids in CFD simulations, dark matter halos in cosmology, and bonds in chemistry. This talk introduces these abstractions and demonstrates their application in a variety of research applications. Biography: Jonas Lukasczyk is a Staff Scientist at TU Kaiserslautern. He received his Ph.D. degree from the Visual Information Analysis Group, Technische Universitat Kaiserslautern, Germany, where he also studied applied computer science and mathematics. His recent work focuses on topologybased characterization of features and their evolution in largescale simulations. He is an active contributor to one of the flagship software libraries for topological analysis of scientific data within Visualization domain called Topology Toolkit (TTK). 
12:30 – 13:30  Lunch at Visualization center restaurant. 
13:40 – 17:00  Afternoon session 
13:40 – 15:10  Introduction to TDA software libraries: GUDHI, Ripser, etc. Hands on session led by Talha Bin Masood (Linköping University) and Farhan Rasheed (Linköping University) 
Coffee break  
15:30 – 17:00  Introduction to Topology Toolkit (TTK) Hands on session led by Jonas Lukasczyk (TU Kaiserslautern, Germany) 
Wednesday, Aug 23
09:00 – 12:20  Morning session 
09:00 – 10:30  Scalar Field Topology II: MorseSmale complexes Vijay Natarajan (Indian Institute of Science, Bangalore) Abstract: This lecture will continue the discussion on topological descriptors for scalar fields with a focus on the MorseSmale complex. Ideas from Morse theory of smooth scalar functions can be transported to piecewiselinear functions and applied towards the study of scientific data. The MorseSmale (MS) complex represents a partition of the domain of a scalar field into regions that exhibit uniform gradient flow behavior. The domain is partitioned into cells, each of which is defined by a pair of critical points of the scalar field. In this lecture, we will introduce the MS complex, study some characteristic properties, and outline an algorithm for computing the complex. The practical utility of the MS complex depends on the existence of methods for topological simplification that help identify and remove noise. Finally, we will describe some applications of the MS complex. Biography: Vijay Natarajan is a Professor in the Department of Computer Science and Automation at Indian Institute of Science, Bangalore. He received his Ph.D. in computer science from Duke University and holds the B.E. degree in computer science and M.Sc. degree in mathematics from BITS Pilani. His research interests include scientific visualization, computational geometry, and computational topology. In current work, he is developing topological methods for timevarying and multifield data visualization, and studying applications in biology, material science, and climate science. 
Coffee break  
10:50 – 12:20  Parallel and distributed algorithms for topological descriptors Vijay Natarajan (Indian Institute of Science, Bangalore) and Jonas Lukasczyk (TU Kaiserslautern, Germany) 
12:30 – 13:30  Lunch at Visualization center restaurant. 
15:00 – 21:00  Excursion to Söderköping followed by joint dinner.

Thursday, Aug 24
09:00 – 12:20  Morning session 
09:00 – 10:30  Vector and tensor field topology Ingrid Hotz (Linköping University) Abstract:In this lecture, I will discuss two topological concepts for the analysis of vector and tensorvalued fields. I will start with the limitset topology assuming a continuous field as introduced into visualization by Helman and Hesselink, focussing on two and threedimensional fields. As an alternative, we take a brief look at the discrete vector field topology, which is based on Forman's discrete vector fields. Then we will discuss some approaches that have been considered for simplifying vector fields. Finally, I will talk about some challenges and open problems in vector/tensor field visualization. Biography: Ingrid Hotz received her M.S. degree in theoretical Physics from the Ludwig Maximilian University in Munich Germany and the PhD degree from the Computer Science Department at the University of Kaiserslautern, Germany. During 20032006 she worked as a postdoctoral researcher at the Institute for Data Analysis and Visualization (IDAV) at the University of California. Then she was the leader of a research group at the Zuse Institute in Berlin Germany during 20062013. From 2013 to 2015 she was the head of the scientific visualization group at the German Aerospace Center (DLR). Since 2015 she is a Professor in Scientific Visualization at the Linköping University in Scientific Visualization and has an affiliation with the Center for Medical Image Science and Visualization (CMIV) in Linköping. The main focus of her research lies in the area of data analysis and scientific visualization, ranging from basic research questions to effective solutions to visualization problems in applications including flow analysis, engineering and physics, medical applications, and mechanical engineering ranging from small to largescale simulations. 
Coffee break  
10:50 – 12:20 
12:30 – 13:30  Lunch at Visualization center restaurant. 
13:40 – 17:00  Afternoon session 
13:40 – 14:00  Uncertainty visualization and topological characterization of atmospheric rivers Fangfei Lan (University of Utah, USA) Abstract: Atmospheric rivers (ARs) are long, narrow regions in the atmosphere that transport water vapor from the Earth's tropics. ARs have been of great interest to climate scientists because they are responsible for a large percentage of precipitation worldwide. In North America, ARs contribute significantly to water supply and flooding risk, especially in the western regions. However, ARs are difficult to characterize due to the lack of a universal definition and their varying shapes and sizes. Many AR detection tools (ARDTs) have been developed for different purposes, producing distinct AR boundaries. In this work, we study the ARs detected by an ensemble of algorithms. We quantify and visualize the uncertainty that arises due to the differences in these methods. We propose an uncertainty visualization framework that captures both the exterior and interior variability of an ensemble of ARs. For the exterior variation, we apply the principles of contour boxplots. For the interior variations, we introduce a topologybased summary of the AR that characterizes its underlying structure independent of its detection method. We visualize the agreements and disagreements among these summaries using a MetroSetinspired visual encoding. Applying our framework to various representative AR detectors, we visually analyze their variations as part of an ensemble analysis. 
14:00 – 14:20  Defining merge treetailored edit distances for comparative analysis of scalar fields Florian Wetzels (University of KaiserslauternLandau, Germany) Abstract: Merge trees are a powerful abstraction of scalar fields with various applications in topological data analysis. An area of increasing interest is the definition of similarity measures between topological abstractions to speed up the comparison of complex scalar fields. In this talk, we consider different variants of socalled deformationbased edit distances which are based on editing operations tailored specifically to topological/geometrical transformations on merge trees. This stands in contrast to the application of classic tree edit distances not considering that the modified structures adhere to certain properties. We discuss their advantages and disadvantages in terms of complexity and expressiveness as well as possible applications. 
14:20 – 14:40  Tracking extremabased features in climate data Emma Nilsson (Linköping University) Abstract: In visualization applications using scalar field topology, extrema often represent features in the field, where a common task is to track the features over time and analyze their temporal evolution. Moreover, there are many different possibilities for tracking the extremabased features. The talk focuses on gradientbased tracking of extrema, where the gradient is discrete, and the scalar field function is a piecewise linear function defined on the domain. I will cover hierarchies based on extrema, define correspondence probabilities between the extrema and how to compare different tracking approaches with each other within scalar field feature tracking. A realworld climate dataset will be used to illustrate how the presented techniques work in practice and how they compare to previous approaches. 
14:40 – 15:00  Multiscale visual analysis of cycle characteristics in spatiallyembedded graphs Farhan Rasheed (Linköping University) Abstract: We present a visual analysis environment based on a multiscale partitioning of a 2d domain into regions bounded by cycles in weighted planar embedded graphs. The work has been inspired by an application in granular materials research, where the question of scale plays a fundamental role in the analysis of material properties. We propose an efficient algorithm to extract the hierarchical cycle structure using persistent homology. The core of the algorithm is a filtration on a dual graph exploiting Alexander’s duality. The resulting partitioning is the basis for the derivation of statistical properties that can be explored in a visual environment. We demonstrate the proposed pipeline on a few synthetic and one realworld data set. 
Coffee break  
15:30 – 17:00  Topology in InfoVis: Scatterplots, Line Charts, Graphs, and Dimension Reduction Paul Rosen (SCI Institute, University of Utah, USA) Abstract: Insights from data are highly influenced by the shape perceived in the data. However, two problems exist. First, data are often high dimensional, making their shape difficult to visualize. Second, once visualized, they often suffer from scalability and readability issues, even with modest amounts of data. By applying topologybased descriptors to these problems, approaches can utilize the shape for presenting and interacting with data in ways that are mathematically robust and correspond to human perception and cognition. This talk will discuss the applications of topologybased descriptors, namely persistent homology, contour trees, and mapper, in several commonly used visualization types, including scatterplots, line charts, nodelink diagrams, and dimension reduction. Biography: Paul Rosen is an Associate Professor in the Kahlert School of Computing and the Scientific Computing and Imaging Institute at the University of Utah. He received his Ph.D. from Purdue University in 2010. Subsequently, he was a Research Assistant Professor at the University of Utah from 2010 to 2015. He was then an Assistant and Associate Professor at the University of South Florida from 2015 to 2022. He has been the coauthor of over 80 papers, 6 having received best paper awards or honorable mentions in the areas of computer graphics, data visualization, and topological data analysis. His research interests lie at the intersection of scientific and information visualization, where he utilizes a mix of humancentered design and geometry and topologybased approaches to improve the efficacy of visualization tools. His research has been supported by the National Institutes of Health, the National Radio Astronomy Observatory, the Defense Intelligence Agency, and National Science Foundation grants, including an NSF CAREER Award in 2019. He is also one of the General Chairs for IEEE VIS 2024, which will be held in Tampa, Florida. 
17:00 – 17:30  Pitch ideas for project work on Friday. Discussion session involving all teachers and students. 
Friday, Aug 25
9:00 – 12:20  Morning session 
9:00 – 10:30  Comparative Analysis of Topological Structures for Scientific Data Visualization Raghavendra G. Sridharamurthy (SCI Institute, University of Utah, USA) Abstract: Many scientific measurements and/or simulations lead to scalar fields which are real valued functions measured over an area, or a volume. They can be static or dynamic i.e., measured across time, or ensembles i.e., dependent on various parameters and initial conditions. Topological structures provide a combinatorial, abstract, and succinct representation of scalar fields with proven methods to simplify these fields. With availability of such structures which capture varying degrees of information, with or without geometric context, comparison becomes one of the key operations in any data analysis pipeline to gain deeper understanding of the scientific phenomena with appropriate focus and context. Such an understanding might not be possible with just analyzing the individual fields or their corresponding topological structures. In this talk we introduce comparative analysis of topological structures, talk about theoretical and practical considerations involved in designing of such comparative methods, differentiate between global and local methods, and finally show usecases where the comparison forms the foundation of multiple applications like symmetry detection, periodicity detection, temporal summarization, and feature tracking. Biography: Raghavendra Sridharamurthy is a postdoctoral researcher at the Scientific Computing and Imaging Institute (SCI), University of Utah. He completed his PhD and master's in computer science from the Indian Institute of Science (IISc), Bengaluru. His research interests include scientific visualization, computational topology, topological data analysis, and their applications. 
Coffee break  
10:50 – 12:20  Hands on session Work on own projects 
12:30 – 13:30  Lunch at Visualization center restaurant. 
13:40 – 15:45  Afternoon session 
13:40 – 15:15  Hands on session Work on own projects 
15:15 – 15:30  Closing remarks Ingrid Hotz (Linköping University) and Vijay Natarajan (Indian Institute of Science, Bangalore) 
Coffee break 
Photos
Summer school partipants at Söderköping viewpoint Outside Söderköping Brunn after joint dinner
Organizers
 Ingrid Hotz, Linköping University, ingrid.hotz@liu.se
 Talha Bin Masood, Linköping University, talha.bin.masood@liu.se
 Vijay Natarajan, Indian Institute of Science, Bangalore, vijayn@iisc.ac.in
Event Coordinator
 Agne Virsilaite Maras, Linköping University, agne.virsilaite.maras@liu.se