**RESOURCES**

**Informing Communities About Applied Algebraic Topology Research.**

Image: © Connor McCranie and Markus Pflaum: Catastrophe Theory.

**Presented by:** Henry Adams, Johnathan Bush, Mark Heim, and Joshua Mirth (Colorado State University)

**Abstract:** This talk is an introduction to applied and computational topology, in particular as related to the study of energy landscapes arising in chemistry. We give a visual introduction to topology and homology groups. The focus of our talk is on persistent homology, which summarizes the changes in topological features across any increasing sequence of spaces. We emphasize sublevelset persistent homology, which provides a topological summary of real-valued functions, in particular energy functions. We also briefly introduce persistent homology applied to point cloud data, perhaps sampled from the low-energy conformations of a chemical system.

**Presented by:** Bala Krishnamoorthy (Washington State University)

**Abstract:** The Mapper construction has recently become one of the main faces of topological data analysis (TDA), especially from the point of view of diverse applications. It is a data visualization technique that is rather straightforward to describe and implement, and produces insightful visualizations of complex data sets that often reveal "hidden" structure. We start with an overlapping cover of the range of values of a variable (a subset of variables more generally). We then cluster the data points which have that variable value within each cover element. Each cluster is represented by a node. When two cover elements intersect, we draw an edge connecting the corresponding cluster nodes. We draw a triangle when three cover elements intersect, and so on. With correct choices of variables as well as the number of intervals and overlap for each variable, the resulting Mapper simplicial complex provides a highly compact representation of the data that still preserves the underlying structure. Given the ease of interpretability, most applications use just the 1-skeleton (Mapper graph) along with associated visualizations to study the data. Structures such as paths and flares in the Mapper graph have been used to identify subpopulations in the data that show distinct behavior. There is also a growing body of mathematical results specifying robustness of the construction as well as features (e.g., paths) identified by Mapper. Through several examples, we will shed light on how Mapper helps to reveal hidden structure from large data sets.

**Presented by:** Sudhakar Pamidighantam (Indiana University Bloomington)

**Abstract:** This talk describes how state of the art computational and data resources can be accessed using science gateways for molecular sciences to support both research and teaching. Cyberinfrastructure for science and computational science are defined. Science gateway based access to simulation is contrasted with command-line access. Science gateway architectures are described with Apache Airavata based gateways. Recently deployed Django web framework integrated science gateways are described. SEAGrid (Science and Engineering Applications Grid) Science Gateway currently serving the molecular science community is used as example to show how workflows are deployed in Science Gateways. Post processing and visualization integration is described with a docker based automated post processing and indexing model. Support for Science Gateways form XSEDE ECSS program and SGCI EDS program are highlighted. Cyberinfrastructure Integration Research Center at Indiana University’s engagement in science gateway design, deployment, hosting is presented.

**Presented by:** Henry Adams (Colorado State University) and Enrique Alvarado (Washington State University)

**Abstract:** We give an introduction to Morse theory. Given a space equipped with a real-valued function, one can use Morse theory to produce a compact cellular model for that space. Furthermore, the cellular model reflects important properties of the function. We describe CW cell complexes, the Morse lemma, the two main theorems of Morse theory, Morse complexes, Morse-Smale complexes, and simplifications of Morse complexes.

**Presented by:** Conor McCranie and Markus Pflaum (University of Colorado Boulder)

**Abstract:** We given an introduction to catastrophe theory by René Thom. After briefly defining what degenerate and non-degnerate critical points of a smooth function are we introduce the algebra of germs of smooth real-valued functions and describe singular germs in that framework. Using the language of germs we present the classification theorem of critical points up to codimension 4. Next unfoldings are introduced and the universal unfoldings for the seven elementary catastrophes are provided. We also explain in that part the catastrophe set and the bifurcation set. To make the abstract definitions and results more transparent we visualize catastrophes by a few examples. At the end an application of catastrophe theory in chemistry is provided.

**Presented by:** Aurora Clark (Washington State University)

**Abstract:** In recent years the methods associated with topological data analysis have begun to be used to understand the complex inter-relationships and correlations between molecules in the domain of Chemistry. This webinar discusses a few of these applications, with an emphasis upon the motivation of why TDA may be useful and a few of the requirements needed to consider such methods as being able to extract new information from chemical data.

**Presented by:** Julien Tierny (Sorbonne Universite)

**Abstract:** This talk gives a global overview of the Topology ToolKit (TTK), an open-source library for topological data analysis and visualization. After introducing the main concepts behind TTK's usage, I will demo how to use it with simple examples in ParaView and Python. Then, I will present TTK's capabilities, going from simple critical point extraction to more advanced features (Reeb graphs, Morse-Smale complexes, distances and barycenters of persistence diagrams, etc) as well as TTK's current capabilities for processing point cloud data. In the process, I will provide links to numerous online resources containing further tutorials, documentations, examples and exercises. If you have any questions or the slightest issue with TTK, please don't hesitate to send us a message at ttk-users@googlegroups.com. We'll do our best to get back to you as soon as possible.

For more materials and tutorials on the TTK software, please visit:

**Presented by:** Henry Adams (Colorado State University)

**Abstract:** Many of the properties of a chemical system are described by its energy landscape, a real-valued function defined on a high-dimensional domain. I will explain how topology, and in particular persistent homology, can be used in order to describe some of the pertinent features of an energy landscape. Whereas a merge tree encodes how connected components of an energy landscape evolve as the energy level increases, persistent homology can also quantify the shape of these connected components. As a motivating example, we completely describe the sublevelset persistent homology of the n-alkanes. In a recently funded NSF Harnessing the Data Revolution project, the DELTA team is learning how to identify and leverage changing topological features of energy landscapes across a range of chemical conditions in order to predict reactivity.