Ellen Gasparovic - Research in Computational Topology 2

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The second Women in Computational Topology (WinCompTop) Research Collaboration Workshop was held from July 1 to 5, 2019, at the Mathematical Sciences Institute (MSI) at the Australian National University (ANU). In addition to generous support from MSI as part of their special year on computational mathematics, the workshop was partially supported by the National Science Foundation and the Association for Women in Mathematics (AWM). Additional support for participant travel and accommodation was provided by the Australian Mathematical Sciences Institute.

The 30 participants of the workshop included women from pure and applied mathematics, statistics, computer science, and physics. They represented women at all career stages from graduate student to professor, and arrived at the workshop in Canberra from Europe, the United States, and across Australia. The main goal of the workshop was to facilitate the formation of new and lasting research collaborations between junior and senior women working in the field of computational topology, growing and strengthening this cohort of women in the area. To that end, based on their research interests and backgrounds, each participant was assigned to one of four working groups headed by leading researchers in the field of computational and applied topology. The week commenced with a day of events that were open to anyone interested in learning more about computational topology and featured introductory talks from the group leaders and a poster session showcasing work from early-career participants. The remaining days included plenty of intense research time, with regular updates on progress from each group, and an ever-popular excursion to look for kangaroos on the nearby Mt. Ainslie nature reserve.

This single-blind peer-reviewed volume is the proceedings of the second WinCompTop workshop and features accepted submissions from each of the working groups as well as additional papers solicited from the wider community. The topics covered are broad in scope including theoretical developments in persistent homology, topological aspects of graphs and networks, and case studies in topological data analysis.

Computational topology is a modern, vibrant, diverse, and ever-evolving field of research that integrates computer science and algorithms with many areas of mathematics, such as algebraic and differential topology, algebra, geometry, category theory, statistics, probability, and functional analysis. Research directions range from the traditionally pure, such as questions pertaining to the decomposition, stability, and structure of homology groups, to the more applied, such as computational efficiency, parallelization questions, and intersections with statistics and machine learning methods.

Though algorithms in topology are not new, the rise of computational power and an influx of data have led to increased demand for rigorous, provable, and fast algorithms in the area. In recent years, practitioners in topological data analysis have created tools to explore the topology of data. One such tool is persistent homology or persistence, a technique that may be described via a pipeline model from geometry through topology and algebra to discrete mathematics. More precisely, data are first modeled as objects in a metric space. The next step is to filter the data to obtain a family of nested topological spaces that capture the topological information at multiple scales. Homology with field coefficients is then applied to the filtration to obtain a graded module, called a persistence module, consisting of a family of vector spaces and linear maps between them. The final step is to encode the persistence module into a discrete object such as the persistence diagram or barcode. Persistent homology provides a multiscale stable summary of the topological features of the data. Another data analysis tool from computational topology is the mapper algorithm, which is an adaptation of the Reeb graph applicable to point cloud data.