Topological Data Analytic Applications of Persistent Cohomology (Cancelled) (22w5197)


Anthea Monod (Imperial College London)

Vin da Silva (Pomona College)


The Institute for Advanced Study in Mathematics will host the "Topological Data Analytic Applications of Persistent Cohomology" workshop in Hangzhou, China from November 6 - November 11, 2022.

Topological Data Analysis (TDA) is a 21st-century branch of data science that seeks to leverage theory from pure mathematics to tackle modern data-analytic challenges. Today data is vastly more abundant, thanks to new technologies that facilitate measurement and collection. Moreover, the structural forms that these data can take have become increasingly complex. Much of the early success of TDA can be attributed to the persistent homology framework. This can be thought of as a multi-scale version of the classical homology theories developed in algebraic topology to measure the structural features (handles, tunnels, cavities, etc.) of a geometric object. The multi-scale approach is essential for ensuring statistical robustness, which in turn is what allows us to apply these methods meaningfully to real data sets. Persistent homology is by now very well studied, and has been integrated successfully with statistical methodology and machine learning algorithms. There are numerous applications in various fields, including sensor networks (Ghrist & de Silva, 2006), neuroscience (Chung et al., 2009), cosmology (Adler et al., 2017), and cancer research (Crawford et al., 2019). The general principle is to convert the input data into an output barcode which records all the relevant information.

Inspired by the success of persistent homology, the purpose of the present workshop is to replicate this success by developing robust, statistically integrated versions of other constructions from classical algebraic topology. In particular, we are interested in persistent versions of cohomology theory, that combine the robustness of persistence with the various sophisticated algebraic structures associated with cohomology. The research literature contains hints that this is a fertile avenue for exploration. One striking application is the use of cohomology to construct meaningful angle-valued coordinates on a data set (de Silva et al., 2011), in analogy with the construction of meaningful real-valued coordinates produced by dimensionality reduction algorithms. A more sophisticated later development (Perea, 2017) produces projective coordinates that can potentially be exploited in data analysis when considering moduli spaces of complex data objects, for example, in the spirit of (Monod et al., 2018).

We intend that this workshop will kick-start several new directions for theoretical development and will uncover potential new applications. Our list of potential participants includes TDA researchers from various backgrounds including: statistics, homotopy theory, sheaf theory, machine learning, mathematical biology.

The Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).