Cycle Representatives in Applied Homological Algebra (25w5481)

The Banff International Research Station will host the “Cycle Representatives in Applied Homological Algebra” workshop in Banff from August 10 - 15, 2025.

A central problem in data-driven scientific inquiry is how to quantitatively describe the organizational structures intrinsic in large data sets. The field of topological data analysis provides a potential solution via the language of homology, which measures features which are, loosely speaking, “holes” of different dimensions (e.g. connected components, loops, trapped volumes, etc.). In principle, these features can be located and studied explicitly. In practice, fundamental mathematical and computational challenges have restricted most topological analyses to the use of numerical summaries that count the number of features but do not allow them to be explicitly examined or manipulated in the context of the data. This dramatically limits the power of algebraic topology in the testing, modeling, and explanation of scientific data.

This workshop focuses on homological cycle representatives, which provides tools beyond these numerical summaries. Cycle representatives explicitly encode holes in the context of the input data. Recent applications have shown that these cycle representatives reveal scientific knowledge in materials science, neuroscience, and biochemistry, among others. At the same time, computational tools for persistent homological algebra have emerged to make new research directions feasible. This workshop will highlight recent advances in homological analysis of cycle representatives in scientific applications, theoretical explorations, and software for computation. This 5-Day workshop is intended to further applications of homological algebra by building connections and new lines of communication between those who develop the computational tools needed to study cycle representatives in TDA and those who apply them.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides 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 is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.