Grant: Discovery Launch Supplement

Introduction Since its inception, the field of algebraic topology (AT) has grappled with a fundamental trade-off between constructiveness and functoriality. For the last five decades, non-constructive approaches have taken centre stage, producing numerous powerful and elegant results. However, the growing applicability of AT in scientific and computational fields has created the need for more constructive approaches to these developments. Long-term vision I aim to provide effective constructions of sophisticated homotopical invariants and, by doing so, allow for the application of some of the profound ideas in AT to fields beyond pure mathematics, including data science and quantum field theory. Program's objectives (1) Compute primary & secondary cohomology operations of triangulated spaces. (MSc1-2, PhD1-2) (2) Compute Steenrod operations on Khovanov homology of knots and links. (MSc3, PhD3) (3) Compute Steenrod barcodes and persistent cup-lengths of real-world datasets. (MSc4-5, PhD4) (4) Build a bridge between state sum and functorial topological quantum field theories. (MSc6, PostDoc) Key ideas Obj. (1) rests on the constructive description of a complete algebro-homotopical invariant of triangulated spaces, a so-called E-infinity algebra structure on their cochains. The fact that this invariant captures all homotopical information is a jewel of modern homotopy theory due to Mandell. Obj. (2) hinges on a recent result by a couple of teams showing that Khovanov homology, a powerful invariant for knots and links, is derived from a spectrum in the sense of stable homotopy theory. This spectrum can be modelled using an object akin to a triangulated space, allowing for the adaptation of our effective construction from Obj (1). Obj. (3) is based on the adaptation to nested families of the effective constructions of cohomology operations for triangulated spaces of Obj (1). This is because persistent cohomology, a cornerstone of Topological Data Analysis, is based on the study of such nested families. Obj. (4) relies on the free n-category generated by the n-simplex. This object, previously related by the author to cohomology operations, belongs to both; the lattice-theoretic domain of triangulated space-times and the higher categorical realm of the cobordism hypothesis. Significance My program will help bridge the gap between the theory and applications of AT by harmonising this field with the powerful computer tools now ubiquitous in contemporary science. Our novel perspective will illuminate key questions in mathematical physics and produce tangible outcomes. These include high-performance modules for calculating topological invariants of triangulated spaces, knots, and complex datasets. Mentorship With its dual emphasis in pure and applied mathematics, our program will prepare 10 trainees for diverse careers, whether they are exploring the abstract landscapes of academic research or the concrete challenges present in industry and public service.