Applications of Homotopy Theory

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Local to global methods have been present in Geometry and Topology for over a hundred years. Grothendieck and his followers introduced the modern form of the term "local" in the 1960s, on their way to proving the Weil conjectures. This led to an explosion of calculational technique in Algebraic Geometry, the introduction of Topos Theory and its applications in Logic, and to the development of methods to compute invariants of Algebraic K-theory. The K-theory calculations introduced a new level of subtlety, as they involve non-abelian phenomena which are encoded in large structures of objects which originate in Algebraic Topology. The modern form of local homotopy theory was initiated by the Canadian mathematicians Jardine and Joyal in the mid 1980s in separate but complimentary work, as a response to extant problems in K-theory and Topos Theory. The theory has applications in multiple research areas of Mathematics, and in the Mathematical Sciences more generally. It is the basis for the study of motives and motivic homotopy theory in Algebraic Geometry and Number Theory, and is the context for the modern homotopical theory of symmetries which is encoded in stacks and higher stacks. The theory explicitly appears in classical stable homotopy theory, and in its applications in Mathematical Physics.  The main line of research of this proposal investigates local to global calculational methods in the study of large data structures. This work incorporates existing topological models for both parallel processing systems and topological data analysis, with a particular emphasis on developing parallelization techniques for the study of these structures. Existing algorithms calculate invariants of interest (execution paths in concurrency, clusters and persistent homology in topological data analysis) from topological objects associated to data structures. These data structures must be relatively small for these algorithms to be practical, and the goal of this research program is to find techniques that would extend existing algorithms to effective calculational devices for larger objects.  There is a particular focus on stability properties in Topological Data Analysis, in the context of the emerging research area of persistent homotopy theory. Stability results say that if two data sets are close, then the associated topological data analysis invariants are close in a measurable way, using an "interleaving distance". This is a local principle, and the present aim is to globalize it.This work is in Homotopy Theory with applications to Data Analysis, and is of particular interest to mathematicians and to data scientists. The results of this research will be used in academia, industry, and government, and HQP trained under this proposal will be highly suited for work in all of these sectors. The work will solidify Canada's leadership in a vibrant new research area at the boundary between Mathematics and Data Analytics.