Applications of Homotopy Theory

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Local homotopy theory is a study of large structures by examining behaviour in small neighbourhoods on these objects.***Local to global methods have been present in Geometry and Topology for over a hundred years. Grothendieck and his followers introduced new meanings for the term "local" in the 1960s, on their way to proving the Weil conjectures. This variation of a term led to an explosion of calculational technique in Geometry, the introduction of topos theory and its applications in Logic, and the development of an array of methods to compute invariants of algebraic K-theory. The K-theory calculations involved a new level of subtlety, in that they relied on non-abelian phenomena which are encoded in large structures of objects which originate in Algebraic Topology. ******The modern form of local homotopy theory was formulated by Jardine and Joyal in the mid 1980s, as a response to extant problems in K-theory and topos theory. The theory now has applications in multiple parts 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 appears in classical stable homotopy theory in elliptic cohomology theories and topological modular forms, and also in equivariant homotopy theories.******The research of this proposal would build on these successes by acquiring a deeper understanding of calculational methods made possible by local homotopy theory, both in traditional Mathematics and in its applications. The program presented here includes the calculation of cohomological invariants which are associated to classical algebraic groups over number fields and the relation of these invariants with Arithmetic, and the calculation of K-theoretic invariants of algebraic curves and their associated moduli. More abstractly, there is a plan to define and study generalizations of etale homotopy theory which are suggested by Jardine's theory of cocycle categories and by his work on homotopy theories of dynamical systems.******On the applications side, the theoretical behaviour of parallel processing systems is strongly related to higher symmetries. Jardine used coherence theory to find an algorithm that classifies execution paths in concurrency models. This algorithm can only work locally in larger structures: Jardine plans to use the methods of local homotopy theory in the study of large parallel processing models, with a particular emphasis on finding parallelization techniques. This is part of a general attack on the problem of finding local to global methods of analyzing data structures which are too large to study with existing computational techniques. This local to global problem is one of the largest issues in the practical analysis of big data.**