Abstract: Computability theory is the mathematical study of the limits and potentialities of discrete computing devices. Computable analysis is the theory of computing with continuous data such as real numbers. Computable structure theory examines which computability-theoretic properties are possessed by the structures in various classes such as partial orders, Abelian groups, and Boolean algebras. Until recently computable structure theory has focused on classes of countable algebraic structures and has neglected the uncountable structures that occur in analysis such as metric spaces and Banach spaces. However, a program has now emerged to use computable analysis to broaden the purview of computable structure theory so as to include analytic structures. The solutions of some of the resulting problems have involved a blend of methods from functional analysis and classical computability theory. We will discuss progress so far on metric spaces and Banach spaces, in particular $\ell^p$ spaces, as well as open problems and new areas for investigation.
The ISU Mathematics Department Colloquium is co-organized by David P. Herzog (firstname.lastname@example.org) and Pablo Stinga (email@example.com).
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