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This monograph introduces the reader to the increasingly popular topic of computable metric structure theory, a subject which unifies methods from effective analysis and computable algebra into one coherent framework. Computable structure theory had been constrained to the algebraic and discrete realms, but in the past 10 years or so, much work has been done in extending this topic to structures from analysis such as metric spaces, Banach spaces, and operator algebras. This book is the first comprehensive treatment of these basic results and discusses several challenging open problems that have arisen. The book provides a foundation for the study of classic objects from functional analysis using tools from continuous model theory and computable structure theory. It is largely self contained, though it is assumed that the reader is familiar with first-order logic. It should prove useful to students and researchers in continuous logic looking to familiarize themselves with computable structure theory as well as to researchers in the latter field looking to extend their work to structures from analysis. The discussion contains a large number of well-crafted examples and could provide the basis for a course or seminar in this area.
List of contents
Introduction.- An introduction to metric structures and their model theory.- Framework for computability of metric structures.- Metric spaces.- Banach spaces: C(X) spaces.- Banach spaces: L^p spaces.- Operator algebras.- Degrees of theories.
About the author
Johanna Franklin is a Professor of Mathematics at Hofstra University. Her research interests lie in computable structure theory, computable analysis, and algorithmic randomness.
Isaac Goldbring is a Professor of Mathematics at the University of California, Irvine. His research interests are in model theory and nonstandard analysis, with applications to a wide variety of areas of mathematics, including operator algebras, combinatorics, Lie theory, and quantum theory.
Timothy McNicholl is a Professor of Mathematics at Iowa State University. He is an internationally recognized scholar in computable analysis. He has worked extensively on the computable structure theory of Banach spaces. He has also published papers on computable complex analysis, computable structure theory, and computable topology.
Summary
This monograph introduces the reader to the increasingly popular topic of computable metric structure theory, a subject which unifies methods from effective analysis and computable algebra into one coherent framework. Computable structure theory had been constrained to the algebraic and discrete realms, but in the past 10 years or so, much work has been done in extending this topic to structures from analysis such as metric spaces, Banach spaces, and operator algebras. This book is the first comprehensive treatment of these basic results and discusses several challenging open problems that have arisen. The book provides a foundation for the study of classic objects from functional analysis using tools from continuous model theory and computable structure theory. It is largely self contained, though it is assumed that the reader is familiar with first-order logic. It should prove useful to students and researchers in continuous logic looking to familiarize themselves with computable structure theory as well as to researchers in the latter field looking to extend their work to structures from analysis. The discussion contains a large number of well-crafted examples and could provide the basis for a course or seminar in this area.
