Reverse Mathematics of Caristi's Fixed Point Theorem and Ekeland's Variational Principle

Explore the intricacies of Caristi's fixed point theorem and Ekeland's variational principle in this 33-minute lecture from the Hausdorff Trimester Program: Types, Sets and Constructions. Delve into the concept of a 'Caristi system,' defined as a tuple (X,V,f) where X is a complete separable metric space, V is a continuous function from X to non-negative reals, and f is an arbitrary function from X to X satisfying specific conditions. Examine the theorem's application to lower semi-continuous functions and investigate its strengths in relation to other mathematical statements. Analyze the varying complexities of these principles, ranging from WKL0 in special cases to beyond Pi11-CA0, providing a comprehensive understanding of their place within reverse mathematics.