The Reverse Mathematics of the Mountain Pass Theorem

Explore the reverse mathematics analysis of the Mountain Pass Theorem in this 57-minute conference lecture that demonstrates the equivalence between this fundamental result in variational analysis and the weak König's lemma (WKL) over RCA₀. Learn how the Mountain Pass Theorem of Ambrosetti and Rabinowitz, which provides necessary conditions for the existence of critical points of differentiable functionals on Hilbert spaces, can be formalized within the framework of reverse mathematics. Discover the mathematical development required to prove that WKL implies the Mountain Pass Theorem over RCA₀, including the construction of continuous function spaces from [0,1] into separable Banach spaces, formalized proofs of the deformation lemma, and the minimax principle. Examine the theory of ordinary differential equations as it relates to this analysis and understand why this theorem about the existence of an infimum requires only WKL rather than ACA₀. Follow the reversal proof that uses contrapositive reasoning, constructing a smooth function that satisfies all hypotheses of the Mountain Pass Theorem but not its conclusion by assuming the existence of an infinite binary tree with no path.