Ulrich Kohlenbach (Darmstadt)

Fluctuations, Effective Learnability and Metastability in Analysis

Abstract: We discuss what kind of quantitative information one can extract under which circumstances from proofs of convergence statements in analysis but also of general (forall)(exists)(forall)-theorems. We show that from proofs using only a limited amount of the law-of-excluded-middle, one can extract functionals B,L, where L is a learning procedure for a rate of convergence which succeeds after at most B(a)-many mind changes. This (B,L)-learnability provides quantitative information strictly in between a full rate of convergence (obtainable in general only from semi-constructive proofs) and a rate of metastability in the sense of Tao (extractable also from classical proofs). In fact, it corresponds to rates of metastability of a particular simple form. Moreover, if a certain gap condition is satisfied, then B and L yield a bound on the number of possible fluctuations. We explain recent applications of proof mining to ergodic theory in terms of these results (joint work with Pavol Safarik).