Abstract:

During the last two decades a systematic program of `proof mining' emerged as a new applied formof proof theory and has successfully been applied to a number of areas of core mathematics. Thisprogram has its roots in Georg Kreisel's pioneering ideas of `unwinding of proofs' going back tothe 1950's who asked for a `shift of emphasis' in proof theory away from issues of mere consistencyof mathematical theories(`Hilbert's program') to the question `What more do we know if we haveproved atheorem by restricted means than if we merely know that it is true?'We are primarilyconcerned with the extraction of hidden finitary and combinatorial content fromproofsthat make use of infinitary noneffective principles. The main logical tools for this areso-called interpretation. Logical meta-theorems based on such interpretations havebeen applied withparticular success in the context of nonlinear analysis includingfixed point theory, ergodic theory,continuous optimization, game theory and abstractCauchy problems. The combinatorial contentcan manifest itself both in expliciteffective bounds as well as uniformity results.

In this talk we will outline the general background of this logic-based approach and its origin inresearch on the foundations of mathematics and indicate some recent applications in the context ofnonlinear analysis.

About the speaker:

Ulrich Kohlenbach is a German mathematician andprofessor of mathematical logicat the Technische Universität Darmstadt. His research interests lie inproof theory. He is a leading expert inthe field of proof mining.Kohlenbach was President of theGerman Association for Mathematical Logic and theFoundations of Exact Sciences (DVMLG)from 2008 to 2012,and President of the Association for Symbolic Logic from 2016 to 2018.He was the Dean of Department of Mathematics atthe Technische Universität Darmstadt.

In 2011 he received the prestigiousKurt Gödel Research Prize Fellowshipof the Kurt Gödel Society. He was an invited speaker at the 2018 International Congress of Mathematicians(Section: Logic and Foundations).He has published more than 90 research papers in mathematics and logic, and a book titled "Applied Proof Theory: Proof Interpretations and their Use in Mathematics" in the series of "Springer Monographs in Mathematics" in 2008. He is anadvisory editorof Journal of Mathematical Logic, and a coordinating editor of Annals of Pure and Applied Logic.