On March 6, 2024 at 16:00 EET, Pedro Pinto (TU Darmstadt) will give a talk in the Proof Mining Seminar.
Title: On generalizations to Nonlinear Smooth Spaces
Abstract:
We shall discuss the notion of a nonlinear smooth space recently introduced in [3], which generalizes both CAT(0) spaces as well as smooth Banach spaces. Leaning on the proof-theoretical study carried out in [2], we established in [3] a nonlinear generalization of Reich’s theorem. This in turn allowed us to prove the convergence of several other methods. In particular, we obtained a unifying treatment of the previous proof mining studies [1, 4]. Moreover it also allowed for a nonlinear discussion of Chang’s reduction argument.
This talk will be a specialized version of the one delivered at the Oberwolfach Workshop last November. With the aim of identifying further application cases, we discuss the types of proofs that could potentially be generalized within this nonlinear framework. Furthermore, we present open questions associated with emerging concepts.
References:
[1] U. Kohlenbach, Quantitative analysis of a Halpern-type proximal point algorithm for accretive operators in Banach spaces. Journal of Nonlinear and Convex Analysis, 21(9):2125–2138, 2020.
[2] U. Kohlenbach and A. Sipoș, The finitary content of sunny nonexpansive retractions, Communications in Contemporary Mathematics 23.1: 1950093, 63pp, 2021.
[3] P. Pinto, Nonexpansive maps in nonlinear smooth spaces, submitted, 47pp, 2023.
[4] A. Sipoș, Abstract strongly convergent variants of the proximal point algorithm. Computational Optimization and Applications, 83(1):349–380, 2022.
Google Meet link: https://meet.google.com/jpw-hrwe-evu