Old and new challenges in Hadamard spaces

TL;DR

This paper surveys recent advances in Hadamard spaces, highlighting their applications in geometry, optimization, and various scientific fields, while also discussing open problems and correcting misconceptions in existing proofs.

Contribution

It provides a comprehensive overview of recent developments in Hadamard space analysis and optimization, emphasizing new applications and identifying fallacies in prior proofs.

Findings

Gradient flow theorem used to solve a conjecture in Kahler geometry

Applications in metric geometry, submodular function minimization, phylogenetics, and imaging

Identification of fallacies in existing proofs

Abstract

Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and nonlinear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kahler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and imaging. We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs.