Old and new challenges in Hadamard spaces

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 non-linear probability theory. The attractiveness of these emerging subject fields stems...

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Bibliographic Details
Published in:Japanese journal of mathematics Vol. 18; no. 2; pp. 115 - 168
Main Author: Bačák, Miroslav
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.09.2023
Springer Nature B.V
Subjects:
Convex analysis
Geometry
Gradient flow
Group theory
History of Mathematical Sciences
Image processing
Mathematics
Mathematics and Statistics
Metric space
Optimization
Phylogenetics
Probability theory
strongly continuous semigroup
non-positive curvature
submodular function
convex function
Mosco convergence
Bi-Lipschitz embedding
proximal mapping
Primary 46N10, 49J53
gradient flow
Hadamard space
weak convergence
harmonic mapping
Secondary 47H20, 49M27
proximal point algorithm
ISSN:0289-2316, 1861-3624
Online Access:Get full text
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Summary: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 non-linear 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 Kähler 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 image processing. 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.
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ISSN:0289-2316
1861-3624
DOI:10.1007/s11537-023-1826-0