Topological Persistence in Geometry and Analysis

Abstract

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry and analysis. In particular, we present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, we discuss topological function theory which provides a new insight on oscillation of functions. The material should be accessible to readers with a basic background in algebraic and differential topology.

arXiv commentAn Erratum added

References in corpus (8)

• The Theory of the Interleaving Distance on Multidimensional Persistence Modules
• Geometry of contact transformations and domains: orderability versus squeezing
• Persistent homology and Floer-Novikov theory
• Bounds on spectral norms and barcodes
• The action spectrum and C^0 symplectic topology
• Noncontractible periodic orbits in cotangent bundles and Floer homology
• Barcodes and area-preserving homeomorphisms
• p-cyclic persistent homology and Hofer distance

Cited by in corpus (2)

• The moduli space of gradient flow lines and Morse homology
• On $C^0$-persistent homology and trees