Topological Persistence in Geometry and Analysis

ISBN-13: 9781470454951

Leonid Polterovich;Daniel Rosen;Karina Samvelyan

Opis

The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology.

Spis treści

• A primer of persistence modules: Definition and first examples
• Barcodes
• Proof of the isometry theorem
• What can we read from a barcode?
• Applications to metric geometry and function theory: Applications of Rips complexes
• Topological function theory
• Persistent homology in symplectic geometry: A concise introduction to symplectic geometry
• Hamiltonian persistence modules
• Symplectic persistence modules
• Bibliography
• Notation index
• Subject index
• Name index.