Barcodes and bubbles: The role of asphericity in Hamiltonian persistence modules
Barcodes and bubbles: The role of asphericity in Hamiltonian persistence modules. This TFM concerns itself with a presentation of the theory of persistence modules associated to Hamiltonian Floer theory. We concentrate on the case of symplectically aspherical manifolds and present a proof of the non...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/456079 |
| Acceso en línea: | https://hdl.handle.net/2117/456079 |
| Access Level: | acceso abierto |
| Palabra clave: | Symplectic geometry Hamiltonian systems Differential topology Filtered Floer homology symplectic asphericity symplectic monotonicity Hofer metric bubbling phenomenon Hamiltonian persistence module symplectic persistence module Geometria simplèctica Sistemes hamiltonians Topologia diferencial Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::57 Manifolds and cell complexes::57R Differential topology Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | Barcodes and bubbles: The role of asphericity in Hamiltonian persistence modules. This TFM concerns itself with a presentation of the theory of persistence modules associated to Hamiltonian Floer theory. We concentrate on the case of symplectically aspherical manifolds and present a proof of the nondegeneracy of the Hofer metric and its connection to a stability theorem for Floer persistent homology, as well as describe the invariant known as boundary depth. We also compare this aspherical theory to the symplectically monotone case and discuss the relevance of the asphericity assumption. In order to do this, we first describe the mathematical theory of Hamiltonian dynamics, with special attention paid to the development of Floer homology from Morse homology, and we briefly introduce the ideas behind persistent homology and its usefulness. |
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