Topological Persistence in Geometry and Analysis

Preface
I A primer of persistence modules
Definition and first examples
Persistence modules
Morphisms
Interleaving distance
First example: interval modules
Morse persistence modules and approximation
Rips modules and the Gromov-Hausdorff distance
Barcodes
The Normal Form Theorem
Bottleneck distance and the Isometry theorem
Corollary: stability theorems
Persistence modules of locally finite type
Proof of the Isometry theorem
An outline
Matchings for surjections and injections
Monotonicity with respect to injections and surjections
Induced matchings construction
Main lemmas and proof of the theorem
Proofs of Lemma 3.3.1 and Lemma 3.3.2
Proof of Lemma 3.3.1.
Proof of Lemma 3.3.2.
What can we read from a barcode?
Infinite bars and characteristic exponents
Characteristic exponents
Boundary depth and approximation
The multiplicity function
Representations on persistence modules
Theoretical development
Applications in geometry
II Applications to metric geometry and function theory
Applications of Rips complexes
delta - hyperbolic spaces
Cech complex, Rips complex and topological data analysis
Manifold Learning
Topological function theory
Prologue
Invariants of upper triangular matrices
Simplex counting method
A combinatorial lemma
Bars and oscillation
The length of the barcode
The fundamental inequality
The Banach indicatrix
Normal lifts of the level lines
Approximation by trigonometric polynomials
III Persistent homology in symplectic geometry
A concise introduction to symplectic geometry
Hamiltonian dynamics
Symplectic structures on manifolds
Group of Hamiltonian diffeomorphisms
Hofer's bi-invariant geometry
A short tour in coarse geometry
Zoo of symplectic embeddings
Hamiltonian persistence modules
Conley-Zehnder index
Filtered Hamiltonian Floer theory
Constraints on full powers
Non-contractible class version
Barcodes for Hamiltonian homeomorphisms
Symplectic persistence modules
Liouville manifolds
Symplectic persistence module
Examples of SH(U)
Symplectic Banach-Mazur distance
Functorial properties
Applications
Computations
Bibliography