Chapter 1: Basic Topology

a. Topological spaces, metric space topology

b. Maps: homeomorphisms, homotopy equivalence, isotopy

c. Manifolds
d. Functions on smooth manifolds
e. Notes and Exercises

Chapter 2 (i) . Complexes

a. Simplicial complexes
b. Nerves, Cech and VietorisRips complexes
c. Sparse complexes (Delaunay, Alpha, Witness)
d. Graph induced complexes
e. Notes and Exercises

Chapter 2 (ii). Homology

a. Chains, cycles, boundaries, homology groups, Betti numbers
b. Induced maps among homology groups
c. Relative homology groups
d. Singular homology groups
e. Cohomology groups
f. Notes and Exercises

Chapter 3. Topological persistence

a. Filtrations, Persistent homology
b. Persistence diagram
c. Bottleneck distance and its computation
d. Persistence algorithm, matrix reduction, clearing, cohomology algorithm
e. Persistence modules
f. Persistence for PLfunctions
g. Notes and Exercises

Chapter 4. General Persistence (Zigzag)

a. Towers, Persistence modules from simplicial maps
b. Algorithm for towers with annotations
c. Zigzag persistence and algorithms
d. Level Set persistence
e. Extended persistence
f. Notes and Exercises

Chapter 5. Generators and Optimality

a. Greedy algorithm for optimal (H_1)basis
b. Computing optimal cycle in a given class
d. Computing optimal persistent cycles
e. Notes and Exercises

Chapter 6. Topological analysis of point clouds

a. Rips and Cech filtrations on PCD
b. Sparsified Rips filtrations
c. Homology inference from PCD
d. Homology inference for scalar fields
e. Notes and Exercises

Chapter 7. Reeb graphs

a. Reeb graphs: Definitions and properties
b. Algorithms in the PLsetting
c. Homology groups for Reeb graphs
d. Distances for Reeb graphs
e. Notes and Exercises

Chapter 8. Topological analysis of graphs

a. Topological summaries for graphs
b. Graph comparisons
c. Topological invariants for directed graphs
d. Computing persistent path homology
e. Notes and Exercises

Chapter 9. Nerve, Cover, Mapper

a. Cover and Nerve
b. Persistent H_1classes
c. Mapper and multiscale mapper
d. Stability of (multiscale) mapper
e. Approximating multiscale mapper
f. Notes and Exercises

Chapter 10. Discrete Morse Theory and Applications

a. Discrete Morse function
b. Discrete Morse Vector Field (DMVF)
c. Persistence Based DMVF
d. Stable and unstable manifolds
e. Graph reconstruction using DMVF
f. Application to road network and neuronal reconstrcution
g. Notes and Exercises

Chapter 11. Multiparameter persistence and decomposition

a. Multiparameter persistence modules
b. Presentation
c. Presentation matrix: simplification and diagonalization
d. Total diagonalization algorithm
e. Computing presentations
f. Invariants
g. Notes and Exercises

Chapter 12. Multiparameter persistence and distances

a. Persistence modules from categorial viewpoint
b. Interleaving distance
c. Matching distance and its computation
d. Bottleneck distance and its computation
e. Notes and Exercises

Chapter 13. Topological persistence and Machine learning

a. Feature vectorization of persistence diagrams
b. Optimizing topological loss function
c. Statistical treatment of topological summaries
d. Notes and Exercises
