An Approximate Nerve Theorem

Explore an in-depth lecture on the Approximate Nerve Theorem and its applications in topological data analysis. Delve into the concept of epsilon-acyclic covers, which encode the idea of almost-good covers, and learn how the persistent homology of a filtration computed on the nerve approximates the persistent homology of a filtration on the underlying space. Discover a refined notion of interleaving and methods for estimating epsilon in certain cases. Follow the lecture's progression through key topics such as the TDA pipeline, persistent good covers, trivial modules, spectral sequences, and the Homological Nerve Theorem. Gain insights into the proof outline and practical examples, concluding with the theorem's implications and potential applications in the field of applied algebraic topology.

Intro
TDA Pipeline
Definitions
Persistent Good Covers
Goal
Trivial Modules
Refinement
Right & Left Interleavings
Possibilities
Decomposition
Back to the Nerve Theorem
Double Complex
Spectral Sequence
First Page
Second Page
Higher Differentials
Collapse
Homological Nerve Theorem
Outline of Proof
Putting It Together
One More Step
Example
Linking
Conclusion
Result