Morse-Based Fibering of the Rank Invariant

Explore the application of discrete Morse theory in computing the rank invariant for multi-parameter persistence modules in this 54-minute lecture. Delve into how critical points, determined by a discrete Morse function, partition the parameter space into equivalence classes and dictate the behavior of the rank invariant. Learn to deduce persistence diagrams for entire classes of rank invariants from a single representative, and understand the importance of critical values in multi-parameter filtrations. Gain insights into the computation of rank invariants through fibration and the creation of equivalence classes of lines, ultimately establishing bijections between diagrams along equivalent lines.

Intro
Outline
Multi-Parameter Persistence
Multi-Parameter Filtration
Discrete Morse Theory: Compatibility with Filtration
Importance of Critical Values of K
The Set of Critical Values
Partitioning R by C
Computing the Rank Invariant from Critical Values
Computing the Rank Invariant by Fibration
Creating Equivalence Classes of Lines
Using Push to Calculate the Persistence Diagram of a Lil
Equivalences Classes of Lines
Diagrams Along Equivalent Lines are in Bijection