Title:Defining Persistence Diagrams for Cohomology of a Cofiltration Indexed Over a Finite Lattice

Advisor: Dr. Amit Patel

Committee: Dr. Mark Shoemaker, Dr. Dustin Tucker

Persistent homology and cohomology are important tools in topological data analysis, allowing us to track how homological features change as we move through a filtration of a space. Original work in the area focused on filtrations indexed over a totally ordered set, but more recent work has been done to generalize persistent homology. In one avenue of generalization, McCleary and Patel prove functoriality and stability of persistent homology of a filtration indexed over any finite lattice. In this thesis, we show a similar result for persistent cohomology of a cofiltration. That is, for P a finite lattice and $F: P \to \nabla K$ a cofiltration, the nth persistence diagram is defined as the Möbius inversion of the nth birth-death function. We show that, much like in the setting of persistent homology of a filtration, this composition is functorial and stable with respect to the edit distance. With a general definition of persistent cohomology, we hope to discover whether duality theorems from 1-parameter persistence generalize to more general lattices.

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