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Browsing Department of Mathematics by Subject "algebraic topology"
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Item Open Access Defining persistence diagrams for cohomology of a cofiltration indexed over a finite lattice(Colorado State University. Libraries, 2022) Rask, Tatum D., author; Patel, Amit, advisor; Shoemaker, Mark, committee member; Tucker, Dustin, committee memberPersistent 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 → ▽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.