Browsing by Author "Patel, Amit, advisor"

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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 member
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 → ▽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.
Open Access
Generalizations of persistent homology
(Colorado State University. Libraries, 2021) McCleary, Alexander J., author; Patel, Amit, advisor; Adams, Henry, committee member; Ben Hur, Asa, committee member; Peterson, Chris, committee member
Persistent homology typically starts with a filtered chain complex and produces an invariant called the persistence diagram. This invariant summarizes where holes are born and die in the filtration. In the traditional setting the filtered chain complex is a chain complex of vector spaces filtered over a totally ordered set. There are two natural directions to generalize the persistence diagram: we can consider filtrations of more general chain complexes and filtrations over more general partially ordered sets. In this dissertation we develop both of these generalizations by defining persistence diagrams for chain complexes in an essentially small abelian category filtered over any finite lattice.
Open Access
Generalized persistence for discrete dynamical systems
(Colorado State University. Libraries, 2025) Cleveland, Jacob, author; Patel, Amit, advisor; King, Emily, committee member; Dineen, Mark, committee member
We introduce a novel method for extracting persistent topological descriptions of discrete dynamical systems from finite samples in the form of generalized persistence diagrams. These persistence diagrams are decorated with eigenvalues of linear maps associated to a certain local system called the persistent local system. We also prove the stability of our method and provide an example of recovering the induced map on homology from a finite sample.