Topics in applied topology

Persistent homology has emerged as a powerful tool in Topological Data Analysis (TDA) for extracting structural information from complex datasets by tracking topological features across varying scales. While 1-parameter persistent homology offers a mature theory with well-defined invariants like persistence diagrams, many real-world phenomena are inherently characterized by multiple interacting parameters. On the theoretical front, this thesis investigates biparameter persistent homology, a more expressive but significantly more challenging framework due to the absence of a direct analogue to the 1-parameter decomposition theorem and its associated compact representations. We study biparametric persistent homology from a differential topological perspective. We propose a method to define and compute persistence diagrams for generic smooth functions from a manifold to the plane by leveraging Whitney singularity theory, offering an alternative to purely algebraic approaches. We then study the statistical properties of these biparametric persistence structures when applied to smooth Gaussian random fields, deriving expected values for quantities related to Whitney singularities, which provides a foundation for understanding typical topological behavior in random biparameteric data. On the algorithmic front, this thesis explores the application of topological methods to two different problems in data analysis. We develop a computational pipeline utilizing persistent homology for the state space realization of nonlinear dynamical systems, and demonstrate its efficacy in recovering the underlying topology of phase spaces from trajectory data of low dimensional observations. We then establish the NP-hardness of decomposing a density function into a minimal number of unimodal components, an important problem in topological statistics, and extend these results to higher-dimensional simplicial complexes.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Assif Poovan Kavil, Mishal

Contributors dc:contributor

• Baryshnikov, Yuliy
• Ali Belabbas, Mohamed
• Raginsky, Maxim
• Erickson, Jeff

• Topological data analysis
• Multiparameter persistent homology
• Random topology
• Topological statistics

Statement dc:rights

• Copyright 2025 Mishal Assif Poovan Kavil

Language dc:language

en

Handle dc:identifier

https://hdl.handle.net/2142/132631

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/132631