University of Illinois Urbana-Champaign

Topics in applied topology

Assif Poovan Kavil, Mishal

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

Description

Title
Topics in applied topology
Author(s)
Assif Poovan Kavil, Mishal
Issue Date
2025-10-20
Director of Research (if dissertation) or Advisor (if thesis)
Baryshnikov, Yuliy
Doctoral Committee Chair(s)
Baryshnikov, Yuliy
Committee Member(s)
  • Ali Belabbas, Mohamed
  • Raginsky, Maxim
  • Erickson, Jeff
Department of Study
Electrical & Computer Eng
Discipline
Electrical & Computer Engr
Degree Granting Institution
University of Illinois Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • Topological data analysis
  • Multiparameter persistent homology
  • Random topology
  • Topological statistics
Abstract
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.
Graduation Semester
2025-12
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/132631
Copyright and License Information
Copyright 2025 Mishal Assif Poovan Kavil

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Graduate Theses and Dissertations at Illinois