{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/132631"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/132631","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Topics in applied topology","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Assif Poovan Kavil, Mishal"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Electrical & Computer Engr","degree_department":null,"school":null,"contributors":["Baryshnikov, Yuliy","Ali Belabbas, Mohamed","Raginsky, Maxim","Erickson, Jeff"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-22T22:25:07Z","subjects":["Topological data analysis","Multiparameter persistent homology","Random topology","Topological statistics"],"languages":["en"],"rights":["Copyright 2025 Mishal Assif Poovan Kavil"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/132631","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baryshnikov, Yuliy","Ali Belabbas, Mohamed","Raginsky, Maxim","Erickson, Jeff"]},{"key":"dc:creator","label":"Author","values":["Assif Poovan Kavil, Mishal"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12","2025-10-20"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Electrical & Computer Engr"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Topological data analysis","Multiparameter persistent homology","Random topology","Topological statistics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Mishal Assif Poovan Kavil"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/132631"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2027-12-01","The student, Mishal Assif Poovan Kavil, accepted the attached license on 2025-10-20 at 01:52.","The student, Mishal Assif Poovan Kavil, submitted this Dissertation for approval on 2025-10-20 at 02:05.","This Dissertation was approved for publication on 2025-10-20 at 14:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #22827 on 2026-02-19 at 18:45:37"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Topics in applied topology"]}]}],"canonical_facts":{"dc:contributor":["Baryshnikov, Yuliy","Ali Belabbas, Mohamed","Raginsky, Maxim","Erickson, Jeff"],"dc:creator":["Assif Poovan Kavil, Mishal"],"dc:date":["2025-12","2025-10-20"],"dc:description":["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.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2027-12-01","The student, Mishal Assif Poovan Kavil, accepted the attached license on 2025-10-20 at 01:52.","The student, Mishal Assif Poovan Kavil, submitted this Dissertation for approval on 2025-10-20 at 02:05.","This Dissertation was approved for publication on 2025-10-20 at 14:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #22827 on 2026-02-19 at 18:45:37"],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/132631"],"dc:language":["en"],"dc:rights":["Copyright 2025 Mishal Assif Poovan Kavil"],"dc:subject":["Topological data analysis","Multiparameter persistent homology","Random topology","Topological statistics"],"dc:title":["Topics in applied topology"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Electrical & Computer Engr"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:07Z"}