{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120516"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120516","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Applied topology through the lens of Analytic Combinatorics","abstract":"Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-05-01","abstract_html":"Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-05-01","abstract_has_math":false,"creators":["Le, Phuong Ha Hai"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Baryshnikov, Yuliy","DeVille, Lee X","Sowers, Richard B","Quan, Zhiyu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Persistent Homology","Analytic Combinatorics","Probabilistic Automaton"],"languages":["en","eng"],"rights":["Copyright 2023 Phuong Le"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120516","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baryshnikov, Yuliy","DeVille, Lee X","Sowers, Richard B","Quan, Zhiyu"]},{"key":"dc:creator","label":"Author","values":["Le, Phuong Ha Hai"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-04-20"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"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 at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Persistent Homology","Analytic Combinatorics","Probabilistic Automaton"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Phuong Le"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120516"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-05-01","The student, Phuong Le, accepted the attached license on 2023-04-17 at 02:51.","The student, Phuong Le, submitted this Dissertation for approval on 2023-04-17 at 03:09.","This Dissertation was approved for publication on 2023-04-20 at 08:51.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19002 on 2023-09-01 at 17:20:32","Analytic combinatorics provides powerful probabilistic techniques to derive higher-order statistics for complex combinatorial objects with different attribute types. In this thesis work, through the lens of analytic combinatorics, we will explore various problems in applied topology and biology in probabilistic settings. Two central topics studied in our thesis are persistent homology and RNA secondary structure. Persistent homology is a core area in the rapidly emerging field of topological data analysis. Even though persistent homology is often associated with a continuous function, we can use winding numbers and merge trees to translate the study of persistent homology into that of discrete objects. This translation enables us to develop and study stochastic versions of persistent homology through the application of analytic combinatorics. In particular, we consider the random zeroth persistent homology of individual drifted Brownian motions and their stochastic sum. We further leverage tools from analytic combinatorics to understand different loop structures in RNA secondary structure. RNA secondary structure is an important subject not only in biology and mathematics but also in other fields of natural science and drug discovery. By leveraging techniques from analytic combinatorics, we are able to develop new frameworks and concepts that allow us to provide different experimental and theoretical results toward topological data analysis and stochastic and geometric analysis of RNA structures. Revisiting classical concepts in analytics combinatorics, our thesis work contributes several original ideas and setups to stochastic applied topology and biology."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Applied topology through the lens of Analytic Combinatorics"]}]}],"canonical_facts":{"dc:contributor":["Baryshnikov, Yuliy","DeVille, Lee X","Sowers, Richard B","Quan, Zhiyu"],"dc:creator":["Le, Phuong Ha Hai"],"dc:date":["2023-05","2023-04-20"],"dc:description":["Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-05-01","The student, Phuong Le, accepted the attached license on 2023-04-17 at 02:51.","The student, Phuong Le, submitted this Dissertation for approval on 2023-04-17 at 03:09.","This Dissertation was approved for publication on 2023-04-20 at 08:51.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19002 on 2023-09-01 at 17:20:32","Analytic combinatorics provides powerful probabilistic techniques to derive higher-order statistics for complex combinatorial objects with different attribute types. In this thesis work, through the lens of analytic combinatorics, we will explore various problems in applied topology and biology in probabilistic settings. Two central topics studied in our thesis are persistent homology and RNA secondary structure. Persistent homology is a core area in the rapidly emerging field of topological data analysis. Even though persistent homology is often associated with a continuous function, we can use winding numbers and merge trees to translate the study of persistent homology into that of discrete objects. This translation enables us to develop and study stochastic versions of persistent homology through the application of analytic combinatorics. In particular, we consider the random zeroth persistent homology of individual drifted Brownian motions and their stochastic sum. We further leverage tools from analytic combinatorics to understand different loop structures in RNA secondary structure. RNA secondary structure is an important subject not only in biology and mathematics but also in other fields of natural science and drug discovery. By leveraging techniques from analytic combinatorics, we are able to develop new frameworks and concepts that allow us to provide different experimental and theoretical results toward topological data analysis and stochastic and geometric analysis of RNA structures. Revisiting classical concepts in analytics combinatorics, our thesis work contributes several original ideas and setups to stochastic applied topology and biology."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120516"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Phuong Le"],"dc:subject":["Persistent Homology","Analytic Combinatorics","Probabilistic Automaton"],"dc:title":["Applied topology through the lens of Analytic Combinatorics"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}