Applied topology through the lens of Analytic Combinatorics

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Le, Phuong Ha Hai

Contributors dc:contributor

• Baryshnikov, Yuliy
• DeVille, Lee X
• Sowers, Richard B
• Quan, Zhiyu

• Persistent Homology
• Analytic Combinatorics
• Probabilistic Automaton

Statement dc:rights

• Copyright 2023 Phuong Le

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120516

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120516