Uniform bounds in D-minimal structures

O-minimality as a classification of the topological tameness of a structure has been extensively studied since its introduction in 1982. Many generalizations and variations of o-minimality have since been introduced and studied as well. In this thesis we focus on one particular generalization: d-minimality. We establish some of the first results on d-minimality that truly mirror those of o-minimality. The most central of which is the equivalence of d-minimality and strong d-minimality. In the process of proving this equivalence we develop a d-minimal version of cell decomposition, one of the most important results in o-minimality. In Chapter 1 we discuss the overarching project of tame topology, and situate d-minimality within this context. We define a grid, the object that we will use to decompose sets in d-minimal structures, and our main results, including the aforementioned decomposition theorem. Chapter 2 is spent establishing some preliminary facts that are used throughout this thesis. In Chapter 3 we motivate our choice of definition for grids by working through an example of a grid decomposition, and compare this decomposition to a previously known result for strongly d-minimal structures. We develop the theory of grids (and cells and stacks which are used to build grids) in Chapter 4. We put all of these pieces together in Chapter 5 where we prove our main results. In Chapter 6 we extend these results to a more general setting. Lastly, we discuss future applications of our main results in Chapter 7.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Farris, Madie

Contributors dc:contributor

• Hieronymi, Philipp
• van den Dries, Lou
• Castle, Ben
• Miller, Chris

• D-minimality
• Model Theory
• O-minimality

Statement dc:rights

• Copyright 2025 Madie Farris

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129922

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129922