2026-09-24T00:00:59Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129922 2025-10-25 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-20 without embargo terms The student, Madie Farris, accepted the attached license on 2025-07-07 at 09:52. The student, Madie Farris, submitted this Dissertation for approval on 2025-07-07 at 09:58. This Dissertation was approved for publication on 2025-07-08 at 13:32. DSpace SAF Submission Ingestion Package generated from Vireo submission #22423 on 2025-10-20 at 20:15:04 Uniform bounds in D-minimal structures Farris, Madie 2025-07-08 Hieronymi, Philipp van den Dries, Lou Castle, Ben Miller, Chris D-minimality Model Theory O-minimality eng 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. 2025-08 Text https://hdl.handle.net/2142/129922 Copyright 2025 Madie Farris Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation