Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item Faithful sets of topological descriptors and the algebraic K-theory of multi-parameter zig-zag grid persistence modules(Montana State University - Bozeman, College of Letters & Science, 2023) Schenfisch, Anna Katherine; Chairperson, Graduate Committee: Tomas Gedeon and Brittany Fasy (co-chair)Given a geometric simplicial complex, the uncountable set of (augmented) persistence diagrams corresponding to lower-star filtrations taken with respect to all possible directions uniquely correspond to the simplicial complex, i.e., the set is faithful. While this hints towards interesting applications in shape comparison, the set of all possible directions is uncountably infinite, and so has no hope of computability. In practice, one might use a finite approximation, but faithfulness of this approximation is not guaranteed. Motivated by the need for both computability and provable faithfulness, we provide an explicit description of a finite faithful set of augmented persistence diagrams. We then show this construction applies to augmented Euler characteristic curves and augmented Betti curves, and is stable under particular perturbations. In the specific case where the underlying complex is a graph, we provide an improved construction that utilizes a radial binary search. We then shift focus to comparing the cardinalities of minimal faithful sets of descriptors as a way to define and order equivalence classes of topological descriptor types. Focusing on six topological descriptor types commonly used in practice, we give a partial order on their corresponding equivalence classes, as well as give bounds on the sizes of minimum faithful sets for each descriptor type. Next, we broaden our view to zig-zag grid persistence modules, functors whose domain categories are posets with grid-like structure. We begin by explicitly defining such persistence modules in terms of constructible cosheaves over stratified Euclidean space, including a careful treatment of augmented persistence modules, which are analogous to the aforementioned augmented descriptors who played a central role in discussion of faithful sets. Exodromy gives us an equivalence between persistence modules as a functor category and as constructible cosheaves; we furthermore show the equivalence of these categories with a category of constructible functors out of Rd with a fixed stratification and localized at weak equivalences, essentially "standardizing" modules so that the category has a clear monoid structure. We compute the algebraic K-theory of zig-zag grid persistence modules, using a double inductive argument to show the K-theory is additive over strata. Finally, we identify connections to related topics, such as the virtual diagrams of Bubenik and Elchensen, as well as Euler characteristic and Betti curves/surfaces/manifolds. We hope a study of K-groups will provide interesting insights into the nature of persistence modules, and we indicate ways in which the zeroth and first K-groups may be interpretedItem Critically fixed anti-rational maps, tischler graphs, and their applications(Montana State University - Bozeman, College of Letters & Science, 2022) McKay, Christopher Michael; Chairperson, Graduate Committee: Lukas GeyerWe are mainly concerned with maps which take the form of the complex conjugate of a rational map and where all critical points are fixed points, which are known as critically fixed anti-rational maps. These maps have a well understood combinatorial model by a planar graph. We make progress in answering three questions. Using this combinatorial model by planar graph, can we generate all critically fixed anti-rational maps from the most basic example, z -> z 2? Under repeated pullback by a critically fixed anti-rational map, does a simple closed curve off the critical set eventually land and stay in a finite set of homotopy classes of simple closed curves off the critical set? This is known as the global curve attractor problem and has been an area of interest since its introduction by Pilgrim in 2012. Lastly, anti-rational maps can be used to model the physical phenomenon of gravitational lensing, which is where the image of a far away light source is distorted and multiplied by large masses between the light source and the observer. Maximal lensing configurations are where n masses generate 5n - 5 lensed images of a single light source. There are very few known examples of maximal lensing configurations, all generated by Rhie in 2003. Can we use these combinatorial models to inspire new examples of maximal lensing configurations? In this dissertation we show one can generate all critically fixed anti-rational maps from the most basic example, z -> z 2 by a repeated 'blow-up' procedure. We also show that all critically fixed anti-rational maps with 4 or 5 critical points have a finite global curve attractor. Lastly we establish a connection between maximal lensing maps and Tischler graphs and generate new examples of maximal lensing maps.Item Homotopy groups of contact 3-manifolds(Montana State University - Bozeman, College of Letters & Science, 2019) Perry, Daniel George; Chairperson, Graduate Committee: David Ayala; Ryan Grady (co-chair)A contact 3-manifold (M, xi) is an three-dimensional manifold endowed with a completely nonintegrable distribution. In studying such a space, standard homotopy groups, which are defined using continuous/smooth maps, are not useful as they are not sensitive to the distribution. To remedy this, we consider horizontal homotopy groups which are defined using horizontal maps, i.e., smooth maps that lie tangent to the distribution at every point. Due to the distribution being completely nonintegrable, horizontal maps into (M, xi) have rank at most 1. This is used to show that the first horizontal homotopy group is uncountably generated and indicates that the higher horizontal homotopy groups are trivial. We also consider Lipschitz homotopy groups which are defined using Lipschitz maps. We first endow (M, xi) with a metric that is sensitive to the distribution, the Carnot-Caratheodory metric. With respect to this metric structure, the contact 3-manifold is purely 2-unrectifiable. This is used to show that the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Furthermore, over the contact 3-manifold is a metric space, called the universal path space, that acts as a universal cover of the contact 3-manifold in that the universal path space is Lipschitz simply-connected and has a unique lifting property. Homotopy groups, horizontal homotopy groups, and Lipschitz homotopy groups are all instances of homotopy groups of sheaves, which are defined.Item On configuration categories(Montana State University - Bozeman, College of Letters & Science, 2019) Cepek, Anna Beth; Chairperson, Graduate Committee: David Ayala; Jaroslaw Kwapisz (co-chair)We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of infinity-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of the circle and Euclidean space.Item Lower semi-continuous multifunctions and properties of the l and k topology(Montana State University - Bozeman, College of Letters & Science, 1969) Feichtinger, OskarItem The topology of laminations(Montana State University - Bozeman, College of Letters & Science, 2000) Johnson, Luther WilliamItem Linear topologies induced by bilinear forms(Montana State University - Bozeman, College of Letters & Science, 1966) Miller, Vinnie HicksItem On the structure of locally connected topological spaces(Montana State University - Bozeman, College of Letters & Science, 1971) Minear, Spencer EdwardItem Convergence structures on homeomorphism groups(Montana State University - Bozeman, College of Letters & Science, 1971) Park, Wayne Richard