Theses and Dissertations at Montana State University (MSU)

Now showing 1 - 4 of 4

From curves to words and back again: geometric computation of minimum-area homotopy
(Montana State University - Bozeman, College of Engineering, 2024) McCoy, Bradley Allen; Chairperson, Graduate Committee: Brittany Fasy
Let gamma be a generic closed curve in the plane. The area of a homotopy is the area swept by the homotopy. We consider the problem of computing the minimum null-homotopy area of gamma. Samuel Blank, in his 1967 Ph.D. thesis, determined if gamma is self-overlapping by geometrically constructing a combinatorial word from gamma. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of gamma by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve gamma with minimum area. Next, we describe the first polynomial implementation of an algorithm to compute the minimum homotopy area of a piecewise linear closed curve in the plane. We discuss how minimum homotopy area can be used as a similarity measure for curves and include experiments that compare the runtime of our algorithm to an implementation of the Frechet distance. We then extend our algorithm for computing the minimum homotopy area in the plane to homotopic, non-intersecting, non-contractible curves on an orientable surface with positive genus. Finally, we consider the inverse problem of determining which combinatorial Blank words correspond to closed curves in the plane. We solve a special case of this problem and give an exponential algorithm to the general case.
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.
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.
Indicability in products of groups
(Montana State University - Bozeman, College of Letters & Science, 1974) Behrens, Elizabeth Darragh

Theses and Dissertations at Montana State University (MSU)

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