Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item Immersions of surfaces(Montana State University - Bozeman, College of Letters & Science, 2022) Howard, Adam Jacob; Co-chairs, Graduate Committee: David Ayala and Ryan GradyTo determine the existence of a regular homotopy between two immersions, f, g : M --> N, is equivalent to showing that they lie in the same path component of the space Imm(M, N). We identify the connected components, pi 0 Imm(W g, M), of the space of immersions from a closed, orientable, genus-g surface W g into a parallelizable manifold M. We also identify the higher homotopy groups of Imm(W g, M) in terms of the homotopy groups of M and the Stiefel space V 2 (n). We then use this work to characterize immersions from tori into hyperbolic manifolds as self covers of a tubular neighborhood of a closed geodesic up to regular homotopy. Finally, we identify the homotopy-type of the space of framed immersions from the torus to itself.Item Designing pattern formation through anisotropy(Montana State University - Bozeman, College of Letters & Science, 2019) Gaussoin, Anthony Danwayne; Chairperson, Graduate Committee: Scott McCallaWhen governed by appropriate potentials, systems of particles interacting pairwise in three dimensions self assemble into diverse patterns near the surface of a sphere. The resulting structure of these minimal energy states can be altered through anisotropic effects. This leads to the inverse problem of finding anisotropic potentials that produce specific targeted equilibrium structures. To study this problem, continuous versions of the discrete particle interaction equations are employed so that a leading order approximation can be obtained. Linear stability is then determined through a Fourier type analysis in terms of spherical harmonics. This allows us to solve the linearized inverse problem: for a targeted equilibrium structure, where the particles congregate along a finite set of spherical harmonics, construct an anisotropic potential that induces the same finite set of linear instabilities. Several examples of anisotropic potentials that cause known linear instabilities are presented. The resulting minimal energy configurations are approximated through a gradient descent of the discrete particle energy. These numerical experiments corroborate that the linear instabilities can be used to predict the minimal energy structure in the full nonlinear dynamics. Solving the linearized inverse problem yields a clear path to designing pattern formation through anisotropic effects.