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University of Colorado

1. McGregor-Dorsey, Zachary Strider. Some properties of full heaps.

Degree: PhD, Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/28

A full heap is a labeled infinite partially ordered set with labeling taken from the vertices of an underlying Dynkin diagram, satisfying certain conditions intended to capture the structure of that diagram. The notion of full heaps was introduced by R. Green as an affine extension of the minuscule heaps of J. Stembridge. Both authors applied these constructions to make observations of the Lie algebras associated to the underlying Dynkin diagrams. The main result of this thesis, Theorem 4.7.1, is a complete classification of all full heaps over Dynkin diagrams with a finite number of vertices, using only the general notion of Dynkin diagrams and entirely elementary methods that rely very little on the associated Lie theory. The second main result of the thesis, Theorem 5.1.7, is an extension of the Fundamental Theorem of Finite Distributive Lattices to locally finite posets, using a novel analogue of order ideal posets. We apply this construction in an analysis of full heaps to find our third main result, Theorem 5.5.1, an ADE classification of the full heaps over simply laced affine Dynkin diagrams.
*Advisors/Committee Members: Richard M. Green, Nathaniel Thiem, Martin E. Walter, J. M. Douglas, Stephen R. Doty.*

Subjects/Keywords: ADE Classification; Combinatorial Algebra; Dynkin Diagram; Full Heap; Lie Algebra; Minuscule Representation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

McGregor-Dorsey, Z. S. (2013). Some properties of full heaps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/28

Chicago Manual of Style (16^{th} Edition):

McGregor-Dorsey, Zachary Strider. “Some properties of full heaps.” 2013. Doctoral Dissertation, University of Colorado. Accessed October 30, 2020. https://scholar.colorado.edu/math_gradetds/28.

MLA Handbook (7^{th} Edition):

McGregor-Dorsey, Zachary Strider. “Some properties of full heaps.” 2013. Web. 30 Oct 2020.

Vancouver:

McGregor-Dorsey ZS. Some properties of full heaps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2020 Oct 30]. Available from: https://scholar.colorado.edu/math_gradetds/28.

Council of Science Editors:

McGregor-Dorsey ZS. Some properties of full heaps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/28

University of Colorado

2. Krupa, Matthew Gregory. Differential Geometry of Projective Limits of Manifolds.

Degree: PhD, Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/47

The nascent theory of projective limits of manifolds in the category of locally R-ringed spaces is expanded and generalizations of differential geometric constructions, definitions, and theorems are developed. After a thorough introduction to limits of topological spaces, the study of limits of smooth projective systems, called promanifolds, commences with the definitions of the tangent bundle and the study of locally cylindrical maps. Smooth immersions, submersions, embeddings, and smooth maps of constant rank are defined, their theories developed, and counter examples showing that the inverse function theorem may fail for promanifolds are provided along with potential substitutes.
Subsets of promanifolds of measure 0 are defined and a generalization of Sard's theorem for promanifolds is proven. A Whitney embedding theorem for promanifolds is given and a partial uniqueness result for integral curves of smooth vector fields on promanifolds is found. It is shown that a smooth manifold of dimension greater than one has the final topology with respect to its set of C^{1}-arcs but not with respect to its C^{2}-arcs and that a particular class of promanifolds, called monotone promanifolds, have the final topology with respect to a class of smooth topological embeddings of compact intervals termed smooth almost arcs.
*Advisors/Committee Members: Markus J. Pflaum, Jeanne N. Clelland, Martin E. Walter, Jonathan Wise, Alexander Gorokhovsky.*

Subjects/Keywords: Inverse Function Theorem; Normed Spaces; Projective Limits of Smooth Manifolds; Promanifolds; Sard's Theorem; Whitney Embedding Theorem; Geometry and Topology; Mathematics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Krupa, M. G. (2016). Differential Geometry of Projective Limits of Manifolds. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/47

Chicago Manual of Style (16^{th} Edition):

Krupa, Matthew Gregory. “Differential Geometry of Projective Limits of Manifolds.” 2016. Doctoral Dissertation, University of Colorado. Accessed October 30, 2020. https://scholar.colorado.edu/math_gradetds/47.

MLA Handbook (7^{th} Edition):

Krupa, Matthew Gregory. “Differential Geometry of Projective Limits of Manifolds.” 2016. Web. 30 Oct 2020.

Vancouver:

Krupa MG. Differential Geometry of Projective Limits of Manifolds. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2020 Oct 30]. Available from: https://scholar.colorado.edu/math_gradetds/47.

Council of Science Editors:

Krupa MG. Differential Geometry of Projective Limits of Manifolds. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/math_gradetds/47