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Full-Text Articles in Physical Sciences and Mathematics
A Categorical Formulation Of Algebraic Geometry, Bradley Willocks
A Categorical Formulation Of Algebraic Geometry, Bradley Willocks
Doctoral Dissertations
We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a ``spec datum" is introduced, as a certain relation between categories, of which one has been given a Grothendieck topology. A ``geometry" is interpreted as a sub-category of $\Omega$, and a formalism is given by which such a subcategory is to be associated to a spec datum, reflecting the standard construction of the category of schemes from the category of rings by affine charts.
Equations For Nilpotent Varieties And Their Intersections With Slodowy Slices, Benjamin Johnson
Equations For Nilpotent Varieties And Their Intersections With Slodowy Slices, Benjamin Johnson
Doctoral Dissertations
This thesis investigates minimal generating sets of ideals defining certain nilpotent varieties in simple complex Lie algebras. A minimal generating set of invariants for the whole nilpotent cone is known due to Kostant. Broer determined a minimal generating set for the subregular nilpotent variety in all simple Lie algebra types. I extend Broer's results to two families of nilpotent varieties, valid in any simple Lie algebra, that include the nilpotent cone, the subregular case, and usually more. In the first part of my thesis I describe a minimal generating set for the ideal of each of these varieties in the …
Coverings Of Graphs And Tiered Trees, Sam Glennon
Coverings Of Graphs And Tiered Trees, Sam Glennon
Doctoral Dissertations
This dissertation will cover two separate topics. The first of these topics will be coverings of graphs. We will discuss a recent paper by Marcus, Spielman, and Srivastava proving the existence of infinite families of bipartite Ramanujan graphs for all regularities. The proof works by showing that for any d-regular Ramanujan graph, there exists an infinite tower of bipartite Ramanujan graphs in which each graph is a twofold covering of the previous one. Since twofold coverings of a graph correspond to ways of labeling the edges of the graph with elements of a group of order 2, we will generalize …
Dynamical Systems And Zeta Functions Of Function Fields, Daniel Nichols
Dynamical Systems And Zeta Functions Of Function Fields, Daniel Nichols
Doctoral Dissertations
This doctoral dissertation concerns two problems in number theory. First, we examine a family of discrete dynamical systems in F_2[t] analogous to the 3x + 1 system on the positive integers. We prove a statistical result about the large-scale dynamics of these systems that is stronger than the analogous theorem in Z. We also investigate mx + 1 systems in rings of functions over a family of algebraic curves over F_2 and prove a similar result there. Second, we describe some interesting properties of zeta functions of algebraic curves. Generally L-functions vanish only to the order required by their root …
A Seifert-Van Kampen Theorem For Legendrian Submanifolds And Exact Lagrangian Cobordisms, Mark Lowell
A Seifert-Van Kampen Theorem For Legendrian Submanifolds And Exact Lagrangian Cobordisms, Mark Lowell
Doctoral Dissertations
We prove a Seifert-van Kampen theorem for Legendrian submanifolds and exact Lagrangian cobordisms, and use it to calculate the change in the DGA caused by critical Legendrian ambient surgery.