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Contributions to Geometry and Graph Theory

Description: In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, the… more
Date: August 2020
Creator: Schuerger, Houston S
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries

Graded Hecke Algebras for the Symmetric Group in Positive Characteristic

Description: Graded Hecke algebras are deformations of skew group algebras which arise from a group acting on a polynomial ring. Over fields of characteristic zero, these deformations have been studied in depth and include both symplectic reflection algebras and rational Cherednik algebras as examples. In Lusztig's graded affine Hecke algebras, the action of the group is deformed, but not the commutativity of the vectors. In Drinfeld's Hecke algebras, the commutativity of the vectors is deformed, but not … more
Date: August 2020
Creator: Krawzik, Naomi
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries
open access

Results on Non-Club Isomorphic Aronszajn Trees

Description: In this dissertation we prove some results about the existence of families of Aronszajn trees on successors of regular cardinals which are pairwise not club isomorphic. The history of this topic begins with a theorem of Gaifman and Specker in the 1960s which asserts the existence from ZFC of many pairwise not isomorphic Aronszajn trees. Since that result was proven, the focus has turned to comparing Aronszajn trees with respect to isomorphisms on a club of levels, instead of on the entire tre… more
Date: August 2020
Creator: Chavez, Jose
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries
open access

Determinacy of Schmidt's Game and Other Intersection Games

Description: Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our m… more
Date: May 2020
Creator: Crone, Logan
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries
open access

Invariants of Polynomials Modulo Frobenius Powers

Description: Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-… more
Date: May 2020
Creator: Drescher, Chelsea
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries
open access

Winning Sets and the Banach-Mazur-McMullen Game

Description: For decades, mathematical games have been used to explore various properties of particular sets. The Banach-Mazur game is the prototypical intersection game and its modifications by e.g., W. Schmidt and C. McMullen are used in number theory and many other areas of mathematics. We give a brief survey of a few of these modifications and their properties followed by our own modification. One of our main results is proving that this modification is equivalent to an important set theoretic game, … more
Date: May 2020
Creator: Ragland, Robin
Item Type: Refine your search to only Thesis or Dissertation
Partner: UNT Libraries
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