Physical Sciences and Mathematics Commons^™

Making Sandwiches: A Novel Invariant In D-Module Theory, David Lieberman

Department of Mathematics: Dissertations, Theses, and Student Research

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff …

Spreads And Transversals And Their Connection To Geproci Sets, Allison Joan Ganger

Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–

Spreads of [set of prime numbers]³ over finite fields can yield geproci sets. We study the existence of transversals to such spreads, proving that spreads with two transversals exist for all finite fields, before further considering the groupoids coming from spreads when transversals do or do not exist. This is further considered for spreads of higher dimensional projective spaces. We also consider how certain spreads might generalize to characteristic zero and the connection to the previously known geproci sets coming from the root systems D₄ and F₄.

Advisor: Brian Harbourne

A Tensor's Torsion, Neil Steinburg

Department of Mathematics: Dissertations, Theses, and Student Research

While tensor products are quite prolific in commutative algebra, even some of their most basic properties remain relatively unknown. We explore one of these properties, namely a tensor's torsion. In particular, given any finitely generated modules, M and N over a ring R, the tensor product $M\otimes_R N$ almost always has nonzero torsion unless one of the modules M or N is free. Specifically, we look at which rings guarantee nonzero torsion in tensor products of non-free modules over the ring. We conclude that a specific subclass of one-dimensional Gorenstein rings will have this property.

Adviser: Roger Wiegand and Tom …

Knörrer Periodicity And Bott Periodicity, Michael K. Brown

Department of Mathematics: Dissertations, Theses, and Student Research

The main goal of this dissertation is to explain a precise sense in which Knörrer periodicity in commutative algebra is a manifestation of Bott periodicity in topological K-theory. In Chapter 2, we motivate this project with a proof of the existence of an 8-periodic version of Knörrer periodicity for hypersurfaces defined over the real numbers. The 2- and 8-periodic versions of Knörrer periodicity for complex and real hypersurfaces, respectively, mirror the 2- and 8-periodic versions of Bott periodicity in KU- and KO-theory. In Chapter 3, we introduce the main tool we need to demonstrate the compatibility between Knörrer …

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …