Algebra Commons^™

On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

Department of Mathematics: Dissertations, Theses, and Student Research

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …

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 …

Closure And Homological Properties Of (Auto)Stackable Groups, Ashley Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Let G be a finitely presented group with Cayley graph Γ. Roughly, G is a stackable group if there is a maximal tree T in Γ and a function φ, defined on the edges in Γ, for which there is a natural ‘flow’ on the edges in Γ\T towards the identity. Additionally, if graph (φ), which consists of pairs (e; φ(e)) for e an edge in Γ, forms a regular language, then G is autostackable. In 2011, Brittenham and Hermiller introduced stackable groups in [4], in part, as a means …

Embedding And Nonembedding Results For R. Thompson's Group V And Related Groups, Nathan Corwin

Department of Mathematics: Dissertations, Theses, and Student Research

We study Richard Thompson's group V, and some generalizations of this group. V was one of the first two examples of a finitely presented, infinite, simple group. Since being discovered in 1965, V has appeared in a wide range of mathematical subjects. Despite many years of study, much of the structure of V remains unclear. Part of the difficulty is that the standard presentation for V is complicated, hence most algebraic techniques have yet to prove fruitful.

This thesis obtains some further understanding of the structure of V by showing the nonexistence of the wreath product Z wr Z^2 as …

Fan Cohomology And Its Application To Equivariant K-Theory Of Toric Varieties, Suanne Au

Department of Mathematics: Dissertations, Theses, and Student Research

Mu-Wan Huang, Mark Walker and I established an explicit formula for the equivariant K-groups of affine toric varieties. We also recovered a result due to Vezzosi and Vistoli, which expresses the equivariant K-groups of a smooth toric variety in terms of the K-groups of its maximal open affine toric subvarieties. This dissertation investigates the situation when the toric variety X is neither affine nor smooth. In many cases, we compute the Čech cohomology groups of the presheaf K_q^T on X endowed with a topology. Using these calculations and Walker's Localization Theorem for equivariant K-theory, we give explicit formulas …