Algebra Commons^™

Semi-Infinite Flags And Zastava Spaces, Andreas Hayash

Doctoral Dissertations

ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct …

An Exploration Of Absolute Minimal Degree Lifts Of Hyperelliptic Curves, Justin A. Groves

Doctoral Dissertations

For any ordinary elliptic curve E over a field with non-zero characteristic p, there exists an elliptic curve E over the ring of Witt vectors W(E) for which we can lift the Frobenius morphism, called the canonical lift. Voloch and Walker used this theory of canonical liftings of elliptic curves over Witt vectors of length 2 to construct non-linear error-correcting codes for characteristic two. Finotti later proved that for longer lengths of Witt vectors there are better lifts than the canonical. He then proved that, more generally, for hyperelliptic curves one can construct a lifting over …

The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs

Doctoral Dissertations

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, …

Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv

Doctoral Dissertations

Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general …

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

Doctoral Dissertations

Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …

Characteristic Sets Of Matroids, Dony Varghese

Doctoral Dissertations

Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set.

The linear representations and characteristic sets are well studied. But the algebraic representations and …

Dvr-Matroids Of Algebraic Extensions, Anna L. Lawson

Doctoral Dissertations

A matroid is a finite set E along with a collection of subsets of E, called independent sets, that satisfy certain conditions. The most well-known matroids are linear matroids, which come from a finite subset of a vector space over a field K. In this case the independent sets are the subsets that are linearly independent over K. Algebraic matroids come from a finite set of elements in an extension of a field K. The independent sets are the subsets that are algebraically independent over K. Any linear matroid has a representation as an algebraic matroid, but the converse is …

General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul

Doctoral Dissertations

In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.

Sequential Deformations Of Hadamard Matrices And Commuting Squares, Shuler G. Hopkins

Doctoral Dissertations

In this dissertation, we study analytic and sequential deformations of commuting squares of finite dimensional von Neumann algebras, with applications to the theory of complex Hadamard matrices. The main goal is to shed some light on the structure of the algebraic manifold of spin model commuting squares (i.e., commuting squares based on complex Hadamard matrices), in the neighborhood of the standard commuting square (i.e., the commuting square corresponding to the Fourier matrix). We prove two types of results: Non-existence results for deformations in certain directions in the tangent space to the algebraic manifold of commuting squares (chapters 3 and 4), …

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

Doctoral Dissertations

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …

A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, Zhijie Dong

Doctoral Dissertations

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian to a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. The relation between this conjecture and the reduceness …

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 …

Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber

Doctoral Dissertations

In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring R and ideal I the vertices are { …

Construction And Classification Results For Commuting Squares Of Finite Dimensional *-Algebras, Chase Thomas Worley

Doctoral Dissertations

In this dissertation, we present new constructions of commuting squares, and we investigate finiteness and isolation results for these objects. We also give applications to the classification of complex Hadamard matrices and to Hopf algebras.

In the first part, we recall the notion of commuting squares which were introduced by Popa and arise naturally as invariants in Jones' theory of subfactors. We review some of the main known examples of commuting squares such as those constructed from finite groups and from complex Hadamard matrices. We also recall Nicoara's notion of defect which gives an upper bound for the number of …

Generalizations And Variations Of The Zero-Divisor Graph, Grace Elizabeth Mcclurkin

Doctoral Dissertations

We explore generalizations and variations of the zero-divisor graph on commutative rings with identity. A zero-divisor graph is a graph whose vertex set is the nonzero zero-divisors of a ring, wherein two distinct vertices are adjacent if their product is zero. Variations of the zero-divisor graph are created by changing the vertex set, the edge condition, or both. The annihilator graph and the extended zero-divisor graph are both variations that change the edge condition, whereas the compressed graph and ideal-based graph change the vertex set. By combining these concepts, we define and investigate graphs where both the vertex set and …

Equivariant Intersection Cohomology Of Bxb Orbit Closures In The Wonderful Compactification Of A Group, Stephen Oloo

Doctoral Dissertations

This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactifica- tion is equivariant, extending the left and right action of the group on itself, and we focus on the local and global topology of the closures of Borel orbits. It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be inter- esting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out …

The Congruence-Based Zero-Divisor Graph, Elizabeth Fowler Lewis

Doctoral Dissertations

Let R be a commutative ring with nonzero identity and ~ a multiplicative congruence relation on R. Then, R/~ is a semigroup with multiplication [x][y] = [xy], where [x] is the congruence class of an element x of R. We define the congruence-based zero-divisor graph of R ito be the simple graph with vertices the nonzero zero-divisors of R/~ and with an edge between distinct vertices [x] and [y] if and only if [x][y] = [0]. Examples include the usual …

Combinatorics Of Equivariant Cohomology: Flags And Regular Nilpotent Hessenberg Varieties, Elizabeth J. Drellich

Doctoral Dissertations

The field of Schubert Calculus deals with computations in the cohomology rings of certain algebraic varieties, including flag varieties and Schubert varieties. In the equivariant setting, GKM theory turns multiplication in the cohomology ring of certain varieties into a combinatorial computation. This dissertation uses combinatorial tools, including Billey’s formula, to do Schubert calculus computations in several varieties. First we address do computations in the equivariant cohomology of full and partial flag varieties, the classical spaces in Schubert calculus. We then do computations in the equivariant cohomology of a family of non-classical spaces: regular nilpotent Hessenberg varieties. The final chapter gives …

Properties Of Ideal-Based Zero-Divisor Graphs Of Commutative Rings, Jesse Gerald Smith Jr.

Doctoral Dissertations

Let R be a commutative ring with nonzero identity and I an ideal of R. The focus of this research is on a generalization of the zero-divisor graph called the ideal-based zero-divisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zero-divisor graph of R. We begin by classifying all rings which have nontrivial ideal-based zero-divisor graph complete on fewer than 5 vertices. We also classify when such graphs are complete on a prime number of vertices. In addition we classify all rings which …

Multiplicative Sets Of Atoms, Ashley Nicole Rand

Doctoral Dissertations

It is possible for an element to have both an atom factorization and a factorization that will always contain a reducible element. This leads us to consider the multiplicatively closed set generated by the atoms and units of an integral domain. We start by showing that for a nice subset S of the atoms of R, there exists an integral domain containing R with set of atoms S. A multiplicatively closed set is saturated if the factors of each element in the set are also elements in the set. Considering polynomial and power series subrings, we find necessary and sufficient …

On The Homology Of Automorphism Groups Of Free Groups., Jonathan Nathan Gray

Doctoral Dissertations

Following the work of Conant and Vogtmann on determining the homology of the group of outer automorphisms of a free group, a new nontrivial class in the rational homology of Outer space is established for the free group of rank eight. The methods started in [8] are heavily exploited and used to create a new graph complex called the space of good chord diagrams. This complex carries with it significant computational advantages in determining possible nontrivial homology classes.
Next, we create a basepointed version of the Lie operad and explore some of it proper- ties. In particular, we prove a …

Elasticity Of Krull Domains With Infinite Divisor Class Group, Benjamin Ryan Lynch

Doctoral Dissertations

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has …

On Conjectures Concerning Nonassociate Factorizations, Jason A Laska

Doctoral Dissertations

We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

Fractions Of Numerical Semigroups, Harold Justin Smith

Doctoral Dissertations

Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown.

Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the …

On The Irreducibility Of The Cauchy-Mirimanoff Polynomials, Brian C. Irick

Doctoral Dissertations

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.

This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index …

Zero-Divisor Graphs, Commutative Rings Of Quotients, And Boolean Algebras, John D. Lagrange

Doctoral Dissertations

The zero-divisor graph of a commutative ring is the graph whose vertices are the nonzero zero-divisors of the ring such that distinct vertices are adjacent if and only if their product is zero. We use this construction to study the interplay between ring-theoretic and graph-theoretic properties. Of particular interest are Boolean rings and commutative rings of quotients.