Algebra Commons^™

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

Dissertations, Theses, and Capstone Projects

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set EX. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E₂-terms coincide with certain …

Quantifying Separability In Limit Groups, Keino Brown

Dissertations, Theses, and Capstone Projects

We show that for any finitely generated non-abelian subgroup H of a limit group L, there exists a finite-index subgroup K which is fully residually H. This generalizes the result of Wilton that limit groups admit local retractions. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes results of Louder, …

A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

Dissertations, Theses, and Capstone Projects

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

Prime Factors: America’S Prioritization Of Literacy Over Numeracy And Its Relationship To Systemic Inequity, Troy Smith

Dissertations, Theses, and Capstone Projects

For much of American history, literacy has been prioritized in K-12 education and society, at large, at the expense of numeracy. This lack of numerical emphasis has established innumeracy as an American cultural norm that has resulted in America not producing a sufficient number of numerate citizens, and ranking poorly on mathematical performance in international comparisons. This paper investigates the decisions and circumstances that led to this under prioritization, along with the public and cultural impact of said actions. Toward this end, literature regarding contemporary and historical influences on American mathematics education (e.g., civic, policy, and parental) was reviewed. The …

Some Model Theory Of Free Groups, Christopher James Natoli

Dissertations, Theses, and Capstone Projects

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily …

Spectral Sequences For Almost Complex Manifolds, Qian Chen

Dissertations, Theses, and Capstone Projects

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. …

Model Theory Of Groups And Monoids, Laura M. Lopez Cruz

Dissertations, Theses, and Capstone Projects

We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given …

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

Dissertations, Theses, and Capstone Projects

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta functions …

A Differential Algebra Approach To Commuting Polynomial Vector Fields And To Parameter Identifiability In Ode Models, Peter Thompson

Dissertations, Theses, and Capstone Projects

In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a …

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Analysis Of A Group Of Automorphisms Of A Free Group As A Platform For Conjugacy-Based Group Cryptography, Pavel Shostak

Dissertations, Theses, and Capstone Projects

Let F be a finitely generated free group and Aut(F) its group of automorphisms.

In this monograph we discuss potential uses of Aut(F) in group-based cryptography.

Our main focus is on using Aut(F) as a platform group for the Anshel-Anshel-Goldfeld protocol, Ko-Lee protocol, and other protocols based on different versions of the conjugacy search problem or decomposition problem, such as Shpilrain-Ushakov protocol.

We attack the Anshel-Anshel-Goldfeld and Ko-Lee protocols by adapting the existing types of the length-based attack to the specifics of Aut(F). We also present our own version of the length-based attack that significantly increases the attack' success …

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these …

Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak

Dissertations, Theses, and Capstone Projects

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this dissertation, we seek to extend these techniques to finitely presented non-free groups, in particular to polycyclic and metabelian groups that are of interest to non-commutative cryptography.

As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. …

Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of …

Some 2-Categorical Aspects In Physics, Arthur Parzygnat

Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description …

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

Dissertations, Theses, and Capstone Projects

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form …

P-Adic L-Functions And The Geometry Of Hida Families, Joseph Kramer-Miller

Dissertations, Theses, and Capstone Projects

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using …

The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, Rachel A. Oppenheimer

Dissertations, Theses, and Capstone Projects

Nationwide, there is a crisis in math learning and math achievement at all levels of education. Upwards of 80% of students who enter the City University of New York’s community colleges from New York City’s Department of Education high schools fail to meet college level math proficiencies and as a result, are funneled into the system’s remedial math system. Once placed into pre-college remedial arithmetic, pre-algebra, and elementary algebra courses, students fail at alarming rates and research indicates that students’ failure in remedial math has negative ripple effects on their persistence and degree completion. CUNY is not alone in facing …