Master of Arts (MA), Bowling Green State University, 2022, Mathematics/Mathematics (Pure)

In this thesis, we study the free Lie algebra on two generators and a deformation of the free Lie bracket. Our goal is a hands-on derivation of relations which this deformed Lie bracket satisfies. The technical achievement that makes this possible is the identification of a basis for where the relations occur. Using that basis, we verify and extend the calculations found in Schneps (2006). An interesting connection to the Euler polynomials is also discussed.

Committee: Benjamin Ward Ph.D. (Committee Chair); Mihai Staic Ph.D. (Committee Member) Subjects: Mathematics

Master of Arts (MA), Bowling Green State University, 2017, Mathematics

In this paper, we find all the quasiconformal maps on a particular non-rigid 2-step Carnot group. In particular, all quasiconformal maps on this Carnot group preserve the vertical direction. Given that a Carnot group is a Lie algebra with a group structure, we employ concepts from linear algebra and abstract algebra to gain information about the group. Utilizing the theory of Pansu differentiability along with the biLipschitz nature of quasisymmetric maps, we use an analytical approach to help determine the form of any quasiconformal map on the Carnot group. The main result has consequences for the rigidity of quasiisometries of negatively curved solvable Lie groups.

Committee: Xiangdong Xie (Advisor); Kit Chan (Committee Member) Subjects: Mathematics

Master of Arts, The Ohio State University, 1962, Graduate School

Committee: Not Provided (Other) Subjects:

Doctor of Philosophy, The Ohio State University, 2023, Mathematics

This dissertation consists of two self-contained papers from my graduate work at Ohio State University. In Chapter 2, we review the definition of C*-algebras, von Neumann algebras, C*/W* categories/ 2-categories. Some of this background material is taken from [GLR85, CHPJP22]. We also review the 2-categories C*Alg/W*Alg of C*/W*-correspondences and Q-system realization construction taken directly from my articles [CHPJP22, CP22]. Chapter 3 is joint work with Corey Jones and David Penneys [CJP21]. First We discuss the construction of a unitary braided tensor category End_loc(C) from a given W*- category C. When C is the category of finitely generated projective modules over a type II1 factor M, the underlying tensor category of its dualizable part of End_loc(Mod_fgp(M)) is Connes' bimodule version χ (M) due to Popa. Second, for each unitary fusion category C, we construct a II1-factor M such that χ (M) \cong Z(C). Chapter 4 is joint work with Roberto Hern andez Palomares and Corey Jones [CPJ22]. We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on unital AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF- algebras.

Committee: David Penneys (Advisor) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2023, Mathematics

As objects that appear throughout mathematics, integer-valued polynomials have been studied extensively. However, integer-valued rational functions are a much less studied generalization. We consider the set of integer-valued rational functions over an integral domain as a ring and study the ring-theoretic properties of such rings. We explore when rings of integer-valued rational functions are Bezout domains, Prufer domains, and globalized pseudovaluation domains. We completely classify when the ring of integer-valued rational functions over a valuation domain is a Prufer domain and when it is a Bezout domain. We extend the classification of when rings of integer-valued rational functions are Prufer domains. This includes a family of rings of integer-valued rational functions that are Prufer domains, as well as a family of integer-valued rational functions that are not Prufer domains. We determine that the classification of when rings of integer-valued rational functions are Prufer domains is not analogous to the interpolation domain classification of when rings of integer-valued polynomials are Prufer domains. We also show some conditions under which the ring of integer-valued rational functions is a globalized pseudovaluation domain. We also prove that even if a pseudovaluation domain has an associated valuation domain over which the ring of integer-valued rational functions is a Prufer domain, the ring of integer-valued rational functions over the pseudovaluation domain is not guaranteed to be a globalized pseudovaluation domain. Because rings of integer-valued rational functions are rings of functions, we can study their properties with respect to evaluation. These properties include the Skolem property and its generalizations, which are properties concerning when ideals are able to be distinguished using evaluation. We connect the Skolem property to the maximal spectrum of a ring of integer-valued rational functions. This is then generalized using st (open full item for complete abstract)

Committee: K. Alan Loper (Advisor); Cosmin Roman (Committee Member); Ivo Herzog (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2017, Mathematics

Skolem properties describe how well ideals of rings of integer-valued polynomials are characterized by their images under evaluation maps. They are usually defined only for finitely generated ideals. Evaluation is sensible for any ring made of polynomials, and it usually makes sense in the context of rational functions. We generalize the notion of a Skolem property to these broader settings. We give several examples of rings exhibiting these properties, and we extend many of the results about Skolem properties of rings of integer-valued polynomials to rings comprising polynomials. We examine rings in which the interesting values occur at only a finite collection of points. We demonstrate that such rings have the almost strong Skolem property, and we use that result to characterize when they are Prufer domains. We also consider n-generator properties in that setting. Making signicant progress toward classifying the Skolem properties for all integrally closed rings of polynomials, we consider valuation domains on K(x), for some field K, contracted to K[x]. In this setting we characterize the almost Skolem property. We also extend the notion of a Skolem closure so that it is a semistar operation, and we demonstrate that it is more natural to consider the Skolem property as a property of star ideals rather than one of finitely generated ideals. We end with an application of this new perspective to the classical ring of integer-valued polynomials Int(Z), answering the open question: What is the largest class of ideals on which Int(Z) has the (strong) Skolem property?

Committee: K. Alan Loper (Advisor); Ivo Herzog (Committee Member); Cosmin Roman (Committee Member) Subjects: Mathematics

Master of Mathematical Sciences, The Ohio State University, 2014, Mathematics

This paper is an educational approach to group characters through examples which introduces the beginner algebraist to representations and characters of finite groups. My hope is that this exploration might help the advanced undergraduate student discover some of the foundational tools of Character Theory. The prerequisite material for this paper includes some elementary Abstract and Linear Algebra. The basic groups used in the examples are intended to excited a student into exploration of groups they understand from their undergraduate studies. Throughout the section of examples there are exercises used to check understanding and give the reader opportunity to explore further. After taking a course in Abstract Algebra one might find that groups are not concrete objects. Groups model actions, rotations, reflections, movements, and permutations. Group representations turn these abstract sets of objects into sets of n X n matrices with real or complex entries, which can be easily handled by a computer for any number of calculations.

Committee: James Cogdell Dr. (Advisor); Warren Sinnott Dr. (Committee Member) Subjects: Mathematics

MEd, University of Cincinnati, 2002, Education : Curriculum and Instruction

This study analyzes the relationship between an eighth grade teacher's questioning techniques between two classes of differing mathematical achievement. One class was an algebra class and the other a pre-algebra class. An observation instrument was created and used to evaluate eight characteristics of each question asked by the teacher. This instrument was used during two months of observations in both classes. When the results were compiled and analyzed, discrepancies were discovered in some of the question's characteristics. The findings imply that subconscious differences in teaching style might exist when dealing with middle school students on different mathematics tracks.

Committee: Dr. Janet Bobango (Advisor) Subjects:

Doctor of Philosophy, The Ohio State University, 2007, Mathematics

A noncommutative Jordan Algebra, J, of degree two can be constructed from an anticommutative algebra S that has a symmetric associative bilinear form. If additional conditions are put on the algebra S, information about the derivations and automorphisms of J can be obtained. If S is a n+1 dimensional algebra, and T is a nonsingular linear transformation on S, it is of interest to know what multiplications and what nondegenerate symmetric associative bilinear forms, can be put on S so that T(T(x)T(y))=xy for all x,y,z in S, and T is equal to its adjoint. If T has only one Jordan block the question is answered, in the form of conditions that must be satisfied on the multiplication constants. It is shown such algebras exist for all n and it is shown how to obtain the multiplication tables

Committee: Bostwick Wyman (Advisor) Subjects: Mathematics

Doctor of Philosophy (PhD), Ohio University, 2011, Mathematics (Arts and Sciences)

We study various connections between the notions of invertibility of elements and linear independence of subsets of algebras over (not necessarily commutative) rings. The main emphasis is on two such notions: invertible and fluid algebras. We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras. The conditions considered in that hierarchy include the requirement that for an invertible basis B, the set of inverses B-1 be itself a basis, the notion that B be closed under inverses and the idea that B be closed under products under a slight commutativity requirement. Among other results, it is shown that this last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any field F not equal to F2, all semisimple F-algebras are invertible. Semisimple invertible F2-algebras are fully characterized. Likewise, the question of which single-variable polynomials over a field yield invertible quotient rings of the F-algebra F[x] is completely answered. Connections between invertible algebras and S-rings (rings generated by units) are also explored. While group rings are the archetype of invertible algebras, this notion is general enough to include many other families of algebras. For example, field extensions and all crossed products (including in particular skew and twisted group rings) are invertible algebras. We consider invertible bases B such that for any two elements from B, a scalar multiple of their product belongs to B. Alternatively, one may consider invertible bases with the requirement that for every basis element, a scalar multiple of its inverse must also be in the basis. We refer to these algebras, respectively, as being scalarly closed under products and scalarly closed under inverses. We explore connections between these ide (open full item for complete abstract)

Committee: Sergio Lopez-Permouth (Advisor); Dinh Huynh (Committee Member); Franco Guerrerio (Committee Member); Jeffery Dill (Committee Member) Subjects: Mathematics

Master of Arts, The Ohio State University, 1961, Graduate School

Committee: Not Provided (Other) Subjects:

Master of Arts, The Ohio State University, 1935, Graduate School

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Master of Arts, The Ohio State University, 1948, Graduate School

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Master of Arts, The Ohio State University, 1917, Graduate School

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Master of Arts, The Ohio State University, 1960, Graduate School

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Master of Arts, The Ohio State University, 1921, Graduate School

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Master of Science, The Ohio State University, 1957, Graduate School

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Master of Science, The Ohio State University, 1964, Graduate School

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Master of Arts, The Ohio State University, 1920, Graduate School

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Master of Science, The Ohio State University, 2005, Graduate School

Committee: Not Provided (Other) Subjects: