Doctor of Philosophy / Department of Mathematics / Gerald H. Höhn / We classify the genera of doubly even binary codes, the genera of even lattices, and the genera of rational vertex operator algebras (VOAs) arising from the modular tensor categories (MTCs) of rank up to 4 and central charges up to 16. For the genera of even lattices, there are two types of the genera: code type genera and non code type genera. The number of the code type genera is finite. The genera of the lattices of rank larger than or equal to 17 are non code type. We apply the idea of a vector valued modular form and the representation of the modular group SL[subscript]2(Z) in [Bantay2007] to classify the genera of the VOAs arising from the MTCs of ranks up to 4 and central charges up to 16.

Master of Science / Department of Mathematics / Sarah Reznikoff / A (1, ≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1, ≤ 2)-identifiable. As such, we discuss properties that must be satisfied by a valid (1, ≤ 2)-identifying code, characteristics of a graph which preclude the existence of a (1, ≤ 2)-identifying code, and relationships between the maximum degree and order of (1, ≤ 2)-identifiable graphs. Additionally, we show that (1, ≤ 2)-identifiable graphs have no forbidden induced subgraphs and provide a list of (1, ≤ 2)-identifiable graphs with minimum (1, ≤ 2)-identifying codes indicated.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field of positive prime characteristic. Let L([lambda]) and [upside-down triangle]([lambda]) be the simple and induced finite dimensional rational G-modules with p-singular dominant highest weight [lambda]. In this thesis, the concept of Weyl filtration dimension of a finite dimensional rational G-module is studied for some highest weight modules with p-singular highest weights inside the p2-alcove when G is of type B[subscript]2. In chapter 4, intertwining morphisms, a diagonal G-module morphism and tilting modules are used to compute the Weyl filtration dimension of L([lambda]) with [lambda] p-singular and inside the p[superscript]2-alcove. It is shown that the Weyl filtration dimension of L([lambda]) coincides with the Weyl filtration dimension of [upside-down triangle]([lambda]) for almost all (all but one of the 6 facet types) p-singular weights inside the p[superscript]2-alcove. In chapter 5 we study some submodule structures of Weyl (and there translations), Vogan, and tilting modules with both p-regular and p-singular highest weights. Most results are for the p[superscript]2 -alcove only although some concepts used are alcove independent.

Doctor of Philosophy / Department of Mathematics / Yan S. Soibelman / Recall the classical Riemann-Roch theorem for curves: Given a smooth projective complex curve and two holomorphic vector bundles E, F on it, the Euler can be computed in terms of the ranks and the degrees of the vector bundles. Remarkably, there are a number of similarly looking formulas in algebra. The simplest example is the Ringel formula in the theory of quivers. It expresses the Euler form of two finite-dimensional representations of a quiver algebra in terms of a certain pairing of their dimension vectors. The existence of Riemann-Roch type formulas in these two settings is a consequence of a deeper similarity in the structure of the corresponding derived categories - those of sheaves on curves and of modules over quiver algebras. The thesis is devoted to a version of the Riemann-Roch formula for abstract derived categories. By the latter we understand the derived categories of differential graded (DG) categories. More specifically, we work with the categories of perfect modules over DG algebras. These are a simultaneous generalization of the derived categories of modules over associative algebras and the derived categories of schemes. Given an arbitrary DG algebra A, satisfying a certain finiteness condition, we define and explicitly describe a canonical pairing on its Hochschild homology. Then we give an explicit formula for the Euler character of an arbitrary perfect A-module, the character is an element of the Hochschild homology of A. In this setting, our noncommutative Riemann-Roch formula expresses the Euler characteristic of the Hom-complex between any two perfect A-modules in terms of the pairing of their Euler characters. One of the main applications of our results is a theorem that the aforementioned pairing on the Hochschild homology is non-degenerate when the DG algebra satisfies a smoothness condition. This theorem implies a special case of the well-known noncommutative Hodge-to-de Rham degeneration conjecture. Another application is related to mathematical physics: We explicitly construct an open-closed topological field theory from an arbitrary Frobenius algebra and then, following ideas of physicists, interpret the noncommutative Riemann-Roch formula as a special case of the so-called topological Cardy condition.

Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $p$ be an odd prime and $\gamma(k,p^n)$ be the smallest positive integer $s$ such that every integer is a sum of $s$ $k$-th powers $\pmod {p^n}$. We establish $\gamma(k,p^n) \le [k/2]+2$ and $\gamma(k,p^n) \ll \sqrt{k}$ provided that $k$ is not divisible by $(p-1)/2$. Next, let $t=(p-1)/(p-1,k)$, and $q$ be any positive integer. We show that if $\phi(t) \ge q$ then $\gamma(k,p^n) \le c(q) k^{1/q}$ for some constant $c(q)$. These results generalize results known for the case of prime moduli. Next we generalize these results to a number field setting. Let $F$ be a number field, $R$ it's ring of integers and $\mathcal{P}$ a prime ideal in $R$ that lies over a rational prime $p$ with ramification index $e$, degree of inertia $f$ and put $t=(p^f-1)/(p-1,k)$. Let $k=p^rk_1$ with $p\nmid k_1$ and $\gamma(k,\mathcal{P}^n)$ be the smallest integer $s$ such that every algebraic integer in $F$ that can be expressed as a sum of $k$-th powers $\pmod{\mathcal{P}^n}$ is expressible as a sum of $s$ $k$-th powers $\pmod {\mathcal{P}^n}$. We prove for instance that when $p>e+1$ then $\gamma(k,\mathcal{P}^n) \le c(t) p^{nf/ \phi(t)}$. Moreover, if $p>e+1$ we obtain the upper bounds $\ds{\gamma(k,\mathcal{P}^n) \le 2313 \left(\frac{k}{k_1}\right)^{8.44/\log p}+\frac{1}{2}}$ if $f=2$ or $3,$ and $\ds{\gamma(k,\mathcal{P}^n)\le 129 \left(\frac{k}{k_1}\right)^{5.55/ \log p}+\frac{1}{2}}$ if $f\ge4$. We also show that if $\mathcal{P}$ does not ramify then $\ds{\gamma(k,\mathcal{P}^n) \le \frac{17}{2} \left(\frac{k}{k_1}\right)^{2.83/ \log p}+\frac{1}{2}}$ if $f\ge 2$ and $k_1\le p^{f/2}$, and $\ds{\gamma(k,\mathcal{P}^n)\le\left(\frac{f}{p^{f/2-1}}\right)k}$ if $f> 2$ and $k_1> p^{f/2}$.

Doctor of Philosophy / Department of Mathematics / Christopher G. Pinner

Doctor of Philosophy / Department of Mathematics / Andrew Bennett / Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).

Master of Science / Department of Mathematics / Nathan Albin / This work utilizes finite differences to approximate the first derivative of non-periodic smooth functions. Math literature indicates that stabilizing Partial Differential Equation solvers based on high order finite difference approximations of spatial derivatives of a non-periodic function becomes problematic near a boundary. Hagstrom and Hagstrom have discovered a method of introducing additional grid points near a boundary, which has proven to be effective in stabilizing Partial Differential Equation solvers. Hagstrom and Hagstrom demonstrated their method for the case of the one-dimensional advection equation using spatial derivative approximations of even orders up to twenty-second order. In this dissertation, we explore the efficacy of the Hagstrom and Hagstrom method for the same Partial Differential Equation with spatial derivative approximations of odd orders and orders higher than twenty-two and report the number and locations of additional grid points required for stability in each case.

Doctor of Philosophy / Department of Mathematics / Pietro Poggi-Corradini / We consider iteration of an analytic self-map f of the unit ball in the N-dimensional complex space C[superscript]N. Many facts were established about such maps and their dynamics in the 1-dimensional case (i.e. for self-maps of the unit disk), and we generalize some of them in higher dimensions. In one dimension, the classical Denjoy-Wolff theorem states the convergence of forward iterates to a unique attracting fixed point, while backward iterates have much more complicated nature. However, under additional conditions (when the hyperbolic distance between two consecutive points stays bounded), backward iteration sequence converges to a point on the boundary of the unit disk, which happens to be a fixed point with multiplier greater than or equal to 1. In this paper, we explore backward-iteration sequences in higher dimension. Our main result shows that in the case when f is hyperbolic or elliptic, such sequences with bounded hyperbolic step converge to a point on the boundary, other than the Denjoy-Wolff (attracting) point. These points are called boundary repelling fixed points (BRFPs) and possess several nice properties. In particular, in the case when such points are isolated from other BRFPs, they are completely characterized as limits of backward iteration sequences. Similarly to classical results, it is also possible to construct a (semi) conjugation to an automorphism of the unit ball. However, unlike in the 1-dimensional case, not all BRFPs are isolated, and we present several counterexamples to show that. We conclude with some results in the parabolic case.