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

Thirty years ago, Effros and Shen classified the simple dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion. There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group. We also consider a related issue: an Effros-Shen group has been shown to arise in a natural way from the tail equivalence relation on a certain sequence space. We describe a more general class of sequence spaces to which this construction can be applied to obtain other dimension groups, including dimension groups corresponding to multidimensional continued fractions.

In this thesis several classes of directed paths are considered as models of linear polymers in a dilute solution. We obtain the generating functions for each model by considering factorization arguments. Information about the polymer behaviour can be extracted from the singularity structure of the associated generating functions. By using modified versions of these models we study the adsorption and localization of polymer molecules, the behaviour of polymers subject to a tensile force, the effects of stiffness, as well as the behaviour of polymers in confined geometries. In each of these situations the resulting generating functions contain at least two physical singularities. We identify the phase transitions in these systems by a changeover in the dominant singularity of the generating function. In the study of localization and polymers subject to a force, we utilize both homopolymer and random copolymer models. For copolymers, the physically relevant properties are obtained by considering a quenched average of the free energy over all possible monomer sequences. This procedure is intractable even for the simplest models. By considering the Morita approximation for several walk models we obtain results which give a bound on the corresponding features of the quenched system. We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin path in order to study the mechanical unzipping of duplex DNA. From this model, we obtain force-temperature diagrams which show re-entrant behaviour of the force. We also develop a simple low temperature theory to describe the behaviour of the force close to T=0 and find that the shape of the force-temperature curve is associated with entropy in the ground state of the system. We consider the effect of stiffness on polymer adsorption and find that the phase transition is second order for all finite stiffness parameters. For systems of polymers in confined geometries, we find that the behaviour of the polymer depends on the distance between the confining surfaces and the associated interactions with each surface. In this problem, there exist regimes where the polymer exerts a force on the surfaces which can be attractive, repulsive or zero.

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.