The Effros-Shen algebra corresponding to an irrational number θ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of θ encodes the dimensions of the summands, and how the matrix algebras at the nth level fit into the summands at the (n+1)th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the C*-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of θ. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In this thesis, the author defines a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This thesis describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.

Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer group of $E$ is $\Lambda(\Gamma)$-cotorsion when $E$ has good ordinary reduction at all primes of $K$ lying over $p$. The conjecture was proven in the case that $K$ is the field of rationals by Kato, but is known to be false when $E$ has supersingular reduction type. To salvage this result, Kobayashi introduced the signed Selmer groups, which impose stronger local conditions than their classical counterparts. Part of the construction of the signed Selmer groups involves using Honda's theory of commutative formal groups to define a canonical system of points. In this paper I offer an alternate construction that appeals to the Functional Equation Lemma, and explore a possible way of generalizing this method to elliptic curves defined over $p$-adic fields by passing from formal group laws to formal modules.

The author employs bundle theory to investigate dynamics on C*- algebras. Using methods old and new to define dynamics on topological spaces (often with additional structure), implications of the dynamics are investigated in the non-commutative setting, and in suitable situations the dynamics are classified. As a result, new Morita equivalence results are derived and new settings introduced in the study of crossed products, whether by group coactions or by actions of groups and groupoids.

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $R(\sigma)$. Any reduced word of $\sigma$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid moves or only commutation moves. This document will present work towards a conjectured relationship between the number of reduced words and the number of braid classes. The set $R(\sigma)$ can be drawn as a graph, $G(\sigma)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper will present brand new work on subgraph structures in $G(\sigma)$, as well as new formulas to count the number of braid edges and commutation edges in $G(\sigma)$. The permutation $\sigma$ covers $\tau$ in the weak order poset if the length of $\tau$ is one less than the length of $\sigma$, and there exists a simple transposition $s_i$ such that $\sigma = \tau s_i$. This paper will cover new work on the relationships between the size of $R(\sigma)$ and $R(\tau)$, and how this creates a new method of writing reduced decompositions of $\sigma$ as products of permutations $\alpha$ and $\beta$, where both $\alpha$ and $\beta$ have a length greater than one. Finally, this thesis will also discuss how these results help relate the number of reduced words and the number of braid classes in certain cases.

This thesis explores several questions concerning the preservation of geometric structure under the Ricci flow, an evolution equation for Riemannian metrics. Within the class of complete solutions with bounded curvature, short-time existence and uniqueness of solutions guarantee that symmetries and many other geometric features are preserved along the flow. However, much less is known about the analytic and geometric properties of solutions of potentially unbounded curvature. The first part of this thesis contains a proof that the full holonomy group is preserved, up to isomorphism, forward and backward in time. The argument reduces the problem to the preservation of reduced holonomy via an analysis of the equation satisfied by parallel translation around a loop with respect to the evolving metric. The subsequent chapter examines solutions satisfying a certain instantaneous, but nonuniform, curvature bound, and shows that when such solutions split as a product initially, they will continue to split for all time. This problem is encoded as one of uniqueness for an auxiliary system constructed from a family of time-dependent, orthogonal distributions of the tangent bundle. The final section presents some details of an ongoing project concerning the uniqueness of asymptotically product gradient shrinking Ricci solitons, including the construction of a certain system of mixed differential inequalities which measures the extent to which such a soliton fails to split.

The Jordan curve theorem states that any homeomorphic copy of a circle into R2 divides the plane into two distinct regions. This paper reconstructs one proof of the Jordan curve theorem before turning its attention toward generalizations of the theorem and their proofs and counterexamples. We begin with an introduction to elementary topology and the different notions of the connectedness of a space before constructing the first proof of the Jordan curve theorem. We then turn our attention to algebraic topology which we utilize in our discussion of the Jordan curve theorem’s generalizations. We end with a proof of the Jordan-Brouwer theorems, extensions of the Jordan curve theorem to higher dimensions.

In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1).

The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$.

I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.

Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.

Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.

Lights Out is a puzzle game where the goal is to turn off all the lights on a nxn board starting from a random configuration. In order to find the solution of a configuration, the game is constructed using a matrix basis in the span of the field Z mod 2.This the game can be modeled by the system Ap=s which will be the center of the investigation when determining the solvability for any n×n board since A is not always invertable leading to some interesting cases. The goal of this thesis was to construct a model that will allow the player to solve for the pushes to attain the zero-state for an nxn system. Constructing the model gave a procedure that will allow to solve the puzzle game. The procedure presented here first uses a simple clearing technique (valid for any board size) to turn off all the lights except in the last row, which we call the standard-clear. The heart of the technique, is to give a way to use the information about which lights remain lit in the last row to determine which switches in the first row need to be pushed before the standard-clear. This part of the solution algorithm we call the first row adjustment, and it depends heavily on the specific board size n of the problem. Finally, after these first row pushes are made, the standard clear will now turn off all the lights including (seemingly magically) the last row. Thus the solution to the Lights Out puzzle of a given size is reduced to finding a first row adjustment for that size. (Please refer to the actual thesis for the full abstract)