<p> We consider a one dimensional random walk in a random environment (RWRE) with a positive speed lim<i>_n</i>_{&rarr;&infin;} (<i>X_n/</i>) = &upsi;_&alpha; > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities <i>P</i>_&omega;(<i>X_n</i> &lt; <i>xn</i>) with <i> x</i>&isin; (0,&upsi;_&alpha;) decay approximately like exp{-<i> n</i>^1-1/<i>^s</i>} for a deterministic <i> s</i> > 1. More precisely, they showed that <i>n</i>^-&gamma; log <i>P</i>_&omega;(<i>X_n</i> &lt; <i>xn</i>) converges to 0 or -&infin; depending on whether &gamma; > 1 - 1/<i>s</i> or &gamma; &lt; 1 - 1/<i> s</i>. In this paper, we improve on this by showing that <i>n</i>^-1+1/<i>^s</i> log <i>P</i>_&omega;(X_n &lt; <i>xn</i>) oscillates between 0 and -&infin; , almost surely.</p>

<p>In this thesis, I present a combinatorial formula for a symmetric invariant quartic form on a spin module for the simple Lie algebra <b>d</b>₆. This formula relies on a description of this spin module as a vector space with weights, and weight vectors, indexed by ideals of a particular heap. I describe a new statistic, the profile, on pairs of heap ideals. The profile efficiently encodes the shape of the symmetric difference between the two ideals and demonstrates the available actions of the Weyl group and Lie algebra on any given pair. From the profile, I identify a property called a crossing. The actions of the Weyl group and Lie algebra on pairs of weights may be interpreted as adding or removing crossings between the corresponding ideals. Using the crossings, I present a formula for the symmetric invariant quartic form on a spin module for <b>d</b>₆, and discuss potential applications to other closely related minuscule representations. </p>

This paper develops several methods to estimate a future volatility of a stock in order to correctly price corresponding stock options. The pricing model known as Black-Scholes-Merton is presented with a constant volatility parameter and compares it to stochastic volatility models. It mathematically describes the probability distribution of the underlying stock price changes implied by the models and the consequences. Arbitrage opportunities between stock options of various maturities or strike prices are explained from the volatility smile and volatility term structure.

For a positive integer 𝛭 and an odd prime p, Sharifi defined a map 𝜛M from the first homology group of the modular curve X₁(𝛭) with Zₚ-coefficients to a second Galois cohomology group over ℚ(µM) with restricted ramification and Zₚ(2)-coefficients that takes Manin symbols to certain cup products of cyclotomic 𝛭-units. Fukaya and Kato showed that if p|𝛭 and p ≥ 5, then 𝜛Mₚ and 𝜛M are compatible via the map of homology induced by the quotient X₁(𝛭p) -> X₁ (𝛭) and corestriction from ℚ(µMₚ) to ℚ(µM). We show that for a prime 𝓁∤𝛭,𝓁≠p ≥ 5, the maps 𝜛M𝓁 and 𝜛M are again compatible under a certain combination of the two standard degeneracy maps from level 𝛭𝓁 to level 𝛭 and corestriction.

<p> In 1980, Murray Marshall proved that the category of Quaternionic Structures is naturally equivalent to the category of abstract Witt rings. This paper develops a combinatorial theory for finite Quaternionic Structures in the case where 1 = &ndash;1, by demonstrating an equivalence between finite quaternionic structures and Steiner Triple Systems (STSs) with suitable block colorings. Associated to these STSs are Block Intersection Graphs (BIGs) with induced vertex colorings. This equivalence allows for a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associated to the Witt rings of so-called elementary type, by providing necessary and sufficient conditions for a BIG associated to a product or group extension.</p>

<p> We prove that the non-<i>k</i>-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, <i>K</i>_1,3, <i>K</i>_1,4, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.</p>

This thesis explores the construction of shapes and, in particular, fractal-type shapes as fixed points of contractive iterated function systems as discussed in Michael Barnsley's 1988 book ``Fractals Everywhere." The purpose of the thesis is to serve as a resource for an undergraduate-level introduction to the beauty and core ideas of fractal geometry, especially with regard to visualizations of basic concepts and algorithms.

In the study of partial differential equations (PDEs) one rarely finds an analytical solution. But a numerical solution can be found using different methods such as finite difference, finite element, etc. The main issue using such numerical methods is whether the numerical solution will converge to the “real" analytical solution and if so how fast will it converge as we shrink the discretization parameter. In the first part of this thesis discrete versions of well known inequalities from analysis are used in proving the convergence of certain numerical methods for the one dimensional Poisson equation with Dirichlet boundary conditions and with Neumann boundary conditions. A matrix is monotone if its inverse exists and is non-negative. In the second part of the thesis we will show that finite difference discretization of two PDEs result in monotone matrices. The monotonicity property will be used to demonstrate stability of certain methods for the Poisson and Biharmonic equations. Convergence of all schemes is also shown. / October 2016

In this thesis, we discuss several problems in the representation theory of finite groups of Lie type. In Chapter 2, we will give essential background material that will be useful for the entirety of the thesis. We will investigate the construction of groups of Lie type, as well as their representations. We will define the field of values of a character afforded by a representation, and state useful results concerning these fields. In Chapter 3, we examine Zsigmondy primes and their existence, a necessary ingredient in proving our main results. In Chapters 4 and 5, we describe our main results in the ordinary and modular cases, which we now summarize. A finite group G is said to be m-rational, for a fixed positive integer m, if [Q((x)) : Q]|m for any irreducible character x∈Irr(G). In 1976, R. Gow studied the structure of solvable rational groups (i.e. m = 1), and found that the possible composition factors of a solvable rational group are cyclic groups of prime order p ∈ {2,3,5}[22]. Just over a decade later, W. Feit and G. Seitz classified the possible non-abelian composition factors of (non-solvable) rational groups. In 2008, J. Thompson found an upper bound of p ≤ 13 for the order of the possible cyclic composition factors of an arbitrary rational group, and conjectured that the bound can be improved to p ≤ 5. More recently, J. McKay posed the question of determining the structure of quadratic rational groups (i.e. m = 2). J. Tent studied the cyclic composition factors of solvable quadratic rational groups in 2013. In Chapter 4, we answer McKay's question concerning non-abelian composition factors, and generalize our results to non-solvable m-rational groups. Modular character theory was founded by R. Brauer in the 1930's, and has been useful in proving historical results including the classification of finite simple groups. In Chapter 5, we prove the modular version of our results. Though our conclusions are similar to those found in the complex case, the methods for proving the results are typically much more complicated.

A generalization of the weight part of Serre's conjecture asks for which Serre weights a given mod p representation of the absolute Galois group of Q is modular. This set is expected to depend only on the restriction of the representation to the Galois group of Q_p. Let rho be a continuous representation of the absolute Galois group of Q_p into GL_n(F_p) that is moreover semisimple. Gee, Herzig, and Savitt [GHS16] defined a certain set W_expl(rho) of Serre weights (which is defined in a very explicit way) that is conjectured to be the correct set of Serre weights as long as rho is sufficiently generic.However, in the non-generic cases that occur in dimensions greater than three, it is not known whether this set behaves in the way it should under certain functorial operations, like tensor products. This thesis shows that in dimension four, the set of explicit Serre weights W_expl(rho) defined in [GHS16] is closed under taking tensor products of two two-dimensional representations.