This thesis studies the spacing between eigenvalues of random matrices with independent and identically distributed (iid) entries. Tail estimates on the minimum distance between any pair of eigenvalues are proven. In particular, we establish that the spectrum of an iid random matrix is simple with high probability. A key technical result is a new least singular value tail estimate for shifted matrices of the form $A_n-zI_n$, where $A_n$ is an iid random matrix with real entries and $z$ is a complex scalar.

In this dissertation we consider the problem of finding small prime gaps in various sets $\mathcal{C} \subset \N$. To approach this problem we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Following the work of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, we let $q_n$ denote the $n$-th prime in $\mathcal{C}$ and establish that for some ``small" constant $\epsilon>0$, the set $\left\{ q_n| q_{n+1}-q_n \leq \epsilon \log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{C}$ has bounded prime gaps. Specific examples, such as the case where $\mathcal{C}$ is an arithmetic progression have already been studied and so the purpose of this dissertation is to present results for general classes of sets.\

We also prove a second moment estimate for the Maynard-Tao sieve and give an application to Goldbach and de Polignac numbers. We show that at least one of two nice properties holds. Either consecutive Goldbach numbers lie within a finite distance from one another or else the set of de Polignac numbers has full density in $2 \N$.

This thesis treats nonlinear dispersive equations with random initial data. First, we study the defocusing energy-critical nonlinear wave equation on Euclidean space. We prove that the scattering mechanism, which is well-understood for smooth initial data, is stable under rough and random perturbations. The main ingredients are Bourgain's bush argument, flux estimates, and a wave packet decomposition of the random linear evolution. Second, we study the three-dimensional wave equation with a Hartree nonlinearity. The main theorem proves the existence and invariance of the Gibbs measure. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. The argument combines techniques from several areas of mathematics, such as dispersive equations, harmonic analysis, and random matrix theory.

The first part of the thesis focuses on the uniformity of harmonic analysis estimates on curves. We first show a decomposition theorem for polynomial curves on local fields as a bounded number of perturbations of monomial curves. Using this theorem, we extend uniform restriction estimates for real curves to the endpoint case, show uniform decoupling for those curves, and show novel uniform restriction estimates for curves over C, and Qp. We then show uniform estimates for the discrete analog to this problem in a restricted range of exponent.

The second part focuses on decoupling estimates for sets with a product or self-similar structure. A recurring phenomenon for those sets is that functions with constant Fourier transform on their support are far from extremizers. As applications we will show a de- coupling estimate for fractal subsets of the parabola, and study subsets of cubes with high additive energy compared to their cardinality.

We examine the L^p norm dimensional asymptotics of spherical and ball maximal averaging operators on Cartesian powers of finite graphs. We extend the past techniques that prove that the spherical maximal operators on Cartesian powers of finite cliques satisfy L^p bounds independent of dimension for all p ∈ (1, ∞]. We use these tools to prove computable sufficient conditions for dimension-independent L^2 bounds on the ball and spherical maximal operators on powers of some distance regular graphs. For instance, our techniques prove that the ball maximal operators on powers of the 5-cycle graph and the spherical maximal operators on powers of the Petersen graph each satisfy dimension-independent L^2 bounds.

We go on to discuss some less well-behaved asymptotics. We prove that for a substantial family of graphs, including all trees with more than 2 vertices and even some regular graphs, the spherical maximal averaging operators have exponentially growing L^p norms with respect to dimension for all p ∈ [1, ∞). We prove that for another family, the ball maximal averaging operators have exponentially growing L^p norms for some p > 1. Finally, we identify some graphs whose behavior lies between these two extremes and discuss a few related open questions.

A subset A of a locally compact group G is said to be a K-approximate group if it contains the identity, is closed under inverses, and if the product set AA = {ab | a,b \in A} can be covered by K left-translates of A. In this work we prove a structure theorem on open precompact K-approximate groups that describes them as a combination of compact subgroups, convex sets in Lie algebras of dimension bounded in terms of K, and certain generalizations of arithmetic progressions. Our results can naturally be regarded as generalizations of a theorem of Breuillard, Green, and Tao that describes finite approximate groups, and also of a theorem of Gleason and Yamabe, which describes locally compact groups as inverse limits of Lie groups.

Along the way we prove continuous analogues of additive combinatorial results of Sanders, Croot, and Sisask, and model-theoretic results of Hrushovski, all of which were first obtained in the finite context.