This thesis concerns statistical patterns among the zeros of the Riemann zeta function, and conditioned on the Riemann hypothesis proves several related original results. Among these:

By extending a well known result of H. Montgomery, we show, at an only microscopically blurred resolution, that the distance between two randomly selected zeros of the zeta function tends to weakly repel away from the location of low-lying zeros of the zeta function.

For random collections of consecutive zeros that are not so large as to see this resurgence effect, we support the view that they resemble the bulk eigenvalues of a random matrix by in particular proving an analogue of the strong Szego theorem.

Concerning even smaller collections of zeros, we show that a statement that the zeros of the Riemann zeta function locally resemble the eigenvalues of a random matrix (the GUE Conjecture) is logically equivalent to a statement about the distribution of primes. On this basis, we make a conjecture for the covariance in short intervals of integers with fixed numbers of prime factors, weighted by the higher order von Mangoldt function. This is related to the so-called ratio conjecture. The covariance pattern is surprisingly simple to write down.

We finally include a rigorous derivation that uniform variants of the Hardy-Littlewood conjectures agree with the GUE Conjecture. Even thus conditioned, the range of correlation test functions against which we may confirm the GUE pattern for zeta zeros remains limited. We consider in detail the case of two, three, and four point correlations, the two point case being due to Mongtomery.

This thesis concerns the Liouville function, the prime number theorem, the Erd\H{o}s discrepancy problem and related topics. We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce this theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $\kappa-1$-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\e})$ many words of length $n$ where $t = \kappa(\kappa+1)/2$. We prove a variant of the $1$-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension $< 1$. We give a new proof of the prime number theorem. We show how this proof can be interpreted in a dynamical setting. Along the way we give a new and improved version of the entropy decrement argument. We give a quantitative version of the Erd\H{o}s discrepancy problem. In particular, we show that for any $N$ and any sequence $f$ of plus and minus ones, for some $n \leq N$ and $d \leq \exp(N)$ that $| \sum_{i \leq n} f(id) | \geq (\log \log N)^{\frac{1}{484} - o(1)}$.

Consider a velocity field $u$ solving the incompressible Navier-Stokes equations on $[0,T]\times\mathbb R^d$ ($d\geq3$) and satisfying $\|u(t)\|_X\leq A$ for all times, where the norm $X$ is critical with respect to the Navier-Stokes scaling. We prove several theorems to the effect that the regularity of the solution can be controlled explicitly in terms of $A$, building upon Tao's pioneering work on the case $d=3$, $X=L^3(\mathbb R^3)$. First we prove a generalization to the critical Lebesgue space in any number of spatial dimensions ($d\geq4$, $X=L^d(\mathbb R^d)$). Then we show a variety of circumstances under which Tao's bounds can be strengthened, including the case in which the solution is nearly axisymmetric. For exactly axisymmetric solutions, we prove regularity in terms of the weak norm $X=L^{3,\infty}(\mathbb R^3)$ which implies effective bounds on approximately self-similar behavior.

This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications.

We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16].

We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. This is a collaborated work with Arindam Biswas and published in the Journal of London Mathematical Society [BY17].

It is known that in various random matrix models, large perturbations create outlier eigenvalues which lie, asymptotically, in the complement of the support of the limiting spectral density. This thesis studies fluctuations of these outlier eigenvalues of iid matrices $X_n$ under bounded rank and bounded operator norm perturbations $A_n$, namely the fluctuations $\lam(\frac{X_n}{\sqrt{n}}+A_n)-\lam(A_n)$. The perturbations $A_n$ that we consider belong to a large class, where we allow for arbitrary Jordan types and almost minimal assumptions on the left and right eigenvectors. We obtain the joint convergence of the normalized asymptotic fluctuations of the outlier eigenvalues in this setting with a unified approach.

In this dissertation we establish various pointwise ergodic theorems, of both quanitative and qualitative nature. Beyond an introductory chapter, we devote one chapter to establishing quantitative ergodic theorems for $\Z^d$ actions; one chapter to quantitative ergodic theorems for polynomial orbits; one chapter to uniform estimates for various modulated ergodic averages; and the final chapter to random modulated averages.