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This thesis is comprised of two main results. First, we examine the locations of eigenvalues of matrix products as the size of the matrices tend to infinity. From the previous results by O'Rourke, Renfrew, Soshnikov, and Vu, we see that under certain moment assumptions on the atom random variables, as *n* tends to infinity, the empirical spectral measure of the eigenvalues of the rescaled product *n*^{-m/2}*P _{n}* converges to a measure supported on a disk centered at the origin in the complex plane with radius depending on the atom random variables. In this work, we study the asymptotic location of eigenvalues which can fall outside of this disk. These outlying eigenvalues may be present when the random matrix product

The second result of this thesis studies the fluctuation of the eigenvalues of the rescaled product *n*^{-m/2}*P _{n}* . In particular, we define a linear statistic and use this to study the spectrum, or the collection of eigenvalues, of

We begin by generalizing the theory of division polynomials attached to an isogeny of elliptic curves, developed by Mazur and Tate, to isogenies of prinicipally polarized abelian varieties.

As an application, we produce a notion of a *p*-adic sigma function attached to a prinicipally polarized abelian variety of good ordinary reduction over a complete non-archimedean field of residue characteristic *p*.

Furthermore, we derive some the properties of the sigma function, many of which uniquely characterize the function.

Independently, a notion of a pair of *p*-adic Weierstrass zeta functions is produced for a smooth projective curve *C* of genus two with invertible Hasse--Witt matrix over a *p*-adically complete field of characteristic zero.

Using the explicit function theory afforded by Jacobians of genus two, general results about *p-adic sigma functions are made more descriptive and the zeta functions on C are compared to the second logarithmic derivatives of the sigma function on the Jacobian of C.*

We end with providing numerical evidence that the sectional curvature of the group D(S^{1})/S^{1}, the diffeomorphism group of the circle modulo its rotations, given the H^{1/2} metric is non-negative. We leave the proof, or search for counterexample, as open as well as discuss other possible topics of study which can follow as a result of this thesis.

This thesis contains many new results for several of these equations. We begin by outlining the original analytic theory as well as the Euler-Arnold theory which studies these equations as geodesic equations on infinite dimensional manifolds. We build on the work of Castro-Cόrdoba and Bauer-Kolev-Preston to show that every solution to the Wunsch equation, a special case of the generalized Constantin-Lax-Majda equation, blows up in finite time. This result also applies to the Constantin-Lax-Majda equation itself. We also investigate the Euler-Weil-Petersson equation which has significant links to the Wunsch equation in the context of Teichmüller theory.

Additionally, we lay the foundations for a geometric theory of the surface quasi-geostrophic equation (SQG). Originally discovered in the context of geophysical fluid mechanics (see Pedlosky), SQG was proposed by Constantin-Majda-Tabak as a 2D version of the 1D Constantin-Lax-Majda equation. In a blog post, Tao showed that SQG arises as the critical point of a functional. This discovery naturally leads to the formulation of SQG as an Euler-Arnold equation. In this thesis we show that the associated geometric space has a smooth, non-Fredholm Riemannian exponential map, and has arbitrarily large curvature of both signs.

Finally we discuss the geometric setting for the Axi-symmetric Euler equations. Here we consider a 3D analogue of the 2D flows considered in Preston. Surprisingly, while the 2D flows exhibit negative curvature, we show that the corresponding 3D flows exhibit positive curvature and a rich structure of conjugate points. Such a result may have significant ramifications for our understanding of the nature of stability in 2D and 3D fluids.

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