The first three chapters are mostly expository. Chapter 1 briefly recalls the necessary hyperbolic geometry and a geometric discussion of binary quadratic and Hermitian forms. Chapter 2 briefly recalls the relation between badly approximable systems of linear forms and bounded trajectories in the space of unimodular lattices (the Dani correspondence). Chapter 3 is a survey of continued fractions from the point of view of hyperbolic geometry and homogeneous dynamics. The chapter discusses simple continued fractions, nearest integer continued fractions over the Euclidean imaginary quadratic fields, and includes a summary of A. L. Schmidt’s continued fractions over Q(√−1).

Chapters 4 and 5 contain the bulk of the original research. Chapter 4 discusses a class of dynamical systems on the complex plane associated to polyhedra whose faces are two-colorable (i.e. edge-adjacent faces do not share a color). To any such polyhedron, one can associated a right-angled hyperbolic Coxeter group generated by reflections in the faces of a (combinatorially equivalent) right-angled ideal polyhedron in hyperbolic 3-space. After some generalities, we discuss a simpler system, billiards in the ideal hyperbolic triangle. We then discuss continued fractions over Q(√−1) and Q(√−2) coming from the regular ideal right-angled octahedron and cubeoctahedron.

Chapter 5 gives explicit examples of numbers/vectors in R^{r}×C^{s} that are badly approximable over number fields F of signature (r, s) with respect to the diagonal embedding. One should think of these examples as generalizations of real quadratic irrationalities, which we discuss first as our prototype. The examples are the zeros of (totally indefinite anisotropic F-rational) binary quadratic and Hermitian forms (the Hermitian case arises when F is CM). Such forms can be interpreted as compact totally geodesic subspaces in the relevant locally symmetric spaces SL_{2}(O_{F} )\SL_{2}(F⊗R)/SO_{2}(R)^{r} × SU_{2}(C)^{s}. We discuss these examples from a few different angles: simple arguments stemming from Liouville’s theorem on rational approximation to algebraic numbers, arguments using continued fractions (of the sorts considered in chapters 3 and 4) when they are available, and appealing to the Dani correspondence in the general case. Perhaps of special note are examples of badly approximable algebraic numbers and vectors, as noted in 5.10.

Chapter 6 considers approximation in R^{n} (in the boundary ∂H^{n} of hyperbolic n−space) over “weakly Euclidean” orders in definite Clifford algebras. This includes a discussion of the relevant background on the “SL_{2}” model of hyperbolic isometries (with coefficients in a Clifford algbra) and a discription of the continued fraction algorithm. Some exploration in the case Z^{3} ⊆ R^{3} is included, along with proofs that zeros of anisotropic rational Hermitian forms are “badly approximable,” and that the partial quotients of such zeros are bounded (conditional on increasing convergent denominators).

Chapter 7 considers simultaneous approximation in R^{r} × C^{s} as a subset of the boundary of (H^{2})^{r} × (H^{3})^{s} over a diagonally embedding number field of signature (r, s). A continued fraction algorithm is proposed for norm-Euclidean number fields, but not even convergence is established. Some exploration and experimentation over the norm-Euclidean field Q(√2) is included.

Finally, chapter 8 includes some miscellaneous results re

]]>Simulations show that, in fact, the roots and critical points of *p _{n}* "pair-up" with one another in a nearly one-to-one fashion, a phenomenon which has been initially investigated by Hanin, O'Rourke, Kabluchko and Seidel, the author, and others. This thesis seeks to quantify root-and-critical-point pairing on several scales, including macroscopic comparisons between entire collections of roots and critical points, microscopic examinations of individual critical points that lie near fixed roots, and a "mesoscopic" local law to explain the situation at scales in between.

In Chapter 2, we show that for a deterministic point ξ lying outside the support of μ, almost surely the polynomial *q _{n}*(

In Chapter 3, under a regularity assumption, we show that if the roots of *p _{n}* are iid, the Wasserstein distance between the empirical distributions of roots and critical points of

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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.*