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The locus of periodic cycles is of fundamental importance to any discrete time dynamical system. We introduce a family Cyc_p(F) of marked cycle varieties that parameterize the $p$-cycles of an algebraic family of dynamical systems. We describe a dynamically natural cell structure for marked cycle curves over the moduli space Per(1) of quadratic polynomials and over the moduli space Per(2) of quadratic rational maps with a 2-periodic critical point. We then analyze the combinatorics of the resulting cell structures, obtaining formulas for the number of cells of each dimension.

Visualization of sets in Euclidean space that possess notions of non-integer dimension has lead to a great deal of curiosity and ideas that connect to many traditional fields in and outside of mathematics. Recently, the study of open subsets of R and their boundaries paved the way for a new definition in the work of Lapidus and van Frankenhuijsen for the term `fractal.' This definition reveals the scaling properties of the relevant sets and serves as continuing motivation for the study of zeta functions that encode a notion of dimension of a corresponding set. There are, in addition, measures called multifractals that possess fractal-like properties. It is the purpose of this dissertation to investigate the properties of self-similar multifractal measures through a very intimate connection to the classically studied hypergeometric functions. One may decompose the support of these measures into a collection of fractal sets each of which possesses the same scaling property. Zeta functions associated to each fractal set in this decomposition encode this generalized notion of dimension, yet they are shown in many cases to be exactly of the form of the generalized hypergeometric fuctions. The classical study of hypergeometric functions boasts the attention of mathematicians such as Euler, Gauss, and Riemann, yet as with many special functions the applications continue to be revealed. Some invaluable tools in understanding the generalized hypergeometric functions, as investigated herein, are the associated hypergeometric differential equation and monodromy matrices.

The theme of this dissertation is the study of fundamental groups and classifying spaces in the context of the étale topology of schemes. The main result is the existence of K(pi,1) neighborhoods in the case of semistable (or more generally log smooth) reduction, generalizing a result of Gerd Faltings. As an application to p-adic Hodge theory, we use the existence of these neighborhoods to compare the cohomology of the geometric generic fiber of a semistable scheme over a discrete valuation ring with the cohomology of the associated Faltings topos. The other results include comparison theorems for the cohomology and homotopy types of several types of Milnor fibers. We also prove an l-adic version of a formula of Ogus, describing the monodromy action on the complex of nearby cycles of a log smooth family in terms of the log structure.

In this thesis, we study the compatibility between local and global Langlands correspondences for \(GL_n\). This generalizes the compatibility between local and global class field theory and is related to deep conjectures in algebraic geometry and harmonic analysis, such as the Ramanujan-Petersson conjecture and the weight monodromy conjecture. Let L be a CM field. We consider the case when \(\Pi\) is a cuspidal automorphic representation of \(GL_n(\mathbb{A}_L^\infty)\), which is conjugate self-dual and regular algebraic. Under these assumptions, there is an l-adic Galois representation \(R_l(\Pi)\) associated to \(\Pi\), which is known to be compatible with the local Langlands correspondence in most cases (for example, when n is odd) and up to semisimplification in general. In this thesis, we complete the proof of the compatibility when \(l \neq p\) by identifying the monodromy operator N on both the local and the global sides. On the local side, the identification amounts to proving the Ramanujan-Petersson conjecture for \(\Pi\) as above. On the global side it amounts to proving the weight-monodromy conjecture for part of the cohomology of a certain Shimura variety.

The quantum and semi-quantum mechanics of rigid molecules rotating in static electric and intense nonresonant laser fields have been studied. Basic quantum features including energy level correlation diagrams, eigenfunctions, alignment, orientation, and time-dependent wave functions propagated under pulsed fields are calculated. Energy level nearest neighbor spacing disributions are the primary semi-quantum approach, but Husimi distributions, periodic orbits, and monodromy diagrams are also investigated. We have elucidated these facets of the quantum-classical correspondence in this important rotational model, which is classically nonscaling with mixed phase space.

We investigate the classical mechanics of diatomic and symmetric top molecules in tilted fields. These molecules exhibit regular, chaotic or mixed phase space depending on the tilt angle $\beta$, the energy $E$, and the relative intensity of the fields $\omega/\Delta\omega$. In the integrable collinear problem the projection of the angular momentum into the spatial $z$ axis is a constant of motion, $m$, which allows us to explore the geometry of the phase space, and to use energy momentum diagrams to classify the motions of the rotor. For $\beta \ne 0$ the system is non-integrable showing mostly regular dynamics in the high-energy (free-rotor) and low-energy (pendular) limits; for energy near the tilted fields barrier the phase space is highly chaotic with degree of chaos increasing with $\beta$ between 0 and $\pi/2$. Periodic orbits and bifurcation diagrams are obtained from symmetry lines and their iterations under the Poincar\'{e} map. These bifurcation diagrams are used to observe the changes in the basic structure of the phase space as $\beta$ changes between collinear and perpendicular fields. Some quantum eigenstates are localized near stable or unstable periodic orbits showing tori quantization or scarring respectively. For asymmetric top molecules only the case of collinear fields is treated. In parallel fields $m$ is a constant of the motion and it is possible to define an effective potential $V_m(\theta,\psi)$. In an $E$-$m$ diagram the equilibrium solutions of $V_m(\theta,\psi)$ are curves that enclose regions of qualitatively different accessible $\theta$-$\psi$ configuration space. Interestingly these regions can be used to classify the quantum eigenstates. For plane rotors primitive semiclassical mechanics is used to calculate the rotational excitation caused by laser pulses. Depending on the pulse intensity and duration several methods are employed from the analytical sudden approximation to primitive semiclassical initial value representation (IVR) integrals. The calculated transition probabilities are in good agreement with the quantum probabilities considering the simplicity of the methods. In the case of plane rotors in electric fields we calculate energy spectra, orientation ($\left$), and alignment ($\left$), using the Herman-Kluk propagator in terms of periodic coherent states. These results are in good agreement with the quantum analogues although the number of trajectories used is discouragingly large. For diatomic rotors in tilted fields, the HK propagator was used to calculate energy spectra with good agreement for high-energy and not very dense eigenspectra. Some steps are taken towards the development of HK-type propagator for rotational coherent states.