We build the foundation for a theory of controlled rough paths

on manifolds. A number of natural candidates for the definition of manifold

valued controlled rough paths are developed and shown to be equivalent. The

theory of controlled rough one-forms along such a controlled path and their

resulting integrals are then defined. This general integration theory does

require the introduction of an additional geometric structure on the manifold

which we refer to as a "parallelism." The

transformation properties of the theory under change of parallelisms is

explored. Using these transformation properties, it is shown that the

integration of a smooth one-form along a manifold valued controlled rough path

is in fact well defined independent of any additional geometric structures. We

present a theory of push-forwards and show how it is compatible with our

integration theory. We give a number of characterizations for solving

a rough differential equation when the solution is interpreted as a controlled

rough path on a manifold and then show such solutions exist and are unique.

We develop the notion of parallel translation along a controlled rough path. This lends itself to a theory of rolling and unrolling maps for not only controlled rough paths but

controlled rough one-forms.

This dissertation is divided into two parts. In Part I of this dissertation--- On the Classical Limit of Quantum Mechanics, we extend a method introduced by Hepp in

1974 for studying the asymptotic behavior of quantum expectations in the limit

as Plank's constant ($\hbar$) tends to zero. The goal is to allow for

unbounded observables which are (non-commutative) polynomial functions of the

position and momentum operators. [This is in contrast to Hepp's original paper

where the "observables" were, roughly speaking, required to be bounded functions

of the position and momentum operators.] As expected the leading order

contributions of the quantum expectations come from evaluating the "symbols" of the observables along the classical trajectories while the next order contributions (quantum

corrections) are computed by evolving the $\hbar=1$ observables by a linear canonical

transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.

Part II of the dissertation --- Powers of Symmetric Differential Operators is devoted to operator theoretic properties of a class of linear symmetric differential operators on the real line. In more detail, let $L$ and $\tilde{L}$ be a linear symmetric differential operator with polynomial coefficients on $L^{2}\left(m \right) $ whose domain is the Schwartz test function space, $\mathcal{S}.$ We study conditions on the polynomial coefficients of $L$ and $\tilde{L}$ which implies operator comparison inequalities of the form $\left( \overline{\tilde{L}}+\tilde{C}\right) ^{r}\leq C_{r}\left( \bar{L}+C\right) ^{r}$

for all $0\leq r<\infty.$ These comparison inequalities (along with their generalizations

allowing for the parameter $\hbar>0$ in the coefficients) are used to supply a

large class of Hamiltonian operators which verify the assumptions needed for

the results in Part I of this dissertation.

A combination of the method of perturbation formulas and polynomial approximation is employed to compute the $k^{\text{th}}$ derivatives of (1) maps on symmetrically normed ideals of a unital Banach algebra induced by a holomorphic function, (2) maps on the self-adjoint elements of symmetrically normed ideals of a unital $C^*$-algebra induced by functions of a real variable that are ``slightly better than $C^k$,'' and (3) maps on the self-adjoint elements of integral symmetrically normed ideals of a von Neumann algebra induced by functions of a real variable that are ``slightly better than $C^k$ and Lipschitz.'' Along the way, the ``separation of variables'' approach to defining multiple operator integrals on non-separable Hilbert spaces is developed. As an application to free probability, a free It\^o formula of P. Biane and R. Speicher is extended, reinterpreted, and made more computationally flexible.