This thesis contains research conducted on various topics in quantum Hall physics and deep learning theory. The first chapter studies a particular aspect of quantum Hall systems, namely their behavior around the $\nu = 1/2$ Landau level (LL) state. This work is motivated by the need to understand better this particular state in light of the two proposed distinct theoretical descriptions existing for the same. Specifically, we analyze quantum oscillations around the $\nu = 1/2$ LL state using one of the propositions to support the latter. The second and third chapters study two distinct domains in deep learning, multi and single-objective models. In particular, the second considers a specific type of multi-objective model, zero-sum games, to demonstrate existing issues in training such setups and develop an efficient optimization scheme. The final chapter involves studying a particular aspect of the generalization behavior of deep neural networks (DNNs). Specifically, it attempts to provide a theoretical framework to explain the recently observed phenomenon of "epoch-wise double descent" in such DNNs.

Among the most interesting approaches in building fault--tolerant quantum computation is utilizing non--Abelian topological phases of matter. These phases of matter are capable of storing information that is less susceptible to loss due to the interactions between the system and its environment. Furthermore, their non--Abelian characteristics allows for reliable performance of logical operations on the stored information. One of the leading physical systems that can realize such non--Abelian topological phases is the quantum Hall system at filling fraction $5/2$. Since the discovery of this state, the nature of the ground state of this system has been the subject of debate. An important development was made recently when the thermal Hall conductance of this state was measured to be $K=2.5 \pi^2k_B^2T/3h$ [M. Banerjee \emph{et al.}, Nature {\bf 559}, 205 (2018)]. Taken at face value, this result points to the PH--Pfaffian state as the true ground state of the quantum Hall system at filling fraction $\nu=5/2$. It is the consequence of the assumption that all the modes running along the edge of this system are well--equilibrated with each other. However, as has been pointed out by other authors, this assumption may not be completely justified. In particular, the measured thermal Hall conductance could also be consistent with the anti--Pfaffian state under some experimental conditions. In this thesis, we study those conditions in detail.

To achieve this, we propose new fixed point theories that describe the low--temperature physics of the anti--Pfaffian state. We demonstrate that these proposed theories could be consistent with the parameters describing the experimental conditions. For each of these fixed points, we identify the effective low--temperature edge modes and study the effect of strong short--range Coulomb interaction and an approximate spin symmetry on the interactions between them.

We derive the kinetic equations that describe the hydrodynamic transport of charge and heat in a general quantum Hall state. This is the expansion of the previous studies and includes the description of transport of strongly coupled edge modes. We use these kinetic equations to describe the hydrodynamic transport of heat and charge in our proposed fixed point theories. We estimate the values of physical parameters in our theory based on the previous experimental studies. This enables us to make meaningful comparisons between our theoretical predictions and the experimental measurements of thermal conductance. We show that there exists an experimentally realistic range of parameters that the anti--Pfaffian state is consistent with the thermal Hall conductance $K=2.5 \pi^2k_B^2T/3h$. We identify these ranges of parameters and based upon them, make predictions on the electrical and thermal Hall conductance of the anti--Pfaffian state for a range of temperatures.

RNA editing in plants converts cytidines to uridines (C-to-U) in chloroplast and

mitochondrial transcripts. Pentatricopeptide repeat (PPR) protein family members have

been shown to be required for editing, and possess key characteristics of editing

deaminases found in other organisms. PPRs have an N-terminal RNA binding domain

responsible for RNA recognition and a C-terminal DYW domain that has deaminase-like

characteristics.

The role of several features of the DYW domain was investigated. The DYW domain

includes the HXE motif that provides a glutamate residue that is catalytically required.

Glutamate to alanine substitution ablated editing, and establishes a key characteristic of

the DYW domain that is required for editing. In addition, a highly conserved PG box was

identified between the N- and C-terminal domains that was required for editing in PPRs

that lacked a DYW domain, and the PG box may be required for protein-protein

interactions to recruit a deaminase in trans. These observations led to the development

of the cis and trans-editing models for RNA editing that posits that the deaminase may be

provided in cis from a single PPR or in trans in PPRs that lack a DYW domain. PPRs, such as MEF8, which have a short RNA binding domain and an intact deaminase domain,were identified as candidates for a trans-deaminases. RNAseq analysis was performed

on mef8 null mutants and catalytically ablated variants to examine the role of MEF8 in

mitochondrial editing. Sixty editing sites were affected, and suggests that MEF8 may

participate as a trans-editing deaminase by providing deaminase capability for a large

number of editing sites.

The role of reactive oxygen species (ROS) during editing dysfunction was investigated to

examine the potential mechanisms of sensing and signaling oxidative distress. LPA66

mutant plants fail to edit a photosystem II polypeptide and exhibit a strong phenotype.

Mutant plants produced elevated levels of ROS, and transcriptomic analysis revealed

higher expression of ROS reactive network genes and PPR or editing related genes.

Lipidomics analysis of the mutant indicated highly elevated levels of oxylipins including

arabidoside A and G and phytoprostanes. Possible signaling mechanisms through

oxylipins and jasmonic acid pathways are discussed.