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.