The recent developments in optimization and deep learning research have significantly advanced our understanding and capabilities in these fields. A notable trend is the shift towards continuous-time formulations and theoretical analyses of adaptive optimization algorithms, which provide deeper insights into the training dynamics and generalization properties of modern deep learning models. This approach has led to the discovery of stable hyperparameter regions and the justification of normalization layers, contributing to more principled architectural decisions. Additionally, there is a growing focus on energy-based self-adaptive learning rates and novel optimization methods that enhance stability and convergence speed, particularly in the early stages of training. Distributed and scalable algorithms for linear algebraic equations are also being refined, with a focus on reducing communication bandwidth and improving scalability through scheduling protocols. Furthermore, the field is witnessing advancements in the development of memory-efficient optimization frameworks that balance performance and resource utilization, addressing the challenges posed by large-scale models. Notably, exact and tractable second-order optimization methods are being explored in reversible architectures, revealing insights into generalization properties that were previously obscured by approximations. Overall, these innovations are paving the way for more efficient, stable, and theoretically grounded optimization techniques in deep learning.