First-order methods are the workhorse of modern optimization and machine learning: they only use gradient information, yet they can solve large-scale problems efficiently. This talk introduces three fundamental algorithms - gradient descent, the heavy-ball method, and Nesterov’s accelerated gradient - and explains the main ideas that make them work. 

We start from gradient descent and discuss how step-size choices affect stability and progress, using simple geometric intuition (descent directions, line-search, and smoothness). We then present the continuous-time perspective through the gradient flow, which provides a clean mental model and helps interpret discrete algorithms as numerical discretizations. 

Building on this viewpoint, we introduce momentum methods: the heavy ball (with its mechanical interpretation) and Nesterov acceleration. We emphasize what acceleration means, what it can guarantee in the convex smooth setting, and why momentum may also create oscillations or instability if it is not tuned carefully. The goal is to connect intuition, basic convergence guarantees, and practical takeaways that students can reuse when reading the modern optimization and learning literature.