This talk explores the mathematical underpinnings of DeepSeek R1, a reinforcement learning model tailored for complex reasoning. Unlike conventional supervised fine-tuning approaches, DeepSeek R1 leverages Group Relative Policy Optimization (GRPO), an innovative technique that refines Proximal Policy Optimization (PPO) by eliminating the need for a critic. GRPO enhances chain-of-thought reasoning by structuring problem-solving into sequential steps. Through an analytical perspective, we will examine the theoretical properties of GRPO.