Title: Mathematical Foundation and Extension of Statistical Mechanics
Abstract:  How can simple macroscopic laws emerge from the chaotic motion of astronomically many microscopic degrees of freedom? Statistical mechanics answers this question with a mathematical structure of striking simplicity, built around entropy, Boltzmann weights, and free energy. In this colloquium, I will revisit why that structure appears and why it is so universal. Using simple examples from probability, gases, and random walks, I will argue that equilibrium statistical mechanics is fundamentally a theory of counting, additivity, and typicality. I will then show how the same logic can be extended beyond equilibrium by shifting attention from probabilities of states to probabilities of entire dynamical trajectories. This broader viewpoint provides a natural language for irreversibility, entropy production, and modern fluctuation relations. The emphasis will be on concepts rather than technical detail, with the goal of giving a broad physics audience a unified picture of how statistical mechanics is founded and how it continues to evolve.