In this talk, we present our recent progress on Hilbert's sixth problem—deriving fluid equations from microscopic dynamics. We will provide the necessary physical background and introduce the key techniques developed in our work, including the Feynman diagram representation and the cutting algorithm. 

In the first lecture, we introduce the basic concepts of particle systems and their statistical descriptions. We explain the Boltzmann equation, the fluid equations, and how physicists heuristically derive them. We then state our main theorem, which provides a rigorous justification, and outline its proof.