Abstract：

This is a series of talks on the construction of invariant manifolds for stationary Boltzmann equations. We will start with the basics of the Boltzmann equation, the conservation laws, H-Theorem, and the macro-micro decomposition. The crucial role of the size and sign of the Euler characteristics is then explained from physical considerations as well as from the analysis of the energy estimates. The time asymptotic analysis with the Green's function approach is then presented for the decomposition of the invariant manifolds on the linear level. Knudsen manifolds are next constructed for the near critical Euler characteristic cases. This leads to the set up of the coordinates on critical Euler characteristics. Finally, the fluid nonlinearity is derived for the study of the coupling of Knudsen-type layers and fluid-like waves, and the striking bifurcation phenomenon is explained.