**Title:** Analytical and numerical issues for non-linear kinetic equations

We will present an overview of novel analytical and numerical methods for

nonlinear models of kinetic statistical flow type. These models share the

feature of preservation of flow invariant quantities. These numerical

techniques enable rigorous stability, convergence and error analysis.

Examples to be discussed are the classical non-linear Boltzmann equation

for Coulombic interactions and grazing collision limits to the Landau

equation, computations for the spectral gap of the Boltzmann equation,

under both either integrable or non-integrable angular cross section.