Attend a one-hour seminar on the analysis of partial differential equations, featuring a talk by Luis Silvestre from the University of Chicago. Explore the Boltzmann equation, which models particle density evolution in gases, and its global well-posedness challenges. Delve into Silvestre's research on Gaussian lower bounds for the inhomogeneous non-cutoff Boltzmann equation, focusing on hard and moderately soft potentials with spatial periodic conditions. Learn about the conditional estimates developed under the assumption of bounded hydrodynamic quantities, including local mass, energy, and entropy density. Gain insights into how these lower bounds contribute to the broader program of conditional estimates and their implications for understanding potential singularities in the Boltzmann equation.