Explore the intricacies of Poincaré inequalities on the Hamming cube in this 45-minute lecture by Alexander Volberg at the Hausdorff Center for Mathematics. Delve into the improvement of the constant π/2 in the L^1-Poincaré inequality, comparing it to the known sharp constant in Gaussian space. Examine the work of L. Ben Efraim and F. Lust-Piquard, and discover a new estimate that proves the constant is strictly smaller than π/2. Investigate various proofs and approaches, including boundary edges, geometric estimators, and production spaces. Analyze the relationship between C1 and sqrt(π/2), and explore related topics such as the central limit theorem, random variables, and computer-to-computer interactions. Gain insights into this complex mathematical landscape, bridging analysis, combinatorics, and probability theory.