Explore Fractional Poincaré inequalities for doubling and non-doubling weights in this 50-minute lecture. Delve into the unification and improvement of well-known results concerning Fractional Poincaré-Sobolev inequalities using flexible Harmonic Analysis methods. Learn about oscillation, self-improvement, and functional proofs while examining key points and sketches of general results. Investigate geometric conditions, motivations for new findings, and the main theorem involving non-doubling and reverse doubling weights. Gain insights from recent joint research as the speaker guides you through the proof and its implications in the field of mathematics.