A Geometric Dichotomy for the Discrete Hilbert Transform

Watch a 52-minute mathematics lecture exploring groundbreaking research on finite Hilbert transforms and their relationship to Banach space parameters. Delve into collaborative work that identifies growth rates of finite Hilbert transform norms as a novel parameter spanning from 0 for UMD spaces to 1 for non-super-reflexive spaces. Examine explicitly constructed spaces achieving intermediate rates while maintaining super-reflexivity without UMD properties. Learn how this research provides precise quantitative insights into the deviation between martingale type and Rademacher type in Pisier's classical examples, surpassing previous upper and lower estimates. Discover how finite Hilbert transforms enable efficient indirect bounds for martingale type without requiring explicit martingale examples with extremal behavior.