Michael Wu L&S Math & Physical Sciences
Log-Sobolev Stability Along the Central Limit Theorem
My project investigates a sharper form of Gaussian convergence behind the Central Limit Theorem, one of the foundational results of probability theory. The theorem says that when many independent random effects are averaged together, their normalized sum becomes increasingly bell-shaped. However, weak convergence—that is, convergence in distribution—can miss finer structures, especially rare-event behavior in the tails of a distribution. I will study this phenomenon through the Log-Sobolev inequality, a powerful tool in modern probability, information theory, and analysis that measures how strongly a distribution controls entropy and concentration. A natural guess is that repeated averaging should drive the relevant Log-Sobolev constant toward the Gaussian benchmark, but recent work suggests that the limiting value may be more subtle. My project aims to determine this limiting constant and, if possible, prove its exact value rigorously. A successful resolution would contribute to a finer theory of Gaussian convergence, connecting classical probability with modern questions in entropy, concentration, sampling, and high-dimensional data.
Message To Sponsor
I am sincerely grateful for your support, which gives me the opportunity to focus on research this summer. My project studies how logarithmic Sobolev constants behave under central-limit-type averaging, with implications for concentration, convergence, and high-dimensional probability. I am excited to take on this challenging problem and make meaningful progress through this work. Thank you for making this experience possible.