Taibleson Colloquium: Khintchine Inequalities: Old and New Stories

Abstract: How much cancellation occurs, on average, in a sum with random signs? Khintchine’s inequality gives a striking answer: the average size of such a sum is comparable to the Euclidean length of its coefficients, with constants independent of the number of terms. This principle forms a fundamental connection between probability, harmonic analysis, and the geometry of Banach spaces.

In this talk, I will introduce the classical inequality and explain some of its significance before asking what happens when the coefficients are matrices. The operator Khintchine inequalities of Lust-Piquard and Pisier reveal a richer geometry, in which two different square functions—and decompositions into column and row components—replace the familiar sum of squares. I will discuss applications to random matrices and conclude with recent joint work with Chian Yeong Chuah and Zhen-Chuan Liu. We determine the sharp L_1  operator Khintchine constant for Rademacher sums, answering a question left open by Haagerup and Musat. I will also discuss the connections between sharp constants and the arithmetic structure of frequency sets, if time permits.

Faculty Host: Brett Wick