There are many synergies between probability theory and convex geometry, especially in high-dimensional settings. We discuss some classical examples and then focus on a recent development concerning a famous theorem of von Neumann, which establishes a one-to-one correspondence between unitarily invariant norms on nxn matrices and the class of 1-symmetric norms on Rn. We describe a measure theoretic version of this theorem for the uniform distribution on p-Schatten balls, and some ramifications. A key ingredient of the proof is the identification of a certain Rademacher phase of the unitary group, which may be of independent interest. This is based on joint work with Grigoris Paouris.
Minimal prior knowledge will be assumed: all terms in the title, abstract and beyond will be defined in the talk. The hope is to make much of the lecture accessible to a broad audience, including graduate students.