It has long been known that Gaussian measures possess unique convexity properties within the class of log-concave measures. In particular, a remarkable sharp form of Gaussian convexity was discovered by A. Ehrhard in the early 1980s, but has mostly remained somewhat of a beautiful curiosity. In recent work, however, this inequality, the theory surrounding it, and its utility in applications have become significantly better understood. My aim in these talks is to review and discuss in some detail several recent developments surrounding this theory and its applications to Gaussian concentration phenomena (by us as well as by other authors). I will also highlight some key mysteries that remain.

It has long been known that Gaussian measures possess unique convexity properties within the class of log-concave measures. In particular, a remarkable sharp form of Gaussian convexity was discovered by A. Ehrhard in the early 1980s, but has mostly remained somewhat of a beautiful curiosity. In recent work, however, this inequality, the theory surrounding it, and its utility in applications have become significantly better understood. My aim in these talks is to review and discuss in some detail several recent developments surrounding this theory and its applications to Gaussian concentration phenomena (by us as well as by other authors). I will also highlight some key mysteries that remain.