The normal distribution appears in a wide and disparate set of circumstances, and this ubiquity is explained by the central limit phenomenon. This talk will explore several forms of the central limit theorem, as well as different methods of proof. Highlights include a new method of moments proof for entries on a hypersphere sphere and results for traces of large random matrices utilizing the Malliavin-Stein method.
For a positive integer , define to be the smallest number such that the additive energy of any subset and any is at most . In this talk, I will survey recent results on bounds for , explore the connections with (variants of) the Hausdorff-Young inequality in analysis and with the Balog-Szemeredi-Gowers theorem in additive combinatorics, and then discuss new results on the asymptotic behavior of as .
