A central limit theorem for Gaussian polynomials and deterministic approximate counting for polynomial threshold functions

In this talk, we will continue, the proof of the Central Limit theorem from my last talk. We will show that that the law of "eigenregular" Gaussian polynomials is close to a Gaussian. The proof will be based on Stein's method and will be dependent on using techniques from Malliavin calculus. We will also describe a new decomposition lemma for polynomials which says that any polynomial can be written as a function of small number of eigenregular polynomials. The techniques in the lemma are likely to be of independent interest. Based on joint work with Rocco Servedio.