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

In the last few years, there has been a lot of activity in the area of structural analysis and derandomization of polynomial threshold functions. Tools from analysis and probability have played a significant role in many of these works. The focus of the talk will be towards achieving the following goal: Given a degree-d polynomial threshold function, deterministically approximating the fraction of satisfying assignments up to o(1) error in polynomial time. Along the way, we'll first survey some important existing results in this area. There will be two new contributions in this talk: a) A new central limit theorem for Gaussian polynomials which says 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. b) 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