In this talk I will insult your intelligence by showing a non-original proof of the Central Limit Theorem, with not-particularly-good error bounds. However, the proof is very simple and flexible, allowing generalizations to multidimensional and higher-degree invariance principles. Time permitting, I will also discuss applications to areas of theoretical computer science: property testing, derandomization, learning, and inapproximability.
I will introduce some example of models of Statistical Mechanics that are called 'weak-universal' and I will discuss the role of the extended scaling relations for the critical indexes. Finally I will mention some results and some works in progress.
I'll describe joint work (in progress) with Ghaith Hiary on implementing and running Hiary's O(t^1/3) algorithm for computing the zeta function and give some highlights of recent computations.