New results on random matrices

Let M_n be an n by n random matrix whose entries are i.i.d Bernoulli random variables (which take value 1 or -1 with probability 1/2). This natural random model and has been studied in various fields for many years. On the other hand, very basic questions have been open for a long time. In this talk, I focus on the following two:

(1) What is the typical value of the determinant of M_n ?
(2) What is the probability that M_n is singular ?

I will give a brief history and discuss a few recent results (joint with Terence Tao) on these questions, which, somewhat surprsingly, involve ideas from additive number theory. If time allows, I will disucss some related questions as well.