Van Vu, Combinatorial problems in random matrix theory

I am going to give a survey on several basic problems of combinatorial nature concerning random Bernoulli matrices, including:

(1) The singularity problem: What is the probability that a random Bernoulli matrix is singular?

(2) The determinant problem: What is the typical value of the determinant?

(3) The permanent problem: What is the typical value of the permanent?

(4) The eigenvector problem: How does a typical eigenvector look like?

If time allows, I will discuss connections to other areas of mathematics, most importantly additive combinatorics.