Arithmetic Combinatorics

Let Q(n,p) denote the adjacency matrix of the Erdos-Renyi graph G(n,p), that is to say a symmetric matrix whose entries above the main diagonal are independently set to 1 with probability p and 0 with probability 1-p. We will examine the behavior of the rank of Q(n,p) with an eye on the following questions (whose answer will of course depend on p) 1. What is the probability that Q(n,p) is singular? 2. If Q(n,p) is likely to be singular, can we describe the dependent sets of rows? Joint work with Prof. Van Vu(IAS/Rutgers) and some with Prof. Terence Tao (UCLA)