What lies between structure and randomness

Objects which are either completely structured or completely random are often straightforward to study. Most objects of mathematical interest are neither completely structured nor completely random. However, there are techniques for decomposing an arbitrary object into relatively structured and relatively random pieces. One of the most powerful results in this direction is Szemerédi's theorem, which played a major role in the 2004 proof that the primes contain arbitrarily long arithmetic progressions. We will use arithmetic progressions and graph structure as our central examples in examining the key ideas behind Szemerédi's theorem and will, if time permits, indicate some of the relations with analysis.