  In these lecture notes I will give a pedagogical introduction to some common
aspects of 4 different problems: (i) random matrices (ii) the longest
increasing subsequence problem (also known as the Ulam problem) (iii) directed
polymers in random medium and growth models in (1+1) dimensions and (iv) a
problem on the alignment of a pair of random sequences. Each of these problems
is almost entirely a sub-field by itself and here I will discuss only some
specific aspects of each of them. These 4 problems have been studied almost
independently for the past few decades, but only over the last few years a
common thread was found to link all of them. In particular all of them share
one common limiting probability distribution known as the Tracy-Widom
distribution that describes the asymptotic probability distribution of the
largest eigenvalue of a random matrix. I will mention here, without
mathematical derivation, some of the beautiful results discovered in the past
few years. Then, I will consider two specific models (a) a ballistic deposition
growth model and (b) a model of sequence alignment known as the Bernoulli
matching model and discuss, in some detail, how one derives exactly the
Tracy-Widom law in these models. The emphasis of these lectures would be on how
to map one model to another. Some open problems are discussed at the end.
