As many students of mathematics know, mathematical problems that are simple to state fall into several classes: there are those whose solutions are equally simple; those that seem practically impossible to solve despite their apparent simplicity; those that are solvable but whose solutions nonetheless end up being too complicated to provide much real insight; and finally, there are those rare and magical problems that turn out to have rich solutions that reveal a fascinating and unexpected structure, with surprising connections to other areas that lie well beyond the scope of the original problem. Such problems are hard, but in the most interesting and rewarding kind of way.

The problems that grew out of the study of longest increasing subsequences, which are the subject of this book, belong decidedly in the latter class. As readers will see, starting from an innocent-sounding question about random permutations we will be led on a journey touching on many areas of mathematics: combinatorics, probability, analysis, linear algebra and operator theory, differential equations, special functions, representation theory, and more. Techniques of random matrix theory, a sub-branch of probability theory whose development was originally motivated by problems in nuclear physics, will play a key role. In later chapters, connections to interacting particle systems, which are random processes used to model complicated systems with many interacting elements, will also surface.