Optimization is concerned in general with a wide range of situations in which the maximum or minimum of a function is sought over some set specified by side conditions. It's one of the areas of modern mathematics in which "pure" and "applied" are most closely linked. There is continual interaction between computation, modeling, and frontier developments in analysis --- called "variational analysis" --- which are essential because classical analysis never was confronted with the issues that now arise, and it can't cope with them. Researchers have had to develop properties of convexity, the tangent and normal cone geometry of sets with nonsmooth boundaries, "subgradients" of functions not having gradients, and even the "subdifferentiation" of set-valued mappings.

The 514 course doesn't itself get into much of the "variational analysis" (that comes more in 515), but it does serve as an excellent example of the breadth of the subject while focusing on a particular area of major interest. Networks, also known as directed graphs, are basic tool for modeling many situation such as finding optimal paths from one "state" (or "location") to another under constraints, or even ascertaining the existence of such a path. But they also support a vast array of applications that, often in ingenious ways, can be viewed in terms of the optimization of "flows" or "potentials". The subject revolves in part around discrete analogs of familiar concepts in calculus such as flows across boundaries of regious, or line integrals, but also it relies on modern ideas like duality in optimization.

The organization of the material is largely algorithmic: virtually every proof is articulated in the form of an iterative, constructive procedure. The algorithms fit within each other as modules, just as one theorem can be derived by way of another. Students get valuable experience with the algebraic way of thinking that goes into the statement and justification of algorithms, but also in the way that rigorous mathematical concepts can be applied to a multitude of practical applications. This is a fine way to learn about mathematics in its broader scope of organizing our thinking about complex issues, moreover ones that don't only come out of traditional themes in mathematics itself.

The 514 course uses a textbook: Network Flows and Monotropic Optimization (Rockafellar), which can be perused in the Math Library. A selection of material in Chapters 1 -- 8 is covered. Grades are based on weekly assignments of exercises from this book.

An advantage of 514, especially for those interested in trying something a bit different, is that there are no course prerequisites; all that's needed is the ability for rigorous mathematical thinking along with the desire to understand more deeply what mathematics is all about.