This volume presents proceedings from the AMS short course, Trends in Optimization 2004, held at the Joint Mathematics Meetings in Phoenix (AZ). It focuses on seven exciting areas of discrete optimization.

In particular, Karen Aardal describes Lovasz's fundamental algorithm for producing a short vector in a lattice by basis reduction and H.W. Lenstra's use of this idea in the early 1980s in his polynomial-time algorithm for integer programming in fixed dimension. Aardal's article, "Lattice basis reduction in optimization: Selected Topics", is one of the most lucid presentations of the material. It also contains practical developments using computational tools.

Bernd Sturmfels' article, "Algebraic recipes for integer programming", discusses how methods of commutative algebra and algebraic combinatorics can be used successfully to attack integer programming problems. Specifically, Gröbner bases play a central role in algorithmic theory and practice. Moreover, it is shown that techniques based on short rational functions are bringing new insights, such as in computing the integer programming gap.

Overall, these articles, together with five other contributions, make this volume an impressive compilation on the state-of-the-art of optimization. It is suitable for graduate students and researchers interested in discrete optimization.