Classical algebraic geometry and modern combinatorics

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Enumeration is the study of counting mathematical objects. Some of the most important enumeration problems have their orgins in algebraic geometry. These "enumerative algebraic geometry" problems have a dual nature: they have a combinatorial (discrete) side and geometric (continuous) side. In some cases, algebra can provide a bridge between these two sides. However, there are many counting problems that are fundamental to a number of areas of mathematics, where we do not have a complete picture like this; in fact, there are relatively few instances where we know what all the pieces are, let alone how they fit together. In some cases we have only mysterious combinatorial procedures that seem to give the right answer, as if by accident, with little understanding of why they work. This type of knowledge provides few clues as to how to solve new, related problems.