6 - Explorations

Summary

These conjectures are of such compelling simplicity that it is hard to understand how any mathematician can bear the pain of living without understanding why they are true.

– David Robbins (1991)

If there had been any lingering doubt of the truth of Conjectures 1 and 2, it was dispelled by the proof we have just seen of Macdonald's conjecture. The critical insight that made this proof possible came from the assumption that Conjectures 1 and 2 were correct. No further confirmation was needed, but there was still no proof.

Charting the territory

The alternating sign matrix conjecture appeared at an opportune moment. The publication of George Andrews's The Theory of Partitions (1976) and Ian Macdonald's Symmetric Functions and Hall Polynomials (1979) had created wide interest in plane partitions and Young tableaux. Enough people knew enough about them to recognize the potential importance of proving this conjecture. Because these books could take researchers from disparate backgrounds and bring them quickly up to the edge of what was known, they drew in mathematicians from a variety of fields, including algebra, combinatorics, and analysis. More important, even though it would be ten years before any significant progress would be made on the proof of this conjecture, those who were drawn to it found that it was only one in a constellation of related problems, and that there were many other interesting discoveries to be made.