The Tropical Mirror
A puzzling connection between completely different geometric objects has puzzled string theorists and mathematicians. Can tropical geometry solve the mystery?
In May 1991, attendees at a conference of the "Mathematical Sciences Research Institute" in California had no idea that a big surprise awaited them. British string theorist Philip Candelas, then at the University of Texas at Austin, presented his research there, but the audience, mostly mathematicians, questioned the accuracy of this work.
The physicist claimed to have found a formula that counted "rational curves" on an extremely complicated six-dimensional "surface". So far, mathematicians have only been able to count curves of the first, second and third degree on the strange structures with complex computer support. Also, they hadn't expected a pattern behind these rapidly growing numbers - let alone that a formula could be found to calculate them.
During the lecture, some listeners noticed that Candela's value for the number of third-degree curves differed from the already known result of the Norwegian mathematicians Stein Arild Strømme, then at the University of Utah, and Geir Ellingsrud, then at the University of Bergen. "The algebraic geometers [in the audience] were arrogant, assuming the physicists had made a mistake," writes conference organizer and Fields Medalist Shing-Tung Yau in his book The Shape of Inner Space. While Candelas and his colleagues then feverishly searched for an error, about a month later Strømme and Ellingsrud discovered inconsistencies in their program code - and publicly announced that the physicists were right.
Candelas and his colleagues' finding had tremendous implications that go beyond simply counting curves: it suggested a fundamental connection between two vastly different geometric domains. Nobody expected such a coincidence - and it still puzzles scientists to this day…
