The Fight to Fix Symplectic Geometry

Kevin Hartnett:

In the 1830s, the Irish mathematician William Rowan Hamilton reformulated Newton’s laws of motion, finding deep mathematical symmetries between an object’s position and its momentum. Then in the mid-1980s the mathematician Mikhail Gromov developed a set of techniques that transformed Hamilton’s idea into a full-blown area of mathematical research. Within a decade, mathematicians from a broad range of backgrounds had converged to explore the possibilities in a field that came to be known as “symplectic geometry.”

The result was something like the opening of a gold-rush town. People from many different areas of mathematics hurried to establish the field and lay claim to its fruits. Research developed rapidly, but without the shared background knowledge typically found in mature areas of mathematics. This made it hard for mathematicians to tell when new results were completely correct. By the start of the 21st century it was evident to close observers that significant errors had been built into the foundations of symplectic geometry.