In the field of symplectic geometry, a central issue involves how to count the intersection points of two complicated geometric spaces. This counting question is at the heart of one of the most famous problems in the field, the Arnold conjecture, and it’s also a matter of basic technique: Mathematicians need to know how to make these counts in order to do other kinds of research.
As I describe in my article “A Fight to Fix Geometry’s Foundations,” developing a method for counting these intersection points has been a drawn-out and sometimes contentious process. A reliable, widely understood, error-free approach has presented a challenge for a number of reasons, from the lack of a shared vocabulary when a new field gets started (symplectic geometry only really took off beginning in the 1990s), to the nature of the problem itself: Simply put, it’s hard. … (Quanta Magazine)