“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.”
“The difficulty lies in the fact that for subtle reasons, it’s not possible to count the intersection points all at once. Instead, mathematicians need to break the space down into “local” regions, count intersection points in each region, and add those together to get the “global” count.”
“Piecing together local counts has proved to be a more delicate and technically demanding task than mathematicians realized at first: If you’re not careful about how you draw your local regions, you could easily omit one intersection point or double-count another.”