01 · The idea
In everyday language
A symplectic manifold carries a geometric structure inspired by the phase spaces of mechanics. One way to explore it is to study pseudoholomorphic curves inside it. The collection of all such curves forms a moduli space, but this space may have singularities or more dimensions than a naive count expects. Pardon developed an approach to virtual fundamental cycles that uses algebraic topology to construct the appropriate substitute for a well-behaved counting space. This makes it possible to define geometric invariants consistently, even when direct counting breaks down. His wider work also illuminates group actions, knots and Fukaya categories.
Why it matters
These foundations make curve-counting constructions dependable across important settings in symplectic geometry. They support invariants such as contact homology and provide tools for comparing geometric spaces through the algebra of their curves.
