Abstract

Calabi--Yau threefolds serve as a natural domain for the enumerative geometry of curves. While naive dimension estimates suggest finite curve counts, the actual geometry is far more intricate, driving extensive research over the last 30 years. I will highlight some pivotal developments, beginning with the advent of Gromov--Witten theory and Mirror Symmetry in the 1990s. Of particular interest is the introduction of Donaldson--Thomas theory in the 2000s, which gave rise to the discovery of derived symplectic geometry and led to categorified invariants. The most significant recent milestone is John Pardon's proof of the MNOP conjecture. This work establishes the fundamental equivalence of Gromov--Witten and Donaldson--Thomas theories, and provides a complete classification of curve-counting theories in Calabi--Yau threefolds.