Universally Counting Curves in Calabi-Yau Threefolds

Explore a lecture on enumerating curves in algebraic varieties, focusing on a proposed "universal" enumerative invariant. Delve into the concept of compactification of smooth embedded curves in varieties and its impact on different enumerative invariants. Learn about the innovative approach using a "Grothendieck group of 1-cycles" and discover how the cluster formalism of Ionel and Parker demonstrates that this group is freely generated by local curves in threefolds with nef anticanonical bundle. Examine the implications of this finding on the MNOP conjecture, particularly in cases with nef anticanonical bundle and primary insertions. Gain insights into how this reduces the conjecture to local curves, where it has been proven by Bryan--Pandharipande and Okounkov—Pandharipande. Presented by John Pardon from the Simons Center for Geometry and Physics (SCGP), this one-hour lecture offers a deep dive into advanced mathematical concepts at the intersection of algebraic geometry and enumerative geometry.