Colloquium: How (not) to count curves

Abstract: Given a geometric space X, how many curves of a certain shape lie in X? For instance, Euclid knew that there is a single line in the plane passing through two points. In the 1990s, Kontsevich, using ideas inspired from string theory, generalized Euclid’s count to determine the number of rational plane curves of degree d through 3d-1 points. This was the birth of Gromov-Witten (GW) theory, which offers a robust framework to think about counting curves, and which has seen an explosion of progress in the last three decades. However, the GW machine comes with a serious caveat: it often spits out a count that includes unwanted “excess” contributions that are less geometrically meaningful, and that are hard to control. I will outline a broad program to confront this problem in a particular class of examples, which forms a bridge between classical and modern enumerative geometry. The story is largely complete in the case of curves on projective spaces, but there are many interesting directions which lie beyond.

Host: John Shareshian