Tautological rings and competing conjectures

Abstract: Let M_g be the moduli space of smooth curves of genus g. The tautological ring is a subring of the cohomology of M_g that was introduced by Mumford in the 1980s in analogy with the cohomology of Grassmannians. It is a graded ring with one generator in each degree, but the ideal of relations between these generators is unknown in general. Work of Faber and Faber-Zagier in the 1990s led to two conjectures, each proposing a full description of the structure of the tautological ring. Both conjectures are true for g < 24, but they contradict each other for g >= 24. Although these competing conjectures are both still open, I will discuss some recent evidence favoring one of them over the other.

algebraic geometrycombinatorics

Audience: researchers in the discipline

MAAGC 2023

Series comments: The eighth annual Mid-Atlantic Algebra, Geometry, and Combinatorics (MAAGC) Workshop will take place Friday and Saturday, December 1-2, 2023 at Virginia Commonwealth University in Richmond, Virginia.

The MAAGC Workshop aims to bring together senior researchers and junior mathematicians from the region to exchange ideas and forge collaborations in algebra, geometry, and combinatorics. The conference is funded by National Science Foundation grant DMS-1728937.