Fall 2025 @ Amherst College - November 21, 2025

The decomposition theorem is one of the deepest known facts about the topology of complex projective varieties. Given a map X → Y of complex projective varieties, with X smooth, it implies strong restrictions on the structure of the cohomology H^*(X) as a module over H^*(Y). We show that many of these restrictions are linear-algebraic consequences of classically-known properties of H^*(X). This enables us to deduce these restrictions in situations where one cannot apply the decomposition theorem, such as in combinatorial Hodge theory and for Chow rings modulo numerical equivalence. Joint work with Omid Amini and June Huh.

The moduli space of real (n+1)-marked stable curves admits polytopal decompositions into associahedra (by work of Kapranov and Devadoss) and, dually, into cubes (Davis-Januszkiewicz-Scott). The latter yields a very nice Coxeter-like presentation of the space's S_n-equivariant fundamental group, called the cactus group.

I will describe an analogous decomposition of the real locus of Hassett's moduli space of weighted stable curves. Our decomposition is into products of permutahedra. When the weights are S_n-symmetric, the resulting "weighted cactus group" is obtained from the ordinary cactus group by imposing braid relations.