Rank Conjectures Across Algebra and Topology

Masterclass
University of Copenhagen, 24-28 June 2024 

The masterclass will explore a family of conjectures spanning multiple disciplines: the Rank Conjectures, each making predictions about the sizes of certain homotopically defined objects.

Halperin’s toral rank conjecture predicts a lower bound on dimension of the rational cohomology of a space with a free torus action. Similar conjectures concerning the actions of elementary abelian p-groups have opened up connections with modular representation theory. Within commutative algebra, the Buchsbaum-Eisenbud-Horrocks conjecture predicts a lower bound on the Betti numbers of modules over regular local rings, and has turned out to hold a strikingly close relationship with the toral rank conjecture.

New techniques have recently led to substantial progress on the rank conjectures in algebraic topology, commutative algebra, and modular representation theory. The guest lecturers will share their expertise in these areas, bringing students up to a state of the art understanding of the conjectures, the tools being used to attack them, and the connections between them. The masterclass will highlight open problems and lines of investigation which could potentially set the tone for future work. The lectures will be supplemented by regular problem sessions, bringing the students to the front line of modern research. The masterclass will also function as an opportunity for students to meet other young (and established) researchers in nearby areas, building new academic relationships across disciplines.