Rank Conjectures Across Algebra and Topology

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.



Tackling rank conjectures in algebra using Adams operations

Homological methods for finite group actions

Rational homotopy theory, equivariant cohomology, and the toral rank conjecture








Each speaker will give four one hour lectures, and there will be many problem sessions. Program to be announced.
















We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.













We will be able to offer some early career participants up to 2500 Danish Kroner each to supplement travel costs. While we hope to be able to support most interested participants, the decision to offer funding will be based on the suitability of the masterclass to the applicant. We strongly encourage participants to apply for other sources of funding as well.

If you intend to apply for funding, please register here no later than 1st May 2024.

If you do not intend to apply for funding, you can register at the same link until 1st June 2024.











Ben Briggs bpb@math.ku.dk

Jesper Grodal jg@math.ku.dk