# Betti numbers in commutative algebra and equivariant homotopy theory

Monday 23 September to Friday 27 September 2024

Total Betti numbers appear in related, decades-old rank conjectures in commutative algebra and equivariant topology. On the topological side, Halperin and Carlsson conjectured that the total Betti number of a compact space with a free torus action or $p$-torus action of rank $r$ is bounded below by $2^r$. On the algebraic side, Avramov conjectured a similar lower bound for the total Betti number of finite length modules over a local ring. Recent work of Walker and VandeBogert-Walker resolves this conjecture positively for rings of prime characteristic.

Motivation for the $p$-toral rank conjecture in topology comes from classical Smith theory, settling the rank $1$ case. Chromatic Smith theory concerns total Betti numbers taken with respect to generalized homology theories, and in particular the relationship between those of the underlying space and the fixed points of an action by a compact Lie group $G$. For example, vanishing results for Morava $K$-theory are central to understanding the Balmer spectrum of finite $G$-spectra. A complete description of the Balmer spectrum has been achieved for abelian groups, but is open beyond the abelian case.

This workshop aims to bring together researchers interested in discussing recent advances, open directions and connections between Betti numbers in commutative algebra, equivariant homotopy theory, and representation theory. In addition to research talks, we will start with introductory lectures, and plan more informal research activities.

Location: Bielefeld University

Times: Registration will open at 9:00 on Monday 23 September, the workshop will end on Friday 27 September at 16:30.

Registration Deadline: August 23, 2024 (For participants applying for support: May 31, 2024)

Organisers: Markus Hausmann, Claudia Miller, Marc Stephan, and Mark Walker

For an additional introduction to the rank conjectures, we advertise the
Masterclass
Rank Conjectures Across Algebra and Topology

that is
taking place at the
University of Copenhagen during
24-28 June, 2024.

## Invited Speakers

- Alejandro Adem (Vancouver)
- Luchezar Avramov (Sofia)
- William Balderrama (Charlottesville)
- Petter Andreas Bergh (Trondheim)
- Ben Briggs (London)
- Natàlia Castellana (Barcelona)
- John Greenlees (Warwick)
- Eloísa Grifo (Lincoln)
- Jesper Grodal (Copenhagen)
- Jeremiah Heller (Urbana-Champaign)
- Srikanth Iyengar (Salt Lake City)
- Janina Letz (Bielefeld)
- Clover May (Trondheim)
- Berrin Şentürk (Ankara)
- Keller VandeBogert (Notre Dame)
- Leopold Zoller (Munich)

## Some Excercises

## Funding

This workshop is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the TRR 358 "Integral Structures in Geometry and Representation Theory" (SFB-TRR 358/1 2023 – 491392403) and the U.S. National Science Foundation #2428995.

## Contact

If you have any questions about the workshop, please contact the organisers at marc.stephan.