Cohomology table SL_n(Z)
© Benjamin Brück

Perspectives on the cohomology of general linear groups

6 – 10 October 2025, Münster, Germany

This workshop focuses on recent advances around the (co-)homology of general linear and related groups. These basic topological invariants are, for example, related to questions in algebraic K-theory and conjectures in number theory. Their study can be approached from many different mathematical perspectives, each using their own set of techniques. The goal of this event is to stimulate exchange between these different points of view, to bring together experts, as well as to introduce PhD students and Postdocs to this exciting area of research.

The core of the workshop is formed by several

  • lecture series and
  • invited talks.

These are supplemented by

  • contributed talks (that participants can apply for during the registration process) and,
  • a poster session (in which all participants are invited to present their work).

The poster session is scheduled for Monday evening, and will be combined with a reception. On Thursday evening, participants are invited to join us for a conference dinner.

More details and a precise schedule will be announced as soon as possible.

(Preliminary) List of speakers

Lecture series by:

  • Philippe Elbaz-Vincent (Université Grenoble Alpes)
    Title: "Cohomology of arithmetic groups, Voronoi's complexes and K-theory"
  • Sam Payne (University of Texas at Austin)
    Title: "Hopf structures in the cohomology of \(\operatorname{GL}_n(\mathbb{Z})\) and \(\operatorname{Sp}_{2n}(\mathbb{Z})\)"
  • Dan Petersen (Stockholm University)
    Title: "Homological stability and moments in families of L-functions over function fields"

Invited talks by:

Contributed talks by:

  • Fabio Capovilla-Searle (Purdue University)

  • Karol Janowicz (Warsaw Doctoral School of Mathematics and Computer Science)

  • Anja Meyer (Loughborough University)

  • Piotr Mizerka (Adam Mickiewicz University in Poznań)

  • Vikram Nadig (Bielefeld University)

  • Matthew Scalamandre (University of Toronto)

  • Ismael Sierra (University of Toronto)

  • Markus Szymik (University of Sheffield)

  • Emiliano Torti (University of French Polynesia)

  • Raluca Vlad (Brown University)

Schedule and Abstracts

A precise schedule will be announced as soon as possible. The title and abstracts for the lecture series and talks will be published here.

Lecture Series:

  • Philippe Elbaz-Vincent – "Cohomology of arithmetic groups, Voronoi's complexes and K-theory"

    Abstract: The cohomology of arithmetic groups, and more generally of linear groups, is a rich subject with links to geometry, topology, algebra and number theory.  In this lecture series, I will give an overview on several explicit results on the cohomology of \(\operatorname{SL}_N(\mathbb{Z})\) and related groups, their homologies with coefficients in their Steinberg modules, computations of their geometric models (in particular based on the Voronoi complexes) and related conjectures. I will also give applications to moduli spaces of curves and K-theory of the integers. The talks are based in part on several joint works of the author.

  • Sam Payne – "Hopf structures in the cohomology of \(\operatorname{GL}_n(\mathbb{Z})\) and \(\operatorname{Sp}_{2n}(\mathbb{Z})\)"

    Abstract: I will construct and explain two related Hopf algebra structures, one on the union of the compactly supported cohomology groups of the locally symmetric spaces for \(\operatorname{GL}_n(\mathbb{Z})\) and the other on the union of the weight zero compactly supported cohomology groups of the locally symmetric spaces for \(\operatorname{Sp}_{2n}(\mathbb{Z})\). Both of these Hopf algebras are bigraded (by the cohomological degree and the genus n), connected, and graded co-commutative. In the symplectic case, the weight zero subspace refers to Deligne’s weight filtration induced by viewing the locally symmetric space for \(\operatorname{Sp}_{2n}(\mathbb{Z})\) as the algebraic moduli space of principally polarized abelian varieties of complex dimension n. The structure of these compactly supported cohomology groups are related, via Poincaré duality, to the corresponding cohomology groups with coefficients in the orientation sheaves of the locally symmetric spaces. These talks will be based on recent joint work with Brown, Chan, and Galatius (https://arxiv.org/abs/2405.11528).

  • Dan Petersen – "Homological stability and moments in families of L-functions over function fields"

    Abstract: This is a report of joint work with Bergström–Diaconu–Westerland and Miller–Patzt–Randal-Williams. The goal of our two papers is to prove a theorem in analytic number theory, using homological stability. The proof uses many diverse tools, some of which are close to the topic of the workshop (Borel's results on stable real cohomology of arithmetic groups, high connectivity of simplicial complexes, cellular \(E_k\)-algebras) and some of which are further afield (Segal-style scanning for moduli spaces, logarithmic étale cohomology, Koszul duality of operads, combinatorics of symmetric functions). I will give an overview of the proof, aimed at an audience with no background in number theory or algebraic geometry.

Contact and Organizers

If you have questions about this event, please use the following email address to contact us.

gl.cohomology.muenster@gmail.com

This workshop is organized by

Registration

To participate in this workshop, please register here. You can apply for funding, as well as for contributed talks during the registration process.

  • Deadline to apply for funding: June 30, 2025
  • Deadline to apply for a contributed talk: June 30, 2025
  • Registration will close on September 15, 2025

Support and child care

Childcare is available free of charge for all conference participants.

Accommodation

We kindly ask the participants to arrange their own accommodation. A selection of hotels in Münster can be found here.

SCAM WARNING: Some participants have received scam emails regarding accommodation arrangements. If you are unsure about any emails you received, please contact us.

Venue and Travel Information

The conference takes place in room SRZ 216/217 on the second floor of the seminar building (Seminarraumzentrum, SRZ) next to the Faculty of Mathematics and Computer Science and the Cluster of Excellence Mathematics Münster.

University of Münster
Seminarraumzentrum (SRZ)
Orléans-Ring 12
48149 Münster
Germany

Directions can be found on the University of Münster campus map and on the MM website.

We have also collected practical information in a leaflet: Information for conference guests / Informationsblatt für Tagungsteilnehmer:innen [enIde]

Conference Poster

You are welcome to download the conference poster from this page and display it at your institution.

Sponsors

This conference is supported by the Cluster of Excellence "Mathematics Münster" and the Collaborative Research Centre "Geometry: Deformations and Rigidity".