# Modelling symmetries using Algebra

Seminar organized by Magdalena Kędziorek, Luca Pol and Niall Taggart from March 18-23, 2024.

**Participants**

David BARNES (Queen’s University Belfast – Belfast, UK), Miguel BARRERO (Radboud University Nijmegen – Nijmegen, The Netherlands), Tobias BARTHEL (Max Planck Institute for Mathematics – Bonn, Germany), Anna Marie BOHMANN (Vanderbilt University – Nashville, USA), John GREENLEES (University of Warwick – Coventry, UK), Magdalena KEDZIOREK (Radboud University Nijmegen – Nijmegen, The Netherlands), Clover MAY, (Norwegian University of Science and Technology NTNU – Trondheim, Norway), Luca POL (University of Regensburg – Regensburg, Germany), Constanze ROITZHEIM (University of Kent – Kent, UK), Niall TAGGART (Utrecht University – Utrecht, The Netherlands), Jordan WILLIAMSON (Charles University – Prague, Czech Republic), Emel YAVUZ (Queen’s University Belfast – Belfast, UK).

The group of participants was very diverse: it consisted of 5 women and 7 men; there were 6 professors and 6 early career researchers including 4 postdoctoral researchers and 2 PhD students; 4 participants were from the UK, 7 from continental Europe and 1 from the USA.

**Résumé**

Le but de ce séminaire était de rassembler des experts dans le domaine des modèles algébriques pour les G- spectres rationnels et de faire progresser la recherche sur le sujet. Le séminaire comprenait un mélange de chercheurs confirmés et en début de carrière, originaires de nombreux pays, présentant un bon équilibre entre les sexes. La majeure partie de la rencontre s’est concentrée sur le travail en petits groupes, chacun étudiant des problèmes ouverts liés aux modèles algébriques. Le premier groupe a abordé les problèmes d’un point de vue global lié à la présence d’objets présentant de nombreuses symétries, le deuxième a travaillé sur la détermination des catégories des G-spectres rationnels qui étaient rigides, et le troisième a utilisé des modèles algébriques rationnels pour construire une nouvelle version, cette fois-ci rationnelle, du calcul orthogonal.

**Summary**

The residential study “modelling symmetries using algebra” aimed to utilise methods from algebra to classify the fundamental structures in the world of geometric objects equipped with symmetries. To do this, the study brought together experts in the topic of algebraic models for rational equivariant cohomology theories with the concrete aim of advancing the research on the subject. The seminar included a mix of more established and early-career researchers from many countries with a good gender balance. The key aspect to the structure of the seminar was the participants working in smaller groups, with each group attacking open problems related to algebraic models. The first group approached problems from a global point of view of having objects with many symmetries, the second one used rational algebraic models to build a new, rational, version of orthogonal functor calculus, and the third worked on determining which categories of rational equivariant spectra were rigid. These groups were designed to foster new ideas and collaborations and the seminar was very successful with it.

**Report**

The seminar concentrated on four themes, within a general topic of providing algebraic models for rational G-spectra :

**Theme 1**: An algebraic model for rational equivariant cohomology theories for compact Lie groups

**Theme 2**: An algebraic model for rational global spectra

**Theme 3**: Applications to the calculus of functors

**Theme 4**: Rigidity of rational equivariant cohomology theories

**Theme 1**and started with John GREENLEES giving a talk on

*Algebraic models for rational G-spectra for a compact Lie group G: living history*, detailing the origins of their original conjecture on an algebraic model for rational equivariant cohomology theories, and providing a concrete description of the model in the case of circular symmetries for the participants to keep in mind for the duration of the workshop. This was followed by a talk on the

*Phenomenology of algebraic models for rational equivariant spectra*by Tobias BARTHEL, with a particular focus on how more complicated symmetries yield more complicated algebraic models, e.g., the difference between the finite, profinite and compact Lie models.

The next two talks were concerned with **Theme 2**. Luca Pol provided an insight into New directions in global homotopy theory, which would potentially allow for the construction of global algebraic models which take multiple symmetries into account at once. This was followed up by Jordan WILLIAMSON who provided the participants with the details of the algebraic model for *Rational global spectra for the family of finite groups*.

Niall TAGGART and David BARNES gave talks concerning **Theme 3**. Taggart gave *An introduction to functor calculus* so that the participants would all have the required background to explore algebraic models for calculus of functors. Several potential avenues of exploration *Towards algebraic models of homotopy calculus were then provided by Barnes*.

The sequence of talks concluded with Constanze ROITZHEIM discussing **Theme 4** with the participants through an account of what is known currently about *Rigidity* in homotopy theory, including the initial rigidity results of Schwede, Roitzheim, Ishak and, Patchkoria.

During the afternoon of the second day, there were two additional short talks by early career researchers (both PhD students), which allowed them to communicate their thesis work. Emel Yavuz gave a talk on *C2-equivariant or- thogonal calculus*, a generalisation of a particular variant of orthogonal calculus which takes into account symmetries governed by the group of two elements. Moreover, Miguel Barrero spoke on *Global N∞-operads*, a generalisation of commutativity which take symmetries by group actions into account.

The other three days were spent on the mathematical work in three smaller groups related to **Themes 2-4** mentioned above. The groups consisted of:

**Group 1, Theme 2**: Miguel Barrero, Tobias Barthel, Luca Pol, and Jordan Williamson.

**Group 2, Theme 3**: David Barnes, Magdalena Kędziorek, Niall Taggart, and Emel Yavuz.

**Group 3, Theme 4**: Anna Marie Bohmann, John Greenlees, Clover May, and Constanze Roitzheim.

Below we outline a more detailed progress of each group:

**Group 1**. This group focused on understanding the tensor-triangular geometry of the category of rational global spectra for the family of elementary abelian p-groups. During the stay at FdT they made substantial progress in this area: they described the Balmer spectrum of the category and planned a strategy for calculating the homological spectrum. The collaboration continued after the stay at the Foundation des Treilles and the group is currently writing an article with these results.

**Group 2**. The focus of this group was on providing an algebraic model for orthogonal functor calculus. This is a technical piece of machinery which is heavily built on an infinite hierarchy of geometric information, and has been extensively studied by Barnes [Bar17] and Taggart [Tag22]. To build an algebraic model for this calculus the group focused on constructing models for the geometric information required in the construction of the calculus. Combining work on unstable rational homotopy theory of Quillen [Qui69], Sullivan [Sul77] and recent advances of Felix, Oprea, Tarne [FOT08] and Buijs, Cantero Morán, and Cirici [BCMC20] the group were able to provide a candidate for the first step in the algebraic models for the geometric hierarchy together with three potential candidates for the second step in this hierarchy. These steps greatly inform the rest of the hierarchy and the group is confident that a full algebraic model for orthogonal functor calculus will be a result from the research carried out at the Fondation des Treilles.

**Group 3**. This group worked on rigidity of rational torus-equivariant spectra. The strategy behind this is based on work of Balchin, Roitzheim and Williamson [BRW23] and involves “glueing” the category in question from its local and complete parts. During the week the group drafted a proof for the category of cofree rational circle-spectra and conjectured a strategy to remove the cofree assumption. Furthermore, they are optimistic that their method generalises from the circle group to compact Lie groups of dimension one.

**References**

1. [Bar17] David Barnes. Rational orthogonal calculus. J. Homotopy Relat. Struct., 12(4):1009–1032, 2017.

2. [BCMC20] Urtzi Buijs, Federico Cantero Morán, and Joana Cirici. Weight decompositions of Thom spaces of vector bundles in rational homotopy. J. Homotopy Relat. Struct., 15(1):1–26, 2020.

3. [BRW23] Scott Balchin, Constanze Roitzheim, and Jordan Williamson. Tensor-triangular rigidity in chromatic homotopy theory, 2023.

4. [FOT08] Yves Félix, John Oprea, and Daniel Tanré. Algebraic models in geometry, volume 17 of Oxf. Grad. Texts Math. Oxford: Oxford University Press, 2008.

5. [Qui69] Daniel Quillen. Rational homotopy theory. Ann. Math. (2), 90:205–295, 1969.

6. [Sul77] Dennis Sullivan. Infinitesimal computations in topology. Publ. Math., Inst. Hautes Étud. Sci., 47:269–331, 1977.

7. [Tag22] Niall Taggart. The localization of orthogonal calculus with respect to homology, 2022.

OpenEdition vous propose de citer ce billet de la manière suivante :

ldiebold (17 avril 2024). Modelling symmetries using Algebra. *Les carnets de la Fondation des Treilles*. Consulté le 22 juillet 2024 à l’adresse https://doi.org/10.58079/w8b8