Periodic Reporting for period 1 - QUILCON (Quillen's Conjecture)
Okres sprawozdawczy: 2021-08-01 do 2023-07-31
In this MSCA project, QUILCON, the fellow Joachim Kock worked at the University of Copenhagen under the supervision of Jesper Møller towards a proof of Quillen's conjecture, using higher category theory. In terms of training, the goal was to train the fellow in goal-oriented research, take him to a higher level of research quality and boost his career perspectives.
Quillen's conjecture, formulated by Fields medallist Quillen in 1978, describes a startling interplay between group theory, homotopy theory, and combinatorics: Let G be a finite group and let p be a prime number; consider the poset S_p(G) of nontrivial p-subgroups of G. Quillen observed that if G has a non-trivial normal p-subgroup, then S_p(G) is contractible. His conjecture states the converse implication:
Quillen's conjecture. If S_p(G) is contractible, then G has a non-trivial normal p-subgroup.
Quillen himself proved it for the class of all solvable groups. The general case has remained open, although many top mathematicians have tried to solve it (and have proved that the conjecture holds in many more cases).
It is not easy to explain the importance of all this outside of mathematics. The importance of Quillen's conjecture is not so much the result in itself, but rather the fact that it expresses a surprising connection between algebra, topology and combinatorics, and it serves as a measure for our mathematical understanding of these connections. In a time where the body of mathematical knowledge is ever growing, it is of very high importance to understand connections between fields to keep it all together. While Quillen's conjecture may not be directly applicable to everyday life, history shows that mathematics always ends up being useful. As a standard example: when group theory was invented more than a hundred years ago, nobody could even imagine that it would become a crucial element in quantum theory or that it is today a key ingredient in cryptography.
The QUILCON project proposed a completely new approach to the Quillen conjecture by taking the viewpoint of higher category theory. At the same time the project should serve as important training of the researcher.
The idea was to generalise to certain higher-categorical fibrations, where more information is available to be exploited, namely fusion data, the information about how various subgroups can be conjugated into each other. Quillen's approach was to prove that if a non-trivial normal p-subgroup exists then the Euler characteristic is non-trivial. A key idea in this project was to exploit higher Euler characteristics and certain higher derived spaces starting from the basic fibrations, to overcome Quillen's solvability assumption. A detailed proof strategy was outlined.
Unfortunately, the scientific part of the project did not fully develop according to plans. Some of the basic ideas broke down, some tools were not up to the task, and the work did not come close to the Quillen conjecture. However, some of the scientific ideas were vindicated, and two out of five subprojects were completed, producing important results, described below. The difficulties encountered also led to new ideas, and prompted interesting new research directions.
At the level of training and career development, the project was a big success. The two-way transfer of knowledge between fellow and host was very valuable for both parties, the fellow learned a lot of group theory and got important experience in goal-oriented research, and altogether it resulted in a considerable boost of career perspectives.
Quillen's conjecture, formulated by Fields medallist Quillen in 1978, describes a startling interplay between group theory, homotopy theory, and combinatorics: Let G be a finite group and let p be a prime number; consider the poset S_p(G) of nontrivial p-subgroups of G. Quillen observed that if G has a non-trivial normal p-subgroup, then S_p(G) is contractible. His conjecture states the converse implication:
Quillen's conjecture. If S_p(G) is contractible, then G has a non-trivial normal p-subgroup.
Quillen himself proved it for the class of all solvable groups. The general case has remained open, although many top mathematicians have tried to solve it (and have proved that the conjecture holds in many more cases).
It is not easy to explain the importance of all this outside of mathematics. The importance of Quillen's conjecture is not so much the result in itself, but rather the fact that it expresses a surprising connection between algebra, topology and combinatorics, and it serves as a measure for our mathematical understanding of these connections. In a time where the body of mathematical knowledge is ever growing, it is of very high importance to understand connections between fields to keep it all together. While Quillen's conjecture may not be directly applicable to everyday life, history shows that mathematics always ends up being useful. As a standard example: when group theory was invented more than a hundred years ago, nobody could even imagine that it would become a crucial element in quantum theory or that it is today a key ingredient in cryptography.
The QUILCON project proposed a completely new approach to the Quillen conjecture by taking the viewpoint of higher category theory. At the same time the project should serve as important training of the researcher.
The idea was to generalise to certain higher-categorical fibrations, where more information is available to be exploited, namely fusion data, the information about how various subgroups can be conjugated into each other. Quillen's approach was to prove that if a non-trivial normal p-subgroup exists then the Euler characteristic is non-trivial. A key idea in this project was to exploit higher Euler characteristics and certain higher derived spaces starting from the basic fibrations, to overcome Quillen's solvability assumption. A detailed proof strategy was outlined.
Unfortunately, the scientific part of the project did not fully develop according to plans. Some of the basic ideas broke down, some tools were not up to the task, and the work did not come close to the Quillen conjecture. However, some of the scientific ideas were vindicated, and two out of five subprojects were completed, producing important results, described below. The difficulties encountered also led to new ideas, and prompted interesting new research directions.
At the level of training and career development, the project was a big success. The two-way transfer of knowledge between fellow and host was very valuable for both parties, the fellow learned a lot of group theory and got important experience in goal-oriented research, and altogether it resulted in a considerable boost of career perspectives.
An initial period of approximately 12 months was invested in setting up the basic machinery required, certain 2-categorical fibrations that should play the role of subgroup posets and at the same time include higher dimensional data in the form of fusion. This was a substantial part of the project as the fellow had to become proficient in finite group theory as the overarching training element of the action, and at the same time reinterpret it in more categorical terms. While the foundations for the theory went well and according to plan, it turned out to be too difficult to establish the basic theorems that were supposed to hold at this new level of generality. The difficulties are too technical to explain here, but roughly they have to do with the fact that two of the basic notions in group theory, normalisers and centralisers, are not functorial constructions and therefore not so amenable to categorical interpretation.
From here on, focus was put on the role of Euler characteristics, a general algebraic invariant (number) that can be associated to various geometric objects, and the Crapo formula for computing them via Möbius functions.
The results obtained follow three lines of research, and were synthesised into three research papers, which will be published in high-quality mathematical journals. The third has already been presented at the International Category Theory Meeting, Louvain-la-neuve, July 2023.
Euler characteristics were a key ingredient in the project, but the notion had not been defined for the objects in question, though, and it required work to set this up properly. The first paper, joint with Møller, develops a general theory for Möbius inversion and Euler characteristics for a large class of simplicial spaces called decomposition spaces, which includes ∞-categories. Previous theory had only been able to deal with so-called fine Möbius inversion and only for decomposition spaces assumed to be complete. One achievement was an important comparison theorem between so-called fine and coarse Möbius functions, which crucial needs the non-complete situation. A second contribution of the paper is the development of Möbius inversion in the setting of linear endofunctors instead of the usual setting of convolution algebras.
The project proposal suggested that the Euler characteristics should be calculated by a generalisation of Crapo's classical complementation formula. The second paper, joint with Gálvez and Tonks, proves a Crapo-style formula valid for any decomposition space and any convex subspace. To establish the result in this generality, it was necessary first to develop general theory of convex subspaces of decomposition spaces.
The third paper, joint with Møller, grew out of the work on Euler characteristics. The paper establishes a categorical setting in which signs have objective reality, by working internally to a certain topos of parity structures. In particular it is shown how exterior algebras, determinants, and Euler characteristics can be developed objectively in this framework. This work has a big potential for application in many areas where combinatorial structures interact with algebraic signs. The ideas are the starting point for a new project, which has been funded by the Danish Independent Research Fund, and which is thus an unexpected offshoot of the QUILCON project.
From here on, focus was put on the role of Euler characteristics, a general algebraic invariant (number) that can be associated to various geometric objects, and the Crapo formula for computing them via Möbius functions.
The results obtained follow three lines of research, and were synthesised into three research papers, which will be published in high-quality mathematical journals. The third has already been presented at the International Category Theory Meeting, Louvain-la-neuve, July 2023.
Euler characteristics were a key ingredient in the project, but the notion had not been defined for the objects in question, though, and it required work to set this up properly. The first paper, joint with Møller, develops a general theory for Möbius inversion and Euler characteristics for a large class of simplicial spaces called decomposition spaces, which includes ∞-categories. Previous theory had only been able to deal with so-called fine Möbius inversion and only for decomposition spaces assumed to be complete. One achievement was an important comparison theorem between so-called fine and coarse Möbius functions, which crucial needs the non-complete situation. A second contribution of the paper is the development of Möbius inversion in the setting of linear endofunctors instead of the usual setting of convolution algebras.
The project proposal suggested that the Euler characteristics should be calculated by a generalisation of Crapo's classical complementation formula. The second paper, joint with Gálvez and Tonks, proves a Crapo-style formula valid for any decomposition space and any convex subspace. To establish the result in this generality, it was necessary first to develop general theory of convex subspaces of decomposition spaces.
The third paper, joint with Møller, grew out of the work on Euler characteristics. The paper establishes a categorical setting in which signs have objective reality, by working internally to a certain topos of parity structures. In particular it is shown how exterior algebras, determinants, and Euler characteristics can be developed objectively in this framework. This work has a big potential for application in many areas where combinatorial structures interact with algebraic signs. The ideas are the starting point for a new project, which has been funded by the Danish Independent Research Fund, and which is thus an unexpected offshoot of the QUILCON project.
The action led to important progress and contributions to the state of the art in homotopy combinatorics. All the results are of wider applicability than just the Quillen conjecture that motivated the research, and will find use in other areas too. (None of the results have any socio-economic impact whatsoever, though.)