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.
