Algebraic groups and buildings

There has been a lot of work over recent years around the Centre Conjecture of Jacques Tits concerning the structure of spherical buildings. This conjecture was proved by Muhlherr-Tits and Leeb-Ramos Cuevas for subcomplexes of spherical buildings, but the case of an arbitrary subset remains wide open. As well as having important ramifications in the theory of algebraic groups (particularly for rationality questions in Geometric Invariant Theory), the conjecture is also of great interest in metric geometry. The aim of this project is to study certain naturally arising classes of subset s of spherical buildings which, whilst not being full subcomplexes, still carry some extra structure. The first place we will look is in the theory of "relative complete reducibility" developed by Bate-Martin-Roehrle-Tange. An example in their paper suggests a nice combinatorial structure in the associated spherical building, for which an existing proof of a case of the Centre Conjecture should still go through. Proceeding from such examples, we hope to generate further ideas for an attack on the full conjecture. This will tie in with ongoing work of Bate-Martin-Roehrle which aims to reduce a proof of the Centre Conjecture for algebraic groups to a proof in type A.