Baer Colloquium
June 24, 2017 — Ghent, Belgium

We will introduce the general theme of compactifying buildings attached to semisimple groups over local fields. In the case of a split group, we can identify equivariantly the maximal Satake–Berkovich compactification of the corresponding Euclidean building with the compactification obtained by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat–Tits buildings, can be described.

➡  Related paper on arXiv

Studying regular subdivisions of a class of convex polytopes which are called hypersimplices naturally leads to a new class of matroids, which we call split matroids. Very many matroids are of this type; e.g., the paving matroids arise as special cases. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry. Interestingly, prominent examples of finite geometries, such as the Fano plane, play an important role. Joint work with Benjamin Schröter.

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Amongst the finite simple groups, the Suzuki–Ree groups are peculiar, because they "are not algebraic groups"; more precisely they are not in a natural way (derived subgroups of groups) of the form \( G(K) \), where \( G \) is an algebraic \( k \)-group and \( K \) a \( k \)-algebra. Intuitively though, they should be algebraic groups over the field with \( \sqrt p \) elements.

We will make this intuition precise by constructing the 'field' \( \mathbb F_{\sqrt p} \) and algebro-geometric objects (rings, varieties, schemes, algebraic groups, ...) over it, and we will explain how in this setting Suzuki–Ree groups are algebraic groups. The construction is categorical in nature, and expands the category of schemes over \( \mathbb F_p \) to a category of twisted schemes, which can be interpreted as a category of 'schemes over \( \mathbb F_{\sqrt p} \)'.
This approach also suggests a hidden world of objects over \( \mathbb F_p \) which we relate to Tits's mixed groups and buildings and to exotic pseudo-reductive groups.

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➡  Related paper on arXiv

The following is mainly work of my PhD student Erich Dorner.

The most important compact symmetric spaces of exceptional type have dimensions 16, 32, 64, 128, and their isometry groups are the exceptional groups of type \( F_4 \), \( E_6 \), \( E_7 \), \( E_8 \). The first space is the octonionic projective plane. Hans Freudenthal has developed an own incidence geometry for each of these spaces. But Boris Rosenfeld back in 1956 tried to describe them as projective planes over the algebra \( \mathbb{K} \otimes \mathbb{O} \) where \( \mathbb{O} \) denotes the octonions and \( \mathbb{K} \) one of the 4 normed division algebras \( \mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O} \).

Rosenfeld's approach failed, but these spaces do have a strong relationship to \( (\mathbb{K}\otimes\mathbb{O})^2 \). We show that \( (\mathbb{K}\otimes\mathbb{L})^2 \) for any two normed division algebras \( \mathbb{K}\subset\mathbb{L} \) (with dimension \( k,l \)) carries the spin representation for \( \operatorname{Spin}_{k+l} \) which allows a simple construction of the exceptional Lie algebras and the Rosenfeld planes. We strongly use \( (\mathbb{K}\otimes\mathbb{L})^2 \) also for associative \( \mathbb{L} \) which correspond to certain Grassmannians. They greatly help understanding the structure of the Rosenfeld planes in which they are contained.

➡  [preprint available upon request]