Linus Kramer

Flag-Homogeneous Compact Connected Polygons

All flag-homogeneous compact connected polygons are classified explicitly. It turns out that these polygons are precisely the compact connected Moufang polygons.

Asymptotic cones and ultrapowers of lie groups

§1. Introduction . Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov. §2. Definitions . The intuitive idea behind Gromovs concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/ n for n ϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromovs original definition, the asymptotic cone existed only for a rather restricted class of spaces.

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon.It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL3(K),PSp4(K) orG2(K) for some algebraically closed fieldK.Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we showTheorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field.Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed.We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

On homomorphisms between generalized polygons

We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphism is either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.

Projective planes and isoparametric hypersurfaces

The isoparametric hypersurfaces in spheres with three distinct principal curvatures were classified by Cartan in 1939. We give a new proof for this result by showing that every such hypersurface can be naturally identified with the flag space of a compact connected Moufang plane. Our approach also leads to a uniform and explicit description of these hypersurfaces.

The topology of smooth projective planes

(i) If the (covering) dimension n of the point space ~ is finite, then ~ is a generalized manifold, and N has the same cohomology as one of the four classical point spaces P2 N, P2t~, P2]H or P2 O, and thus n e {2, 4, 8, 16}, cp. L6wen [111. This holds in particular, if the point rows and the pencils of lines are topological manifolds. In this case, the point rows and pencils of lines are n/2-spheres, cp. Salzmann [14, 7.12], Breitsprecher [2, 2.1]. It is not known whether the point rows have to be topological manifolds even if the point space is a topological manifold. (ii) If the (covering) dimension of the point space ~ is 2 or 4, then the point rows and the pencils of lines are topological manifolds, cp. Salzmann [14, 2.0] [15, 1.1], and .~ is homeomorphic to P2 N or PzlE, respectively, cp. Salzmann [14, 2.0], Breitsprecher [2, 2.5].

Robustness versus evolvability: A paradigm revisited

Evolvability is the property of a biological system to quickly adapt to new requirements. Robustness seems to be the opposite. Nonetheless many biological systems display both properties—a puzzling observation, which has caused many debates over the last decades. A recently published model by Draghi et al. [Nature 463, 353–355 (2010)] elegantly circumvents complications of earlier in silico studies of molecular systems and provides an analytical solution, which is surprisingly independent from parameter choice. Depending on the mutation rate and the number of accessible phenotypes at any given genotype, evolvability and robustness can be reconciled. Further research will need to investigate if these parameter settings adequately represent the range of degrees of freedom covered by natural systems and if natural systems indeed assume a state in which both properties, robustness and evolvability, are featured.

Loop Groups and Twin Buildings

In these notes we describe some buildings related to complex Kac–Moody groups. First we describe the spherical building of SLn(ℂ) (i.e. the projective geometry PG(ℂn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field ℂ(z). Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac–Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof which is a variant of the proof by Quillen and Mitchell, of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac–Moody groups.

NON-DISCRETE EUCLIDEAN BUILDINGS FOR THE REE AND SUZUKI GROUPS

We call a non-discrete Euclidean building a {\it Bruhat-Tits space} if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type

Differentiability of continuous homomorphisms between smooth loops

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