Introduction To Spherical and Affine Buildings
Richard M. Weiss
Syllabus: The course has four parts.
A. Introduction to Coxeter groups. We will present
the basic facts about Coxeter groups which are needed to get started in
the theory of buildings.
B. Introduction to buildings. We will introduce
buildings as edge-colored graphs satisfying certain properties and study
the basic structures of a building: residues, roots and apartments.
C. Spherical buildings. These are the buildings
whose apartments are finite. We will study root groups, generalized polygons
and the Moufang property. We will give an overview of the classification
of Moufang polygons and spherical buildings of rank at least three.
D. Affine buildings (also known as Euclidean buildings).
These are the buildings whose apartments have a natural representation
as a tiling of a Euclidean space. We will study the building at infinity
(a spherical building), "tree-preserving" isomorphisms and root data with
valuation. We will give an overview of the classification of affine buildings
whose building at infinity satisfies the Moufang property.
Among Jacques Tits's most remarkable accomplishments are
his classifications results for spherical and affine buildings. The goal
of this course is to leave the students with some appreciation of this
work.
References:
1. K. Brown, Buildings, Springer 1989.
2. F. Bruhat and J. Tits, Groupes réductifs sur
un corps local, I. Données radicielles valués, Publ. Math.
I.H.E.S. 41 (1972), 5-252.
3. M. Ronan, Lectures on Buildings, Academic Press,
New York 1989.
4. J. Tits, Buildings of Spherical Type and Finite
BN-Pairs, Lecture Notes in Math. 386, Springer, 1974.
5. J. Tits, Immeubles de type affine, in Buildings
and the Geometry of Diagrams (Como 1984), pp. 159-190, Lecture Notes
in Math. 1181, Springer, 1986.
6. J. Tits and R. M. Weiss, Moufang Polygons, Springer,
2002.
7. R. M. Weiss, The Structure of Spherical Buildings,
Princeton, 2003.
8. R. M. Weiss, The Structure of Affine Buildings,
in preparation.
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