Cornell Math - MATH 757, Fall 1999

MATH 757 — Fall 1999
Topics in Topology: Buildings

Instructor: Ken Brown

Time: TR 8:40-9:55

Room: Malott 206

Buildings are simplicial complexes that were introduced by Jacques Tits in an attempt to give a unified geometric interpretation of the simple Lie groups, especially the exceptional ones. This course will give an introduction to buildings. Topics include:

  • Introduction and first examples.

  • Coxeter groups and Coxeter complexes.

  • Buildings as W-metric spaces.

  • Buildings and BN-pairs.

  • Spherical buildings associated to classical groups.

  • Affine buildings associated to p-adic groups.

  • Analogies between affine buildings and symmetric spaces.

  • Root groups.

  • Introduction to twin buildings and Kac--Moody groups.

The main references are my book and my Trieste lectures, with supplementary material (especially for the last two topics) taken from the other references.

An attempt will be made to minimize the prerequisites. The course should be accessible to any graduate student who knows elementary group theory and topology; and very little topology is needed.

Students who enroll in the course for credit will be expected to do a project (typically an expository paper and/or a lecture).

Bibliography

Nicolas Bourbaki, Groupes et algèbres de Lie, Chapitre 4–6, Masson, Paris, 1981.

Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989.

Kenneth S. Brown, Five lectures on buildings, Group theory from a geometrical viewpoint (Trieste, 1990), World Sci. Publishing, River Edge, NJ, 1991, pp. 254–295.

Paul Garrett, Buildings and classical groups, Chapman & Hall, London, 1997.

James E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.

Mark Ronan, Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press Inc., Boston, MA, 1989.

Rudolf Scharlau, Buildings, Handbook of incidence geometry, North-Holland, Amsterdam, 1995, pp. 477–645.

Jacques Tits, Buildings of spherical type and finite BN-pairs, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 386.

Jacques Tits, Twin buildings and groups of Kac-Moody type, Groups, combinatorics & geometry (Durham, 1990), Cambridge Univ. Press, Cambridge, 1992, pp. 249–286.