# Mathematics Research

My mathematical research is mainly in the area of group theory and geometry. Some of my early work was on geometries for the sporadic groups, and along with my colleague Stephen Smith in Chicago, we produced the first geometry for the Monster, assuming of course that it existed. Subsequent joint work with Gernot Stroth in Berlin led to a uniform analysis of all sporadic groups, some of which have very interesting geometries, though they do not fit into a general theory like those for the groups of Lie type.

The discovery of these groups (sporadic plus Lie type) yielded a complete list of all finite simple groups. The quest to find them all, and show there are no more, is described in my book *Symmetry and the Monster*.

My later work has been on geometries for the groups of Lie type, and groups of Kac-Moody type. These geometries are called buildings (nothing to do with buildings in the usual sense), and my book, *Lectures on Buildings* gives the basics of the subject. It was originally published in 1989 by Academic Press, and the present version — revised and updated — was published in 2009 by the University of Chicago Press.

Buildings are, so to speak, ‘multi-crystals’, composed in a very elegant way from crystal structures called apartments.

Jacques Tits first developed the theory of buildings in the late 1950s and 1960s to give a geometric basis for the groups of Lie type. His work led to buildings of “spherical type”, whose apartments are tilings of spheres in n-dimensions. In the early 1970s buildings of “affine type” appeared, with apartments that are tilings of Euclidean space. Affine buildings are used in studying groups over fields having a discrete valuation; fields such as the *p*-adic numbers, or just the rational numbers with a *p*-adic valuation.

In the early 1990s, Tits’ work on Kac-Moody groups (which are infinite dimensional analogues of the simple Lie groups) led to a theory of twin buildings. A twin building is a pair of buildings twinned with one another in a way made precise by a “codistance function” between the two buildings. A spherical building is automatically twinned with itself, and the class of twin buildings is a vast generalization of the class of spherical buildings.

For a more detailed discussion on buildings, click here.

### Some selected research papers.

- “Topological groups of Kac-Moody type, right-angled twinnings and their lattices” (with B. Rémy),
*Comment. Math. Helv.*81 (2006), 191–219. - “Multiple Trees”,
*J. Algebra*271 (2004), 673–697. - “Affine Twin Buildings”,
*J. of London Maths. Soc.*68 (2003), 461–476. - “Local Isometries of Twin Buildings,”
*Math. Zeitschrift*234 (2000), 435–455. - “Twin Trees II: Local Structure and a Universal Construction” (with J. Tits),
*Israel J. Math.*109 (1999), 349–377. - “Local to Global Structure of Twin Buildings” (with B. Mühlherr),
*Inventiones Math.*122 (1995), 71–81. - “Twin Trees I” (with J. Tits),
*Inventiones Math.*116 (1994), 463–479. - “Buildings and their applications, II. Affine buildings and symmetric spaces”,
*Bull. London Math. Soc.*24 (1992), 97–126. - “Building buildings” (with J. Tits),
*Math. Annalen*278 (1987), 291–306.