Mark Ronan's website
Mathematics Research
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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 in 2009
the University of Chicago Press are publishing an updated and revised version
in paperback. 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. Muehlherr), 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. For a complete list of mathematics papers, click here. |
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