William Walker Professor Emeritus of Mathematics, Research Professor
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
Group theory, especially buildings and other geometric aspects of group theory
Dr.rer.nat. Technische Universität Berlin
Awards and Honors:
- Honorary Professor, University of Birmingham, UK (2005-2020)
- Humboldt Research Prize (2003)
- 86th Kuwait Foundation Lecture, Cambridge (2008)
- Mercator Guest Professor, Giessen (winter
- Simons Collaboration Grant (2017-2022)
- Fellow of the AMS (2018)
I work in group theory. Group theory is the mathematical theory of symmetry.
Symmetry is a basic notion which plays a unifying role in both mathematics
and theoretical physics. The study of simple groups -- those from which all other groups are
assembled -- reveals uncanny connections to geometrical structures of various
sorts. In fact, it is impossible for me to say
whether the group theory I study is more a branch of algebra or more a branch of geometry, and
it is precisely this ambiguity that I find particularly fascinating.
My current expertise is in the theory of buildings. Buildings are geometric structures
discovered and studied over a lifetime by the great Jacques Tits at the Collège de France.
Spherical buildings provide a systematic description of a fundamental class of simple groups
which include all the finite simple groups of Lie type. I am especially interested in
the "exceptional groups," whose spherical buildings
are particularly intricate and beautiful objects.
Buildings are made up of of substructures called apartments which are glued together
according to certain algebraic rules.
The apartments of a spherical building, and hence spherical buildings themselves,
are crystal-like in nature.
Affine buildings, on the other hand, have apartments
which consist simply of dots spread systematically
across ordinary space, so in some sense they should be more familiar things.
Instead the algebraic rules governing the structure of affine buildings turn out to
depend on properties of the prime numbers! Thus in the study of affine buildings,
we see the intertwining not just of group theory and geometry, but algebra and
number theory as well.
"Moufang Polygons" (co-author: Jacques Tits), Springer Monographs in
Mathematics, 2002, pp. 535.
"The Structure of Spherical Buildings," Princeton University Press,
2004, pp. 140.
"Quadrangular Algebras," Mathematical Notes 46, Princeton University
Press, 2005, pp.140.
"The Structure of Affine Buildings," Annals of Mathematics Studies
168, Princeton University Press, 2008, pp.365.
"Descent in Buildings" (co-authors: Holger P. Petersson and Bernhard
Mühlherr), Annals of Mathematics Studies 190, Princeton University
Press, 2015, pp.336.
Galois involutions and exceptional groups (co-author: Bernhard
Mühlherr), L'Enseign. Math., 62 (2016), 207–260.
Rhizospheres in spherical buildings (co-author: Bernhard Mühlherr),
Math. Annalen 369 (2017), 839-868.
Tits endomorphisms and buildings of type F_4 (co-authors:
Tom De Medts and Yoav Segev), Ann. Inst. Fourier 67 (2017), 2349-2421.
Freudenthal triple systems in arbitrary characteristic (co-author: Bernhard Mühlherr), J. Algebra 520 (2019), 237-275.
Root graded groups of rank 2 (co-author: Bernhard Mühlherr), J.
Comb. Alg. 3 (2019), 189-214.
Isotropic quadrangular algebras (co-author: Bernhard Mühlherr), J.
Math. Soc. Japan, to appear.
Tits triangles (co-author: Bernhard Mühlherr), Canad. Math. Bull., to appear.
Tits polygons (co-author: Bernhard Mühlherr), Mem. Amer. Math. Soc.,
The exceptional Tits quadrangles (co-author: Bernhard Mühlherr),
Transform. Groups, to appear.