- Department of Mathematics
William Walker Professor Emeritus of Mathematics, Research Professor
Dr.rer.nat. Technische Universität Berlin
Group theory, especially buildings and other geometric aspects of group theory
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.
Selected Honors and Awards
Honorary Professor, University of Birmingham, UK (2005-2020)
Humboldt Research Prize (2003)
86th Kuwait Foundation Lecture, Cambridge (2008)
Mercator Guest Professor, Giessen (winter semester 2012-2013)
Simons Collaboration Grant (2017-2022)
Fellow of the AMS (2018)
Selected Publications and Presentations
"Moufang Polygons" (co-author: Jacques Tits), Springer Monographs in Mathematics, 2002.
"The Structure of Spherical Buildings," Princeton University Press, 2004.
"Quadrangular Algebras," Mathematical Notes 46, Princeton University Press, 2005.
"The Structure of Affine Buildings," Annals of Mathematics Studies 168, Princeton University Press, 2008.
"Descent in Buildings" (co-authors: Holger P. Petersson and Bernhard Mühlherr), Annals of Mathematics Studies 190, Princeton University Press, 2015.
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.
Tits triangles (co-author: Bernhard Mühlherr), Canad. Math. Bull., 62 (2019), 583-601.
Isotropic quadrangular algebras (co-author: Bernhard Mühlherr), J. Math. Soc. Japan 71 (2019), 1321-1380.
Exceptional groups of relative rank one and Galois involutions of Tits quadrangles (co-author: Bernhard Mühlherr), Pacific J. Math. 307 (2020), 391-454.
The exceptional Tits quadrangles (co-author: Bernhard Mühlherr), Transform. Groups 25 (2020), 1289-1344.
Dagger-sharp Tits octagons (co-author: Bernhard Mühlherr), J. Korean Math. Soc. 58 (2021), 173-205.
Tits polygons (co-author: Bernhard Mühlherr), Mem. Amer. Math. Soc., to appear.