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Richard Weiss
William Walker Professor Emeritus of Mathematics, Research Professor

Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
Bromfield-Pearson
Room 213
Medford, MA 02155

Email @tufts.edu:
rweiss
Phone: 617-627-3802

Expertise:

Group theory, especially buildings and other geometric aspects of group theory

Degrees:

Dr.rer.nat. Technische Universität Berlin

Major Awards:

• Honorary Professor, University of Birmingham (UK)
• Humboldt Research Prize (2003)
• 86th Kuwait Foundation Lecture, Cambridge (2008)
• Mercator Guest Professor, University of Giessen (winter semester 2012-13)

Research:

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.

Books:

"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.

Recent Articles:

Moufang quadrangles of type E_6 and E_7, J. Reine Angew. Math. (Crelle) 590 (2006), 189-226.

Moufang sets and Jordan division algebras (co-author: Tom De Medts), Math. Annalen 335 (2006), 415-433.

On the action of the Hua group in special Moufang sets (co-author: Yoav Segev), Math. Proc. Cambridge Phil. Soc. 144 (2008), 77-84.

Non-discrete Euclidean buildings for the Ree and Suzuki groups (co-authors: Petra Hitzelberger and Linus Kramer), Amer. J. Math., 132 (2010), 1113-1152.

The group E_6(q) and graphs with a locally linear group of automorphisms (co-author: Vladimir Trofimov), Math. Proc. Cambridge Phil. Soc. 148 (2010), 1-32.

The norm of a Ree group (co-author: Tom De Medts ), Nagoya Math. J., 199 (2010), 15-41.

On the existence of certain affine buildings of type E_6 and E_7, J. Reine Angew. Math. (Crelle) 653 (2011), 135-147.

The Kneser-Tits conjecture for groups with Tits-index E_{8,2}^{66} over an arbitrary field (co-authors: R. Parimala and Jean-Pierre Tignol), Transf. Groups, 17 (2012), 209-231.

Compact totally disconnected Moufang buildings (co-authors: Theo Grundhöfer, Linus Kramer and Hendrik Van Maldeghem), Tohoku Math. J. 64 (2012), 333-360.

Webs of Lagrangian tori in projective symplectic manifolds (co-author: Jun-Muk Hwang), Invent. Math. 192 (2013), 83-109.

Local identification of spherical buildings and finite simple groups of Lie type (co-authors: Ulrich Meierfrankenfeld and Gernot Stroth), Math. Proc. Cambridge Phil. Soc. 154 (2013), 527-547.

Receding polar regions of a spherical building and the center conjecture (co-author: Bernhard Mühlherr), Ann. Inst. Fourier 63 (2013), 479-513.

Coarse equivalences of Euclidean buildings (co-author: Linus Kramer), Adv. Math. 253 (2014) 1-49.

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, to appear.

Tits endomorphisms and buildings of type F_4 (co-authors: Tom De Medts and Yoav Segev), Ann. Inst. Fourier, to appear.