Abstract
The ‘Erlangen Program’ of Felix Klein worked by ‘reducing’ problems in geometry to the study of their symmetry groups—thereby algebraizing geometry. Jacques Tits’s work goes in the opposite direction—he made fundamental contributions to the abstract theory of groups via geometric methods. His geometric techniques apply to not only finite groups, but also to rather diverse situations such as groups defined over the p-adic numbers, and to the so-called arithmetic groups etc. Tits’s ideas have enriched many of the important advances in group theory and geometry in the last six decades. He designed the theory of so-called ‘buildings’ which incorporates geometrically the algebraic structure of linear groups. Amazingly, these ideas have also led to applications in subjects like the study of Riemannian manifolds of higher rank that are seemingly remote from the original developments.
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Suggested Reading
The Abel Prize 2008–2012, Helge Holden and Ragni Piene (Editors), Springer-Verlag, Berlin Heidelberg 2014.
E8, The Most Exceptional Group, Skip Garibaldi, Bull. Amer. Math. Soc., Vol.53, pp.643–671, October 2016.
Interview with John G Thompson and Jacques Tits by Martin Raussen and Christian Skau, Notices Amer. Math. Soc., Vol.54, pp.471–478, 2009.
Buildings: Theory and Applications, Peter Abramenko & Kenneth S Brown, Graduate Texts in Mathematics, Springer Science + Business Media LLC, Vol.248, 2008.
On buildings and their applications, J. Tits, Proceedings of the International Congress of Mathematicians, Vol.1, Canadian Mathematical Congress, Montreal, Que., pp.209–220, 1975.
Buildings of Spherical Type and Finite BN-Pairs, Jacques Tits, Springer-Verlag, Vol.386, Berlin-Heidelberg 1974, 311 pages.
Conformal Invariance and String Theory, edited by P. Dita and V. Georgescu, Academic Press, 1989.
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Sury is with the Indian Statistical Institute in Bangalore since 1999. He learnt some of the rich mathematics talked about in this article at the Tata Institute in Bombay during 1981–1999. He was the National co-ordinator for the Mathematical Olympiad Programme in the country and likes to communicate with young students at the school as well as college levels. He introduces this article with the limerick: There once was a Belgian who talked of pairs BN. He was the king of many a building. We can only say ‘Très Bien!’
The spherical Tits building of a finite group of Lie type is homotopically a bouquet of spheres.
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Sury, B. The Mathematical Legacy of Jacques Tits. Reson 27, 1687–1702 (2022). https://doi.org/10.1007/s12045-022-1464-5
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DOI: https://doi.org/10.1007/s12045-022-1464-5