Providence, RI---Geordie Williamson of the University of Sydney and the Max Planck Institute for Mathematics, will receive the inaugural AMS Claude Chevalley Prize in Lie Theory. Williamson is honored "for his work on the representation theory of Lie algebras and algebraic groups [which includes] proofs and reproofs of some longstanding conjectures as well as spectacular counterexamples to the expected bounds in others." Williamson is the inaugural recipient of the Chevalley Prize, which was established through a donation by MIT mathematician George Lusztig.
Lie theory is the study of Lie groups and Lie algebras, which are mathematical objects that highlight the structural and symmetric properties inherent in geometric objects. While Lie groups were invented by the Norwegian mathematician Sophus Lie (1842-1899), most of the development of Lie theory came in the 20th century. Lie algebras have become an essential tool in modern physics, where they are used to represent symmetries in properties of elementary particles.
In 1979, two mathematicians, David Kazhdan and George Lusztig, made conjectures about the nature of certain Lie algebras. The conjectures stimulated a great deal of research. Several proofs emerged, but they all relied on translating the problem, in several steps, out of its natural algebraic context and into different areas of mathematics. One of Geordie Williamson's outstanding contributions, achieved in joint work with Ben Elias, was to produce the first algebraic proof of the Kazhdan-Lusztig conjectures, which in turn resulted in a great simplification of the whole theory.
Williamson's next breakthrough concerned another conjecture, made in 1980 by Lusztig. The conjecture represented major progress in representation theory, and experts worked hard to prove it in the 1980s and 1990s. Finally, in 1994 it was proved for all very large values of a parameter arising in the conjecture; however it remained a mystery as to how small this parameter could be. Williamson stunned the experts by exhibiting infinite families of counterexamples that defied expectations about the range of validity of the conjecture.
"More importantly, Williamson did not just provide counterexamples," the prize citation says. "[H]e provided a new framework for thinking about these conjectures---a framework that revealed how inadequate the numerical evidence for these conjectures really had been. Williamson's work has re-opened the field of modular representations to new ideas, in a sense taking it beyond a focus on the famous conjectures."
Presented in even-numbered years, the Chevalley Prize recognizes notable work in Lie theory published during the preceding six years by an individual who is at most twenty-five years past the Ph.D. The prize will be presented on Thursday, January 7, 2016, at the Joint Mathematics Meetings in Seattle.