Geordie Williamson

Geordie Williamson, Oberwolfach 2012

Geordie Williamson (* 1981 in Bowral , Australia ) is an Australian mathematician.

Life

Williamson grew up in Canyonleigh near Colo Vale south of Sydney in rural solitude near the Belanglo State Forest . His father sold vegetables in markets and they lived in a self-built house in the Australian bush. His mother was an elementary school teacher and died in a bicycle accident in 2003. Williamson attended Steiner School and Chevalier College in Bowral , received top grades near the maximum of the ATAR points and stood out for his mathematical talent, but initially wanted to study English as a major. He studied from 1999 at the University of Sydney with a bachelor's degree in 2003, where he won the university medal, and then at the Albert-Ludwigs-Universität Freiburg , where he received his doctorate in 2008 with Wolfgang Soergel ( Singular Soergel Bimodules ). As a post-doctoral student he was at Oxford University (St. Peter's College) and from 2011 he was at the Max Planck Institute for Mathematics (advanced researcher with a research professorship ). Since 2017 he has been a professor at the University of Sydney and also a visiting researcher at the Hausdorff Center for Mathematics in Bonn. In 2018 he became the founding director of a national mathematics center at the University of Sydney, which, following the example of the Max Planck Institute in Bonn, is to bundle mathematical talent in Australia and is endowed with 5 million dollars from the Simon Marais Foundation.

plant

Williamson deals with geometrical representation theory of groups . With Ben Elias he succeeded in the first purely algebraic proof and a simplification of the theory of the Kazhdan-Lusztig conjectures (previously proven in 1981 by Jean-Luc Brylinski and Masaki Kashiwara , Alexander Beilinson and Joseph Bernstein ). To do this, they built on the work of Wolfgang Soergel and developed a purely algebraic Hodge theory of Soergel bimodules over polynomial rings . In this context, they also succeeded in proving the long open positivity conjecture for the coefficients of the Kazhdan-Lusztig polynomials for Coxeter groups . David Kazhdan and George Lusztig already succeeded in doing this for Weyl groups (special Coxeter groups that are connected to Lie groups ) by interpreting the polynomials as invariants ( local cut cohomology ) of singularities of Schubert varieties . Elias and Williamson also succeeded in using this line of evidence for more general reflection groups (Coxeter groups), although, in contrast to the case of the Weyl groups, there is no geometric interpretation there.

He is also known for some counterexamples. In 1980 Lusztig suggested a character formula for simple modules of reductive groups over fields of finite characteristics . The conjecture was proven in 1994 by HH Andersen, Jens Carsten Jantzen and Soergel for sufficiently large group-specific characteristics (without an explicit limit) and later by Peter Fiebig for a very high, explicitly specified limit. Williamson found several infinitely large families of counterexamples to Lusztig's conjecture in 2013, contrary to popular belief. He also found counterexamples to a 1990 Gordon James conjecture about symmetric groups . His work also provided new perspectives on the respective conjectures. Together with George Lusztig, he made new conjectures about the characters of symmetry groups, which are geometrically represented as a result of the interpretation as a discrete dynamic system (billiards). ${\displaystyle p}$

Honors and memberships

In 2016 he received the AMS Chevalley Prize and the Clay Research Award . He is invited speaker at the European Congress of Mathematicians in Berlin 2016 ( Shadows of Hodge theory in representation theory ). In 2016 he received the EMS Prize and in 2017 the New Horizons in Mathematics Prize . In 2018 he was plenary speaker at the ICM in Rio, was elected to the Royal Society ( Parity sheaves and the Hecke category ) and received the medal of the Australian Mathematical Society .

Others

He speaks fluent German and French and does yoga and rock climbing. His brother James (1983-2010) was 2008 winner of the World Solo 24-hour mountain bike championships and Australian champion. He died of an undiagnosed heart condition after a competition in South Africa.

Fonts

• Modular intersection cohomology complexes on flag varieties , Mathematische Zeitschrift 272 (2012), 697–727 (with appendix by Tom Braden), Arxiv
• with Ben Elias: Kazhdan-Lusztig conjectures and shadows of Hodge theory , Arbeitstagung Bonn 2013, Arxiv
• On an analogy of the James conjecture , Representation Theory 18 (2014), 15-27, Arxiv
• with Daniel Juteau, Carl Mautner: Parity sheaves , Journal of the AMS 27 (2014), 1169–2012, Arxiv
• with Ben Elias : The Hodge Theory of Soergel bimodules , Annals of Mathematics 180 (2014), 1089–1136, Arxiv
• Local Hodge theory of Soergel bimodules. Acta Mathematica 217 (2016), 341-404.
• Schubert calculus and torsion explosion , (with appendix by A. Kontorovich, P. McNamara, G. Williamson), Journal of the American Mathematical Society, 30 (2017), 1023-1046, Arxiv 2013
• Algebraic representations and constructible sheaves , Takagi Lectures 2016, Arxiv
• Parity sheaves and the Hecke category , ICM 2018, Arxiv

Web links

• Homepage at the University of Sydney
• Notices AMS, 2016, No 4, Chevalley Prize for Williamson, pdf
• Maths prodigy comes home to establish $ 5 million world-class maths center (The Sydney Morning Herald, July 13, 2018)

Individual evidence

1. ↑ Geordie Williamson in the Mathematics Genealogy Project (English)
2. ^ Williamson, Schubert calculus and torsion explosion, Arxiv 2013
3. ↑ Lusztig, Williamson: Billiards and Tilting Characters for ${\displaystyle SL_{3}}$ , SIGMA 14, 2018
4. ^ Preprint as Hodge Theory and the Hecke Category , Arxiv , 2016