mathematics

Geordie Williamson

Geordie Williamson is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Geordie Williamson rather than just read about it. In short: Geordie Williamson (born 1981 in Bowral, Australia) is an Australian mathematician at the University of Sydney. He was the youngest living Fellow of the Royal Society when he was elected in 2018 at the age of 36.

Key takeaways

Geordie Williamson belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Geordie Williamson to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Geordie Williamson from memory before moving on to harder problems.

Reference excerpt

Geordie Williamson (born 1981 in Bowral, Australia) is an Australian mathematician at the University of Sydney. He was the youngest living Fellow of the Royal Society when he was elected in 2018 at the age of 36.

Education Educated at Chevalier College, Williamson graduated in 1999 with a UAI of 99.45. He studied at the University of Sydney and graduated with a Bachelor's degree in 2003 and then at the Albert-Ludwigs University of Freiburg, where he received his doctorate in 2008 under the supervision of Wolfgang Soergel. Williamson is the brother of the late James Williamson, a World Solo 24-hour mountain bike champion who died while competing in South Africa in 2010.

Research and career After his PhD, Williamson was a post-doctoral researcher at the University of Oxford, based at St. Peter's College, Oxford, and from 2011 until 2016 he was at the Max Planck Institute for Mathematics. Williamson deals with a geometric representation of group theory. With Ben Elias, he gave a new proof and a simplification of the theory of the Kazhdan–Lusztig conjectures (previously proved in 1981 by both Beilinson–Bernstein and Brylinski–Kashiwara). For this purpose, they built on works by Wolfgang Soergel and developed a purely algebraic Hodge theory of Soergel bimodules about polynomial rings. In this context, they also succeeded in proving the long-standing positive presumption of positivity for the coefficients of the Kazhdan–Lusztig polynomials for Coxeter groups. For Weyl groups (special Coxeter groups, which are connected to Lie groups), David Kazhdan and George Lusztig succeeded in doing so by identifying the polynomials with certain invariants (local intersection cohomology) of Schubert varieties. Elias and Williamson were able to follow this path of proof also for more general groups of reflection (Coxeter groups), although there is no geometrical interpretation in contrast to the case of the Weyl groups. He is also known for several counterexamples. In 1980, Lusztig suggested a character formula for simple modules of reductive groups over fields of finite characteristic p. The conjecture was proved in 1994-95 by a combination of three papers, one by Henning Haahr Andersen, Jens Carsten Jantzen, and Wolfgang Soergel, one by David Kazhdan and George Lusztig and one by Masaki Kashiwara and Toshiyuki Tanisaki for sufficiently large group-specific characteristics (without explicit bound) and later by Peter Fiebig for a very high explicitly stated bound. Williamson found several infinite families of counterexamples to the generally suspected validity limits of Lusztig's conjecture. He also found counterexamples to a 1990 conjecture of Gordon James on symmetric groups. His work also provided new perspectives on the respective conjectures. In 2023 he was awarded an Australian Laureate Fellowship to further his research into fundamental symmetries.

Publications Ben Elias; Geordie Williamson (2014), "The Hodge Theory of Soergel bimodules", Annals of Mathematics, 180 (3): 1089–1136, arXiv:1212.0791 Williamson, Geordie (2017), "Schubert calculus and torsion explosion (With Appendix by A. Kontorovich, P. McNamara, G. Williamson)", Journal of the American Mathematical Society, 30: 1023–1046, arXiv:1309.5055, doi:10.1090/jams/868 Williamson, Geordie (2012), "Modular intersection cohomology complexes on flag varieties (With Appendix by Tom Braden)", Mathematische Zeitschrift, 272: 697–727, arXiv:0709.0207, doi:10.1007/s00209-011-0955-y Williamson, Geordie (2014), "On an analogue of the James conjecture", Representation Theory, 18 (2): 15–27, arXiv:1212.0794, doi:10.1090/S1088-4165-2014-00447-3 Ben Elias; Geordie Williamson (2016), "Kazhdan-Lusztig conjectures and shadows of Hodge theory", in Werner Ballmann; Christian Blohmann; Gerd Faltings; Peter Teichner; Don Zagier (eds.), Arbeitstagung Bonn 2013: In Memory of Friedrich Hirzebruch, Progress in Mathematics, vol. 319, Birkhäuser, pp. 105–126, arXiv:1403.1650, doi:10.1007/978-3-319-43648-7_5, ISBN 978-3-319-43646-3 Daniel Juteau; Carl Mautner (2014), "Parity sheaves", Journal of the American Mathematical Society, 27 (4): 1169–1212, arXiv:0906.2994, doi:10.1090/S0894-0347-2014-00804-3

Awards and honours In 2016, he received the Chevalley Prize of the American Mathematical Society and the Clay Research Award. He is an invited speaker at the European Congress of Mathematicians in Berlin 2016 (Shadows of Hodge theory in representation theory). In 2016 he was awarded the EMS Prize, for 2017 he was awarded the New Horizons in Mathematics Prize. In 2018, he was plenary speaker at the International Congress of Mathematicians in Rio de Janeiro and was elected a Fellow of the Royal Society (FRS) and the Australian Academy of Science. Williamson was awarded the 2018 Australian Mathematical Society Medal, the NSW Premier's Prizes for Science & Engineering: Excellence in Mathematics, Earth Sciences, Chemistry or Physics in 2022 and the Max Planck-Humboldt Research Award in 2024.

References

Illustrations

Worked examples

Example 1 — a first encounter with Geordie Williamson

Start with the simplest possible case. Write down what Geordie Williamson claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Geordie Williamson before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Geordie Williamson ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Geordie Williamson

In research: Geordie Williamson appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Geordie Williamson in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Geordie Williamson is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1981 births, 21st-century Australian mathematicians, Australian fellows of the Royal Society, so understanding it makes those chapters shorter.
In everyday life: Look for Geordie Williamson outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Geordie Williamson in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Geordie Williamson means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Geordie Williamson in simple terms?

Why does Geordie Williamson matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Geordie Williamson?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Geordie Williamson.