mathematics

Thomas Callister Hales

Thomas Callister Hales 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 Thomas Callister Hales rather than just read about it. In short: Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu).

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Reference excerpt

Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings, the honeycomb conjecture, and the dodecahedral conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture.

Biography He received his Ph.D. from Princeton University in 1986 with a dissertation titled The Subregular Germ of Orbital Integrals. Hales taught at Harvard University and the University of Chicago, and from 1993 and 2002 he worked at the University of Michigan. In 1998, Hales submitted his paper on the computer-aided proof of the Kepler conjecture, a centuries-old problem in discrete geometry which states that the most space-efficient way to pack spheres is in a tetrahedron shape. He was aided by graduate student Samuel Ferguson. In 1999, Hales proved the honeycomb conjecture, and also stated that the conjecture may have been in the minds of mathematicians before Marcus Terentius Varro. The conjecture is mentioned by Pappus of Alexandria in his Book V. After 2002, Hales became the University of Pittsburgh's Mellon Professor of Mathematics. In 2003, Hales started work on Flyspeck to vindicate his proof of the Kepler conjecture. His proof relied on computer calculation to verify conjectures. The project used two proof assistants, HOL Light and Isabelle. Annals of Mathematics accepted the proof in 2005; but was only 99% sure of the proof. In August 2014, the Flyspeck team's software finally verified the proof to be correct. In 2017, he initiated the Formal Abstracts project which aims to provide formalised statements of the main results of each mathematical research paper in the language of an interactive theorem prover. The goal of this project is to benefit from the increased precision and interoperability that computer formalisation provides while circumventing the effort that a full-scale formalisation of all published proofs currently entails. In the long term, the project hopes to build a corpus of mathematical facts which would allow for the application of machine learning techniques in interactive and automated theorem proving. Hales worked on a conjecture of Karl Reinhardt with Koundinya Vajjha, that the smoothed octagon has the lowest maximum packing density of all centrally symmetric convex shapes in the plane. Although they failed to prove Reinhardt's conjecture, in 2024 they claim to have proved a related conjecture of Kurt Mahler:

It seems highly probable from the convexity condition, that the boundary of an extreme convex domain consists of line segments and arcs of hyperbolae. Hales retired in May of 2025.

Awards Hales was an invited speaker at the International Congress of Mathematicians in 2002. He won the Chauvenet Prize in 2003, the R. E. Moore Prize in 2004, a Lester R. Ford Award in 2008, and a Fulkerson Prize in 2009. He was awarded the inaugural Robbins Prize of the American Mathematical Society in 2007. In 2012 he became a fellow of the American Mathematical Society. He was invited to give the Tarski Lectures in 2019. His three lectures were titled "A formal proof of the Kepler conjecture", "Formalizing mathematics", and "Integrating with Logic". He was awarded the Senior Berwick Prize of the London Mathematical Society in 2020.

Publications Hales, Thomas C. (1994). "The status of the Kepler conjecture". The Mathematical Intelligencer. 16 (3): 47–58. doi:10.1007/BF03024356. ISSN 0343-6993. MR 1281754. S2CID 123375854. Hales, Thomas C. (2001). "The Honeycomb Conjecture". Discrete and Computational Geometry. 25 (1): 1–22. arXiv:math/9906042. doi:10.1007/s004540010071. MR 1797293. S2CID 14849112. Hales, Thomas C. (2005). "A proof of the Kepler conjecture". Annals of Mathematics. 162 (3): 1065–1185. arXiv:math/9811078. doi:10.4007/annals.2005.162.1065. Hales, Thomas C. (2006). "Historical overview of the Kepler conjecture". Discrete & Computational Geometry. 36 (1): 5–20. doi:10.1007/s00454-005-1210-2. ISSN 0179-5376. MR 2229657. Hales, Thomas C.; Ferguson, Samuel P. (2006). "A formulation of the Kepler conjecture". Discrete & Computational Geometry. 36 (1): 21–69. arXiv:math/9811078. doi:10.1007/s00454-005-1211-1. ISSN 0179-5376. MR 2229658. S2CID 6529590. Hales, Thomas C.; Ferguson, Samuel P. (2011), The Kepler Conjecture: The Hales-Ferguson Proof, New York: Springer, ISBN 978-1-4614-1128-4 Hales, Thomas C.; Adams, Mark; Bauer, Gertrud; Dang, Tat Dat; Harrison, John; Hoang, Truong Le; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; An Hoai Thi Ta; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Vu, Ky; Zumkeller, Roland (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi. 5 e2. arXiv:1501.02155. doi:10.1017/fmp.2017.1.

Notes

External links Thomas Callister Hales at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Thomas Callister Hales

Start with the simplest possible case. Write down what Thomas Callister Hales 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 Thomas Callister Hales 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 Thomas Callister Hales 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 Thomas Callister Hales

In research: Thomas Callister Hales 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 Thomas Callister Hales 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: Thomas Callister Hales is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1958 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Thomas Callister Hales 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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Frequently asked questions

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