mathematics

Richard Palais

Richard Palais 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 Richard Palais rather than just read about it. In short: Richard Sheldon Palais (May 22, 1931 – May 13, 2026) was an American mathematician working in differential geometry. Education and career Palais studied at Harvard University, where he obtained a B.A. in 1952, an M.A. in 1954 and a Ph.D. in 1956.

Key takeaways

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

Reference excerpt

Richard Sheldon Palais (May 22, 1931 – May 13, 2026) was an American mathematician working in differential geometry.

Education and career Palais studied at Harvard University, where he obtained a B.A. in 1952, an M.A. in 1954 and a Ph.D. in 1956. His PhD thesis, entitled A Global Formulation of the Lie Theory of Transformation Groups, was supervised by Andrew M. Gleason and George Mackey. Palais was a postdoctoral researcher at University of Chicago from 1956 to 1958 and at the Institute for Advanced Study from 1958 to 1960. He moved then to Brandeis University, where he worked as assistant professor in 1960–1962, as associate professor in 1962-1965 and as full professor from 1965 until his retirement in 2003. From 2004 he was an adjunct professor at the University of California, Irvine. Palais was awarded a Sloan Fellowship in 1965. In 1970, he was an invited speaker at the International Congress of Mathematicians in Nice. From 1965 to 1982 he was an editor for the Journal of Differential Geometry and from 1966 to 1969 an editor for the Transactions of the American Mathematical Society. In 1979 he co-founded the TeX Users Group and became its first president. In 1980, Palais was elected a fellow of the American Association for the Advancement of Science and in 2012 a fellow of the American Mathematical Society. In 2006 he was awarded the first prize, together with Luc Bénard, for the NSF/Science Visualization Challenge and in 2010 he received a Lester R. Ford Award. Palais died on May 13, 2026, at the age of 94.

Research Palais' research interests included differential geometry, the theory of compact differentiable groups of transformations, the geometry of submanifolds, Morse theory and non-linear global analysis. In particular, he is known for the principle of symmetric criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, and the Palais–Smale compactness condition. Since the 1990s he worked on the theory of solitons and mathematical visualization. His doctoral students include Edward Bierstone, Leslie Lamport, Jill P. Mesirov, Chuu-lian Terng, and Karen Uhlenbeck.

Selected publications

Books As editor:

Seminar on the Atiyah-Singer Index Theorem, Annals of Mathematics Studies, no. 4, Princeton Univ. Press, 1964 As author:

A Global Formulation of the Lie Theory of Transformation Groups, Memoirs AMS 1957 The classification of G-Spaces, Memoirs AMS 1960 Foundations of Global Nonlinear Analysis, Benjamin 1968 The geometrization of physics, Tsinghua University Press 1981 Real algebraic differential topology, Publish or Perish 1981 with Chuu-Lian Terng: Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol.1353, Springer 1988 with Robert A. Palais: Differential Equations, Mechanics, and Computation, AMS 2009

Articles Richard Palais and Stephen Smale, A generalized Morse theory, Research Announcement, Bulletin of the American Mathematical Society 70 (1964), 165–172 R. Palais, Morse Theory on Hilbert Manifolds, Topology 2 (1963), 299–340. R. Palais, Linear and Nonlinear Waves and Solitons, in The Princeton Companion to Mathematics, T. Gower Ed., Princeton Univ. Press 2008, 234-239 ([1]) R. Palais, The Symmetries of Solitons, Bulletin. Amer. Math. Soc., New Series 34, No. 4, 339-403 (1997) [ISSN 0273-0979], ([2]) R. Palais, The Visualization of Mathematics: Towards a Mathematical Exploratorium, Notices Amer. Math. Soc., 46, No. 6 (June–July 1999, ([3]) R. Palais, A Simple Proof of the Banach Contraction Principle, The Journal for Fixed Point Theory and its Applications, 2 (2007) 221–223, ([4]) A nearly complete list of all papers authored or co-authored by Richard Palais is available for downloading as PDF files at his website.

References

External links Richard Palais at the Mathematics Genealogy Project Home page Curriculum Vitae Homepage of 3D-XplorMath, Mathematical Visualization software developed by R. Palais Interview with the TeX Users Group

Illustrations

Worked examples

Example 1 — a first encounter with Richard Palais

Start with the simplest possible case. Write down what Richard Palais 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 Richard Palais 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 Richard Palais 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 Richard Palais

In research: Richard Palais 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 Richard Palais 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: Richard Palais is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1931 births, 2026 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard Palais 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 Richard Palais in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richard Palais 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.
Explain Richard Palais out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richard Palais in simple terms?

Richard Sheldon Palais (May 22, 1931 – May 13, 2026) was an American mathematician working in differential geometry. Education and career Palais studied at Harvard University, where he obtained a B.A. in 1952, an M.A. in 1954 and a Ph.D. in 1956.

Why does Richard Palais 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 Richard Palais?

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 Richard Palais.