mathematics

Peter Shalen

Peter Shalen 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 Peter Shalen rather than just read about it. In short: Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology.

Key takeaways

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

Reference excerpt

Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

Life He graduated from Stuyvesant High School in 1962, before earning a B.A. from Harvard College in 1966 and his Ph.D. from Harvard University in 1972. Following academic appointments at Columbia University, Rice University, and the Courant Institute of Mathematical Sciences, he joined the faculty of the University of Illinois at Chicago. Shalen was a Sloan Foundation Research Fellow in mathematics (1977–1979). In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley, California. He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to three-dimensional topology and for exposition".

Work His work with Marc Culler related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. Based on this work, Culler, Cameron Gordon, John Luecke, and Shalen proved the cyclic surgery theorem. An important corollary of the theorem is that at most one nontrivial Dehn surgery (+1 or −1) on a knot can result in a simply-connected 3-manifold. This was an important piece of the Gordon–Luecke theorem that knots are determined by their complements. This paper is often referred to as "CGLS". With John W. Morgan, he generalized his work with Culler, and reproved several foundational results of William Thurston.

Selected publications Jaco, William H. & Shalen, Peter B. (1979). Seifert fibered spaces in 3-manifolds. Providence: American Mathematical Society. ISBN 0-8218-2220-9. Shalen, Peter B. Separating, incompressible surfaces in 3-manifolds. Inventiones Mathematicae 52 (1979), no. 2, 105–126. Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds. Annals of Mathematics (2) 117 (1983), no. 1, 109–146. Culler, Marc; Gordon, C. McA.; Luecke, J.; Shalen, Peter B. Dehn surgery on knots. Annals of Mathematics (2) 125 (1987), no. 2, 237–300. Morgan, John W.; Shalen, Peter B. Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math. (2) 120 (1984), no. 3, 401–476. Morgan, John W.; Shalen, Peter B. Degenerations of hyperbolic structures. II. Measured laminations in 3-manifolds. Annals of Mathematics (2) 127 (1988), no. 2, 403–456. Morgan, John W.; Shalen, Peter B. Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston's compactness theorem. Annals of Mathematics (2) 127 (1988), no. 3, 457–519.

References

External links Shalen's home page at UIC Art Rothstein's Stuyvesant Math Team page Peter Shalen at the Mathematics Genealogy Project

Worked examples

Example 1 — a first encounter with Peter Shalen

Start with the simplest possible case. Write down what Peter Shalen 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 Peter Shalen 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 Peter Shalen 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 Peter Shalen

In research: Peter Shalen 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 Peter Shalen 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: Peter Shalen is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1940s births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Peter Shalen 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Peter Shalen” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Peter Shalen in 20 minutes

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

Frequently asked questions

What is Peter Shalen in simple terms?

Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology.

Why does Peter Shalen 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 Peter Shalen?

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 Peter Shalen.