mathematics

Richard Swan

Richard Swan 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 Swan rather than just read about it. In short: Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an l-adic projective representation of a Galois group. His work has mainly been in the area of algebraic K-theory.

Key takeaways

Richard Swan 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 Swan to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Richard Swan from memory before moving on to harder problems.

Reference excerpt

Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an l-adic projective representation of a Galois group. His work has mainly been in the area of algebraic K-theory.

Education and career As an undergraduate at Princeton University, Swan was one of five winners in the William Lowell Putnam Mathematical Competition in 1952. He earned his Ph.D. in 1957 from Princeton University under the supervision of John Coleman Moore. In 1969 he proved in full generality what is now known as the Stallings–Swan theorem. He is the Louis Block Professor Emeritus of Mathematics at the University of Chicago. His doctoral students at Chicago include Charles Weibel, also known for his work in K-theory. Together with Otto Forster he proved the Forster–Swan theorem.

Awards and honors In 1970 Swan was awarded the American Mathematical Society's Cole Prize in Algebra.

Books Swan, R. G. (1964). The Theory of Sheaves. Chicago lectures in mathematics. Chicago: The University of Chicago Press. Swan, R. G. (1968). Algebraic K-theory. Lecture Notes in Mathematics. Vol. 76. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0080281. ISBN 978-3-540-04245-7. MR 0245634. Swan, Richard G. (1970). K-Theory of Finite Groups and Orders. Lecture Notes in Mathematics. Vol. 149. Notes by E. Graham Evans. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0059150. ISBN 978-3-540-04938-8. MR 0308195.

References

External links Swan's homepage at Chicago.

Worked examples

Example 1 — a first encounter with Richard Swan

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

In research: Richard Swan 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 Swan 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 Swan is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1933 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard Swan 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 Swan in 20 minutes

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

Frequently asked questions

What is Richard Swan in simple terms?

Why does Richard Swan 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 Swan?

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 Swan.