mathematics

George Bergman

George Bergman 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 George Bergman rather than just read about it. In short: George Mark Bergman (born 22 July 1943) is an American mathematician. He attended Stuyvesant High School in New York City and received his PhD from Harvard University in 1968, under the direction of John Tate.

Key takeaways

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

Reference excerpt

George Mark Bergman (born 22 July 1943) is an American mathematician. He attended Stuyvesant High School in New York City and received his PhD from Harvard University in 1968, under the direction of John Tate. The year before he had been appointed assistant professor of mathematics at the University of California, Berkeley, where he has taught ever since, being promoted to associate professor in 1974 and to professor in 1978. His primary research area is algebra, in particular associative rings, universal algebra, category theory and the construction of counterexamples. Mathematical logic is an additional research area. Bergman officially retired in 2009, but is still teaching. His interests beyond mathematics include subjects as diverse as third-party politics and the works of James Joyce. He was designated a member of the inaugural class of fellows of the American Mathematical Society in 2013.

Selected bibliography An Invitation to General Algebra and Universal Constructions, Universitext, Springer, 2015, doi:10.1007/978-3-319-11478-1, ISBN 978-3-319-11477-4 (updated 2016) Bergman, George M. (2011), "Homomorphic images of pro-nilpotent algebras", Illinois Journal of Mathematics, 55 (3): 719–748, arXiv:0912.0020, doi:10.1215/ijm/1369841782 Bergman, George M. (2006), "Generating infinite symmetric groups", Bulletin of the London Mathematical Society, 38 (3): 429–440, arXiv:math/0401304, doi:10.1112/S0024609305018308, S2CID 1892679 (with Adam O. Hausknecht) Co-groups and co-rings in categories of associative rings, Mathematical Surveys and Monographs, vol. 45, American Mathematical Society Providence, RI, 1996, ISBN 0-8218-0495-2 Bergman, George M. (1983), "Embedding rings in completed graded rings 4. Commutative algebras", Journal of Algebra, 84 (1): 62–106, doi:10.1016/0021-8693(83)90068-6 Bergman, George M. (1978), "The diamond lemma for ring theory", Advances in Mathematics, 29 (2): 178–218, doi:10.1016/0001-8708(78)90010-5 Bergman, George (1976), "Rational relations and rational identities in division rings. II", Journal of Algebra, 43 (1): 267–297, doi:10.1016/0021-8693(76)90160-5 Bergman, George M. (1974), "Coproducts and some universal ring constructions", Transactions of the American Mathematical Society, 200: 33–88, doi:10.1090/S0002-9947-1974-0357503-7

References

External links George Bergman at the Mathematics Genealogy Project Wall Street Journal article, Oct 27, 2009 UC Berkeley website

Illustrations

Worked examples

Example 1 — a first encounter with George Bergman

Start with the simplest possible case. Write down what George Bergman 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 George Bergman 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 George Bergman 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 George Bergman

In research: George Bergman 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 George Bergman 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: George Bergman is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1943 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for George Bergman 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 George Bergman in 20 minutes

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

Frequently asked questions

What is George Bergman in simple terms?

Why does George Bergman 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 George Bergman?

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 George Bergman.