mathematics

Halsey Royden

Halsey Royden 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 Halsey Royden rather than just read about it. In short: Halsey Lawrence Royden, Jr. (September 26, 1928 – August 22, 1993) was an American mathematician, specializing in complex analysis on Riemann surfaces, several complex variables, and complex differential geometry.

Key takeaways

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

Reference excerpt

Halsey Lawrence Royden, Jr. (September 26, 1928 – August 22, 1993) was an American mathematician, specializing in complex analysis on Riemann surfaces, several complex variables, and complex differential geometry. Royden is the author of a popular textbook on real analysis.

Education and career After study at Phoenix College, Royden transferred in 1946 to Stanford University, where he received his bachelor's degree in 1948 and his master's degree in 1949, with a master's thesis written under the supervision of Donald Spencer. Royden received his Ph.D. in 1951 at Harvard University under the supervision of Lars Ahlfors with thesis Harmonic functions on open Riemann surfaces. At Stanford University he became an assistant professor in 1951, an associate professor in 1953, and a full professor in 1958. In addition to serving on the faculty of the mathematics department, for Stanford's School of Humanities and Sciences he was in 1962–1965 associate dean, in 1968–1969 executive dean (acting dean until the vacancy was resolved), and in 1973–1981 dean. In 1981 he resigned as dean to work full-time as a mathematics professor. He was on the editorial board of the Pacific Journal of Mathematics for the five years from 1956 to 1960. Royden was a visiting scholar at the Institute for Advanced Study in Princeton for 3 months in the fall of 1969, 3 months in the spring of 1974, and for the academic year 1982–1983. In 1970, he showed the equivalence of the Kobayashi metric and the Teichmüller metric on Teichmüller space. Royden was a Guggenheim Fellow for the academic year 1973–1974. In 1974 he was an Invited Speaker (Intrinsic metrics on Teichmüller space) at the International Mathematical Congress in Vancouver. Upon his death he was survived by his wife (the mathematician Virginia "Jinx" Voegeli), two daughters (one, Leigh Royden, a noted geologist), a son, and several grandchildren. His doctoral students include Alan Huckleberry, Peter A. Loeb and John Wetzel.

Selected publications

Books Real Analysis. Macmillan. 1963. 2nd edition. 1968. 3rd edition. 1988. 4th edition. 2010.

Papers Royden, H. L. (November 1949). "The Coefficient Problem for Bounded Schlicht Functions". Proc Natl Acad Sci U S A. 35 (11): 657–662. Bibcode:1949PNAS...35..657R. doi:10.1073/pnas.35.11.657. PMC 1063103. PMID 16578322. with P. R. Garabedian: Garabedian, P. R.; Royden, H. L. (January 1952). "A Remark on Cavitation Flow". Proc Natl Acad Sci U S A. 38 (1): 57–61. Bibcode:1952PNAS...38...57G. doi:10.1073/pnas.38.1.57. PMC 1063498. PMID 16589052. Royden, H. L. (1952). "Harmonic functions on open Riemann surfaces". Trans. Amer. Math. Soc. 73: 40–94. doi:10.1090/s0002-9947-1952-0049396-8. MR 0049396. Royden, H. L. (1952). "On the regularity of boundary points in potential theory". Proc. Amer. Math. Soc. 3: 82–86. doi:10.1090/s0002-9939-1952-0048639-x. MR 0048639. Royden, H. L. (1953). "Some counterexamples in the classification of open Riemann surfaces". Proc. Amer. Math. Soc. 4 (3): 363–370. doi:10.1090/s0002-9939-1953-0054056-x. MR 0054056. Royden, H. L. (1954). "The conformal rigidity of certain subdomains on a Riemann surface". Trans. Amer. Math. Soc. 76: 14–25. doi:10.1090/s0002-9947-1954-0059377-8. MR 0059377. Royden, H. L. (1954). "A property of quasi-conformal mapping". Proc. Amer. Math. Soc. 5 (2): 266–269. doi:10.1090/s0002-9939-1954-0060598-4. MR 0060598. Royden, H. L. (1956). "Rings of analytic and meromorphic functions". Trans. Amer. Math. Soc. 83 (2): 269–276. doi:10.1090/s0002-9947-1956-0089908-5. MR 0089908. Royden, H. L. (1958). "Rings of meromorphic functions". Proc. Amer. Math. Soc. 9 (6): 959–965. doi:10.1090/s0002-9939-1958-0103974-7. MR 0103974. Royden, H. L. (1963). "Function algebras". Bull. Amer. Math. Soc. 69 (3): 281–298. doi:10.1090/s0002-9904-1963-10900-3. MR 0149327. Royden, H. L. (1974). "The extension of regular holomorphic maps". Proc. Amer. Math. Soc. 43 (2): 306–310. doi:10.1090/S0002-9939-1974-0335851-X. MR 0335851. Royden, H. L. (1974). "Holomorphic fiber bundles with hyperbolic fiber". Proc. Amer. Math. Soc. 43 (2): 311–312. doi:10.1090/S0002-9939-1974-0338465-0. MR 0338465. Royden, H. L. (1984). "The Picard theorem for Riemann surfaces". Proc. Amer. Math. Soc. 90 (4): 571–574. doi:10.1090/S0002-9939-1984-0733408-6. MR 0733408. A History of Mathematics at Stanford in A century of mathematics in America, American Mathematical Society, 1989, vol. 2.

References

Worked examples

Example 1 — a first encounter with Halsey Royden

Start with the simplest possible case. Write down what Halsey Royden 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 Halsey Royden 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 Halsey Royden 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 Halsey Royden

In research: Halsey Royden 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 Halsey Royden 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: Halsey Royden is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1928 births, 1993 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Halsey Royden 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 Halsey Royden in 20 minutes

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Frequently asked questions

What is Halsey Royden in simple terms?

Why does Halsey Royden 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 Halsey Royden?

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 Halsey Royden.