mathematics

Irwin Kra

Irwin Kra 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 Irwin Kra rather than just read about it. In short: Irwin Kra (born January 5, 1937) is an American mathematician, who works on the function theory in complex analysis. Life and work Kra studied at Polytechnic Institute of Brooklyn (bachelor's degree in 1960) and at Columbia University, where he graduated in 1964 and received his doctorate in 1966 under supervision of Lipman Bers (Conformal Structure and Algebraic Structure).

Key takeaways

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

Reference excerpt

Irwin Kra (born January 5, 1937) is an American mathematician, who works on the function theory in complex analysis.

Life and work Kra studied at Polytechnic Institute of Brooklyn (bachelor's degree in 1960) and at Columbia University, where he graduated in 1964 and received his doctorate in 1966 under supervision of Lipman Bers (Conformal Structure and Algebraic Structure). After that, he was from 1966 to 1968 a C.L.E. Moore instructor at Massachusetts Institute of Technology and then at the State University of New York at Stony Brook, where he chaired from 1975 to 1981 the Faculty of Mathematics. From 1991 to 1996, there, he was Dean of the Division of Physical Sciences and Mathematics. Since 2004 he has been Professor Emeritus. He was a visiting professor at the Hebrew University in Jerusalem (where he collaborated with Hershel M. Farkas), the University of Perugia in Santiago de Chile, the Tohoku University, the Fudan University in Shanghai (1987). From 2004 to 2008 he was Director of Math for America, a private organization that is dedicated to the improvement of college education in mathematics in the United States. He currently lives in New York City. In 2010 he taught at the Northwestern University. From 1972 to 1973 he was a Guggenheim Fellow. In 2012 he became a fellow of the American Mathematical Society. Kra studies Riemann surfaces and their moduli spaces (Teichmüller spaces) and their connection with Kleinian groups and associated automorphic forms and their applications for example in number theory. With Bernard Maskit in 1998, he published the collected works of Lipman Bers on function theory.

Family relations Irwin Kra has three children. He is the father of mathematician Bryna Kra, climate tech venture capitalist Gabriel Kra, and Douglas Kra.

Writings

Books With Jane P. Gilman, Rubí E. Rodríguez Complex Analysis in the spirit of Lipman Bers, Springer Verlag, 2007 With Hershel M. Farkas Riemann Surfaces, Springer Verlag, 1980, 2nd edn. August 1992 With Farkas: Theta constants, Riemann surfaces, and the modular group: an introduction with applications to uniformization theorems, partition identities and combinatorial number theory, Providence, American Mathematical Society 2001 With Lipman Bers (editors): Crash course on Kleinian Groups, Springer 1974 Automorphic forms and Kleinian Groups, Benjamin 1972 Lipman Bers, a Life in Mathematics, Eds. I. Kra, R. Rodriguez, L. Keen, Amer. Math. Soc., Providence, RI, 2015.

Selected articles "On the ring of functions on an open Riemann surface". Trans. Amer. Math. Soc. 132: 231–244. 1968. doi:10.1090/s0002-9947-1968-0226043-5. MR 0226043. Kra, I. (July 1969). "Cohomology of Kleinian groups". Proc Natl Acad Sci U S A. 63 (3): 655–660. Bibcode:1969PNAS...63..655K. doi:10.1073/pnas.63.3.655. PMC 223501. PMID 16591774. Kra, Irwin (1971). "A generalization of a theorem of Poincaré". Proc. Amer. Math. Soc. 27 (2): 299–302. doi:10.1090/s0002-9939-1971-0301189-7. MR 0301189. with B. Maskit: Kra, I.; Maskit, B. (1972). "Involutions on Kleinian groups". Bull. Amer. Math. Soc. 78 (5): 801–805. doi:10.1090/s0002-9904-1972-13042-8. MR 0301190. Kra, Irwin (1981). "Canonical mappings between Teichmüller spaces". Bull. Amer. Math. Soc. (N.S.). 4 (2): 143–179. doi:10.1090/s0273-0979-1981-14870-9. MR 0598682. Kra, Irwin (1983). "On the vanishing of Poincaré series of rational functions". Bull. Amer. Math. Soc. (N.S.). 8: 63–66. doi:10.1090/s0273-0979-1983-15085-1. MR 0682823. Kra, Irwin (1989). "Accessory parameters for punctured spheres". Trans. Amer. Math. Soc. 313 (2): 589–617. doi:10.1090/s0002-9947-1989-0958896-0. MR 0958896. "Horocyclic coordinates for Riemann surfaces and moduli spaces. I: Teichmüller and Riemann spaces of Kleinian groups". J. Amer. Math. Soc. 3: 499–578. 1990. doi:10.1090/s0894-0347-1990-1049503-1. MR 1049503. with Hershel M. Farkas: Farkas, Hershel M.; Kra, Irwin (1999). "On the quintuple product identity". Proc. Amer. Math. Soc. 127 (3): 771–778. doi:10.1090/S0002-9939-99-04791-7. MR 1487364. "Cusp forms associated to rank 2 subgroups of Kleinian groups". Proc. Amer. Math. Soc. 111: 803–814. 1991. doi:10.1090/s0002-9939-1991-1056679-1. MR 1056679. with C. J. Earle and S. L. Krushkal´: Earle, C. J.; Kra, I.; Krushkalь, S. L. (1994). "Holomorphic motions and Teichmüller spaces". Trans. Amer. Math. Soc. 343 (2): 927–948. doi:10.1090/s0002-9947-1994-1214783-6. MR 1214783.

References

portrait on the sides of Math for America The original article was a translation of the corresponding German article.

Illustrations

Worked examples

Example 1 — a first encounter with Irwin Kra

Start with the simplest possible case. Write down what Irwin Kra 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 Irwin Kra 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 Irwin Kra 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 Irwin Kra

In research: Irwin Kra 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 Irwin Kra 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: Irwin Kra is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1937 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Irwin Kra 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 Irwin Kra in 20 minutes

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Frequently asked questions

What is Irwin Kra in simple terms?

Irwin Kra (born January 5, 1937) is an American mathematician, who works on the function theory in complex analysis. Life and work Kra studied at Polytechnic Institute of Brooklyn (bachelor's degree in 1960) and at Columbia University, where he graduated in 1964 and received his doctorate in 1966 u…

Why does Irwin Kra 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 Irwin Kra?

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 Irwin Kra.