mathematics

Peter Orlik

Peter Orlik 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 Orlik rather than just read about it. In short: Peter Paul Nikolas Orlik (born 12 November 1938, in Budapest) is an American mathematician, known for his research on topology, algebra, and combinatorics. Orlik earned in 1961 his bachelor's degree from the Norwegian Institute of Technology in Trondheim and in 1966 his Ph.D. from the University of Michigan under Frank Raymond with thesis Necessary conditions for the homeomorphism of Seifert manifolds.

Key takeaways

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

Reference excerpt

Peter Paul Nikolas Orlik (born 12 November 1938, in Budapest) is an American mathematician, known for his research on topology, algebra, and combinatorics. Orlik earned in 1961 his bachelor's degree from the Norwegian Institute of Technology in Trondheim and in 1966 his Ph.D. from the University of Michigan under Frank Raymond with thesis Necessary conditions for the homeomorphism of Seifert manifolds. He became in 1966 an assistant professor and in 1973 a full professor at the University of Wisconsin–Madison. Orlik was in the academic year 1971/72 a visiting professor in Oslo. From 1967 to 1969 he was a visiting scholar at the Institute for Advanced Study. Orlik is the author of over 70 publications. He works on Seifert manifolds, singularity theory, braid theory, reflection groups, invariant theory, and hypergeometric integrals. He was, with Louis Solomon and Hiroaki Terao, a pioneer of the theory of arrangements of hyperplanes in complex space. In 2012 he was elected a Fellow of the American Mathematical Society.

Selected publications

Books Seifert Manifolds, Lecture Notes in Mathematics 291, Springer Verlag, 1972 as editor: Singularities, 2 vols., American Mathematical Society, 1983 Introduction to Arrangements, CBMS Regional Conference Series, American Mathematical Society, 1989 Orlik, Peter; Terao, Hiroaki (1992). Arrangements of Hyperplanes. Grundlehren der mathematischen Wissenschaften. Berlin, Heidelberg: Springer. doi:10.1007/978-3-662-02772-1. ISBN 978-3-642-08137-8. ISSN 0072-7830. MR 1217488. Orlik, Peter; Terao, Hiroaki (2007) [2001]. Arrangements and hypergeometric integrals. MSJ Memoirs. Vol. 9 (2nd ed.). Tokyo: Mathematical Society of Japan. ISBN 978-4-931469-10-5. MR 2325213. with Volkmar Welker: Algebraic Combinatorics. Lectures at a Summer School in Nordfjordeid, Norway, June 2003, Universitext, Springer Verlag, 2007

Articles with Colin P. Rourke: "Free involutions on homotopy (4k + 3)-spheres". Bulletin of the American Mathematical Society. 74: 949–953. 1968. doi:10.1090/s0002-9904-1968-12101-9. MR 0229245. with John Milnor: Milnor, John; Orlik, Peter (1970). "Isolated singularities defined by weighted homogeneous polynomials". Topology. 9 (4): 385–393. doi:10.1016/0040-9383(70)90061-3. with Frank Raymond: Orlik, Peter; Raymond, Frank (1970). "Actions of the torus on 4-manifolds. I". Transactions of the American Mathematical Society. 152 (2): 531–559. doi:10.1090/s0002-9947-1970-0268911-3. MR 0268911. with Philip Wagreich: Orlik, Peter; Wagreich, Philip (1971). "Isolated singularities of algebraic surfaces with C*-action". Annals of Mathematics. 93 (2): 205–228. doi:10.2307/1970772. JSTOR 1970772. The multiplicity of a holomorphic map at an isolated critical point, in Real and complex singularities, Oslo: Sijthoff and Noordhoff, 1977, 405–474 Orlik, Peter (1979). "Singularities and group actions". Bulletin of the American Mathematical Society. 1 (5): 703–720. doi:10.1090/s0273-0979-1979-14643-3. MR 0537624. with Louis Solomon: Orlik, Peter; Solomon, Louis (1980). "Combinatorics and topology of complements of hyperplanes". Inventiones Mathematicae. 56 (2): 167–189. Bibcode:1980InMat..56..167O. doi:10.1007/bf01392549. S2CID 123685579. with Daniel C. Cohen: Cohen, Daniel C.; Orlik, Peter (2005). "Gauss–Manin connections for arrangements, III Formal connections". Transactions of the American Mathematical Society. 357 (8): 3031–3050. arXiv:math/0105063. doi:10.1090/s0002-9947-04-03621-9. MR 2135734. S2CID 2173664.

References

Worked examples

Example 1 — a first encounter with Peter Orlik

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

In research: Peter Orlik 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 Orlik 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 Orlik is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1938 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Peter Orlik 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 Peter Orlik in 20 minutes

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

Frequently asked questions

What is Peter Orlik in simple terms?

Why does Peter Orlik 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 Orlik?

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 Orlik.