mathematics

Stephen Halperin

Stephen Halperin 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 Stephen Halperin rather than just read about it. In short: John Stephen Halperin (born 1 February 1942 in Kingston, Ontario) is a Canadian mathematician who deals with differential geometry and algebraic topology. A son of the mathematician Israel Halperin, Stephen Halperin studied at the University of Toronto with a bachelor's degree in 1966 and a master's degree in 1967.

Key takeaways

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

Reference excerpt

John Stephen Halperin (born 1 February 1942 in Kingston, Ontario) is a Canadian mathematician who deals with differential geometry and algebraic topology. A son of the mathematician Israel Halperin, Stephen Halperin studied at the University of Toronto with a bachelor's degree in 1966 and a master's degree in 1967. He received in 1970 his PhD from Cornell University under the supervision of Hsien Chung Wang with thesis Real Cohomology and Smooth Transformation Groups. He then became an assistant professor and in 1979 a full professor at the University of Toronto. Halperin was a visiting scholar in 1981 at the University of Bonn, in 1986 at the University of Nice, and in 1995 at the University of Lille. His research deals with homotopy theory and homology of loop spaces with applications in geometry. He wrote a three-volume textbook on differential geometry with Werner H. Greub and Ray Vanstone. In 1984 Halperin was elected a Fellow of the Royal Society of Canada. In 1997 he received the Jeffery–Williams Prize. In 1999 he became a full professor at the University of Maryland, College Park, where he served as the dean of the College of Computer, Mathematical and Physical Sciences.

Selected publications with Ray Vanstone and Werner H. Greub: Connections, Curvature and Cohomology , 3 volumes (Volume 1: De Rham Cohomology of Manifolds and Vector Bundles, Volume 2 Lie Groups, Principal Bundles and Characteristic Classes, Volume 3 Cohomology of principal bundles and homogeneous spaces), Academic Press 1972, 1973, 1976 with Yves Félix: Rational L.-S. category and its applications , Transactions of the American Mathematical Society, vol. 273, 1982, 1–38 with Karsten Grove: Contributions of rational homotopy theory to global problems in geometry , Publications Mathématiques de l'IHÉS, vol. 56, 1982, 171–177 doi:10.1007/BF02700465 Lectures on minimal models, Mémoires de la Société Mathématique de France, 2e série, tome 9–10, 1983, 1–261 with Yves Félix and Jean-Claude Thomas: Differential graded algebras in topology, in I.M. James (editor) Handbook in Algebraic Topology , Elsevier Science 1995, Chapter 16, pp. 829–865. with Yves Félix and Jean-Claude Thomas: Rational Homotopy Theory, Graduate Texts in Mathematics, Springer Verlag 2001

References

External links Homepage

Worked examples

Example 1 — a first encounter with Stephen Halperin

Start with the simplest possible case. Write down what Stephen Halperin 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 Stephen Halperin 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 Stephen Halperin 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 Stephen Halperin

In research: Stephen Halperin 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 Stephen Halperin 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: Stephen Halperin is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1942 births, 20th-century Canadian mathematicians, 21st-century Canadian mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Stephen Halperin 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 Stephen Halperin in 20 minutes

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

Frequently asked questions

What is Stephen Halperin in simple terms?

Why does Stephen Halperin 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 Stephen Halperin?

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 Stephen Halperin.