mathematics

Richard Lashof

Richard Lashof 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 Richard Lashof rather than just read about it. In short: Richard K. Lashof (November 9, 1922 – February 4, 2010) was an American mathematician.

Key takeaways

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

Reference excerpt

Richard K. Lashof (November 9, 1922 – February 4, 2010) was an American mathematician. He contributed to the field of geometric and differential topology, working with Shiing-Shen Chern, Stephen Smale, among others. Lashof was regarded as "the key figure in sustaining the Chicago Mathematics Department as an international center for research and the training of topologists" by Melvin Rothenberg. He received the Quantrell Award. Born in Philadelphia, Pennsylvania, Lashof earned a Bachelor of Science degree in chemical engineering from the University of Pennsylvania in 1943 and was a communications officer in the United States Navy from 1943 to 1946. He earned his doctoral degree in mathematics from Columbia University in 1954 under the supervision of Richard Vincent Kadison. In 1971 he was an invited to speak at the International Congress of Mathematicians in Nice.

Personal life and death Lashof was married to Joyce Lashof, a physician, and had three children, Judith, who died in 2018, Carol, and Dan. Lashof died of heart failure on February 4, 2010, after a long illness. He was 87.

References

External links Richard Lashof at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Richard Lashof

Start with the simplest possible case. Write down what Richard Lashof 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 Richard Lashof 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 Richard Lashof 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 Richard Lashof

In research: Richard Lashof 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 Richard Lashof 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: Richard Lashof is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1922 births, 2010 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard Lashof 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 Richard Lashof in 20 minutes

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

Frequently asked questions

What is Richard Lashof in simple terms?

Richard K. Lashof (November 9, 1922 – February 4, 2010) was an American mathematician.

Why does Richard Lashof 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 Richard Lashof?

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 Richard Lashof.