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Paul A. Schweitzer

Paul A. Schweitzer 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 Paul A. Schweitzer rather than just read about it. In short: Paul Alexander Schweitzer SJ (born July 21, 1937) is an American mathematician specializing in differential topology, geometric topology, and algebraic topology. Schweitzer has done research on foliations, knot theory, and 3-manifolds.

Key takeaways

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

Reference excerpt

Paul Alexander Schweitzer SJ (born July 21, 1937) is an American mathematician specializing in differential topology, geometric topology, and algebraic topology. Schweitzer has done research on foliations, knot theory, and 3-manifolds. In 1974, he found a counterexample to the Seifert conjecture that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995, he demonstrated that Sergei Novikov's compact leaf theorem cannot be generalized to manifolds with dimension greater than 3. Specifically, Schweitzer proved that a smooth, compact, connected manifold with Euler characteristic zero and dimension > 3 has a C1 codimension-one foliation that has no compact leaf.

Life and career Schweitzer was raised in New York. After high school, he graduated from the College of the Holy Cross with a Bachelor of Science (B.S.) in mathematics in 1958. As an undergraduate at Holy Cross, he wrote a senior thesis titled, "Hereditary and Productive Properties in Topological Spaces". He then earned his Ph.D. from Princeton University in 1962 under the supervision of Norman Steenrod. His doctoral dissertation was titled, "Secondary cohomology operations induced by the diagonal mapping". In 1963, Schweitzer became a member of the Society of Jesus (Jesuits). He earned a degree in philosophy (Ph.L.) from Weston College in Weston, Massachusetts, in 1966 and a Bachelor of Divinity (B.Div.) from the Weston Jesuit School of Theology in Cambridge, Massachusetts, in 1970. He was ordained in 1970 as a Catholic priest. In 1971, he became a professor extraordinarius at the Pontifical Catholic University of Rio de Janeiro, and, in 1980, he became a professor ordinarius there. Schweitzer has been a visiting professor at the University of Notre Dame, the Fairfield University, Northwestern University, Boston College, Harvard University, and the University of Strasbourg. For the academic years 1970–1971 and 1981–1982 he was at the Institute for Advanced Study. Since 1978 he has been on the board of the Brazilian Mathematical Society. He was elected a Fellow of the American Mathematical Society in 2012. He was an Invited Speaker at the ICM in 1974 in Vancouver.

References

External links Paul A. Schweitzer, S.J., Full Professor, Dept. of Mathematics, PUC-Rio

Worked examples

Example 1 — a first encounter with Paul A. Schweitzer

Start with the simplest possible case. Write down what Paul A. Schweitzer 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 Paul A. Schweitzer 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 Paul A. Schweitzer 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 Paul A. Schweitzer

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

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Paul A. Schweitzer means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Paul A. Schweitzer out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paul A. Schweitzer in simple terms?

Paul Alexander Schweitzer SJ (born July 21, 1937) is an American mathematician specializing in differential topology, geometric topology, and algebraic topology. Schweitzer has done research on foliations, knot theory, and 3-manifolds.

Why does Paul A. Schweitzer 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 Paul A. Schweitzer?

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 Paul A. Schweitzer.

Tags

  • 1937 births
  • 20th-century American Jesuits
  • 20th-century American mathematicians
  • 21st-century American Jesuits
  • 21st-century American mathematicians
  • Academic staff of the Pontifical Catholic University of Rio de Janeiro
  • American topologists
  • Boston College School of Theology and Ministry alumni
  • College of the Holy Cross alumni
  • Fellows of the American Mathematical Society
  • Living people
  • Mathematicians from New York (state)

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