mathematics

Tim Cochran

Tim Cochran 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 Tim Cochran rather than just read about it. In short: Thomas "Tim" Daniel Cochran (April 7, 1955 – December 16, 2014) was a professor of mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra. Education and career Tim Cochran was a valedictorian for the Severna Park High School Class of 1973.

Key takeaways

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

Reference excerpt

Thomas "Tim" Daniel Cochran (April 7, 1955 – December 16, 2014) was a professor of mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.

Education and career

Tim Cochran was a valedictorian for the Severna Park High School Class of 1973. Later, he was an undergraduate at the Massachusetts Institute of Technology, and received his Ph.D. from the University of California, Berkeley in 1982 (Embedding 4-manifolds in S5). He then returned to MIT as a C.L.E. Moore Postdoctoral Instructor from 1982 to 1984. He was an NSF postdoctoral fellow from 1985 to 1987. Following brief appointments at Berkeley and Northwestern University, he started at Rice University as an associate professor in 1990. He became a full professor at Rice University in 1998. He died unexpectedly, aged 59, on December 16, 2014, while on a year-long sabbatical leave supported by a fellowship from the Simons Foundation.

Research contributions With his coauthors Kent Orr and Peter Teichner, Cochran defined the solvable filtration of the knot concordance group, whose lower levels encapsulate many classical knot concordance invariants. Cochran was also responsible for naming the slam-dunk move for surgery diagrams in low-dimensional topology.

Awards and honors While at Rice, he was named an Outstanding Faculty Associate (1992–93), and received the Faculty Teaching and Mentoring Award from the Rice Graduate Student Association (2014) He was named a fellow of the American Mathematical Society in 2014, for contributions to low-dimensional topology, specifically knot and link concordance, and for mentoring numerous junior mathematicians.

Selected publications Cochran, T. (1984). "Four-manifolds which embed in R 6 {\displaystyle \mathbb {R} ^{6}} but not in R 5 {\displaystyle \mathbb {R} ^{5}} and Seifert manifolds for fibered knots". Inventiones Mathematicae. 77: 173–184. doi:10.1007/BF01389141. S2CID 121286879. Cochran, Tim D. (1985). "Geometric invariants of link cobordism". Commentarii Mathematici Helvetici. 60: 291–311. doi:10.1007/BF02567416. S2CID 120444453. Cochran, Tim D. (1990). "Derivatives of links: Massey products and Milnor's concordance invariants". Memoirs of the American Mathematical Society. 84 (427). doi:10.1090/memo/0427. Cochran, Tim D.; Orr, Kent E. (1993). "Not all links are concordant to boundary links". Annals of Mathematics. 138 (3): 519–554. doi:10.2307/2946555. JSTOR 2946555. Cochran, Tim D.; Orr, Kent E.; Teichner, Peter (2003). "Knot Concordance, Whitney Towers and L 2 {\displaystyle L^{2}} -signatures". Annals of Mathematics. 157 (2): 433–519. arXiv:math/9908117. doi:10.4007/annals.2003.157.433. Cochran, Tim D.; Orr, Kent E.; Teichner, Peter (2004). "Structure in the Classical Knot Concordance Group". Commentarii Mathematici Helvetici. 79 (1): 105–123. doi:10.1007/s00014-001-0793-6. Cochran, Tim D. (2004). "Noncommutative Knot Theory". Algebraic and Geometric Topology. 4: 347–398. arXiv:math/0206258. doi:10.2140/agt.2004.4.347. Cochran, Tim D.; Teichner, Peter (2007). "Knot Concordance and von Neumann ρ {\displaystyle \rho } -invariants". Duke Mathematical Journal. 137 (2): 337–379. doi:10.1215/S0012-7094-07-13723-2. S2CID 119495376. Cochran, Tim D.; Harvey, Shelly (2008). "Homology and Derived Series of Groups II: Dwyer's Theorem". Geometry and Topology. 12 (1): 199–232. arXiv:math/0609484. doi:10.2140/gt.2008.12.199. Cochran, Tim D.; Harvey, Shelly; Leidy, Constance (2009). "Knot concordance and Higher-order Blanchfield duality". Geometry and Topology. 13 (3): 1419–1482. arXiv:0710.3082. doi:10.2140/gt.2009.13.1419. Cochran, Tim D.; Harvey, Shelly; Leidy, Constance (2011). "Primary decomposition and the fractal nature of knot concordance". Mathematische Annalen. 351 (2): 443–508. arXiv:0906.1373. doi:10.1007/s00208-010-0604-5. S2CID 7556758. Cochran, Tim D.; Davis, Christopher William (2015). "Counterexamples to Kauffman's conjectures on slice knots". Advances in Mathematics. 274: 263–284. arXiv:1303.4418. doi:10.1016/j.aim.2014.12.006.

References

External links Tim Cochran's home page.

Illustrations

Worked examples

Example 1 — a first encounter with Tim Cochran

Start with the simplest possible case. Write down what Tim Cochran 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 Tim Cochran 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 Tim Cochran 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 Tim Cochran

In research: Tim Cochran 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 Tim Cochran 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: Tim Cochran is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1955 births, 2014 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Tim Cochran 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 Tim Cochran in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tim Cochran 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.
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Frequently asked questions

What is Tim Cochran in simple terms?

Why does Tim Cochran 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 Tim Cochran?

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 Tim Cochran.