mathematics

Lenhard Ng

Lenhard Ng 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 Lenhard Ng rather than just read about it. In short: Lenhard Ng (born 1976) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.

Key takeaways

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

Reference excerpt

Lenhard Ng (born 1976) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.

Background and education Lenhard Ng is an American of Chinese descent. His father, Jack Ng, is a professor of physics at University of North Carolina Chapel Hill. Lenhard earned his B.A. summa cum laude in Mathematics and Physics from Harvard University in three years and his Ph.D. in Mathematics from the Massachusetts Institute of Technology in 2001. He is married to Astrid Giugni.

Child prodigy Ng was a child prodigy who was once thought to be the "smartest kid in America". At age 10, he earned a perfect score of 800 on the math portion of the SAT, which made him the youngest person to have achieved this feat on his first try. At the age of 11, he earned a perfect score on the College Board Test of Standard Written English. He earned a perfect score on the American High School Mathematics Examination in all 4 years of high school at Chapel Hill High School (Chapel Hill, North Carolina). He attended the Johns Hopkins Center for Talented Youth and was one of the gifted children included in the Study of Mathematically Precocious Youth longitudinal cohort. He was estimated to be 1 in approximately 30 million of his age-mates. At the age of 12, he began taking courses (on a part-time basis) at the University of North Carolina, Chapel Hill. He was not yet 13 when he won the Written Round of the MATHCOUNTS competition. At the age of 14, he participated in the International Mathematical Olympiad and earned a Silver medal. He participated in this competition for the next two years and earned Gold medals. He entered college (Harvard University) full-time at the age of 16 and majored in Mathematics and Physics, graduating summa cum laude in three years. He competed in the William Lowell Putnam Mathematical Competition while at Harvard University and was a three-time fellow (in 1993, 1994, and 1995), one of only 18 people to have achieved this feat since 1938. The first time he became a Putnam Fellow was at the age of 16, making him one of only 6 people (the 5 others being Arthur Rubin, Noam Elkies, Don Zagier, David Ash and John Tillinghast) in the history of the competition to have achieved this feat.

Mathematical work Ng works in contact and symplectic geometry. His Ph.D. thesis and several other papers concern Legendrian knots, and his best-known work applies symplectic field theory to derive invariants of (topological) knots. More precisely, the conormal bundle of a knot embedded in the three-sphere is a Legendrian torus inside the three-sphere's unit ecosphere bundle (a contact five-manifold). Relative contact homology produces symplectic invariants of this pair, which give topological invariants of the knot. Ng computed the linearized contact homology in this case, providing an entirely combinatorial model for it which is a powerful knot invariant.

Recognition He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to Floer homology and low-dimensional topology and service to the mathematical community".

References

Ng, Lenhard. Knot and braid invariants from contact homology I. Geom. Topol. 9 (2005), 247–297. Ng, Lenhard. Knot and braid invariants from contact homology II. Geom. Topol. 9 (2005), 1603–1637.

External links Lenny Ng's home page Lenny Ng: AMS-MAA-SIAM Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student (Honorable Mention) Archived 2012-12-28 at the Wayback Machine Lenny Ng's Putnam Exam Record Archived 2000-02-29 at the Wayback Machine Lenhard Ng's results at International Mathematical Olympiad Lenhard Ng at the Mathematics Genealogy Project

Worked examples

Example 1 — a first encounter with Lenhard Ng

Start with the simplest possible case. Write down what Lenhard Ng 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 Lenhard Ng 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 Lenhard Ng 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 Lenhard Ng

In research: Lenhard Ng 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 Lenhard Ng 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: Lenhard Ng is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1976 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Lenhard Ng 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 Lenhard Ng in 20 minutes

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

Frequently asked questions

What is Lenhard Ng in simple terms?

Lenhard Ng (born 1976) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.

Why does Lenhard Ng 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 Lenhard Ng?

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 Lenhard Ng.