mathematics

Ian Agol

Ian Agol 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 Ian Agol rather than just read about it. In short: Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego.

Key takeaways

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

Reference excerpt

Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.

Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego. His faculty advisor was Michael Freedman and his thesis was Topology of Hyperbolic 3-Manifolds. He is a professor at the University of California, Berkeley and a former professor at the University of Illinois at Chicago.

Contributions In 2004, Agol proved the Marden tameness conjecture, a conjecture of Albert Marden. It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was also independently proven by Danny Calegari and David Gabai, and implies the Ahlfors measure conjecture. In 2012, he announced a proof of the virtually Haken conjecture, which was published a year later. The conjecture (now theorem) states that every aspherical 3-manifold is finitely covered by a Haken manifold. In 2022, he posted on the ArXiv a proof of Cameron Gordon's 1981 conjecture on knot theory saying that ribbon concordance forms a partial ordering on the set of knots.

Awards and honors Agol, Calegari, and Gabai received the 2009 Clay Research Award for their proof of the Marden tameness conjecture. In 2005, Agol was a Guggenheim Fellow. In 2012 he became a fellow of the American Mathematical Society. In 2013, Agol was awarded the Oswald Veblen Prize in Geometry, along with Daniel Wise. In 2015, he was awarded the 2016 Breakthrough Prize in Mathematics, "for spectacular contributions to low dimensional topology and geometric group theory, including work on the solutions of the tameness, virtually Haken and virtual fibering conjectures." In 2016, he was elected to the National Academy of Sciences.

Personal His identical twin brother, Eric Agol, is an astronomy professor at the University of Washington in Seattle.

References

External links

Ian Agol publications indexed by Google Scholar

Illustrations

Worked examples

Example 1 — a first encounter with Ian Agol

Start with the simplest possible case. Write down what Ian Agol 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 Ian Agol 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 Ian Agol 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 Ian Agol

In research: Ian Agol 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 Ian Agol 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: Ian Agol is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1970 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Ian Agol 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Ian Agol in 20 minutes

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

Frequently asked questions

What is Ian Agol in simple terms?

Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of Ca…

Why does Ian Agol 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 Ian Agol?

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 Ian Agol.