mathematics

Larry Guth

Larry Guth 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 Larry Guth rather than just read about it. In short: Lawrence David Guth (; born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale University in 2000 with a BS in mathematics.

Key takeaways

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

Reference excerpt

Lawrence David Guth (; born 1977) is a professor of mathematics at the Massachusetts Institute of Technology.

Education and career Guth graduated from Yale University in 2000 with a BS in mathematics. In 2005, he received his PhD in mathematics from the Massachusetts Institute of Technology (MIT), where he studied geometry of objects with random shapes under the supervision of Tomasz Mrowka. After MIT, Guth went to Stanford as a postdoc and later to the University of Toronto as an assistant professor on a tenure track. In 2011, New York University's Courant Institute of Mathematical Sciences hired Guth as a professor, listing his areas of interest as "metric geometry, harmonic analysis, and geometric combinatorics." In 2012, Guth moved to MIT, where he is Claude Shannon Professor of Mathematics.

Research In his research, Guth has strengthened Gromov's systolic inequality for essential manifolds and, along with Nets Katz, found a solution to the Erdős distinct distances problem. His interests include the Kakeya conjecture and the systolic inequality.

Doctoral students Guth has supervised graduate students at the University of Toronto and the Massachusetts Institute of Technology. His other doctoral students have included Dominic Dotterrer, Hannah Alpert, Ben Yang, Thao Do, Aleksandr Berdnikov, Alexey Balitskiy, Sahana Vasudevan, Yuqiu Fu, Shengwen Gan, Lingxian Zhang, and Sarah Tammen. His doctoral student Hong Wang received her PhD from MIT in 2019 under his supervision and was awarded the Fields Medal in 2026 for her work in harmonic analysis and geometric measure theory.

Recognition Guth won an Alfred P. Sloan Fellowship in 2010. He was an invited speaker at the International Congress of Mathematicians in India in 2010, where he spoke about systolic geometry. In 2013, the American Mathematical Society awarded Guth its annual Salem Prize, citing his "major contributions to geometry and combinatorics." In 2014 he received a Simons Investigator Award. In 2015, he received the Clay Research Award. He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to harmonic analysis, combinatorics and geometry, and for exposition of high level mathematics". On February 20, 2020, the National Academy of Sciences announced that Guth is the first winner of their new $20,000 Maryam Mirzakhani Prize in Mathematics for mid-career mathematicians. The citation states that his award is "for developing surprising, original, and deep connections between geometry, analysis, topology, and combinatorics, which have led to the solution of, or major advances on, many outstanding problems in these fields." He was one of three recipients of the 2020 Bôcher Memorial Prize. In 2021, he was elected member of the US National Academy of Sciences.

Personal He is the son of Alan Guth, a theoretical physicist known for the theory of inflation in cosmology.

Work Metaphors in systolic geometry: the video Guth, Larry (2011), "Volumes of balls in large Riemannian manifolds", Annals of Mathematics, 2nd ser., 173 (1): 51–76, arXiv:math.DG/0610212, doi:10.4007/annals.2011.173.1.2, MR 2753599, S2CID 1392012. Guth, Larry; Katz, Nets Hawk (2010), "Algebraic methods in discrete analogs of the Kakeya problem", Advances in Mathematics, 225 (5): 2828–2839, arXiv:0812.1043, doi:10.1016/j.aim.2010.05.015, MR 2680185, S2CID 15590454. Guth, Larry (2010), "Systolic inequalities and minimal hypersurfaces", Geometric and Functional Analysis, 19 (6): 1688–1692, arXiv:0903.5299, doi:10.1007/s00039-010-0052-0, MR 2594618, S2CID 17827200. Guth, Larry (2010), "The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture", Acta Mathematica, 205 (2): 263–286, arXiv:0811.2251, doi:10.1007/s11511-010-0055-6, MR 2746348, S2CID 16258342. Guth, Larry (2009), "Minimax problems related to cup powers and Steenrod squares", Geometric and Functional Analysis, 18 (6): 1917–1987, arXiv:math/0702066, doi:10.1007/s00039-009-0710-2, MR 2491695, S2CID 10402235. Guth, Larry (2008), "Symplectic embeddings of polydisks", Inventiones Mathematicae, 172 (3): 477–489, arXiv:0709.1957, Bibcode:2008InMat.172..477G, doi:10.1007/s00222-007-0103-9, MR 2393077, S2CID 18065526. Guth, Larry (2007), "The width-volume inequality", Geometric and Functional Analysis, 17 (4): 1139–1179, arXiv:math/0609569, doi:10.1007/s00039-007-0628-5, MR 2373013, S2CID 16014518. Guth, Larry; Katz, Nets Hawk (2015), "On the Erdős distinct distance problem on the plane", Annals of Mathematics, 181 (1): 155–190, arXiv:1011.4105, doi:10.4007/annals.2015.181.1.2, MR 3272924, S2CID 43051852 Guth, Larry (2016). Polynomial Methods in Combinatorics. American Mathematical Society. ISBN 978-1-4704-2890-7.

References

Illustrations

Worked examples

Example 1 — a first encounter with Larry Guth

Start with the simplest possible case. Write down what Larry Guth 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 Larry Guth 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 Larry Guth 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 Larry Guth

In research: Larry Guth 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 Larry Guth 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: Larry Guth is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1977 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Larry Guth 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 Larry Guth in 20 minutes

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

Frequently asked questions

What is Larry Guth in simple terms?

Lawrence David Guth (; born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale University in 2000 with a BS in mathematics.

Why does Larry Guth 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 Larry Guth?

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 Larry Guth.