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Larry Guth, MIT professor of mathematics, was awarded the 2015 Clay Research Prize jointly with collaborator Netz Katz, a professor of mathematics at the California Institute of Technology.
The Clay Prize, awarded yearly by the Clay Mathematics Institute, is one of the most prestigious prizes in mathematics awarded for a single piece of research. Among the past winners are eight Fields medallists.
Guth and Katz were awarded the Clay Research Prize for their solution to the Erdős distinct distance problem, and for other joint and separate contributions to the field of combinatorial incidence geometry. In awarding the pair the prize, the institute described their work as “an important contribution to the understanding of the interplay between combinatorics and geometry.”
“It was great to be able to win the prize as a pair of collaborators,” said Guth. “Math is often very collaborative, so it’s a nice thing to be able to win the prize as a team.”
The pair solved what is known as the “distinct distances” problem, posed by Paul Erdős in 1946. Paul Erdős was a Hungarian mathematician famous for posing deceptively simple sounding problems whose solutions often required new and creative ideas. At that time, most fields of mathematics were moving towards questions that were more abstract, so that it took a lot of thought to even understand the statements of the questions. Erdős instead sought out questions that were as concrete and as simple as possible. For example, he discovered many interesting unsolved questions remaining in Euclidean geometry.
The distinct distances problem was Erdős’s favourite problem of this sort. Imagine a farmer is planting an orchard of apples trees. In order to water the orchard, he needs to run a straight pipe between each pair of trees. He would like to minimize the number of different length pipes he needs to use. The naïve farmer would likely plant the trees in a square grid and Erdős conjectured this was optimal.
Since 1946 many mathematicians have worked on this problem, including Abel Prize-winner Endre Szemeredi, but the results proven previously were far from sharp. Finally in 2010, 64 years after it was initially posed, Guth and Katz announced their surprising proof of the Erdős conjecture, building on a novel approach to the problem suggested by mathematicians György Elekes and Micha Sharir.
The main point of a problem like the distinct distance problem is not so much the direct applications of the result, but that solving the problem will require new ideas and that those ideas may lead somewhere interesting. In this case, important new ideas in the proof came from the theory of error-correcting codes in computer science.
Computer scientists have found and studied interesting error-correcting codes based on polynomials. The idea of an error-correcting code is that we store data in a redundant way, so that even after errors are introduced in the data, the original information can still be recovered. In polynomial codes, the data stored is a polynomial, and one needs tools to recognize when a given set of data is close to a polynomial. Guth and Katz adapted these tools to the distinct distances problem.
Sets with few distinct distances are rare and hard to find. They seem to have some type of special structure, but people in the field had found it hard to articulate what type of structure. Using tools from error correcting codes, Guth and Katz proved that a set with few distinct distances needs to have polynomial structure – various aspects of the set are well approximated by polynomials. This was the key new insight that led to the sharp estimate.
2014 wasn’t a bad year for Guth either. He received the Salem Prize in Mathematics for outstanding contributions to Fourier analysis, as well as being named a Simons Investigator by the Simons Foundation
The work with Katz was a big change in mathematical direction for Guth, who was already well known for his contributions to Riemannian geometry, symplectic geometry, and harmonic analysis. Guth is now working on a book on the Erdős distance problem, and more generally on polynomial methods in combinatorial geometry.
Guth began studying geometry as a graduate student at MIT under the supervision of professor of mathematics Tomasz Mrowka. After completing his PhD, he moved as a postdoc to Stanford University. After taking positions at the University of Toronto, the Institute for Advanced Studies, and the Courant Institute, Guth returned to MIT in 2012 as a full professor. Larry is the son of physicist Alan Guth, a winner of the inaugural Breakthrough Prize in Fundamental Physics. Larry is happy to be back at MIT, where he fondly remembers getting Mountain Dew with his father and is no longer worried about finding his father’s office.