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Guth won an Alfred P. Sloan Fellowship in 2010. [9] He was an invited speaker at the International Congress of Mathematicians in India in 2010, where he spoke about systolic geometry.
Erdős also considered the higher-dimensional variant of the problem: for let () denote the minimal possible number of distinct distances among points in -dimensional Euclidean space.
Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are ...
A variation of the polynomial method, often called polynomial partitioning, was introduced by Guth and Katz in their solution to the Erdős distinct distances problem. [4] Polynomial partitioning involves using polynomials to divide the underlying space into regions and arguing about the geometric structure of the partition.
The Erdős Distance Problem consists of twelve chapters and three appendices. [5]After an introductory chapter describing the formulation of the problem by Paul Erdős and Erdős's proof that the number of distances is always at least proportional to , the next six chapters cover the two-dimensional version of the problem.
Guth (2011) and Ambrosio & Katz (2011) developed approaches to the proof of Gromov's systolic inequality for essential manifolds. ... Guth, Larry (2011), ...
The main conjecture of Vinogradov's mean value theorem was that the upper bound is close to this lower bound. More specifically that for any > we have , + + (+) +. This was proved by Jean Bourgain, Ciprian Demeter, and Larry Guth [4] and by a different method by Trevor Wooley.
However, according to the 2015 solution of the Erdős distinct distances problem by Larry Guth and Nets Katz, the distance set of any finite collection of points in the Euclidean plane is only slightly sublinear, nearly as large as the given collection. [5] In particular, only a finite collection of points can have a finite distance set.