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Jean Louis, baron Bourgain (French:; () 28 February 1954 – () 22 December 2018) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics.
As reformulated, it became the "paving conjecture" for Euclidean spaces, and then a question on random polynomials, in which latter form it was solved affirmatively. 2015: Jean Bourgain, Ciprian Demeter, and Larry Guth: Main conjecture in Vinogradov's mean-value theorem: analytic number theory: Bourgain–Demeter–Guth theorem, ⇐ decoupling ...
Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ F n be a Kakeya set, i.e. for each vector y ∈ F n there exists x ∈ F n such that K contains a line {x + ty : t ∈ F}. Then the set K has size at least c n |F| n where c n >0 is a constant that only depends on n. Zeev Dvir proved this conjecture in 2008, showing that the ...
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.
Further conjectures reduce that value, in the case of all sufficiently large denominators. [5] Jean Bourgain and Alex Kontorovich have shown that A can be chosen so that the conclusion holds for a set of denominators of density 1. [6]
Ronen Eldan – "For the creation of the stochastic localization method, that has led to significant progress in several open problems in high-dimensional geometry and probability, including Jean Bourgain's slicing problem and the KLS conjecture."
Theorem (Bourgain, Katz, Tao (2004)): [23] Let p be prime and let A ⊂ 픽 p with p δ < | A | < p 1−δ for some 0 < δ < 1. Then max(| A + A |, | AA |) ≥ c δ | A | 1+ε for some ε = ε(δ) > 0. Bourgain, Katz, and Tao extended this theorem to arbitrary fields. Informally, the following theorem says that if a sufficiently large set does ...
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf [1] about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis .