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Misha Rudnev awarded Leverhulme Project Grant

15 December 2017

Congratulations to Misha Rudnev who has been awarded a Leverhulme Project Grant titled "Geometry, Combinatorics and Alegbra of Sum-products".

The Erdös-Szemerédi sum-product problem came about in 1983 as a conjecture, based on partial results about the integers and then reals. In the early 2000s first qualitative sum-product estimates were also obtained for the prime residue field, leading to striking applications. The sum-product phenomenon became a fountainhead of the vast theme of growth, expansion and deterministic randomness  throughout many areas of mathematical research, computer science and complexity theory, having provided the local onset of growth. 

However, strong quantitative sum-product bounds remain conjectural. The phenomenon itself appears to be universal, regardless of the underlying field, but for a few categorisable exceptional cases. The result of my recent work was establishing some sort of interim general state of the art, i.e., proving comparable strength quantitative estimates over all fields. Prior to this sum-product estimates over general fields had been far behind the case of the reals, where proofs could rely on their order properties. This progress was rooted in (basic) algebraic geometry; along the way some interesting connections of the sum-product phenomenon with such fundamental geometric concepts as the symplectic form, cross-ratio, and more generally geometric group action invariants came about. These connections gave rise to a series of new questions, whose investigation, aiming at achieving further progress in the sum-product theory as a geometric incidence phenomenon, the Trust has magnanimously agreed to support.

Further information

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