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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - ErgComNum (Ergodic theory and additive combinatorics)

Teaser

Summary

One of the central problems in additive combinatorics is finding effective bounds for the so called inverse theorems for the Gowers norms. The inverse theorem for the Gowers norms in the cyclic setting, proved by the PI in joint work with Green and Tao, attracted much attention, since it was shown by Green and Tao that combined with their earlier work on arithmetic progressions in primes and a correlation estimate for the mobius function with nilsequences, it provides a proof for the well known Hardy-Littlewood conjectures for systems of finite complexity. The theorem in the finite field geometry, proved by the PI in joint work with Bergelson and Tao, attracted much attention in theoretical computer science. One of the main goals of the project was to find alternative proofs for these theorems. Much progress has been made in the finite field geometry context and new connection with central questions in algebraic geometry have been discovered,

Work performed

In joint work with T. Tao we proved asymptotics for some polynomial configurations in the prime numbers. this was done via a new theorem in additive combinatorics. In joint work with D. Kazhdan we are working on connections between questions in algebraic geometry in high dimensional spaces and questions in additive combinatorics.

Final results

We expect to make progress on one of the central problems in additive combinatorics, which is finding effective bounds for the so called inverse theorems for the Gowers norms. In the process we expect to form new connections between additive combinatorics and high rank varieties in algebraic geometry.