OOEW04 | Prof. Terence Tao | Marton’s Polynomial Freiman-Ruzsa conjecture

OOEW04 | Prof. Terence Tao | Marton’s Polynomial Freiman-Ruzsa conjecture Speaker: Professor Terence Tao (University of California, Los Angeles) Date: 12th Apr 2024 - 15:30 to 16:30 Venue: INI Seminar Room 1 Title: Marton’s Polynomial Freiman-Ruzsa conjecture Event: (OOEW04) Structure and Randomness - a celebration of the mathematics of Timothy Gowers Abstract: The Freiman-Ruzsa theorem asserts that if a finite subset $A$ of an $m$-torsion group $G$ is of doubling at most $K$ in the sense that $|A A| \leq K|A|$, then $A$ is covered by at most $m^{K^4 1} K^2$ cosets of a subgroup $H$ of $G$ of cardinality at most $|A|$. Marton’s Polynomial Freiman-Ruzsa conjecture asserted (in the $m=2$ case, at least) that the constant $m^{K^4 1} K^2$ could be replaced by a polynomial in $K$. In joint work with Timothy Gowers, Ben Green, and Freddie Manners, we establish this conjecture for $m=2$ with a bound of $2K^{12}$ (later improved to $2K^{11}$ by Jyun-Jie Liao by a modification of the method), and for arbitrary $m$ with a bound of $(2K)^{O(m^3 \log m)}$. Our proof proceeds by passing to an entropy-theoretic version of the problem, and then performing an iterative process to reduce a certain modification of the entropic doubling constant, taking advantage of the bounded torsion to obtain a contradiction when the (modified) doubling constant is non-trivial but cannot be significantly reduced. By known implications, this result also provides polynomial bounds for inverse theorems for the $U^3$ Gowers uniformity norm, or for linearization of approximate homomorphisms. In a collaborative project with Yael Dillies and many other contributors, the proof of the $m=2$ result has been completely formalized in the proof assistant language Lean. In this talk we will present both the original human-readable proof, and the process of formalizing it into Lean. Workshop LINK: ------------------- FOLLOW US 🌐| Website: 🎥| Main Channel: @isaacnewtoninstitute 🐦| Twitter: 💬| Facebook: 📷| Instagram: 🔗| LinkedIn: SEMINAR ROOMS 🥇| INI Seminar Room 1: @iniseminarroom1 🥈| INI Seminar Room 2: @iniseminarroom2 🛰️| INI Satellite Events: @inisatellite ABOUT The Isaac Newton Institute is a national and international visitor research institute. It runs research programmes on selected themes in mathematics and the mathematical sciences with applications over a wide range of science and technology. It attracts leading mathematical scientists from the UK and overseas to interact in research over an extended period. 👉 Learn more about us and our events here: