# Reading Seminar on Polynomial Freiman Ruzsa (PFR)

This reading group will cover the recent breakthrough solving Marton's conjecture (also known as the Polynomial Freiman Ruzsa (PFR) Conjecture). We will meet in Room C3-005 at Campus Nord (usual seminar room) on Thursdays at 14.30. Talks will be planned to be 1 hour with space for questions and discussion after.

**Reading Seminar on Polynomial Freiman Ruzsa (PFR):**

This reading group will cover the recent breakthrough solving Marton's conjecture (also known as the Polynomial Freiman Ruzsa (PFR) Conjecture). The conjecture is a cornerstone in Additive Combinatorics and asserts that one can deduce strong structural information about a set that has bounded doubling. The resolution was achieved by Gowers, Green, Manners and Tao using a translation of the problem that utilises entropy. We will cover this proof as well as the relevant context and tools that are used.

We will meet in

**Room C3-005**at Campus Nord (usual seminar room) on**Thursdays at 14.30.**Talks will be planned to be 1 hour with space for questions and discussion after.**Provisional Schedule:**

**Thursday 29th February:**Introduction to topic, statement of theorem and discussion of some classical tools*(Oriol Serra)***Thursday 7th March:**Entropy methods, definition and basic properties of entropic Ruzsa distance, entropic formulation of PFR*(Clément Requilé)***Thursday 14th March:**More entropy: proof that 'entropic PFR' implies 'PFR', Covering lemma, and possibly Balog–Szemerédi–Gowers lemma*(Miquel Ortega)***Thursday 21st March:**Balog–Szemerédi–Gowers lemma. Classical version and implications/context. Entropic version with proof.*(Tássio Naia)***Thursday 11th April:**Proof of PFR conjecture - big picture and scheme of proof. Fibring Lemma.*(Miquel Ortega)***Thursday 18th April:**Outline of cases with one in detail. Endgame.*(Patrick Morris)*

**References:**

- W. T. Gowers, B. Green, F. Manners and T. Tao, On a conjecture of Marton, prepint 2024
- B. Green, F. Manners and T. Tao Sumsets and entropy resvisted, preprint 2023
- B. Green, Notes on the polynomial Freiman–Ruzsa conjecture, unpublished notes
- Y. Zhao, Graph Theory and Additive Combinatorics, Lecture Notes (Chapter 7 on Freiman's Theorem)
- Quanta Article on PFR Conjecture Breakthrough

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