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Ergodic Ramsey Theory

This is the page for a TCC module in Ergodic Ramsey Theory taught in Term 1 (Autumn) of 2021.

Information:

Lectures are 16:00-18:00 Thursdays starting October 14, held via MS Teams.

TCC modules are open to students from Bath, Bristol, Imperial, Oxford, and Warwick. If you want to attend, please email gradstud@maths.ox.ac.uk. More information about the TCC in their webpage.

I will be teaching from room MB0.08 in the Mathematical Sciences building at Warwick, and local atendees are welcome to join in person.

Assessment:

If you are taking the module for assessment, please email me. Assessment will be based on exercises from the worksheets provided after each lecture. You should aim to complete an average of at least one exercise per week.

Please submit the solutions to my email joel.moreira@warwick.ac.uk within 2 weeks of each class.

Overview:

Ergodic Ramsey theory is a relatively new subject which applies ideas and techniques from ergodic theory to problems arising in Ramsey theory, combinatorics and number theory. The module will introduce the basics of this subject and build up to the ergodic theoretic proof of Szemeredi's theorem on arithmetic progressions discovered by Furstenberg. Depending on time and students preferences, the last half/third of the module will cover more recent developments in the field, including connections to multiplicative number theory, additive combinatorics or Higher Order Fourier Analysis.

Prerequisites:

We will liberally make use of basic tools from functional analysis, topology, analysis and measure theory (eg. the Riesz Representation theorem, Jensen's inequality, the Stone-Weierstrass theorem, the Dominated Convergence Theorem,...).

Familiarity with Ergodic Theory can be helpful but is not necessary to follow the module. In fact one of the goals of this module is to serve as a gateway into ergodic theory for people interested in Ramsey theory or additive combinatorics.

Familiarity with Ramsey Theory can equally be helpful but is not necessary since all Ramsey theoretic statements can (and will) be formulated in very basic terms.

Classes

Class 1 (14/10):

Class notes, Extended Notes, Exercises Sheet [[EDIT on 23/10: Changed numbering to match the notes]] (due October 29th)

Class 2 (21/10):

Class notes, Extended Notes, Exercises Sheet (due November 5th)

Class 3 (28/10):

Class notes, Extended Notes, Exercises Sheet (due November 12th)

Class 4 (4/11):

Class notes, Extended Notes, Exercises Sheet [[EDIT on 12/11: Fixed typos in Exercise 5.26 (2)]] (due November 19th)

Class 5 (11/11):

Class notes, Extended Notes, Exercises Sheet [[EDIT on 21/11: Fixed numbering]] (due November 26th)

Class 6 (18/11):

Class notes, Extended Notes, Exercises Sheet [[EDIT on 28/11: Fixed numbering]] (due December 3rd)

Class 7 (25/11):

Class notes, Extended Notes, Exercises Sheet (due December 10th)