ART in Manchester

Monday 10th September - Thursday 13th September 2018

University of Manchester, UK

Organisers: Sean Prendiville (Manchester), Julia Wolf (Bristol) and Jacob Fox (Stanford).

Whilst in Manchester, Paul Erdős co-authored two formative papers in Ramsey theory:

• 'A combinatorial theorem in geometry' (with G. Szekeres, 1935), giving a new proof of Ramsey's theorem.
• 'On some sequences of integers' (with P. Turán, 1936), laying the foundation for density results over arithmetic sets.

Some 80 years later, we would like to commemorate this and subsequent discoveries in additive combinatorics, continuing the celebration of the return of the number theory group to Manchester initiated by Diophantine Problems.

The central topic of the conference concerns the existence of structure within large arithmetic sets (broadly interpreted). In particular:

• Density bounds for sets lacking arithmetic configurations (Roth's theorem, Szemerédi's theorem, the cap-set problem and the polynomial method).
• Existence of arithmetic configurations in relatively dense sets (Szemerédi's theorem in the primes, combinatorial theorems in sparse (pseudo)random sets).
• Partition regularity of arithmetic configurations (monochromatic sums and products, regularity of non-linear equations).
• Applications of higher-order Fourier analysis to all of the above, including counting solutions to equations in arithmetic sets of interest.

Enquiries can be sent to the organisers at arithmeticramsey@gmail.com.