The workshop on additive combinatorics will take place October 2-6, 2017 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Additive combinatorics is a mathematical area bordering on number theory, discrete mathematics, harmonic analysis and ergodic theory. It has achieved a number of successes in pure mathematics in the last two decades in quite diverse directions, such as:
- The first sensible bounds for Szemerédi’s theorem on progressions (Gowers);
- Linear patterns in the primes (Green, Tao, Ziegler);
- Construction of expanding sets in groups and expander graphs (Bourgain, Gamburd);
- The Kakeya Problem in Euclidean harmonic analysis (Bourgain, Katz, Tao).
Ideas and techniques from additive combinatorics have also had an impact in theoretical computer science, for example
- Constructions of pseudorandom objects (eg. extractors and expanders);
- Constructions of extremal objects (eg. BCH codes);
- Property testing (eg. testing linearity);
- Algebraic algorithms (eg. matrix multiplication).
The main focus of this workshop will be to bring together researchers involved in additive combinatorics, with a particular inclination towards the links with theoretical computer science. Thus it is expected that a major focus will be additive combinatorics on the boolean cube (Z/2Z) n , which is the object where the exchange of ideas between pure additive combinatorics and theoretical computer science is most fruitful. Another major focus will be the study of pseudorandom phenomena in additive combinatorics, which has been an important contributor to modern methods of generating provably good randomness through deterministic methods. Other likely topics of discussion include the status of major open problems (the polynomial Freiman-Ruzsa conjecture, inverse theorems for the Gowers norms with bounds, explicit correlation bounds against low degree polynomials) as well as the impact of new methods such as the introduction of algebraic techniques by Croot–Pach–Lev and Ellenberg–Gijswijt.