During the last few decades the polynomial method has developed into a powerful tool in combinatorics. It involves encoding combinatorial problems to nonvanishing of some polynomials, and then investigating the resulting polynomial question. Usually a common step is to show that a low-degree (non-zero) polynomial can not vanish on a certain set (which is sufficiently large). We will start with exploring several applications of the Combinatorial Nullstellensatz, including classical problems like the Cauchy-Davenport theorem and other questions. The solution of the finite field Kakeya conjecture is also going to be discussed. Finally, we will study a new variant of the polynomial method developed more recently (in 2016) which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like Z_4^n and Z_3^n are exponentially small (compared to the size of the group). We will also explore some further applications, for instance, the solution of the Erdős-Szeméredi sunflower conjecture.

Organized by MTA-ELTE CoGe as part of the AlgoMaNet – Algorithms and Mathematics Network courses.When: 2022 September 5-9, each day from 13:00-14:30.

Where: ELTE Déli Tömb 3-306

Credits: ELTE students can take the course in Neptun to receive credits.