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 studying 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 also some previously unsolved questions, for instance from combinatorial geometry. 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 very 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). In this method linear algebra plays an important role, which is nicely captured by the slice rank formulation of it. Therefore, we will also look at this formulation and explore some further applications, for instance, the solution of the Erdos-Szemeredi sunflower conjecture.

The plan for the 5 lectures:

1. Combinatorial Nullstellensatz and its applications.
2. Finite field Kakeya conjecture.
3. Progression-free sets, cap set problem. 
4. Slice Rank Method.
5. Further applications.