In-Person Talk 

In 1961, Erdos, Ginzburg and Ziv showed that among any 2n-1 numbers one can find n whose sum is divisible by n. Consider a higher dimensional generalization of this fact: what is the smallest N such that among any N points in Z^d one can find n whose sum is zero mod n? This turns out to be a much more difficult question and the exact answer is only known when d =1 or 2 or n is a power of 2. I will talk about 2 new upper bounds substantially improving previously known results for most parameters (d, n). Proofs use a variety of tools coming from additive combinatorics, algebraic methods and convex geometry. Joint work with Lisa Sauermann.