assume $v_1,...,v_n$ are unit vectors in $d$-dimensional space whose sum is zero. can you reorder these vectors as $v_{i_1},...,v_{i_n}$ so that each partial sum $s_k=\sum _{j=1}^k v_{i_j}$ is bounded by a constant that depends only on dimension? the answer is yes, you can. this is going to be the topic of the lecture. the proof is based on linear algebra. further applications of the proof method will also be presented.