The previous talk in the series is here. The next talk in the series is here.
How many vectors can you find in ℝn if the angle between any two of them is at least a right angle? It’s easy to see that one can find 2n such vectors, but can one do any better than this? And what if the vectors have to have all coordinates equal to 1 or -1? This video contains answers to these questions, though not a full answer to the last one, since that would require answering a famous open problem. We consider vectors in ℝn that make no acute angles, then Hadamard matrices and Walsh matrices, then Paley matrices.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.