The Kakeya conjecture

Abstract: How large area is needed to rotate a needle? How small a
hedgehog can be? Are lines much bigger than line segments? What do
these questions have to do with the Kakeya conjecture, which claims
that if a compact set in R^n has unit line segments in every direction
then the set must have Hausdorff/Minkowski dimension n? Why is this
conjecture so important to some of the leading mathematicians? What
partial results could they prove? How do I fail to achieve
breakthrough?

You can watch the lecture here:

https://www.youtube.com/watch?v=oS-p22i69Gk&list=PL_0phSnA7tyQJDn-5hacAu...