The joints problem for varieties

We generalize the Guth-Katz joints theorem from lines to varieties. A special case of our result says that N planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects. Joint work with Jonathan Tidor and Hung-Hsun Hans Yu