The Kakeya problem is an important open question about the possible overlap patterns of thin cylindrical tubes in R^n. If we replace these thin tubes by straight lines, we can formulate analogous questions of pure incidence geometry. In the past six years, there has been a lot of progress in incidence geometry. In particular, some of these analogous questions about lines have been resolved. The Kakeya problem over finite fields was resolved by Zeev Dvir, and a problem about lines in R^3 in a similar spirit was resolved by Nets Katz and myself. These proofs use polynomials in a crucial way to get at the behavior of the lines.
How much can this breakthrough about the incidence geometry of lines tell us about the incidence geometry of tubes? I will discuss some small partial results - estimates about tubes proven by adapting the polynomial techniques mentioned above. I will also discuss some of the issues and difficulties with adapting these arguments from lines to tubes.
Back to Workshop III: The Kakeya Problem, Restriction Problem, and Sum-product Theory