From incidence geometry of lines towards incidence geometry of tubes.

Larry Guth
Massachusetts Institute of Technology

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