48 research outputs found
The flecnode polynomial: a central object in incidence geometry
We give a brief exposition of the proof of the Cayley-Salmon theorem and its
recent role in incidence geometry. Even when we don't use the properties of
ruled surfaces explicitly, the regime in which we have interesting results in
point-line incidence problems often coincides with the regime in which lines
are organized into ruled surfaces.Comment: 12 pages. An expository note submitted to ICM proceeding
A Kakeya maximal function estimate in four dimensions using planebrushes
We obtain an improved Kakeya maximal function estimate in
using a new geometric argument called the planebrush. A planebrush is a higher
dimensional analogue of Wolff's hairbrush, which gives effective control on the
size of Besicovitch sets when the lines through a typical point concentrate
into a plane. When Besicovitch sets do not have this property, the existing
trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch
set. In particular, we establish a maximal function estimate in
at dimension . As a consequence, every Besicovitch set in
must have Hausdorff dimension at least .Comment: 40 pages 2 figures. v2: revised based on referee's comments. In v1,
the Nikishin-Pisier-Stein factorization theorem was stated (and used)
incorrectly. This version corrects the problem by introducing several new
arguments. The new argument leads to a Kakeya maximal function estimate at
dimension 3.059, which is slightly worse than the previously claimed exponent
3.085