I will discuss a class of maximal operators that arise from
averaging functions over thin neighborhoods of curves in the plane.
Examples of such operators are the Kakeya maximal function and the Wolff
and Bourgain circular maximal functions. To understand the behavior of
these operators, we need to study the possible intersection patterns for
collections of curves in the plane: how often can these curves intersect,
how often can they be tangent, and how often can they be tangent to higher
order?
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