Bobby Wilson (U Chicago):

Title: Tangents of $\sigma$-finite curves and scaled oscillation

Abstract: In this talk we will discuss two questions.  The first question
involves necessary regularity properties for continuous functions on [0,1]^d
for which oscillation can be controlled locally, but not necessarily at
every scale or at every point.  It is known that if the function is defined
on [0,1] and the set on which this "lower scaled oscillation" of the
function is not finite is at most countable, then the function is always
differentiable on a set of positive measure.  We will show that we can
extend the same result to functions defined on [0,1]^d and discuss
conditions on the exceptional set that preserve the existence of the set of
differentiability.

The second question is whether continuous simple curves with $\sigma$-finite
length have tangents at positively many points.  The results on
$\sigma$-finite curves that we will discuss were initiated by the
observation that the graph of a continuous function on [0,1] with the
properties of the functions from the first question has $\sigma$-finite
1-Hausdorff measure. We will discuss our conclusion that every
$\sigma$-finite curve has a tangent, in the pointwise sense, on a set of
positive measure. Surprisingly, we show that this conjecture is false for
d>1. 

This is joint work with M. Cs\"ornyei.