Title: Differentiability of Lipschitz functions and
tangents of sets.
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Abstract: We will show how elementary product decompositions of
measures can detect directionality in sets, and show how this can be used
to describe non-differentiability sets of
Lipschitz functions on R^n, and to understand the phenomena
that occur because of behaviour of Lipschitz functions around the
points of null sets.
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In order to prove this we will need to prove results about the geometry
of sets of small Lebesgue measure: we show that sets of small measure are
always contained in a "small" collection of Lipschitz surfaces.
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The talk is based on a joint work
with G. Alberti, P. Jones and D. Preiss.