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PRODID:-//talks.bham.ac.uk//v3//EN
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CATEGORIES:Analysis seminar
SUMMARY:Differentiability of Lipschitz Functions inside Ne
gligible Sets - Michael Dymond (Birmingham)
DTSTART:20130220T160000Z
DTEND:20130220T170000Z
UID:TALK1004AT
URL:/talk/index/1004
DESCRIPTION:Rademacher’s Theorem states that Lipschitz functio
ns on Euclidean spaces are differentiable almost e
verywhere with respect to the Lebesgue measure. Mo
reover\, in $\\mathbb{R}$\, any set $N$ of Lebesg
ue measure zero admits a Lipschitz function on $\\
mathbb{R}$\, nowhere differentiable on $N$. Howeve
r\, the situation is vastly different in Euclidean
spaces of dimension higher than one. In 1990\, Pr
eiss gave an example of a Lebesgue null subset of
the plane containing a differentiability point of
every Lipschitz function on $\\mathbb{R}^2^$. This
set contains every line segment between points of
$\\mathbb{R}^2^$ with rational co-ordinates. In s
ome sense\, Preiss’ set is still rather large\; it
s closure is the whole of $\\mathbb{R}^2^$. Thus\,
it is natural to ask whether we can force differe
ntiability of every Lipschitz function inside much
smaller sets. In recent work of Maleva and Dor\\’
{e}\, the existence of a compact universal differe
ntiability set with Hausdorff dimension one is ver
ified. The Hausdorff dimension is bounded above by
the (upper and lower) Minkowski dimensions\, and
the question of whether there exists a universal d
ifferentiability set with upper or lower Minkowksi
dimension one has remained open. We discuss some
new results including a construction of a universa
l differentiability set having both upper and lowe
r Minkowski dimension one\, and a general property
of universal differentiability sets. We also disc
uss possible directions for future research in thi
s area.
LOCATION:R17/18
CONTACT:Neal Bez
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