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CATEGORIES:Analysis seminar
SUMMARY:Differentiability of typical Lipschitz functions -
Olga Maleva (University of Birmingham)
DTSTART:20211129T150000Z
DTEND:20211129T160000Z
UID:TALK4740AT
URL:/talk/index/4740
DESCRIPTION:This talk is devoted to differentiability properti
es of Lipschitz functions.\n\nThe classical Radema
cher Theorem guarantees that every Lipschitz funct
ion between finite-dimensional spaces is different
iable almost everywhere. This means that for every
set T of positive Lebesgue measure and for every
Lipschitz function f defined on the whole space th
e set of points from T where f is differentiable i
s non-empty and is 'much larger' than the set of p
oints where it is not differentiable.\n\nA major d
irection in geometric measure theory research of t
he last two decades has been to explore to what ex
tent this is true for Lebesgue null sets. Even for
real-valued Lipschitz functions\, there are null
subsets S of R^n (with n>1) such that every Lipsch
itz function on R^n has points of differentiabilit
y in S\; one says that S is a universal differenti
ability set (UDS).\n\nSome sets T which are not UD
S still have the property that a *typical* Lipschi
tz function has points of differentiability in T.
We characterise such sets completely in the langua
ge of Geometric Measure Theory: these are exactly
the sets which cannot be covered by an F-sigma 1-p
urely unrectifiable set. We also show that for all
remaining sets a typical 1-Lipschitz function is
nowhere differentiable\, even directionally\, at e
ach point.\n\nSurprisingly though\, no matter how
good the set T is\, it turns out that a typical 1-
Lipschitz function is non-differentiable at a typi
cal point of T. As above\, 'typical' is used in th
e sense of Baire category.\n\nThe results in the t
alk are based on two joint papers with Michael Dym
ond.
LOCATION:Lecture Theatre C\, Watson Building
CONTACT:Jon Bennett
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