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UID:news428@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20181217T192947
DTSTART;TZID=Europe/Zurich:20170405T141500
SUMMARY:Seminar Analysis: Filip Rindler (University of Warwick)
DESCRIPTION:The classical Rademacher Theorem asserts that every Lipschitz f
 unction is differentiablea lmost everywhere with respect to Lebesgue measu
 re.  On the other hand\, Preiss (’90) gave a surprising example of a nul
 lset in the plane such that every Lipschitz function is differentiable at 
 at least one point of this set.  Thus\, it is a natural question to ask wh
 ether there exists a singular measure such that all Lipschitz functions ar
 e differentiable with respect to this singular measure.  It turns out that
  this question has an intricate connection to the geometric structure  of 
  normal  one-currents.   In  this  talk  I  will  present  a  converse  to
   Rademacher’s Theorem\, which settles the question in the negative in a
 ll dimensions:  if a positive measure μ has the property that all Lipschi
 tz functions are μ-a.e.  differentiable\, then μ is absolutely continuou
 s with respect to Lebesgue measure (in the plane\, this question was alrea
 dy solved by Alberti\, Csornyei and Preiss in ’05). In a geometric conte
 xt\, Cheeger conjectured in ’99 that in all Lipschitz differentiability 
 spaces  (which  are  essentially  Lipschitz  manifolds  in  which  Rademac
 her’s  Theorem  holds) likewise  there  is  a  “functional  converse
 ”  to  Rademacher’s  Theorem.   As  the  second  main result\, I will 
 present a recent solution to this conjecture.Technically\, the proofs of b
 oth of these theorems are based on a recent structure result for the singu
 lar parts of PDE-constrained measures\, its corollary on the structure of 
 normalone-currents\, and the powerful theory of Alberti representations.\\
 r\\nThis is a joint work with A. Marchese and G. De Philippis
X-ALT-DESC:\nThe classical Rademacher Theorem asserts that every Lipschitz 
 function is differentiablea lmost everywhere with respect to Lebesgue meas
 ure.  On the other hand\, Preiss (’90) gave a surprising example of a nu
 llset in the plane such that every Lipschitz function is differentiable at
  at least one point of this set.  Thus\, it is a natural question to ask w
 hether there exists a singular measure such that all Lipschitz functions a
 re differentiable with respect to this singular measure.  It turns out tha
 t this question has an intricate connection to the geometric structure  of
   normal  one-currents.   In  this  talk  I  will  present  a  converse  t
 o  Rademacher’s Theorem\, which settles the question in the negative in 
 all dimensions:  if a positive measure μ has the property that all Lipsch
 itz functions are μ-a.e.  differentiable\, then μ is absolutely continuo
 us with respect to Lebesgue measure (in the plane\, this question was alre
 ady solved by Alberti\, Csornyei and Preiss in ’05). In a geometric cont
 ext\, Cheeger conjectured in ’99 that in all Lipschitz differentiability
  spaces  (which  are  essentially  Lipschitz  manifolds  in  which  Radema
 cher’s  Theorem  holds) likewise  there  is  a  “functional  converse
 ”  to  Rademacher’s  Theorem.   As  the  second  main result\, I will 
 present a recent solution to this conjecture.Technically\, the proofs of b
 oth of these theorems are based on a recent structure result for the singu
 lar parts of PDE-constrained measures\, its corollary on the structure of 
 normalone-currents\, and the powerful theory of Alberti representations.\n
 This is a joint work with A. Marchese and G. De Philippis
DTEND;TZID=Europe/Zurich:20170405T151500
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