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UID:6533@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20210218T140000
DTEND;TZID=Europe/Paris:20210218T150000
DTSTAMP:20250118T133212Z
URL:https://www.i2m.univ-amu.fr/evenements/c%e1%b5%90-solutions-of-semialg
 ebraic-equations-and-the-whitney-extension-problem/
SUMMARY:Jean-Baptiste Campesato (University of Toronto): Cᵐ solutions of 
 semialgebraic equations and the Whitney extension problem
DESCRIPTION:Jean-Baptiste Campesato: We address the question of whether geo
 metric conditions on the given data can be preserved by a solution in\n(1)
  the Whitney extension problem\, which consists in determining whether a f
 unction g:X→ℝ defined on a closed subset X⊂ℝⁿ admits a Cᵐ exte
 nsion on ℝⁿ\, and\,\n(2) the Brenner-Fefferman-Hochster-Kollár proble
 m\, about the existence of a Cᵐ solution to A(x)G(x)=F(x)\, where A is a
  matrix of functions on ℝⁿ\, and the unknown is a vector-valued functi
 on G.\nIn a joint work with E. Bierstone and P.D. Milman\, we prove that\,
  for both problems\, when the data are semialgebraic (or\, more generally\
 , definable in a suitable o-minimal structure)\, the existence of a soluti
 on implies the existence of a semialgebraic (or definable) solution. Our r
 esults involve a certain loss of differentiability.\nMore precisely\, for 
 (1)\, we prove that given a semialgebraic closed subset X⊂ℝⁿ\, there
  exists r:ℕ→ℕ such that if a semialgebraic function g:X→ℝ is the
  restriction of a Cʳ⁽ᵐ⁾ function then it is the restriction of a se
 mialgebraic Cᵐ function.\nFor (2)\, we prove that given A a matrix of se
 mialgebraic functions\, there exists r:ℕ→ℕ such that if F is semialg
 ebraic and A(x)G(x)=F(x) admits a Cʳ⁽ᵐ⁾ solution\, then there exist
 s a Cᵐ solution which is semialgebraic.\nSlides: http://www.math.toronto
 .edu/campesat/docs/krakow.pdf\n
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Jean-Baptiste_Campesato.jpg
CATEGORIES:Groupe de travail,Singularités,Virtual event
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