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DESCRIPTION:Title:  A Liouville type theorem in sub-Riemannian geometry\, a
 nd applications to several complex variables\n	Abstract:  The Riemann mappi
 ng theorem tells us that any simply connected planar domain is conformally
  equivalent to the disk. This provides a classification of simply connecte
 d domains via conformal maps. This classification fails in higher dimensio
 nal complex spaces\, as already Poincare' had proved that bi-discs are not
  bi-holomorphic to the ball. Since then\, mathematicians have been looking
  for criteria that would allow to tell whether two domains are bi-holomorp
 hic equivalent. In the early 70's\, after a celebrated result by Moser and
  Chern\, the question was reduced to showing that any bi-holomorphism betw
 een smooth\, strictly pseudo-convex domains extends smoothly to the bounda
 ry. This was established by Fefferman\, in a 1974 landmark paper. Since th
 en\, Fefferman's result has been extended and simplified in a number of wa
 ys. About 10 years\, ago Michael Cowling conjectured that one could prove 
 the smoothness of the extension by using minimal regularity hypothesis\, t
 hrough an argument resting on ideas from the study of quasiconformal maps.
  In its simplest form\, the proposed proof is articulated in two steps: (1
 ) prove that any bi-holomorphism between smooth\, strictly pseudoconvex do
 mains extends to a homeomorphisms between the boundaries that is 1-quasico
 nformal with respect to the sub-Riemannian metric associated to the Levi f
 orm\; (2) prove a Liouville type theorem\, i.e. any $1-$quasiconformal hom
 eomorphism between such boundaries is a smooth diffeomorphism. In this tal
 k I will discuss recent work with Le Donne\, where we prove the first step
  of this program\, as well as joint work with Citti\, Le Donne and Ottazzi
 \, where we settle the second step\, thus concluding the proof of Cowling'
 s conjecture. The proofs draw from several fields of mathematics\, includi
 ng nonlinear partial differential equations\, and analysis in metric space
 s. \n
DTSTART:20180214T183000Z
DTEND:20180214T193000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Luca Capogna (Worcester Polytechnic Institute) 
URL:https://www.mcgill.ca/mathstat/channels/event/luca-capogna-worcester-po
 lytechnic-institute-285080
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