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SUMMARY:Geometric results on linear actions of reductive Lie groups for ap
 plications to homogeneous dynamics - Rodolphe Richard (Cambridge)
DTSTART:20181019T124500Z
DTEND:20181019T134500Z
UID:TALK112471@talks.cam.ac.uk
CONTACT:Richard Webb
DESCRIPTION:Several problems in number theory when reformulated in terms o
 f homogeneous dynamics involve the study of limiting distributions of tran
 slates of algebraically defined measures on orbits of reductive groups. Th
 e general non-divergence and linearization techniques\, in view of Ratner'
 s measure classification for unipotent flows\, reduce such problems to dyn
 amical questions about linear actions of reductive groups on finite-dimens
 ional vector spaces.\n\nWe will limit ourselves to the treatment of the ar
 chimedean place (i.e. over the field of real numbers)\, taken from the epo
 nymous article https://arxiv.org/abs/1305.6557 (Ergodic Theory and Dynamic
 al Systems (this month issue)).\n\nWe will address these finite-dimensiona
 l problems\, namely: in which directions does the trajectory of a bounded 
 subset get arbitrarily small\, or stay bounded. This involves the geometry
  of the concerned real algebraic groups (Mostow decomposition\, convexity 
 arguments on the associated symmetric spaces)\, and some representation th
 eory.\n\nThe problem treated and the methods used give an example of the r
 elevance\, in homogeneous dynamics\, of the notion of stability (in the Mu
 mford sense) taken in an archimedean context and in an arithmetic context.
LOCATION:CMS\, MR13
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