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SUMMARY:Joinings of higher rank diagonalizable actions - Elon Lindenstraus
 s\, Einstein Institute\, Jerusalem
DTSTART:20190321T160000Z
DTEND:20190321T170000Z
UID:TALK93553@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:In the 1950's\, Cassels and Swinnerton-Dyer conjectured that t
 he values of products of n linear forms in n variables at integer points b
 ehave rather differently in n>=3 variables than in the case of 2 variables
 . The reason these two questions behave differently is that the symmetry g
 roup of such forms is an (n-1)-dimensional diagonal group\, and (as conjec
 tured independently from a different point of view by Furstenberg in 1967)
  higher rank diagonalizable actions\, unlike one-parameter diagonalizable 
 actions\, have subtle rigidity properties which are still quite mysterious
 . One aspect of this question where the current state of knowledge is quit
 e satisfactory is the study of joinings of such actions\, where Einsiedler
  and I have a rather general classification of ergodic joinings.\n\nThis c
 lassification has several striking number theoretic applications\, and I w
 ill describe two. Both relate to work of Linnik from around 70 years ago r
 egarding the distribution of integer points of the sphere. In this directi
 on\, Aka\, Einsiedler\, and Shapira studied joint distribution of  integer
  points on a two dimensional sphere together with the shape of its orthogo
 nal lattice\, and work of Khayutin regarding orbits of the class group of 
 a number field on pairs of CM points. 
LOCATION:MR2\, CMS
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