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SUMMARY:Approximate groups and projective geometries.  -  Emmanuel Breuill
 ard (Cambridge)
DTSTART:20190517T120000Z
DTEND:20190517T125000Z
UID:TALK125161@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:The structure of finite subsets A of an ambient algebraic grou
 p G\, which do not grow much under multiplication\, say |AA|<|A|^{1+\\epsi
 lon}\, is well understood after the works of Hrushovski\, Pyber-Szabo and 
 Breuillard-Green-Tao on approximate subgroups of algebraic groups. A more 
 general question\, tackled by Elekes and Szabo\, asks for the structure of
  Cartesian products A_1 \\times ... \\times A_n of finite subsets of size 
 N of an arbitrary d-dimensional algebraic variety W\, with large (i.e. >N^
 {\\dim V/d}) intersection with a given subvariety V \\leq W^n (the case n=
 3\, W=G\, A_i=A\, V={(x\,y\,xy)} corresponds to the above mentioned approx
 imate group problem). In joint work with Martin Bays\, we completely chara
 cterize the algebraic varieties V that can admit a (general position) fami
 ly of such finite Cartesian products with large intersection. We show that
  they are in algebraic correspondence with a subgroup of a commutative alg
 ebraic group endowed with an extra structure arising from a certain divisi
 on ring of group endomorphisms. The proof makes use of the Veblen-Young th
 eorem on abstract projective geometries\, generalized Szemeredi-Trotter bo
 unds and Hrushovski's formalism of pseudo-finite dimensions. 
LOCATION:MR3 Centre for Mathematical Sciences\, level -1
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