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CATEGORIES:SEEMOD Workshop 9
SUMMARY:Approximate groups and projective geometries. -
Emmanuel Breuillard (Cambridge)
DTSTART;TZID=Europe/London:20190517T130000
DTEND;TZID=Europe/London:20190517T135000
UID:TALK125161AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/125161
DESCRIPTION:The structure of finite subsets A of an ambient al
gebraic group G\, which do not grow much under mul
tiplication\, say |AA|<|A|^{1+\\epsilon}\, is well
understood after the works of Hrushovski\, Pyber-
Szabo and Breuillard-Green-Tao on approximate subg
roups of algebraic groups. A more general question
\, tackled by Elekes and Szabo\, asks for the stru
cture of Cartesian products A_1 \\times ... \\time
s 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 subva
riety V \\leq W^n (the case n=3\, W=G\, A_i=A\, V=
{(x\,y\,xy)} corresponds to the above mentioned ap
proximate group problem). In joint work with Marti
n Bays\, we completely characterize the algebraic
varieties V that can admit a (general position) fa
mily of such finite Cartesian products with large
intersection. We show that they are in algebraic c
orrespondence with a subgroup of a commutative alg
ebraic group endowed with an extra structure arisi
ng from a certain division ring of group endomorph
isms. The proof makes use of the Veblen-Young theo
rem on abstract projective geometries\, generalize
d Szemeredi-Trotter bounds and Hrushovski's formal
ism of pseudo-finite dimensions.
LOCATION:MR3 Centre for Mathematical Sciences\, level -1
CONTACT:HoD Secretary\, DPMMS
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