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SUMMARY:Functional transcendence and equidistribution - Harry Schmidt\, Un
 iversity of Basel
DTSTART:20221116T141500Z
DTEND:20221116T151500Z
UID:TALK192260@talks.cam.ac.uk
CONTACT:Dhruv Ranganathan
DESCRIPTION:A little over a year ago Kühne\, building on work of Dimitrov
 \, Gao and Habegger\, proved a uniform version of the Manin-Mumford conjec
 ture (a theorem of Raynaud). His proof uses various ingredients\, among th
 em equi-distribution on families of abelian varieties. The latter ingredie
 nt was generalized by Yuan and Zhang to families of dynamical systems (amo
 ng other things)\, with related work by Gauthier. Another ingredient in hi
 s proof\, that is also central to the work of Dimitrov\, Gao and Habegger\
 , is the so-called Ax-Schanuel theorem. This is a powerful functional tran
 scendence statement\, motivated by the André-Oort conjecture\, that was o
 btained over several works in the past 20 years and characterizes the inte
 rsection of algebraic varieties with the graph of the uniformization map o
 f mixed Shimura varieties. It is however particular to those and not appli
 cable to dynamical systems in general. \n\nAfter giving a brief historical
  overview\, I will explain how Myrto Mavraki and I overcame the restrictio
 ns imposed by a reliance on Ax-Schanuel in recent work and proved uniform 
 versions of the dynamical Manin-Mumford conjecture for one-parameter famil
 ies of endomorphisms of the projective line. I will explain aspects of our
  new strategy that also applies in the abelian context and generalizes res
 ults of Habegger as well as results of Baker and Call-Silverman. Time perm
 itting\, I will also report on very recent results for higher dimensional 
 bases. 
LOCATION:CMS MR13
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