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Some Problems in Geometry of Shimura Varieties

Final Report Summary - SPGSV (Some Problems in Geometry of Shimura Varieties)

The crux of the proposal was to apply techniques of Model theory (o-minimality) and Ergodic theory to problems in Diophantine geometry, namely the Zilber-Pink conjecture.
It is currently a hot topic in Pure Mathematics. In 2011 and 2012 - just before the start of the project - two major workshops/schools have been held in Luminy and Jussieu respectively on the subject.
Since then, spectacular progress has been made and ERC funding of this project certainly directly and indirectly contributed to it. A whole semester was held in the second half
of 2017 at Fields Institute in Toronto devoted in part to these topics. Some of the outcomes of the project were presented by the Principal Investigator there in February and June 2017.

Coming out directly from the project are the following achievements: proof of the full Ax-Lindemann-Wieierstass conjecture (Klingler, Ullmo, Yafaev), complete resolution of the problem of bounding the height of pre-special points (Daw and Orr), opening of the new avenue of research on flows on abelian and Shimura varieties and obtaining a number of results (Ullmo and Yafaev), bounds on the height of elements in Siegel sets and results in the direction of the Zilber-Pink conjecture (Orr), results on Andre-Pink-Zannier conjecture and beyond (Richard and Yafaev).
Furthermore, the funding of the project indirectly contributed to results explicitly featured in the initial proposal, notably the elaboration of a definite strategy for proving the Zilber-Pink conjecture (Daw and Ren) as well as results on algebraic flows on abelian varieties (Peterzil and Starchenko) and results on equidistribution of special subvarieties (Daw, Gorodnik and Ullmo – work in progress).