This book considers the so-called Unlikely Intersections, a topic that
embraces well-known issues, such as Lang's and Manin-Mumford's,
concerning torsion points in subvarieties of tori or abelian varieties.
More generally, the book considers algebraic subgroups that meet a given
subvariety in a set of unlikely dimension. The book is an expansion of
the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute
for Advanced Study in Princeton in May 2010.
The book consists of four chapters and seven brief appendixes, the last
six by David Masser. The first chapter considers multiplicative
algebraic groups, presenting proofs of several developments, ranging
from the origins to recent results, and discussing many applications and
relations with other contexts. The second chapter considers an analogue
in arithmetic and several applications of this. The third chapter
introduces a new method for approaching some of these questions, and
presents a detailed application of this (by Masser and the author) to a
relative case of the Manin-Mumford issue. The fourth chapter focuses on
the André-Oort conjecture (outlining work by Pila).