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).