Rational Points and Heights

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The project addresses questions in Arithmetic Algebraic Geometry.It is proposed to study the distribution of rational points onalgebraic varieties over number fields, and to explore the interplaybetween global geometric invariants and arithmetic properties ofvarieties. The focus is on proving potential density for familiesof varieties, such as Fano varieties or Calabi-Yau varieties, andon establishing asymptotic formulas for the number of rational pointsof bounded height on equivariant compactifications of groups and homogeneous spaces.The proposed research ranges fromexplicit numerical experiments to advanced problems inmodern and rapidly developing areas of mathematics.This work will have a broader impact on the education of studentsand training of young mathematicians, as there is a rich supply ofconcrete questions and examples, consolidating our understanding of thesubject and leading to new structures.Increasingly, applications of Arithmetic Algebraic Geometryare spreading to vital areas of information storage, processing andtransmission; these applications rely on further advances onfundamental problems to be addressed in this proposal.