Heights and unlikely intersections

• Applicant: Professor Dr. Philipp Habegger
• Subject Area: Mathematics
• Term: from 2012 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 223746744

We will investigate various aspects of conjectures on unlikely intersections as formulated by Boris Zilber and Richard Pink on semi-abelian varieties and Shimura varieties. They imply classical results such as Faltings's Theorem, formerly known as the Mordell Conjecture, but go beyond such results. These conjectures transcend the boundaries of number theory into algebraic geometry and model theory. The former provides an appropriate framework and powerful techniques. Zilber's motivation for formulating his conjecture came from potential applications to the model theory of C with the exponential function. The concept of an o-minimal structure, originating in model theory, has recently proved itself to be a formidable tool for studying unlikely intersections. In this project we will emphasize height functions used in diophantine geometry. They are powerful tools in this circle of problems and beyond. Approaches towards proving conjectures on unlikely intersections are linked to problems of independent interest which we also pursue. These include algebraic independence statements related to Schanuel's Conjecture. The study of unlikely intersections has also opened up new avenues to open problems such as Bogomolov's Conjecture over function fields

• DFG Programme: Research Grants
• International Connection: Switzerland