Using heights and other methods, for example tools from o-minimality, we study various problems in  diophantine and arithmetic geometry. The main emphasis is on conjectures on unlikely intersections  generalizing the classical Andr\'e-Oort, Manin-Mumford, and Mordell-Lang Conjectures.

Subprojects directly involving the PI:

- Unlikely Intersections and o-Minimality (cooperation with J. Pila)

- Reduction properties of curves in genus 2 with complex multiplication and bad reduction (cooperation with F. Pazuki)

- Degenerate cases in the Zilber-Pink Conjecture / CIT (cooperation with M. Bays)

- Interplay between group structure and the Bogomolov property involving small height (cooperation with R. Grizzard and L. Pottmeyer)

Linda Frey is writing a Ph.D. thesis in the context of this project. Lukas Pottmeyer is a postdoc.

Supported by the DFG, German Research Foundation.