"We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."

I will discuss a couple of techniques often useful to prove quasi-isometric rigidity results for isometry groups. I will then sketch how these were used by B. Kleiner and B. Leeb to obtain quasi-isometric rigidity for the class of fundamental groups of closed locally symmetric spaces of noncompact type.

It is unknown whether a bound on axion field ranges exists within quantum gravity. We study axion field ranges using extended supersymmetry, in particular allowing an analysis within strongly coupled regions of moduli space. We apply this strategy to Calabi-Yau compactifications with one and two Kähler moduli. We relate the maximally allowable decay constant to geometric properties of the underlying Calabi-Yau geometry. In all examples we find a maximal field range close to the reduced Planck mass (with the largest field range being 3.25 $M_P$). On this perspective, field ranges relate to the intersection and instanton numbers of the underlying Calabi-Yau geometry.