Differential modular forms: an overview

If classical algebraic geometry is enriched by ``adjoining'' a Fermat quotient operator $\delta$ one obtains a new geometry referred to as $\delta$ geometry. Modular forms in $\delta$-geometry are called differential modular forms. The main motivation for the theory comes from the existence of certain special differential modular forms having surprising properties. These properties do not have an analogue in the classical theory. On the other hand these special differential modular forms can be used to prove purely diophantine results in the classical setting. The talk will be an overview of the subject; it will briefly present some of the basic results, constructions, applications, and open problems.