Modular-type functions attached to mirror quintic Calabi-Yau threefolds

In this talk we introduce a differential algebra  of modular-type
functions attached to the periods of a one parameter family
of Calabi-Yau varieties which is mirror dual to the universal family of
quintic threefolds. Such an algebra is generated by seven functions
satisfying functional and differential equations in parallel to  the
modular functional equations of classical Eisenstein series and the
Ramanujan differential equation. Our result is the first
example of automorphic-type functions attached to varieties whose period
domain is not Hermitian symmetric.