E-functions and geometry

After recalling the conjectural relation between G-functions and periods of families of algebraic varieties, I will explain why every exponential period function of the form $\int_\sigma e^{-f} \omega$, where $f$ is a regular function on an algebraic variety $X$ defined over the field of algebraic numbers, $\omega$ is an algebraic differential form on $X$, and $\sigma$ is a rapid decay cycle on $X(\mathbb{C})$, is a linear combination of E-functions "with monodromy” with coefficients in the field generated by usual periods, special values of the gamma function and Euler’s constant. This is how E-functions arise from geometry and gives some intuition of why a positive answer to Siegel’s question whether all E-functions are polynomial expressions in hypergeometric E-functions was extremely unlikely (joint work with Peter Jossen).