Frank Calegari of the University of Chicago presents a Joint IAS/Princeton Number Theory seminar talk on G-functions, power series with rational coefficients whose denominators grow at most exponentially while also satisfying a linear differential equation over the rationals. Calegari lays out the tension Siegel first identified between these arithmetic and analytic properties, then walks through work done with Vesselin Dimitrov and Yunqing Tang proving holonomy bounds that force certain power series to be G-functions of low order. He surveys where the method currently stops, poses a set of open questions about the structure of G-functions, and points to applications outside the immediate subject, including the effective S-unit equation and a new argument for the transcendence of pi. The talk is pitched at a specialist audience already fluent in algebraic number theory and differential equations, delivered on a whiteboard-style blackboard format typical of the IAS seminar series, with no padding beyond the mathematics itself.