Abstract:
Given a transcendental holomorphic function f, what kinds of varieties V and W satisfy that f(V) is contained in W? In a sense, answering this question involves classifying the algebraic properties of f. For the exponential function, the modular j function, and other solutions of differential equations, the question has been answered, and even stronger results are known. Some of these results have played a key role in solutions to diophantine problems, including important cases of the André-Oort conjecture.
For differentially transcendental functions, comparatively little is known. In this talk, I will show how to answer some cases of the question for Euler’s Gamma function, which interpolates the factorial and is differentially transcendental.