Functional transcendence theorems, such as Ax--Lindemann--Weierstrass and Ax--Schanuel, characterize the algebraic independence of analytic functions in terms of their arguments. These results have played a crucial role in Diophantine geometry, most notably in the proof of the Andre--Oort conjecture and related unlikely intersection problems.
In this talk, we present results towards an Ax--Lindemann--Weierstrass theorem for the Euler Gamma function. Historically, such theorems have been proven for the exponential function, modular j-function, and general classes of functions satisfying algebraic differential equations. As the Gamma function is known to not satisfy any algebraic differential equation, it falls outside the scope of previous differential-algebraic methods. We will discuss our methods and also highlight several new peculiarities that are not present in prior results.