Pure Math Colloquium

Mahler's method in number theory is an area wherein one answers questions surrounding the transcendence and algebraic independence of both power series $F(z)$, which satisfy the functional equation $$a_0(z)F(z)+a_1(z)F(z^k)+\cdots+a_d(z)F(z^{k^d})=0$$ for some integers $k\geqslant 2$ and $d\geqslant 1$ and polynomials $a_0(z),\ldots,a_d(z)$, and their special values $F(\alpha)$, typically at algebraic numbers $\alpha$. The most important examples of Mahler functions arise from important sequences in theoretical computer science and dynamical systems, and many are related to digital properties of sets of numbers. For example, the generating function $T(z)$ of the Thue-Morse sequence, which is known to be the fixed point of a uniform morphism in computer science or equivalently a constant-length substitution system in dynamics, is a Mahler function. In 1930, Mahler proved that the numbers $T(\alpha)$ are transcendental for all non-zero algebraic numbers $\alpha$ in the complex open unit disc. With digital computers and computation so prevalent in our society, such results seem almost second nature these days and thinking about them is very natural. But what is one really trying to communicate by proving that functions or numbers such as those considered in Mahler’s method? 

In this talk, highlighting work from the very beginning of Mahler’s career, we speculate---and provide some variations---on what Mahler was really trying to understand.

MC 5501