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. This talk will combine modern and historical methods and will be accessible to students.

In this talk, I will outline my interest in, and results towards, the Erdős Discrepancy Problem (EDP). I came about this problem as a PhD student sometime around 2007. At the time, many of the best number theorists in the world thought that this problem would outlast the Riemann hypothesis. I had run into some interesting examples of some structured sequences with very small growth, and in some of my early talks, I outlined a way one might be able to attack the EDP. As it turns out, the solution reflected quite a bit of what I had guessed. And I say 'guessed' because I was so young and naïve that my guess was nowhere near informed enough to actually have the experience behind it to call it a conjecture. In this talk, I will go into what I was thinking and provide proof sketches of what turn out to be the extremal examples of EDP.

I will talk a bit about the benefits of a regular outlook.

I will survey some recent and not-so-recent results surrounding the areas of Diophantine approximation and Mahler's method related to variations of the Chomsky-Schützenberger hierarchy.

I will survey my career both mathematically and personally offering advice and opinions, which should probably be taken with so many grains of salt that it makes you nauseous. (Note: Please bring with you a sense of humour and all of your preconceived notions of how your life will turn out. It will be more fun for everyone that way.)

In this talk, we will show that a D-finite Mahler function is necessarily rational. This gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our method of proof, we also provide a new proof of a Pólya-Carlson type result for Mahler functions due to Randé; that is, a Mahler function which is meromorphic in the unit disk is either rational or has the unit circle as a natural boundary. This is joint work with Jason Bell and Eric Rowland.

Let $F(z)$ be a power series, say with integer coefficients. In the late 1920s and early 1930s, Kurt Mahler discovered that for $F(z)$ satisfying a certain type of functional equation (now called Mahler functions), the transcendence of the function $F(z)$ could be used to prove the transcendence of certain special values of $F(z)$. Mahler's main application at the time was to prove the transcendence of the Thue-Morse number $\sum_{n\geq 0}t(n)/2^n$ where $t(n)$ is either 0 or 1 depending on the parity of the number of 1s in the base 2 expansion of $n$. In this talk, I will talk about some of the connections between Mahler functions and finite automata and highlight some recent approaches to large problems in the area. If time permits, I will outline a new proof of a version of Carlson's theorem for Mahler functions; that is, a Mahler function is either rational or it has the unit circle as a natural boundary.