## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Monday, November 5, 2018 — 4:00 PM EST

**Michael Coons, University of Newcastle**

"Mahler’s methods: theorems, speculations and variations"

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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.