Affiliation: University of Newcastle, Australia

Title Of Talk: On certain irrational values of the logarithm: beyond the 1979 limitations

Abstract: In a note On certain irrational values of the logarithm" by Alladi and Robinson, a family of integrals was introduced to investigate the irrationality of logarithms of some (simple) algebraic numbers. (The integrals were inspired by Beukers' proof of Ap\'ery's theorem about the irrationality of $\zeta(2)$ and $\zeta(3)$.) The note concluded with a discussion of natural limitations of the method, for example, its failure to approach the irrationality of $\log(3)$ and $\pi$. In my talk I will explain how the Alladi--Robinson integrals, without any modification, are used in establishing that the latter two numbers are irrational.

Title of Special Colloquium: Short random walks and Mahler measures

An $n$-step uniform random walk is a walk that starts at the origin and consists of $n$ steps of length 1 each taken into a uniformly random direction. It is particularly interesting for $n=2,3,4,5$ because of its beautiful links to modular and hypergeometric functions. The Mahler measure of an $n$-variable polynomial is its geometric mean over the $n$-dimensional torus. There are several cases when $n$-variate Mahler measures are known or conjectured to be linked to hypergeometric functions and noncritical $L$-values, for small $n$ as well. In the talk I will outline the links above and indicate some new interconnections between the short random walks and Mahler measures. These novel results are from joint work in progress with Armin Straub.