Title: Tiling and exponential bases in ${\Bbb Z}_p \times {\Bbb Z}_p$
Speaker: Alex Iosevich (University of Rochester)
Abstract: The celebrated Fuglede Conjecture says that if $\Omega$ is a domain in a locally compact abelian group, then $L^2(\Omega)$ possesses an orthogonal basis of exponentials (characters) if and only if $\Omega$ tiles by translation. This conjecture was disproved by Tao in ${\Bbb Z}_p^d$ and ${\Bbb R}^d$ in dimensions $5$ and higher and counter-example were extended (in both directions) to dimension $4$ by Kolountzakis, Matolcsi and others. We will prove that the conjecture does hold in ${\Bbb Z}_p^2$, $p$ prime. The argument is a blend of analysis, combinatorics and elementary number theory. The result opens up a rather interesting question of what happens in ${\Bbb R]^2$ and, more generally, in other locally compact abelian groups.
Seminar URL: https://research.math.osu.edu/pde/
