# Mathematics Colloquia and Seminars

Ever since Mark Kac's important paper Can one hear the shape of the drum?'' there has been interest in finding compact Riemannian manifolds $X$ and $Y$ which are isospectral (i.e. the spectrum of the Laplacian, $\Delta$, acting on $L^2(X)$ is equal to the spectrum of $\Delta$ acting on $L^2(Y)$), but not isomorphic. Most known isospectral pairs which are not isomorphic are commensurable (i.e. they share a common finite cover). For $d \geq 3$, $G = PGL_d(F)$, $K$, a maximal compact subgroup of $G$, $S= G/K$ and for any $m \in \N$, we will present a family of $m$ co-compact arithmetic lattices $\{\Gamma_i\}$ in $G$ such that $\Gamma_i \backslash S$ are isospectral and not commensurable. Here $F = \R, \C$, or a local field of positive characteristic.