I will begin with an introduction to various spectra like the eigenvalue and geodesic length spectra. The primary focus will be constructions producing examples of manifolds with the same spectra. I will also discuss higher dimensional versions of the geodesic length spectrum and some work with Alan Reid in this direction. In the second lecture, I will report on recent work with Benjamin Linowitz, Paul Pollack, and Lola Thompson on effective versions of rigidity results due to Reid and Chinburg-Hamilton-Long-Reid. These rigidity results assert that arithmetic hyperbolic 2- and 3-manifolds are determined, up to commensurability, by the eigenvalue or geodesic length spectrum. Time permitting, I will discuss some counting results that are central in establishing our effective results.
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