On Indefinite Integral Ternary Quadratic Forms

We present solutions to two problems on indefinite integral ternary quadratic forms. The first, highlighted by Margulis in 1990, concerns the distribution of the ternary Markoff spectrum associated with minima of forms. The second, initiated by Serre in 1990, is on the asymptotic growth of the number of isotropic forms of bounded height. To overcome difficulties involving high ramification, we develop a tool that proves useful for other distribution problems as well. Joint work with Alexander Gamburd, Amit Ghosh, and Peter Sarnak.