Dubi Kelmer (Boston College and Princeton)
Following Margulis's proof of the Oppenheim conjecture we know that integer values of an irrational indefinite quadratic form in n >= 3 variables are dense on the real line. The same is true for an inhomogeneous form obtained by shifting values by a fixed vector if either the form or the shift is irrational. In this talk, I will describe several approaches to this problem that give effective results that hold for a fixed rational form Q and almost all shifts, by reducing it to the density of certain orbits of a discrete group acting on the torus. I will then describe different approaches using dynamics, representation theory, and estimates on exponential sums for this problem.