|Date||Dec 02, 2022|
The homogeneous space SL(d,R)/SL(d,Z) can be identified with the space of lattices, whose covolume is one, in the d-dimensional real vector space. Based on this identification, one can use many dynamical properties to study many problems related to the integral lattice. In this talk, we want to recall some equidistribution theorems starting from the famous Ratner’s theorems, and introduce how we can apply these equidistribution theorems to solve the Oppenheim conjecture-typed problems, especially about the distribution of images of integral vectors under a system of a quadratic form and a linear form. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.