Moments of Margulis Functions and Values of Ternary Quadratic Forms

In this lecture from the Joint IAS/PU Groups and Dynamics Seminar, Wooyeon Kim from the Korea Institute for Advanced Study explores the moments of Margulis functions and values of ternary quadratic forms. Delve into the Oppenheim conjecture, which was proven by Margulis in 1986 and states that for a non-degenerate indefinite irrational quadratic form Q in n≥3 variables, the image set Q(Zn) of integral vectors forms a dense subset of the real line. Learn about the quantitative Oppenheim conjecture, which aims to determine the asymptotic distribution of values of indefinite quadratic forms at integral points. While this conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in n≥4 variables, this talk specifically addresses the case of ternary quadratic forms (n=3). Discover how the proof relies on a uniform boundedness result for the moments of Margulis functions over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit. The seminar takes place at 4:30pm in Simonyi 101 on March 18, 2025.