Determinant values on lattices

The determinant is one of the most fundamental polynomial functions on the space of matrices. Given a lattice of real matrices, one may ask how its determinant values are distributed: are they confined to a discrete set, or do they spread throughout the real line? I will discuss recent joint work with Wooyeon Kim on the quantitative distribution of determinant values on lattices, recently featured by MathInstitutes.org. For two-by-two matrices, our theorem recovers the Eskin–Margulis–Mozes quantitative Oppenheim theorem for quadratic forms of signature (2,2) and in higher dimensions, it provides a higher-degree analogue.