Distribution of determinants at lattice points of matrices

In this talk, we study the distribution of determinant values taken by lattice points in the space of d by d real matrices. Unless the lattice has an additional multiplicative structure like the lattice of integer matrices, it turns out that the determinant values are dense in the real line, as a consequence of Ratner's theorem. A natural question is how these determinant values are distributed in the real line. We give a complete answer to this question for d=3. For d=2 our approach gives an alternative proof for the quantitative version of the Oppenheim conjecture for quadratic forms of signature (2,2), obtained by Eskin-Margulis-Mozes (2005). This is joint work with Hee Oh.

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.