Indefinite Integral Quadratic Forms Beyond Reduction Theory

The classical reduction theory of integral quadratic forms was developed by Hermite, Minkowski, Siegel and many others. It is known that a non-degenerate integral quadratic form in n-variables is integrally equivalent to a form whose height (the maximum value of the coefficients) is less than its determinant (up to a multiple constant), and whose value at (1, 0,…0) is less than the n-th root of its determinant. However, for indefinite forms in at least 3 variables it turns out that neither of the estimate is optimal. In this talk we will discuss some classical results and recent effort in improving these estimates. This is a joint work with Prof. Margulis.