Abstract: A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent.
Stated in terms matrices the problem asks for given symmetric n-by-n integral matrices A and B whether there is a unimodular integral matrix X
satisfying A=X'BX, where X' is the transpose of X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly,
no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s D. W. Masser conjectured
for n at least 3, there exists a polynomial search bound for X in terms of the heights of A and B. In this talk we shall discuss our recent
resolution of this conjecture based on a joint work with Professor Gregory A. Margulis.