Lattice angles and quadratic forms (Lenny Fukshansky, CMC)

What are the possible angles between two integer vectors in R^n? If we fix one such possible angle and one integer vector x, is there always another integer vector y that makes this angle with x? Assuming that x makes a given angle with some vector, how can we find the shortest such vector y? What if we designate a forbidden set of vectors, what is the shortest y making a given angle with x outside of this forbidden set? It turns out that all of these questions can be reformulated in terms of a search for zeros of integral quadratic forms, a rich arithmetic theory. We will give an introduction to this research direction and also show some of our recent results. Joint work with Sehun Jeong (CGU).