Many questions in number theory lead to the study of lattices and regions in real vector spaces. For example, the two square theorem can be viewed as an assertion about the existence of integer points on circles in the plane. Minkowski's theorem asserts that a region in R^n satisfying certain conditions must contain an integer point, helping solve these problems geometrically.
Time permitting, I will also explain the theoretical connection between Minkowski's theorem and two of the fundamental theorems of algebraic number theory: the finiteness of the ideal class group, and the finite generation of the unit group.