Jeff Vaaler (University of Texas - Austin) will present "Diophantine inequalities for height functions." Abstract: Diophantine inequalities for height functions This will be a mostly expository talk about recent results and open problems in the theory of height functions. For example, the basic Weil height is defined on Q^x, the multiplicative group of nonzero algebraic numbers. We will describe a Banach space that is naturally determined by this height. And we will describe how this Banach space leads to a generalization of the Weil height from elements of Q^x to nitely generated subgroups of Q^x. The height on subgroups turns out to be equal to the volume of a related convex symmetric subset of a Euclidean space. This height-volume connection leads to a bound on the norm of small integer vectors that provide multiplicative dependencies among finite sets of algebraic numbers. An unusual feature of our approach is that the inequalities we obtain are independent of number fields that contain the initial set of algebraic numbers.