The Sage mathematics software project (http://www.sagemath.org) aims to "Create a viable free open source alternative to Magma, Maple, Mathematica and Matlab."

This hands-on introduction to Sage will get new users solving their computational problems quickly. Emphasis will be placed on using Sage for current research and for using Sage in teaching calculus to undergraduate students.

We will use Sage on the web (http://www.sagenb.org); please bring your laptop if you have one.

The Hodge group (aka special Mumford-Tate group) of a complex abelian variety $X$ is a certain linear reductive algebraic group over the rationals that is closely related to the endomorphism ring of $X$. (For example, the Hodge group is commutative if and only if $X$ is an abelian variety of CM-type.) In this talk I discuss" lower bounds" for the center of Hodge groups of superelliptic jacobians. (This is a joint work with Jiangwei Xue.)

The values of the partial zeta functions for an abelian extension of number fields at non-positive integers are rational numbers with known bounds on their denominators. David Hayes conjectured that when the associated fields satisfy certain algebraic conditions, the bound at s=0 can be sharpened. I will present a counterexample to Hayes's conjecture. I will then propose a new conjecture sharpening the bounds at arbitrary non-positive integers that implies a weaker version of Hayes conjecture at s=0. I will conclude by proving that the new conjecture is a consequence of the Coates-Sinnott conjecture.