Title: The Milnor Ring and Special Values of $L$-functions
This will be an elementary talk aimed at a general audience. We begin with an overview of the theory of modular symbols for $GL_2(\Q)$. The beauty of the theory is in the simplicity of the approach it give us into the theory of special values of $L$-functions. We proceed by giving a sequence of increasingly more interesting - but always concrete and explicit - examples, explaining how the examples illuminate the arithmetic theory of special values of $L$-functions attached to modular forms. Our final example will be a modular symbol taking values in Milnor's $K$-theoretic Ring associated to a certain ring of trigonometric functions. This example generalizes nicely to $GL_n(\Q)$.
As a simple application, we will see how the theory leads to a simple proof of the classical Dedekind Reciprocity formula.