Title:
The
Milnor Ring and Special Values of $L$-functions

** **

**Abstract:**

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.