Algebraic K-theory and the Cyclotomic Trace

Projective modules over rings are the algebraic analogs of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some number rings they need not be free. Even for rings like Z, over which all finitely generated projective modules are free, the category of such modules contains a wealth of interesting information. In this talk I will introduce algebraic K-theory, which encodes this information. I will also explain why one would try to use some kind of trace from K-theory to simpler invariants, outline the cyclotomic trace and briefly show how it is used in calculations.