The mod $p$ Steenrod algebra is the (Hopf) algebra of
stable operations on mod $p$ cohomology, and in part
measures the subtle behavior of $p$-local homotopy theory
(as opposed rational homotopy theory, which is much simpler).
A classical theorem of Dold-Thom tells us that the infinite
symmetric power of the $n$-dimensional sphere is the Eilenberg-Maclane
space K(Z, n),and one can use an appropriate modification of this
construction to compute the dual Steenrod algebra. The infinite
symmetric power of the sphere spectrum has a filtration whose
$k$-th cofiber miraculously turns out to be the Steinberg summand
(from modular representation theory of GL_k(F_p)) of the
classifying space of (Z/p)^k. This opens the door for slick
computations - for example, the Milnor indecomposables can
be picked out as explicit cells.
In this talk, I will introduce the concepts and results
chronologically. I will also include hands-on homotopy
theory computations as time permits.