#### Abstract

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.