Historically, Adams realized Whithead's $J$-homomorphism $J: \pi_i(SO)\to \pi^s_i$ can be used to understand the stable homotopy groups of spheres, since $\pi_i(SO)$ is known by Bott periodicity. The Adams conjecture then serves a central role in the identification of the image of the J-homomorphism. 
Multiple proofs of the Adams conjecture were completed around 1970. Notably, Quillen's (second) proof by computing the cohomology of $BGL(\mathbb{F}_q)$ led to his later construction of higher algebraic $K$-theory. In this talk, we present Sullivan's proof using profinite completion and \'etale homotopy theory, which turns the Adams conjecture into a beautiful case of Galois symmetry of algebraic varieties.
 
The talk notes are as follows: Etale Homotopy Theory and Adams Conjecture.pdf