  We prove a thick subcategory theorem for the category of $d$-excisive
functors from finite spectra to spectra. This generalizes the Hopkins-Smith
thick subcategory theorem (the $d=1$ case) and the $C_2$-equivariant thick
subcategory theorem (the $d=2$ case). We obtain our classification theorem by
completely computing the Balmer spectrum of compact $d$-excisive functors. A
key ingredient is a non-abelian blueshift theorem for the generalized Tate
construction associated to the family of non-transitive subgroups of products
of symmetric groups. Also important are the techniques of tensor triangular
geometry and striking analogies between functor calculus and equivariant
homotopy theory. In particular, we introduce a functor calculus analogue of the
Burnside ring and describe its Zariski spectrum \`{a} la Dress. The analogy
with equivariant homotopy theory is strengthened further through two
applications: We explain the effect of changing coefficients from spectra to
${\mathrm{H}\mathbb{Z}}$-modules and we establish a functor calculus analogue
of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work
offers a new perspective on functor calculus which builds upon the previous
approaches of Arone-Ching and Glasman.
