  A major open problem in the theory of twisted commutative algebras (tca's) is
proving noetherianity of finitely generated tca's. For bounded tca's this is
easy, in the unbounded case, noetherianity is only known for
Sym(Sym^2(C^\infty)) and Sym(\wedge^2(C^\infty)). In this paper, we establish
noetherianity for the skew-commutative versions of these two algebras, namely
\wedge(Sym^2(C^\infty)) and \wedge(\wedge^2(C^\infty)). The result depends on
work of Serganova on the representation theory of the infinite periplectic Lie
superalgebra, and has found application in the work of Miller-Wilson on
"secondary representation stability" in the cohomology of configuration spaces.
