Abstract:  In this talk I will consider algebraic structures such as lie, Hopf, and Frobenius algebras. I will show that under certain assumptions such structures can be reconstructed from the scalar invariants they define. I will then show how one can interpolate the category of representation of the automorphism groups of the structures by interpolation of the invariants of the algebraic structures. In this way we recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t, and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring. If time permits we will also show how the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov arise in this context.