Orthogonal polynomials and algebraic curves
Families of polynomials orthogonal on the unit circle can be defined by a recurrence relation whose. In fact, such polynomials provide a one-to-one correspondence between sequences of complex numbers lying in the unit disk and probability measures on the unit circle. When the sequence of coefficients of the recurrence relation is periodic, there is an analogue of the Bloch-Floquet theory for Hill's equation that allows the correspondence to be formulated in terms of an algebraic curve and geometric data on that curve. In this talk I will discuss this algebrogeometric picture with an eye towards taking a long-period limit to relate this formulation of the spectral problem to the scattering-theoretic formulation introduced by Geronimo and Case to handle decaying sequences of coefficients.