Abstract: In this series of talks we will review Quillen's celebrated determinant line bundle construction on the space of Fredholm operators and study the geometry of this line bundle over the space of Cauchy-Riemann operators on a Riemann surface. Quillen defines a Hermitian metric using zeta regularized determinants on this line bundle and computes its curvature. This computation is then used to define a holomorphic determinant for Cauchy-Riemann operators. It is fairly easy to see that one cannot define a determinant function which is both holomorphic and gauge invariant (conformal anomaly).

Then we will move to a noncommutative setting and review our recent work, with A. Fathi and A. Ghorbanpour, in which we studied the curvature of the determinant line bundle over a space of Dirac operators on the noncommutative two torus. We developed the tools that are needed in our computation of the curvature, including an algebra of logarithmic pseudodifferential symbols and a Konstsevich-Vishik type trace on this algebra. These talks will move slowly and the idea is to develop the necessary tools for further study of the determinant line bundle in noncommutative geometry.