Noncommutative Geometry
Speaker: Masoud Khalkhali (Western University)
"Curvature of the determinant line bundle for noncommutative tori I"
Time: 11:00
Room: MC 106
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.
Mathematics Calendar 
Week of January 4, 2015
