Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / We investigate aspects of differential K-theory. In particular, we give a direct proof that the Freed-Lott differential analytic index is well defined, and a short proof of the differential Grothendieck-Riemann-Roch theorem in the setting of Freed-Lott differential K-theory. We also construct explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory, define the Simons-Sullivan differential analytic index, and prove the differential Grothendieck-Riemann-Roch theorem in the setting of Simons-Sullivan differential K-theory. / 2031-01-02

For T the circle group, we construct a differential refinement of T-equivariant K-theory. We first construct a de Rham model for delocalized equivariant cohomology and a delocalized equivariant Chern character based on [19] and [14]. We show that the delocalized equivariant Chern character induces a complex isomorphism. We then construct a geometric model for differential T-equivariant K-theory analogous to the model of differential K-theory in [27] and deduce its basic properties. / text