Caleb Suan, Department of Pure Mathematics, University of Waterloo
"Differential Operators on Manifolds with $G_2$-Structure"
The group $G_2$ is one of the possible holonomy groups according to the Berger classification. In this talk, we look at differential operators defined on manifolds with $G_2$-structure, in particular, the Dirac and twisted Dirac operators on the spinor bundle and bundle of spinor-valued 1-forms respectively. We develop results regarding these operators using a bundle isomorphism between the spinor bundle and the octonion bundle $\mathbb{R} \oplus TM$, as well as define and study the properties of extensions of the divergence, gradient, and curl operators. Using this methodology, we compute the kernel of the twisted Dirac operator and its dimension.
Please contact Spiro Karigiannis at karigiannis@uwaterloo.ca if you are interested in attending the presentation.