Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo
"Calibrated subbundles of $\mathbb R^7$"
One can view $\mathbb R^7$ as the total space of the bundle $E = \Lambda^2_- (\mathbb R^4)$ of anti-self-dual 2-forms on $\mathbb R^4$. In this way we can describe the standard flat $G_2$-structure in terms of 4-dimensional geometry. Given an oriented surface $M^2$ in $\mathbb R^4$, the restriction of $E$ to $M$ decomposes as a direct sum of a line bundle and a rank 2 bundle. We determine conditions on the immersion of $M^2$ in $\mathbb R^4$ that are equivalent to the total spaces of the subbundles being (respectively) associative and coassociative submanifolds. This is (very old) work of myself, Ionel, and Min-Oo from 2005. I will also discuss several ways in which it has already been generalized and ways in which it can potentially still be generalized further.