Sebastian Goette (Freiburg) will speak in the geometry seminar at 2pm on 14th April, in the Salle des Profs (9th floor building NO). Sebastian’s title is “A cobordism approach to calibrated submanifolds” and his abstract is below.
Calibrations provide us with an easy way to prove minimality of certain submanifolds of Riemannian manifolds. Counting such calibrated submanifolds can lead to interesting differential topological invariants like Gromov-Witten invariants. However, calibrated submanifolds of certain types are hard to find. In this talk, we introduce a much weaker notion of “homotopy calibrated submanifolds”, for which existence and uniqueness can be treated by purely topological means. For this, we need the calibration to arise from a special geometric structure, for example, from a Riemannian metric of special holonomy.
In the second half of the talk, we plan to focus on homotopy associative submanifolds in $G_2$-manifolds. If $\varphi_t$ for $t\in[0,1]$ is a family of $G_2$-structures, one can sometimes find an associative submanifold $A_t$ for each $t$ that varies continuously in $t$. Some of the $A_t$ could be singular. We will see that our approach allows us to detect certain singularities in the family $A_t$ by considering only $A_0$ and $A_1$.