IMPA - O Instituto de Matemática Pura e Aplicada

In this talk, I will discuss a series of works on Gromov's $p$-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index $p$. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the $p$-widths and how surfaces provide a convenient setting to investigate this.

The Obata connection on a hypercomplex manifold is the unique torsion-free connection that preserves the hypercomplex structure. Up to taking the product with a torus, Joyce constructed left-invariant hypercomplex structures on compact semisimple Lie groups and certain related homogeneous spaces. Soldatenkov showed in 2011 that every Joyce hypercomplex structure on $SU(3)$ has Obata holonomy equal to the quaternionic general linear group and it was expected that the same should hold for all Joyce hypercomplex manifolds. For all such group manifolds except for $SU(2n+1)$, we will show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $SU(2n+1)$ is more subtle, however for every $n > 1$, there still exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in the quaternionic general linear group and at least one with full holonomy on $SU(5)$. This talk is based on a joint work with Udhav Fowdar, Giovanni Gentili and Luigi Vezzoni.