29 May 2018; Dept Math & Stat (UiT), room U1=A228. Tuesday 14:00-15:30

Travis Willse (University of Vienna)

"Special geometries via projective holonomy".

Restricting the usual action of $\mathrm{SL}(n+1,R)$ on the projective $n$-sphere to a subgroup $H$ canonically induces natural geometric structures on each of the $H$-orbits. We can extend this picture to the curved setting: Given a projective structure, a holonomy reduction of the canonical Cartan connection to $H$ determines a partition of the projective manifold into so-called "curved orbits", each equipped with a geometric structure of a type that occurs in the flat (projective $n$-sphere) setting. Among other consequences, this establishes relationships among different types of geometry that may otherwise not readily apparent; in some cases this leads to canonical notions of compactification of one type of geometry by another.

After reviewing the general setup and the prototypical example that $H$ is an orthogonal group, we'll examine in some detail the particular cases that (1) $n$ is odd and $H$ is the special unitary group, in which case the resulting geometric structures are Sasaki-Einstein and Fefferman conformal structures, and (2) $n=6$ and $H=G_2$, which yields various exceptional structures. The former case is joint work with A.R.Gover and K.Neusser; the latter is joint with A.R.Gover and R.Panai.