In this note we show that the property of having only vanishing triple Massey products in the equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular, it can be viewed as a partial answer to the Allday-Puppe question about finding conditions ensuring the "formality" of G-actions.
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left i…
We show how a polar representation of a compact connected Lie group can be linearly determined from its dimension and isotropy subgroup data in the general reducible case.
These notes transcribe a workshop about the notion of total positivity and $Θ$-positivity and its relation to Higher Teichmüller Theory. $Θ$-positivity is a notion of positivity in semisimple Lie groups and was recently introduced by Guichard and Wienhard as a generalization of Lusztig's total positivity. It is believed to be the cathartic notion to classify higher Teichmüller spaces. Without doubt, substantial progress will be achieved in the near future on the study of $Θ$-positive structures. These notes provide an account of the state of the art as of 2021.
