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I will speak on the classification problem of compact homogeneous Einstein spaces G/H. Recall that a G-invariant Einstein metric can be characterized as a critical point of the scalar curvature function restricted to the space of G-invariant metrics of volume one. To this end, I will introduce a new simplicial complex (defined by certain intermediate subgroups H

In this talk, I will discuss concentration of measure phenomena for the Gibbs of the focusing, mass-subcritical Schrödinger equation on the real line. I will first discuss how these results connect to the soliton resolution conjecture for NLS, and then explain how to approach this family of problems, combining the variational representation of solitons with more probabilistic techniques. This talk is based on joint works with T. Oh (Edinburgh), M. Okamoto (Osaka), H. Weber (Münster), J. Forlano (Monash), and F. Höfer (Münster).

Introduced by Ozawa, W*-bundles can be understood as continuous fields of tracial von Neumann algebras over a compact Hausdorff space. To study W*-bundles, we thus need to analyze von Neumann algebraic phenomena in a way that varies continuously. This work proves several C*-algebraic regularity properties, such as real rank zero, stable rank one, and comparison of projections for a class of W*-bundles that includes all locally trivial II_1-factor bundles. Namely, we consider W*-bundles which are selfless, or which admit a sequence of asymptotically free Haar unitaries, where the freeness occurs with respect to the trace in each fiber, uniformly over all fibers of the bundle. The proofs use Baire-category arguments where the density of each open set is demonstrated using perturbations by an approximately free element. For instance, to prove stable rank, or density of invertible elements, we consider perturbing a given x by a free circular operator, which will regularize it to something with trivial kernel, which we show admits a polar decomposition in the bundle. This is based on joint work, to appear soon, with Stuart White and Maxwell Ryder

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