Termine

We describe a geometric and homotopical perspective on higher categories: higher categories are built by iterated attachment of directed cells in the spirit of CW-complexes and admit Postnikov towers giving rise to homotopy posets. These carry the obstruction classes controlling extensions from one skeleton to the next. The associated graded is given by wedges of spheres and directed spheres, leading to stabilization phenomena and categorical analogues of spectra. This is joint work with David Gepner.

The Weyl energy on four-manifolds is a quadratic curvature functional related to the Chern-Gauss-Bonnet formula. It is conformally invariant, and its critical points are called Bach-flat metrics and provide an interesting generalization of Einstein metrics or self-dual and anti-self-dual metrics. We will show that, under suitable conditions, it is possible to decrease the total Weyl energy under connected sum between two manifolds. These gluing results might be particularly useful for understanding the compactness properties of Weyl-minimizing sequences of metrics. This is based on joint works with Andrea Malchiodi.

In these two talks I will present my current project on endowing a torsor structure on L-packet of the covering tori. In the first half I will talk about how to construct gerbes on the groupoid of isocrystals, and a formulation of the Langlands correspondence of sharp covering tori on the level of gerbes. In the second half I will talk about how the groupoid of isocrystals of the sharp tori can help to interpolate the L-packets of different extended pure inner forms of the covering tori, which gives rise to the desired torsor structure.

A C*-algebra is called pure if it has good comparison and divisibility properties; more precisely, its Cuntz semigroup is almost unperforated and almost divisible. A natural question, raised at several conferences, is whether the minimal tensor product $A \otimes B$ remains pure whenever $A$ is. In this talk, we shall see results addressing this question when $B = C(X)$. We will also discuss applications to crossed products. This talk is based on two joint works: one with Vilalta, and another with Archey, Buck, Mohammadkarimi, and Phillips.