Noncommutative Geometry, Semiclassical Analysis, and Weak Schatten p-Classes

The first part of the lecture will survey the main facts regarding Connes’ integration, Weyl’s laws for compact operators and their relationships with semiclassical analysis. In particular, we will explain the link between Connes’ integration formula and semiclassical Weyl’s laws. This will include some background on Schatten p-classes and the Birman-Schwinger principle. The 2nd part will present new results regarding semiclassical Weyl’s laws and integration formulas for noncommutative mani folds (i.e., spectral triples). This improves and simplifies recent results of McDonald-Sukochev-Zanin and Kordyukov-Sukochev-Zanin. For the Dirichlet and Neumann problems on Euclidean domains and closed Riemannian manifolds this enables us to recover the semiclassical Weyl’s laws in those settings from old results of Minakshisundaram and Pleijel from the late 40s. For closed manifolds this also allows us to recover the celebrated Weyl’s laws of Birman-Solomyak for negative-order pseudodifferential operators. A further set of examples is provided by Schrödinger operators associated to sub-Laplacians on sub-Riemannian manifolds, including contact manifolds and Baouendi Grushin example. Finally, we will explain how this framework enables us to get semiclassical Weyl’s laws for noncommutative tori. This will solve conjectures by Edward McDonald and the speaker.

More information about the Analysis Seminar can be found here.