Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. In this series of lectures we will explain some of these links.
In the first part of the lecturewewill survey the main facts regarding Connes’ quantized calculus and its relationship 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 operator ideals and the Birman-Schwinger principle. The 2nd part will present semiclassical Weyl’s laws and integration formulas for noncommutative manifolds (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.
Raphael Ponge教授是法国人，在四川大学数学学院工作，四川省千人计划入选者。他于2000年在巴黎11大获得博士学位，师从菲尔兹奖得主Alain Connes。Ponge教授的研究领域是非交换几何，曾在美国，加拿大，日本，韩国工作，并在Adv. Math.、 J. Funct. Anal.等期刊发表了40余篇学术论文。