Noncommutative geometry and semiclassical analysis

Abstract: In this talk, I will present new results regarding semiclassical Weyl’s laws in the setup of Connes’ noncommutative geometry. They provide precise asymptotics for the counting functions of Schroedinger operators under the semiclassical limit. This improves and simplifies previous results of McDonald-Sukochev-Zanin. This provides a bridge between semiclassical analysis and noncommutative geometry. Thanks to the Birman-Scwhinger principle and old results of Birman-Solomyak this reduces to establishing various weak Schatten class properties for the operators at stake. This has a number of applications. We shall present two of them. First, we recover previously known semiclassical Weyl’s laws on Euclidean domains and closed manifolds. These results were proved in 60s and 70s. However, thanks to our setup, they can be deduced results of Minakshisundaram and Pleijel on short time heat kernel asymptotics for Laplacians that were established in the late 40s. Second, we obtain semiclassical Weyl laws for noncommutative tori for any dimension. These laws were conjectured by Ed McDonald and the speaker.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( video )

---

Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

***** We're transitioning to a new platform google meet. Please bear with us and we apologize for the inconvenience! ****

---