Session S33 - Spectral Geometry

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An introduction to Toeplitz quantization

Alix Deleporte

Universite Paris Saclay, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Toeplitz quantization associates, to a function on a phase space, a family of self-adjoint operators, indexed by a semiclassical parameter. The resulting Berezin-Toeplitz operators encompass usual pseudodifferential operators as well as quantum spin systems and the quantization of Arnold's catmap on the torus.

The geometric ingredients of Toeplitz quantization are a symplectic manifold (phase space) with a "compatible" Riemannian metric (or, equivalently, a compatible complex structure), and an associated magnetic Laplacian. The lowest eigenvalue of this magnetic Laplacian is highly degenerate, and the associated spectral projector, named the Szegö or Bergman projector, allows one to define Toeplitz quantization.

In this introductory talk, I will introduce the Toeplitz picture and its modern challenges; this will be motivated by applications to spectral theory, geometry, PDEs, and theoretical physics.

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